{"id": "2606.04307", "title": "Folded Transport MCMC: Eliminating Label Switching by Sampling on a Fundamental Domain", "abstract": "In Bayesian mixture models and other exchangeable-component models, the posterior is invariant under permutation of component labels, creating m! equivalent modes-the label-switching problem. Standard MCMC methods either mix poorly across these modes or rely on post-hoc relabelling that cannot guarantee the sampler has converged. We propose Folded Transport MCMC (FolT-MCMC), which eliminates label switching before sampling by restricting the Markov chain to a fundamental domain-a sorted or reflected subspace containing exactly one representative from each symmetric mode. The proposal is a learned normalising flow whose density is symmetrised over the group orbits, ensuring correct targeting on the reduced space. We show that this construction preserves a computable convergence diagnostic based on the oscillation of the log-density ratio, and that the diagnostic becomes sharper on the fundamental domain whenever the original-space flow under-covers one or more symmetric modes. Experiments on Gaussian mixtures (d=2-20), label-switching targets (up to 24 equivalent modes), a standard Bayesian three-component mixture posterior, and real accelerometer data from a supertall building show improvement ratios of 2x to 145x, with the folded diagnostic stable across dimensions while the unfolded diagnostic collapses.", "date": "2026-06-03", "categories": []}
{"id": "2606.10358", "title": "KG-SoftMAP: Soft Knowledge-Graph Priors for Bayesian Network Structure Learning from Sparse Discrete Data", "abstract": "Learning Bayesian network (BN) structure from sparse discrete data is hard: when each instance records only a few variables, most variable pairs lack the joint observations needed for reliable scoring, and data-only methods recover little structure. However, imperfect domain knowledge, expressible as a weighted directed knowledge graph (KG), is often available. We propose KG-SoftMAP, which encodes such a KG as a finite-strength, confidence-weighted edge prior and maximizes a MAP objective combining the BDeu score with a logit-form prior; the KG may be expert-curated or LLM-extracted. On synthetic benchmarks with known DAGs, KG-SoftMAP reaches Directed-F1 (DF1) $0.19$--$0.32$ at observation rate $\\rho=0.05$ and DF1 $0.44$--$0.97$ at $\\rho\\geq0.2$, while every data-only learner tested stays near zero under the same sparse masks. Recovery tracks KG quality: controlled corruption degrades it smoothly, a zero-signal KG yields DF1 $0.00$, and a blindly LLM-extracted KG with imperfect precision and recall still drives substantial recovery. On three real sparse educational datasets, the learned BN acts as a concept-level posterior model: on SAF it matches logistic regression (LR) within $0.03$ F1_FAIL while providing an inspectable concept graph, calibrated Fail probabilities, and tractable posterior queries from partial observations.", "date": "2026-06-09", "categories": []}
{"id": "2606.20538", "title": "Multi-Task Bayesian In-Context Learning", "abstract": "Bayesian predictive inference provides a principled framework for uncertainty quantification, data efficiency, and robust generalization. However, exact inference is often intractable, and scalable approximations may remain computationally expensive or require restrictive modeling assumptions that degrade predictive performance. Prior-Data Fitted and in-context models have recently emerged as an amortized alternative by learning to map datasets directly to predictive distributions, but existing approaches are tightly coupled to the support of the training prior and lack explicit mechanisms for adapting to new priors at test time, resulting in limited robustness under distribution shift. We introduce a multi-task in-context learning framework for amortized hierarchical Bayesian predictive inference that explicitly represents prior information as a prefix of in-context datasets. A transformer trained on sequences of prior and target tasks learns to adapt its predictions across families of priors. On a suite of evaluations with increasing difficulty, including out-of-meta-distribution priors and priors with high-dimensional latent structures, our method matches oracle Bayesian predictors while being orders of magnitude faster. We further demonstrate its practical relevance on a real-world spatiotemporal temperature prediction benchmark. Code is available at https://github.com/martianmartina/multi-task-bayesian-icl/.", "date": "2026-06-18", "categories": []}
{"id": "2602.21160", "title": "Not Just How Much, But Where: Decomposing Epistemic Uncertainty into Per-Class Contributions", "abstract": "In safety-critical classification, the cost of failure is often asymmetric, yet Bayesian deep learning summarises epistemic uncertainty with a single scalar, mutual information (MI), that cannot distinguish whether a model's ignorance involves a benign or safety-critical class. We decompose MI into a per-class vector $C_k(x)=\\sigma_k^{2}/(2\\mu_k)$, with $\\mu_k{=}\\mathbb{E}[p_k]$ and $\\sigma_k^2{=}\\mathrm{Var}[p_k]$ across posterior samples. The decomposition follows from a second-order Taylor expansion of the entropy; the $1/\\mu_k$ weighting corrects boundary suppression and makes $C_k$ comparable across rare and common classes. By construction $\\sum_k C_k \\approx \\mathrm{MI}$, and a companion skewness diagnostic flags inputs where the approximation degrades. After characterising the axiomatic properties of $C_k$, we validate it on three tasks: (i) selective prediction for diabetic retinopathy, where critical-class $C_k$ reduces selective risk by 34.7\\% over MI and 56.2\\% over variance baselines; (ii) out-of-distribution detection on clinical and image benchmarks, where $\\sum_k C_k$ achieves the highest AUROC and the per-class view exposes asymmetric shifts invisible to MI; and (iii) a controlled label-noise study in which $\\sum_k C_k$ shows less sensitivity to injected aleatoric noise than MI under end-to-end Bayesian training, while both metrics degrade under transfer learning. Across all tasks, the quality of the posterior approximation shapes uncertainty at least as strongly as the choice of metric, suggesting that how uncertainty is propagated through the network matters as much as how it is measured.", "date": "2026-02-24", "categories": []}
{"id": "2606.16214", "title": "Calibrated Sampling-Free Uncertainty Estimation in Bayesian Deep Learning", "abstract": "Modern deep learning models remain notoriously prone to overconfidence, limiting their reliability in high-stakes applications. Bayesian methods aim to counter this by learning a distribution over model parameters, and recent advances now make this feasible for large-scale architectures at costs comparable to AdamW. However, a challenge remains at test time: predictions must be averaged across many forward passes with weights sampled from the posterior, which is prohibitively expensive. Variance propagation offers an efficient alternative, computing layer-wise analytical approximations of uncertainty in a single forward pass. While such techniques are effective for MLPs, their extension to modern architectures remains challenging, due to increased depth and diversity of layer types. To fill this gap, we propose Calibrated Variance Propagation (CVP), which introduces a new propagation method for normalization layers, combines it with recent techniques for handling activation functions, and absorbs residual error through a light calibration step. CVP yields comparably accurate uncertainty estimates to MC sampling across transformers and CNNs, at a fraction of the cost. Against prior variance propagation work, CVP improves coverage at $0.5\\%$ risk from $8.2\\%$ to $14.6\\%$ with BEiT-3 on Visual Reasoning (NLVR2) and from $2.6\\%$ to $10.8\\%$ with ViLT on VQAv2, with gains extending to convolutional architectures.", "date": "2026-06-15", "categories": []}
{"id": "2606.17491", "title": "A Bayesian Boolean Matrix Factorization with Application to Copy Number Analysis in Cancer", "abstract": "Binary data factorization is common, but real-valued methods ignore discreteness and yield hard-to-interpret factors. Boolean Matrix Factorization (BooMF) instead decomposes a binary matrix into two lower-rank binary matrices via logical AND and OR, expressing the data as a Boolean disjunction of interpretable patterns. In cancer genomics, BooMF can reveal coordinated feature changes that may drive tumor evolution, unlike rotational or additive decompositions. Most existing BooMF methods are heuristic, greedy, sensitive to initialization, prone to local optima, and do not support principled model selection or uncertainty quantification. We introduce Bayesian Boolean Matrix Factorization (BBMF), a fully conjugate generative model with sparsity-inducing priors. It enforces Boolean constraints, yields interpretable latent factors with coherent uncertainty quantification, and admits Gibbs sampling with closed-form full conditionals. Because cancer evolution often involves widespread, near-simultaneous chromosome-number changes (e.g., whole-genome duplication followed by instability and selection), Boolean factorizations capture these patterns more naturally than additive models. Applied to arm-level copy-number alteration data in multiple myeloma, where entries indicate presence/absence of chromosomal-arm amplifications, BBMF finds a small set of interpretable bicliques linking patient subsets to recurrently co-altered chromosomal arms, providing a compact, biologically meaningful summary of tumor heterogeneity and demonstrating BBMF's utility for uncovering discrete latent structure in complex binary data.", "date": "2026-06-16", "categories": []}
{"id": "2606.19105", "title": "Smoothness-Based Derandomization of PAC-Bayes Bounds", "abstract": "We study PAC-Bayes derandomization for smooth loss functions. Our goal is to obtain generalization bounds that hold with high probability for deterministic predictors by exploiting smoothness properties of both the loss and the predictor class. We show that passing from the Gibbs predictor to the deterministic predictor at the posterior mean has a precise cost, given by the generalization gap of the Jensen gap class. We control this class through its Rademacher complexity, leading to bounds for deterministic predictors that involve flatness quantities expressed in terms of parameter Jacobians and Hessians of the score map. The framework applies to both bounded and unbounded smooth loss functions, and we specialize the results to linear predictors and smooth neural networks. Finally, the Jacobian and Hessian quantities appearing in the theory motivate a practical regularizer. For BatchNorm networks, we compute this regularizer with respect to effective BatchNorm weights obtained by folding the BatchNorm transformation into the adjacent affine weights. Experiments on CIFAR-10 illustrate the behavior of this regularizer under different batch sizes.", "date": "2026-06-17", "categories": []}
{"id": "2606.15458", "title": "Structured Nonparametric Variational Inference for Dependent Latent Modeling", "abstract": "Variational inference (VI) is a core engine of modern AI, enabling scalable approximate Bayesian learning and uncertainty-aware training of large probabilistic and generative models. In this paper, we propose Structured Nonparametric Variational Inference (SN-VI), a novel framework for modeling complex dependencies among latent variables in posterior approximation, leveraging multivariate spline techniques. Unlike traditional methods that rely on the mean-field assumption, SN-VI preserves intricate latent variable dependencies, providing a flexible and accurate approximation of posteriors with arbitrary shapes. We establish rigorous theoretical guarantees, including the derivation of the lower bound for the variational objective and proof of asymptotic consistency in posterior estimation. To facilitate practical implementation, we develop an algorithm that automatically identifies dependent latent variables and their underlying dependence structure, without requiring manual specification. Simulation studies validate the effectiveness of SN-VI in approximating posterior distributions with bounded support and complex dependencies. The proposed method has been successfully applied to high-dimensional structured data, including computer vision datasets and spatial transcriptomics. In these applications, SN-VI demonstrates improved generative model performance and effectively uncovers coupled biological signals through the learned dependency structure.", "date": "2026-06-13", "categories": []}
{"id": "2606.15479", "title": "Bayesian 3D Steerable CNNs: Enabling Equivariance and Uncertainty Quantification Simultaneously", "abstract": "Steerable convolutional neural networks (Steerable-CNNs) guarantee SE(3)-equivariance by parameterizing kernels as linear combinations of steerable basis functions, but their deterministic nature precludes uncertainty quantification - limiting their use in settings where confidence estimates are essential. We propose a Bayesian Steerable-CNN that places posterior distributions over the basis coefficients, yielding stochastic kernels while preserving equivariance exactly. The loss function of the model is obtained via variational inference and minimized by Bayes-by-Backpropagation. The framework admits a decomposition of predictive uncertainty into epistemic and aleatoric components. Empirically, the model attains competitive classification accuracy alongside an expected calibration error of 0.0263 and outperforms its deterministic counterpart by up to 6.17% under distributional shift induced by additive Gaussian noise. Furthermore, we leverage the model's uncertainty estimates to enhance its performance significantly, achieving a notable gain - approximately 4% higher accuracy across 84% of the test dataset. A statistically significant negative correlation between epistemic uncertainty and prediction error confirms that the learned posterior variance is semantically meaningful. The framework unifies Bayesian uncertainty quantification with the inductive bias of equivariant CNNs.", "date": "2026-06-13", "categories": []}
{"id": "2606.15784", "title": "Bayesian Networks with Latent Time Embedding for Stage-Aware Causal Modeling of Alzheimer's Disease Progression", "abstract": "Alzheimer's disease (AD) progression is often described through the amyloid-tau-neurodegeneration, or AT(N), cascade. However, most longitudinal models represent this cascade either as a fixed sequence of biomarkers or as a black-box forecasting task. This makes it difficult to determine when biologically guided biomarker relationships influence future regional pathology. In this study, we introduce Bayesian Networks with Latent Time Embedding (BN-LTE), a Bayesian structural framework for stage-aware modeling of AD progression. BN-LTE estimates disease pseudotime from baseline biomarker profiles and constrains directed dependencies according to biologically plausible AT(N) ordering. Posterior spline-varying structural equations are then used to link initial multimodal measurements with future annualized regional tau-PET change. Across repeated subject-disjoint evaluations using ADNI data, BN-LTE shows strong spatial reconstruction of tau progression compared with the included forecasting baselines. Beyond spatial reconstruction, BN-LTE recovers posterior stage-varying AT(N)-constrained effects and identifies a mid-pseudotime window of amyloid sensitivity. This window is supported by model-implied g-formula contrasts, root-adjusted AIPW, mechanism-sensitive ablations, and robustness analyses across spline and prior specifications. Overall, these findings position BN-LTE as a Bayesian structural framework for forecasting tau progression while examining stage-dependent AT(N)-cascade mechanisms in observational longitudinal neuroimaging data. Our code is available at https://github.com/danleneurocom/BN-LTE.", "date": "2026-06-14", "categories": []}
{"id": "2606.15871", "title": "Amortized mean-shift interacting particles", "abstract": "Bayesian inference for inverse problems is run to evaluate integrals -- posterior expectations, tail probabilities, and risks -- across a stream of observations. The standard estimate averages the integrand over posterior samples, a Monte-Carlo average whose error decays only as the square root of the sample size, so accuracy demands many samples -- prohibitive when each one calls a partial-differential-equation forward model. Mean-shift interacting particles need far fewer: they return a small set of signed-weight nodes -- a deterministic quadrature whose weighted averages estimate those integrals. Finding the nodes, however, is a per-observation optimization that, in its most accurate form, reads the posterior score at every step -- returning the cost it meant to save. We introduce amortized mean-shift interacting particles, a learned map that emits the weighted nodes from an observation and a few posterior samples in a single forward pass. Training asks only for joint parameter-observation samples and a posterior to draw from -- a conditional normalizing flow, an empirical conditional, or any reference the user can sample -- and the map learns to integrate that posterior from samples alone, evaluating neither its density nor its score. Once trained, it generalizes to unseen observations and integrands at any node budget and improves on independent samples in two ways: by reweighting them, provably no worse than the equal weights of Monte-Carlo; and by moving them, which empirically lowers it further. Across closed-form, sampled, learned, and physics-based posteriors -- up to a thousand-coefficient groundwater field -- it integrates more accurately than the same number of samples at every budget, and a posterior-whitened, dimension-aware kernel removes the high-dimensional wall. The result is a Pareto improvement on Monte-Carlo integration, not a competitor to drawing more samples.", "date": "2026-06-14", "categories": []}
{"id": "2606.16524", "title": "Neural Bayesian Anomaly Mitigation: A Robust Loss that Doubles as an Unsupervised Contamination Classifier", "abstract": "Engineered robust losses such as Huber, Student-$t$, and generalised cross-entropy make supervised models tolerant of contamination but cannot answer which observations are corrupted. We introduce Neural Bayesian Anomaly Mitigation (NBAM), a general-purpose drop-in loss derived from a Bayesian latent-switch mixture model: the marginal likelihood defines a robust supervised loss, and the associated posterior defines an unsupervised contamination classifier. Like Huber or Student-$t$, NBAM can replace the standard training loss in any supervised pipeline; unlike them, it additionally learns a structured contamination model and returns a calibrated per-sample contamination posterior. A learned input-dependent prior $\\pi_\\phi(x)$ captures the spatial locality of contamination, so that samples near known corruptions are more likely to be flagged, while an Occam penalty emerges automatically and regularises against over-flagging. On CIFAR-10 with asymmetric label contamination, NBAM recovers the structure of the corruption process without supervision: the contamination posterior separates clean from corrupted samples, and the learned anomaly head identifies the direction of every label-flip pair. Alongside these capabilities, NBAM outperforms the four robust-loss baselines considered here at contamination rates 0.2-0.6.", "date": "2026-06-15", "categories": []}
{"id": "2606.13818", "title": "Uncertainty Estimation and Generalization Bounds for Modern Deep Learning", "abstract": "This thesis investigates how Bayesian principles can deepen our understanding of modern deep learning systems. While neural networks achieve remarkable predictive performance, their ability to generalize and to quantify uncertainty remains only partly understood. This thesis approaches this challenge from both methodological and theoretical angles: unifying Bayesian inference, function-space modeling, and large-deviation theory under a common probabilistic perspective.\n  On the methodological side, the thesis introduces the Deep Variational Implicit Process (DVIP), a scalable Bayesian framework that extends implicit processes to deep architectures. Complementing this, two post-hoc methods -- the Variational Linearized Laplace Approximation (VaLLA) and the Fixed-Mean Gaussian Process (FMGP) -- are proposed to equip pretrained deterministic networks with calibrated uncertainty estimates.\n  The theoretical contributions focus on one of the central open questions in modern machine learning: why do large, over-parameterized neural networks generalize so well? To address this, the thesis develops a unified probabilistic framework that connects three key mechanisms -- diversity, smoothness, and stochasticity -- within the language of PAC-Bayesian and large-deviation theory.", "date": "2026-06-11", "categories": []}
{"id": "2606.14235", "title": "Implicit Variational Rejection Sampling", "abstract": "Variational Inference (VI) is a fundamental inference technique in Bayesian machine learning for approximating complex posterior distributions. Traditional VI often relies on the mean-field factorization, which can inadequately capture true posterior complexity. Recent advancements have leveraged neural networks to model implicit distributions, offering increased flexibility. However, the practical constraints of neural network architectures still produces inaccuracies. In this paper, we propose a method called Implicit Variational Rejection Sampling (IVRS), which integrates implicit distributions with rejection sampling to improve the posterior approximation. Our method uses neural networks to construct implicit proposal distributions, and rejection sampling with a discriminator network that estimates the density ratio between the implicit proposal and the true posterior for refining the approximation. Towards this end, we introduce the Implicit Resampling Evidence Lower Bound (IR-ELBO) as a metric to characterize the resampled distribution's quality and derive a tighter variational lower bound. Experimental results demonstrate that our method outperforms traditional variational inference techniques.", "date": "2026-06-12", "categories": []}
{"id": "2606.14313", "title": "Nonlocal Bayesian Modeling of Continuous Spatio-Temporal Dynamics", "abstract": "Real-world spatio-temporal forecasting must handle irregular time points, spatially sparse observations, and the need for uncertainty quantification. This setting is often further compounded by nonlocal interactions (long-range spatial coupling). Modeling continuous-space, continuous-time nonlocal dynamics naturally leads to infinite-dimensional integro-differential equations (IDEs), making principled Bayesian inference intractable. We propose the NonLocal Bayesian Spatio-Temporal model (NLBST), a hierarchical Bayesian framework for continuous spatio-temporal fields that learns explicit nonlocal coupling while retaining tractable inference. NLBST represents the latent field via a coordinate-based spatial basis expansion and models the coefficient process with a continuous-time ODE whose learnable linear operator corresponds to a Galerkin reduction of a nonlocal IDE; a Neural ODE residual captures additional nonlinear dynamics. A linear-Gaussian observation model enables Kalman-style sequential updates under missing and irregular observations, while the spatial basis representation enables inductive prediction at unmeasured locations without retraining. Global parameters are learned via variational inference, and uncertainty is handled through a Bayesian hierarchy. Experiments on synthetic and real-world datasets demonstrate strong forecasting and spatial generalization with well-calibrated uncertainty, yielding substantial gains over baselines in strongly nonlocal and partially observed regimes.", "date": "2026-06-12", "categories": []}
{"id": "2411.18385", "title": "Federated Learning with Uncertainty and Personalization via Efficient Second-order Optimization", "abstract": "Federated Learning (FL) has emerged as a promising method to collaboratively learn from decentralized and heterogeneous data available at different clients without the requirement of data ever leaving the clients. Recent works on FL have advocated taking a Bayesian approach to FL as it offers a principled way to account for the model and predictive uncertainty by learning a posterior distribution for the client and/or server models. Moreover, Bayesian FL also naturally enables personalization in FL to handle data heterogeneity across the different clients by having each client learn its own distinct personalized model. In particular, the hierarchical Bayesian approach enables all the clients to learn their personalized models while also taking into account the commonalities via a prior distribution provided by the server. However, despite their promise, Bayesian approaches for FL can be computationally expensive and can have high communication costs as well because of the requirement of computing and sending the posterior distributions. We present a novel Bayesian FL method using an efficient second-order optimization approach, with a computational cost that is similar to first-order optimization methods like Adam, but also provides the various benefits of the Bayesian approach for FL (e.g., uncertainty, personalization), while also being significantly more efficient and accurate than SOTA Bayesian FL methods (both for standard as well as personalized FL settings). Our method achieves improved predictive accuracies as well as better uncertainty estimates as compared to the baselines which include both optimization based as well as Bayesian FL methods.", "date": "2024-11-27", "categories": []}
{"id": "2606.07841", "title": "Large-scale empirical tuning and comparison of default optimizers for variational inference", "abstract": "Black-box variational inference (BBVI) is a methodology for posterior approximation that relies on stochastic optimization. In practice, the stochastic optimizers underpinning BBVI generally require extensive problem-specific tuning, which undermines its promise as a truly \"black box\" inference algorithm. However, over the past decade, many new adaptive stochastic optimization algorithms have been developed that reduce or remove entirely the need for tuning. In this work, we investigate this new collection of adaptive methods in the context of BBVI, with the goal of establishing the current state of the art in tuning-free optimization-based inference. In particular, we present a large-scale empirical evaluation of 56 stochastic gradient-based optimization algorithms applied to 1092 Bayesian inference optimization problems, involving over 550,000 individual optimization runs and 15 core-years of compute. The optimization algorithms we evaluate are chosen to represent a wide spectrum of recent approaches and the benchmark problems are chosen to span a range of difficulty, with posterior target dimension 1-10^4, condition number 1-10^8, and a range of variational families. Our results show that no single method dominates, but running a selection of 5 algorithms suffices to reliably get close to the best-possible observed performance. We thus provide a strong baseline for applications where expert tuning is not possible and for comparison when developing new stochastic optimization algorithms.", "date": "2026-06-05", "categories": []}
{"id": "2606.08218", "title": "How Deep Are Deep GPs, Really? A Sharp Threshold and a Non-Gaussian Limit for Compositional GPs", "abstract": "Compositional priors describe the generic properties of layered functions in deep Bayesian models, where deep neural networks with random weights are a canonical example.In the wide-network limit, the prior is a Gaussian process with a depth-dependent kernel, and its behaviour as depth grows has been extensively studied through this kernel. Here, we study another case, where each layer itself is a vector valued Gaussian process, and our aim is similarly to understand the limiting behaviour of the prior as depth grows.\n  Previous GP work has established that for the RBF kernel and a certain range of bandwidths $r$, the prior degenerates in the limit, converging to the set of constant functions -- which is not useful as a probabilistic model. In this paper we establish several new results. First, we identify a sharp bandwidth threshold $r_c(d) = \\Theta(\\sqrt{d})$ above which the limit is degenerate, strengthening the earlier bounds. Second, and more importantly, we show that for $r$ below the threshold $r_c(d)$ the prior converges to a limit distribution $\\pi_{\\bar{Z}}$. We also prove that these distributions are non-degenerate and non-Gaussian, with non-vanishing dependence between coordinates. In contrast to the previously known degenerate regime, deep Gaussian process priors can therefore admit non-trivial limits.\n  Empirically, we verify the threshold across a range of dimensions $d$, and demonstrate a complex multimodal behaviour of the limit distributions $\\pi_{\\bar{Z}}$ -- a regime that becomes increasingly narrow with $d$ and would be hard to identify without knowing the threshold.", "date": "2026-06-06", "categories": []}
{"id": "2404.02141", "title": "Robustly estimating heterogeneity in factorial data using Rashomon Partitions", "abstract": "In both observational data and randomized control trials, researchers select statistical models to articulate how the outcome of interest varies with combinations of observable covariates. Choosing a model that is too simple can obfuscate important heterogeneity in outcomes between covariate groups, while too much complexity risks identifying spurious patterns. In this paper, we propose a novel Bayesian framework for model uncertainty called Rashomon Partition Sets (RPSs). The RPS consists of all models that have posterior density close to the maximum a posteriori (MAP) model. We construct the RPS by enumeration, rather than sampling, which ensures that we explore all models with high evidence in the data, even if they offer dramatically different substantive explanations. We use a l0 prior, which allows the allows us to capture complex heterogeneity without imposing strong assumptions about the associations between effects, showing this prior is minimax optimal from an information-theoretic perspective. We characterize the approximation error of (functions of) parameters computed conditional on being in the RPS relative to the entire posterior. We propose an algorithm to enumerate the RPS from the class of models that are interpretable and unique, then provide bounds on the size of the RPS. We give simulation evidence along with three empirical examples: price effects on charitable giving, heterogeneity in chromosomal structure, and the introduction of microfinance.", "date": "2024-04-02", "categories": []}
{"id": "2507.12878", "title": "Bayesian Modeling and Estimation of Linear Time-Varying Systems using Neural Networks and Gaussian Processes", "abstract": "The identification of Linear Time-Varying (LTV) systems from input-output data is a fundamental yet challenging ill-posed inverse problem. This work introduces a unified Bayesian framework that models the system's impulse response, $h(t, \\tau)$, as a stochastic process. We decompose the response into a posterior mean and a random fluctuation term, a formulation that provides a principled approach for quantifying uncertainty, unifies intrinsic channel variability and epistemic uncertainty through a common posterior representation, and naturally defines a new, useful system class we term Linear Time-Invariant in Expectation (LTIE). To perform inference, we leverage modern machine learning techniques, including Bayesian neural networks and Gaussian Processes, using scalable variational inference. We demonstrate through a series of experiments that our framework can infer the properties of an LTI system from a single noisy input-output pair, including under deliberate additive-noise misspecification, achieve a lower overall error floor than the classical CCF stacking baseline in a simulated ambient noise tomography setting, and track a continuously varying LTV impulse response by using a structured Gaussian Process prior. This work provides a flexible and robust methodology for uncertainty-aware system identification in dynamic environments.", "date": "2025-07-17", "categories": []}
{"id": "2510.10968", "title": "Blade: A Derivative-free Bayesian Inversion Method using Diffusion Priors", "abstract": "Derivative-free Bayesian inversion arises in science and engineering applications, particularly when forward model is costly or infeasible to differentiate through. Existing derivative-free methods collapse the posterior to a point estimate or return severely over-confident uncertainty on high-dimensional, nonlinear problems. We introduce Blade, which produces accurate and well-calibrated posteriors using an ensemble of interacting particles. Blade leverages diffusion models as data-driven priors, and only queries the forward model through forward evaluations (i.e., derivative-free). Theoretically, we show the convergence and stability of Blade under forward model approximation and prior score estimation error. Empirically, on nonlinear fluid dynamics, Blade produces well-calibrated posterior samples that existing derivative-free methods cannot, as measured by CRPS, the spread-skill ratio, and the rank histogram. Its accuracy and calibration improve consistently with more iterations and particles, backed by our convergence and stability analysis and empirical experiments.", "date": "2025-10-13", "categories": []}
{"id": "2606.06351", "title": "Function-Space Priors for Bayesian Neural ODEs with Application to Vessel Trajectory Prediction", "abstract": "Vessel trajectory prediction from Automatic Identification System (AIS) data is essential for maritime situational awareness, yet it remains challenging due to irregular sampling, missing reports, and complex dynamics. Beyond accurate point forecasts, maritime applications also demand well-calibrated uncertainty estimates for reliable decision-making. Bayesian Neural Ordinary Differential Equations (ODEs) offer a principled framework for continuous-time trajectory modeling with uncertainty quantification by placing a prior over the neural vector field parameters. However, the commonly used isotropic Gaussian weight prior fails to encode informative structural properties of vessel dynamics, such as smoothness and locality. Existing function-space Bayesian neural network methods address this limitation for static mappings, but do not transfer directly to Neural ODEs, where the primary quantity of interest is the trajectory rather than the vector field itself. In principle, one could place a Gaussian process (GP) prior directly over ODE solutions, but this requires propagating distributions through a nonlinear ODE solver, which is analytically intractable. To address this challenge, we adopt a practical approach that imposes a GP-kernel-based prior directly on the vector field evaluated at a finite set of measurement points. Specifically, we augment the standard weight-space variational objective with a kernel-based regularizer that penalizes deviations of the vector field from the structure implied by a GP prior. To handle long and irregular AIS trajectories, we further combine this function-space regularization with probabilistic multiple shooting, which decouples inference across temporal segments while maintaining global consistency.", "date": "2026-06-04", "categories": []}
{"id": "2602.20651", "title": "Sparse Bayesian Deep Functional Learning with Structured Region Selection", "abstract": "In modern applications such as ECG monitoring, neuroimaging, wearable sensing, and industrial equipment diagnostics, complex and continuously structured data are ubiquitous, presenting both challenges and opportunities for functional data analysis. However, existing methods face a critical trade-off: conventional functional models are limited by linearity, whereas deep learning approaches lack interpretable region selection for sparse effects. To bridge these gaps, we propose a sparse Bayesian functional deep neural network (sBayFDNN). It learns adaptive functional embeddings through a deep Bayesian architecture to capture complex nonlinear relationships, while a structured prior enables interpretable, region-wise selection of influential domains with quantified uncertainty. Theoretically, we establish rigorous approximation error bounds, posterior consistency, and region selection consistency. These results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor. Empirically, comprehensive simulations and real-world studies confirm the effectiveness and superiority of sBayFDNN. Crucially, sBayFDNN excels in recognizing intricate dependencies for accurate predictions and more precisely identifies functionally meaningful regions, capabilities fundamentally beyond existing approaches.", "date": "2026-02-24", "categories": []}
{"id": "2606.04324", "title": "Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries", "abstract": "One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function.\n  Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition.\n  The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.", "date": "2026-06-03", "categories": []}
{"id": "2605.29058", "title": "Parallel Adaptive Multi-Objective Evolutionary Learning of Discretized Bayesian Network Classifiers for Clinical Data", "abstract": "Bayesian Networks (BNs) are of interest from an explainable AI viewpoint, offering transparent probabilistic models for decision support. Baymex is a recently introduced multi-objective evolutionary algorithm for learning discretized BNs, enabling experts to trade-off different objectives of interest, such as likelihood, model complexity, and prior beliefs. While Baymex has been shown to outperform state-of-the-art BN learning approaches, Baymex still 1) requires a lot of computation time and 2) has only been evaluated on synthetic data. To improve scalability, we introduce a parallelization strategy as well as a mechanism that enables adaptively steering optimization toward networks that overfit less. We furthermore reconfigure Baymex to train a BN classifier through multi-objective optimization of cross-entropy loss and the BIC complexity term so as to evaluate its performance on real-world clinical classification tasks. Besides observing speedups up to over 54 times on a 16-core CPU, comparisons against clinically familiar baselines (decision trees, logistic regression, naive Bayes, and random forests) on two open-source (RADCURE and SUPPORT) and one in-house dataset, show that Baymex obtains statistically similar or better predictive performance while producing compact, clinically inspectable BNs. Importantly, Baymex finds multiple plausible BN classifiers that contain predictors consistent with established clinical factors.", "date": "2026-05-27", "categories": []}
{"id": "2606.01340", "title": "Sample Complexity and Decision-Theoretic Guarantees for Bayesian Model Averaging over Decision Trees with Catalan-Exponential Priors", "abstract": "We ask: when do Bayesian model averaging (BMA) weights over decision trees carry sufficient epistemic information to justify committed exploitation of the averaging distribution? We answer this question in closed form for Bayesian decision trees (BDTs) with Dirichlet-Multinomial leaf models and a Catalan-exponential tree-size prior (Schetinin&Jakaite, 2025), establishing a complete non-asymptotic theory of rational commitment thresholds.", "date": "2026-05-31", "categories": []}
{"id": "2602.19126", "title": "Robust Predictive Uncertainty and Double Descent in Contaminated Bayesian Random Features", "abstract": "We propose a robust Bayesian formulation of random feature (RF) regression that accounts explicitly for prior and likelihood misspecification via Huber-style contamination sets. Starting from the classical equivalence between ridge-regularized RF training and Bayesian inference with Gaussian priors and likelihoods, we replace the single prior and likelihood with $\\epsilon$- and $\\eta$-contaminated credal sets, respectively, and perform inference using pessimistic generalized Bayesian updating. We derive explicit and tractable bounds for the resulting lower and upper posterior predictive densities. These bounds show that, when contamination is moderate, prior and likelihood ambiguity effectively acts as a direct contamination of the posterior predictive distribution, yielding uncertainty envelopes around the classical Gaussian predictive. We introduce an Imprecise Highest Density Region (IHDR) for robust predictive uncertainty quantification and show that it admits an efficient approximation via an adjusted Gaussian credible interval. We further obtain predictive variance bounds (under a mild truncation approximation for the upper bound) and prove that they preserve the leading-order proportional-growth asymptotics known for RF models. Together, these results establish a robustness theory for Bayesian random features: predictive uncertainty remains computationally tractable, inherits the classical double-descent phase structure, and is improved by explicit worst-case guarantees under bounded prior and likelihood misspecification.", "date": "2026-02-22", "categories": []}
{"id": "2603.14798", "title": "Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces", "abstract": "We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a $64\\times64$ posterior sample in $\\sim 10^{-3}$s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries.", "date": "2026-03-16", "categories": []}
{"id": "2606.00309", "title": "Large-scale Uncertainty Quantification for Latent Variable Models Using Subsampling Markov Chain Monte Carlo", "abstract": "Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD--Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD--Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while each latent variable converges to a jump process, reflecting the use of intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to propose explicit guidance on hyperparameter tuning for SGLD--Gibbs that ensures meaningful uncertainty quantification. Numerical experiments show that SGLD--Gibbs with our tuning guidance leads to better parameter estimates, uncertainty quantification, and predictive performance than stochastic variational inference.", "date": "2026-05-29", "categories": []}
{"id": "2606.01954", "title": "Flow-Transformed Implicit Processes for Function-Space Variational Inference", "abstract": "Implicit-process priors define distributions over functions through flexible generative mechanisms, making them attractive for Bayesian function-space modelling. However, performing posterior inference with such priors is challenging because their induced function-space distributions are typically not available in closed form. One practical strategy is to approximate the prior using a finite collection of sampled functions, and then represent posterior functions as learned combinations of these samples. Existing approaches commonly place a Gaussian variational distribution over the combination weights. While tractable, this choice limits the shapes of posterior uncertainty that can be represented, especially when the true posterior is asymmetric, heavy-tailed, or multimodal. We propose Flow-Transformed Implicit Processes (FTIP), a variational inference method that makes this finite-dimensional function-space approximation more expressive. Instead of using a Gaussian distribution over the combination weights, FTIP uses a normalizing flow to define a richer variational distribution. This induces a flexible posterior distribution over functions while preserving tractable optimization. We train the model using a Black-Box {\\alpha} objective, allowing us to compare mass-covering and mode-seeking variational behaviour. Experiments show that FTIP captures asymmetric and multimodal posterior structure in function space that Gaussian coefficient approximations tend to smooth or collapse.", "date": "2026-06-01", "categories": []}
{"id": "2606.02228", "title": "Bayesian meta-learning for modeling Alzheimer's disease progression", "abstract": "Predicting whether an individual with Alzheimer's disease will experience mild or severe disease progression is essential for personalized treatment. Typically, practitioners seek to predict the distribution of a discrete disease score, conditional on an individual's current MRI volume and their historical disease trajectory. Classical statistical regression models and single-task neural networks are not well-suited for this purpose because fitting separate models is infeasible (since each individual typically has few observations), while ignoring individual-level correlation leads to poor generalization. Meta-learning, in contrast, provides a natural avenue to dynamically predict distributions without retraining and model nonlinear relationships between the outcome and covariates. Motivated by this, we propose a Bayesian meta-learner that is trained on multiple individuals but tailors the predictive disease score distribution to each individual's historical data. Our model predicts on unseen individuals without retraining, scales linearly with the number of historical observations, and is guaranteed to be less overconfident when predicting long-term disease scores compared to its deterministic counterpart. On real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, our model achieves performance competitive with both single-task models and deterministic meta-learners, while substantially improving performance when predicting long-term disease progression.", "date": "2026-06-01", "categories": []}
{"id": "2603.09453", "title": "Variational Routing: A Scalable Bayesian Framework for Calibrated Mixture-of-Experts Transformers", "abstract": "Foundation models are increasingly being deployed in contexts where understanding the uncertainty of their outputs is critical to ensuring responsible deployment. While Bayesian methods offer a principled approach to uncertainty quantification, their computational overhead renders their use impractical for training or inference at foundation model scale. State-of-the-art models achieve parameter counts in the trillions through carefully engineered sparsity including Mixture-of-Experts (MoE) layers. In this work, we demonstrate calibrated uncertainty at scale by introducing Variational Mixture-of-Experts Routing (VMoER), a structured Bayesian approach for modelling uncertainty in MoE layers. VMoER confines Bayesian inference to the expert-selection stage which is typically done by a deterministic routing network. We instantiate VMoER using two inference strategies: amortised variational inference over routing logits and inferring a temperature parameter for stochastic expert selection. Across fine-tuning tested foundation models, VMoER improves routing stability under noise by 38\\%, reduces calibration error by 94\\%, and increases out-of-distribution AUROC by 12\\%, while incurring less than 1\\% additional FLOPs. These results suggest VMoER offers a scalable path toward robust and uncertainty-aware foundation models.", "date": "2026-03-10", "categories": []}
{"id": "2605.29373", "title": "Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems", "abstract": "Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.", "date": "2026-05-28", "categories": []}
{"id": "2605.30860", "title": "Bayesian Inference with Shaped Deep Non-linear MLPs", "abstract": "A central aim of deep learning theory is to characterize how neural networks make predictions in the regime of simultaneously large model and training set size. Since the limits of diverging number of model parameters and dataset size do not commute it is not clear a priori what limits exist. In this work, we shed new light on these questions by studying Bayesian inference in deep non-linear MLPs in the regime where the number of training samples ($P$), the input dimension ($N_0$), the hidden layer width ($N$), and the number of hidden layers ($L$) can all be large. We build on the Neural Covariance SDE (Li et al., 2022) to analyze predictive posteriors in the regime where $LP/N\\in\\Theta(1)$, playing the role of an effective network depth. Our framework covers both smooth and ReLU activation functions and applies to arbitrary temperature. We find to first order in $LP/N$ a simple criterion for which data generating processes benefit from depth in the sense that larger $LP/N$ increases the Bayesian model evidence. We also give a novel derivation of a prior result from the physics literature that at least to first order in $LP/N$, the Bayesian predictive posterior is remarkably simple and is simply equivalent to that of a data-dependent kernel method.", "date": "2026-05-29", "categories": []}
{"id": "2505.02743", "title": "Cooperative Variance Estimation and Bayesian Neural Networks for Disentangling Aleatoric and Epistemic Uncertainties", "abstract": "Real-world data contains aleatoric uncertainty - irreducible noise arising from imperfect measurements or from incomplete knowledge about the data generation process. Mean-variance estimation networks can learn this type of uncertainty but require ad-hoc regularization strategies to avoid overfitting and are unable to predict epistemic uncertainty (model uncertainty). Conversely, Bayesian neural networks predict epistemic uncertainty but are notoriously difficult to train due to the approximate nature of Bayesian inference. We propose to cooperatively train a variance estimation network with a Bayesian neural network and empirically demonstrate that the resulting model disentangles aleatoric and epistemic uncertainties while improving the mean estimation. We demonstrate the effectiveness and scalability of this method across a diverse range of datasets, including a time-dependent heteroscedastic regression dataset we created where the aleatoric uncertainty is known. The proposed method is straightforward to implement, robust, and adaptable to various model architectures.", "date": "2025-05-05", "categories": []}
{"id": "2511.16815", "title": "BITS for GAPS: Bayesian Information-Theoretic Sampling for hierarchical GAussian Process Surrogates", "abstract": "We introduce Bayesian Information-Theoretic Sampling for hierarchical GAussian Process Surrogates (BITS for GAPS), a framework enabling information-theoretic experimental design of Gaussian process-based surrogate models. Unlike standard methods, which use fixed or point-estimated hyperparameters in acquisition functions, our approach propagates hyperparameter uncertainty into the sampling criterion through Bayesian hierarchical modeling. In this framework, a latent function receives a Gaussian process prior, while hyperparameters are assigned additional priors to capture the modeler's knowledge of the governing physical phenomena. Consequently, the acquisition function incorporates uncertainties from both the latent function and its hyperparameters, ensuring that sampling is guided by both data scarcity and model uncertainty. We further establish theoretical results in this context: a closed-form approximation and a lower bound of the posterior differential entropy.\n  We demonstrate the framework's utility for hybrid modeling with a vapor-liquid equilibrium case study. Specifically, we build a surrogate model for latent activity coefficients in a binary mixture. We construct a hybrid model by embedding the surrogate into an extended form of Raoult's law. This hybrid model then informs distillation design. This case study shows how partial physical knowledge can be translated into a hierarchical Gaussian process surrogate. It also shows that using BITS for GAPS increases expected information gain and predictive accuracy by targeting high-uncertainty regions of the Wilson activity model. Overall, BITS for GAPS is a generalized uncertainty-aware framework for adaptive data acquisition in complex physical systems.", "date": "2025-11-20", "categories": []}
{"id": "2605.12208", "title": "Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification", "abstract": "Approximate Bayesian inference typically revolves around computing the posterior parameter distribution. In practice, however, the main object of interest is often a model's predictions rather than its parameters. In this work, we propose to bypass the parameter posterior and focus directly on approximating the posterior predictive distribution. We achieve this by drawing inspiration from self-training within self-supervised and semi-supervised learning. Essentially, we quantify a Bayesian model's predictive uncertainty by refitting on self-predicted data. The idea is strikingly simple: If a model assigns high likelihood to self-predicted data, these predictions are of low uncertainty, and vice versa. This yields a deterministic, sampling-free approximation of the posterior predictive. The modular structure of our Self-Supervised Laplace Approximation (SSLA) further allows us to plug in different prior specifications, enabling classical Bayesian sensitivity (w.r.t. prior choice) analysis. In order to bypass expensive refitting, we further introduce an approximate version of SSLA, called ASSLA. We study (A)SSLA both theoretically and empirically in regression models ranging from Bayesian linear models to Bayesian neural networks. Across a wide array of regression tasks with simulated and real-world datasets, our methods outperform classical Laplace approximations in predictive calibration while remaining computationally efficient.", "date": "2026-05-12", "categories": []}
{"id": "2605.29580", "title": "On the Construction and Implications of Low-Loss Valleys in LoRA-based Bayesian Inference", "abstract": "While parameter-efficient fine-tuning methods like low-rank adaptation (LoRA) are standard for large language models, principled estimation of epistemic uncertainty remains challenging. Recent results in the LoRA regime suggest that discrete multi-mode approaches such as deep ensembles offer little benefit over single-mode methods. This contradicts broader observations in deep learning, where ensembling independent optima typically improves generalization, and linking these modes through continuous low-loss valleys further enhances Bayesian model averaging (BMA). Whether such structure exists in the LoRA space and whether it yields functional diversity missed by local or discrete methods has not been studied. We introduce LoRA-Curve, a segmented B\\'ezier curve parameterization in the LoRA space, with two variants: a free configuration that jointly optimizes all control points, and an anchored configuration that connects independently fine-tuned LoRA optima. We prove pathwise continuity and Lipschitz regularity of the loss along the curve and empirically show, across reasoning and classification benchmarks with Qwen2.5 7B, that linear interpolation encounters loss barriers, while our anchored multi-segment curves connect independent optima through continuous low-loss valleys. Combined with flat-minima perturbations and a Jensen-Shannon divergence regularizer, LoRA-Curve yields measurably higher mutual information of the predictive distribution without sacrificing performance, and links continuous parameter-space traversal to functional diversity.", "date": "2026-05-28", "categories": []}
{"id": "2502.08695", "title": "A Bayesian Nonparametric Perspective on Mahalanobis Distance for Out of Distribution Detection", "abstract": "Bayesian nonparametric methods are naturally suited to the problem of out-of-distribution (OOD) detection. However, these techniques have largely been eschewed in favor of simpler methods based on distances between pre-trained or learned embeddings of data points. Here we show a formal relationship between Bayesian nonparametric models and the relative Mahalanobis distance score (RMDS), a commonly used method for OOD detection. Building on this connection, we propose Bayesian nonparametric mixture models with hierarchical priors that generalize the RMDS. We evaluate these models on the OpenOOD detection benchmark and show that Bayesian nonparametric methods can improve upon existing OOD methods, especially in regimes where training classes differ in their covariance structure and where there are relatively few data points per class.", "date": "2025-02-12", "categories": []}
{"id": "2605.26147", "title": "Neural Bayesian Sequential Routing", "abstract": "Human decision-making is sequential and uncertainty-aware, yet standard neural networks often rely on static, dense forward computation with limited visibility into evidence acquisition, uncertainty evolution, or when computation should stop. We introduce \\textbf{Neural Bayesian Sequential Routing (NBSR)}, a framework that models neural inference as active evidence accumulation over a hierarchical Directed Acyclic Graph (DAG). Within a Dirichlet--Categorical conjugate framework, neural experts query a persistent global knowledge oracle to extract positive evidence vectors, which act as pseudo-counts and update a Dirichlet belief state by exact conjugate addition. Coupled with a Gumbel-Softmax Straight-Through estimator, this update enables hard, path-dependent routing while preserving surrogate gradients for end-to-end training. The resulting Dirichlet precision and entropy provide mechanisms for uncertainty quantification, entropy-based early exiting, OOD abstention, and cost-aware evidence acquisition. We prove that, under strictly positive evidence extraction, total Dirichlet precision increases monotonically along any valid trajectory and marginal predictive variance is bounded, formalizing sequential ``hypothesis sharpening''; under idealized capacity and optimization assumptions, the terminal Dirichlet expectation recovers the Bayes-optimal conditional distribution. Empirical evaluations across visual categorization, structured medical diagnosis, language modeling, partially observable control, and cost-aware Bayesian experimental design show that NBSR achieves competitive predictive performance while providing transparent routing traces, path-dependent evidence attribution, uncertainty-aware decision control, and resource-rational inference. Overall, NBSR offers a mathematically grounded framework for interpretable, modular, and resource-rational agentic AI.", "date": "2026-05-22", "categories": []}
{"id": "2605.26477", "title": "Variational Inference for Evidential Deep Learning", "abstract": "While Deep Neural Networks (DNNs) achieve remarkable performance, their tendency to produce overconfident predictions. Evidential Deep Learning (EDL) mitigates this by formulating predictions as a Dirichlet distribution over class probabilities to explicitly quantify epistemic uncertainty. However, we found that the conventional EDL suffers from two fundamental limitations: a Kullback-Leibler (KL) penalty that only suppresses the evidence of negative classes, producing excessively high evidence therefore decreasing the model's ability to quantify uncertainty, and an absence in theoretical guarantee of setting Dirichlet parameter $\\alpha=e+1$. In this paper, we propose a mathematically principled framework, Variational Inference Evidential Deep Learning (VI-EDL). By reformulating evidential learning through the lens of variational inference, we derive an Evidence Lower Bound (ELBO), which prevents the evidence from growing excessively. Theoretically, we rigorously establish a generalization bound and reveal how the predicted uncertainty, feature and network complexity affect this bound, and why setting $\\boldsymbol{\\alpha} = \\mathbf{e} + \\mathbf{1}$ can minimize it. Extensive experiments on standard visual and medical datasets demonstrate that VI-EDL achieves state-of-the-art performance, showing excellent performance in out-of-distribution detection, noise detection and autonomous driving scenario. The code is available in https://github.com/seutjw/VI-EDL.", "date": "2026-05-26", "categories": []}
{"id": "2605.26509", "title": "SIKA-GP: Accelerating Gaussian Process Inference with Sparse Inducing Kernel Approximations for Bayesian Deep Learning", "abstract": "Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.", "date": "2026-05-26", "categories": []}
{"id": "2605.26990", "title": "Constrained Bayesian Experimental Design via Online Planning", "abstract": "Bayesian experimental design (BED) is a principled framework for data-efficient design of sequential experiments. However, existing BED methods are unable to adapt to dynamic constraints inherent in real-world tasks due to budget limitations, varying costs, or physical constraints that restrict how designs evolve over time. In this paper, we introduce a novel approach to BED that enables constrained optimization of experimental designs by combining offline pre-training of an amortized policy and a posterior network with online multi-step lookahead planning using scenario trees. We empirically demonstrate that our method yields substantially more informative design sequences than existing methods across a range of constrained BED tasks, while incurring only a modest additional computational overhead.", "date": "2026-05-26", "categories": []}
{"id": "2605.27093", "title": "Gaussian Process-based learning with new MCMC-based implementation of Wishart prior on correlation matrix", "abstract": "In probabilstic supervised learning of an input-output relationship - as a sample function of a Gaussian Process (GP) - priors are typically specified for the hyperparameters of the kernel that parametrises the covariance function of the GP, where the induced covariance matrix of the (resulting multivariate Normal) likelihood, governs the learning and prediction. When the sought function is highly multivariate, multiple lengthscale parameters must be learnt simultaneously, making inference difficult. We develop a ``self-assembled'' Wishart prior for the covariance matrix, while undertaking Bayesian inference on the kernel hyperparameters using MCMC. The construction uses a look-back window over recent MCMC iterations to define a time-step dependent scale matrix, thereby introducing adaptiveness to the chain. Results suggest that direct prior specification on the covariance matrix can be useful for diagnosing weakly informative inputs within the GP-based learning paradigm. We support our prior development with two distinct empirical illustrations - one on synthetic data, and another on a real-world dataset.", "date": "2026-05-26", "categories": []}
{"id": "2605.26059", "title": "Accelerating Bayesian inverse design in computational fluid dynamics using neural operators", "abstract": "Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.", "date": "2026-05-25", "categories": []}
{"id": "2601.22367", "title": "Amortized Simulation-Based Inference in Generalized Bayes via Neural Posterior Estimation", "abstract": "Generalized Bayesian Inference (GBI) tempers a loss with a temperature $\\beta > 0$ to mitigate overconfidence and improve robustness under model misspecification, but existing GBI methods typically rely on costly MCMC or SDE-based samplers and must be re-run for each new dataset and each $\\beta$ value. We give the first fully amortized variational approximation for the tempered posterior family by training a single data- and $\\beta$-conditioned neural posterior estimator that enables sampling in a single forward pass, without simulator calls or inference-time MCMC. We introduce two complementary training routes: one synthesizes off-manifold samples from the tempered joint distribution, and the other reweights a fixed base dataset using self-normalized importance sampling (SNIS). We show that the SNIS-weighted objective provides a consistent forward-KL fit to the tempered posterior with finite weight variance. Across four standard simulation-based inference benchmarks, including the chaotic Lorenz-96 system, our $\\beta$-amortized estimator achieves competitive posterior approximations, in standard two-sample metrics, matching non-amortized MCMC-based power-posterior samplers over a wide range of temperatures.", "date": "2026-01-29", "categories": []}
{"id": "2605.23434", "title": "Onsager-Machlup Posterior Transport for Deep Gaussian Processes", "abstract": "Approximate inference over inducing variables is the central computational bottleneck of Deep Gaussian Processes (DGPs). Existing methods either fit an explicit density $q_\\phi(\\bU)$ by an ELBO (DSVI, IPVI, DDVI, DBVI) or sample by MCMC (SGHMC). We instead frame DGP inference as \\emph{posterior transport}: learn a deterministic sampler that maps a tractable reference measure to posterior-relevant inducing variables, regularised by a path prior derived from the Doob-bridged reference diffusion. Our realisation, \\textbf{OM-Path} (formally FBVI-bridge-Path), uses Song's probability-flow ODE applied to DBVI's Doob-bridged forward SDE; the reference drift is closed-form from the bridge marginal coefficients (no score matching) and the path regulariser is the \\textbf{Onsager--Machlup action}. At the finite-$\\epsilon$ value used at training, the objective is the negative log unnormalised density of a tempered Doob-bridge path posterior, and Theorem 1 identifies it with the same posterior's small-noise MAP path via the Freidlin--Wentzell LDP. Two strict path-space ELBO variants on the same bridge backbone (FFJORD log-det; OM-regularised CNF) are derived as ablations. Under a matched-seed paired Wilcoxon test against DBVI on seven UCI regression benchmarks, OM-Path delivers statistically significant wins on the two largest datasets (\\textit{power}: $p\\!=\\!0.014$, NLL $\\mathbf{0.012}$ matching the DSVI baseline of $0.017$; \\textit{protein}: $p\\!=\\!0.002$, RMSE $\\mathbf{0.716}$ vs.\\ $0.764$, NLL $\\mathbf{1.086}$ vs.\\ $1.149$), statistical ties on \\textit{yacht} / \\textit{qsar}, and concedes \\textit{boston} / \\textit{energy} / \\textit{concrete} to DBVI on small-$N$ noisy data. The strict-ELBO variants do not clear DBVI on any UCI metric: in this regime, reducing the variance of the path objective dominates exact-density tracking.", "date": "2026-05-22", "categories": []}
{"id": "2602.01279", "title": "Richer Bayesian Last Layers with Subsampled NTK Features", "abstract": "Bayesian Last Layers (BLLs) provide a convenient and computationally efficient way to estimate uncertainty in neural networks. However, they underestimate epistemic uncertainty because they apply a Bayesian treatment only to the final layer, ignoring uncertainty induced by earlier layers. We propose a method that improves BLLs by leveraging a projection of Neural Tangent Kernel (NTK) features onto the space spanned by the last-layer features. This enables posterior inference that accounts for variability of the full network while retaining the low computational cost of inference of a standard BLL. We show that our method yields posterior variances that are provably greater or equal to those of a standard BLL, correcting its tendency to underestimate epistemic uncertainty. To further reduce computational cost, we introduce a uniform subsampling scheme for estimating the projection matrix and for posterior inference. We derive approximation bounds for both types of subsampling. Empirical evaluations on UCI regression, contextual bandits, image classification, and out-of-distribution detection tasks in image and tabular datasets, demonstrate improved calibration and uncertainty estimates compared to standard BLLs and competitive baselines, while reducing computational cost.", "date": "2026-02-01", "categories": []}
{"id": "2602.05873", "title": "Large-scale Score-based Variational Posterior Inference for Bayesian Deep Neural Networks", "abstract": "Bayesian (deep) neural networks (BNN) are often more attractive than the vanilla point-estimate deep learning in various aspects including uncertainty quantification, robustness to noise, resistance to overfitting, and more. The variational inference (VI) is one of the most widely adopted approximate inference methods. Whereas the ELBO-based variational free energy method is a dominant choice in the literature, in this paper we introduce a score-based alternative for BNN variational inference. Score-based VI can address the known issue of mode collapsing in ELBO-based VI. Although several score-based VI methods have been proposed in the community, most are not adequate for large-scale BNNs for various computational and technical reasons. We propose a novel scalable VI method where the learning objective combines the score matching loss and the proximal penalty term in iterations, which helps our method avoid the reparametrized sampling, and allows for noisy unbiased mini-batch scores through stochastic gradients. This in turn makes our method scalable to large-scale neural networks including Vision Transformers. On several benchmarks including visual recognition and time-series forecasting with large-scale deep networks, we empirically show the effectiveness of our approach.", "date": "2026-02-05", "categories": []}
{"id": "2603.21610", "title": "Rule-State Inference (RSI): A Bayesian Framework for Compliance Monitoring in Rule-Governed Domains", "abstract": "Compliance monitoring in rule-governed domains (tax administration, clinical protocol adherence, environmental regulation) faces three structural obstacles that standard machine learning does not simultaneously address: the absence of labeled outcomes at deployment, strategically missing observations where non-compliant entities selectively withhold evidence, and a regulatory environment that changes faster than any supervised model can be retrained. We introduce Rule-State Inference (RSI), a Bayesian framework that reverses the usual paradigm. Rather than learning rules from data, RSI treats an authoritative, formalized rule set as structured Bayesian priors and infers the latent compliance state of a population through mean-field variational inference with exact coordinate-ascent updates. The central modeling object is a joint latent state per regulatory period: a global compliance-culture factor eta and per-rule components for activation, population compliance level, and parametric drift. RSI delivers three formal guarantees: O(n_k + K) regulatory adaptability per rule update; Bernstein-von Mises consistency for the identifiable continuous components; and monotone ELBO convergence at every iteration. We instantiate RSI on the Togolese fiscal system on a benchmark of 2,000 synthetic enterprises grounded in official regulatory law; full numerical validation is forthcoming. The framework is designed for direct extension to Sequential RSI, a state-space formulation where the posterior from one regulatory period becomes the prior for the next, yielding an exact Kalman filter for compliance-trajectory tracking and entity-level Bayesian scoring.", "date": "2026-03-23", "categories": []}
{"id": "2605.21765", "title": "Position: The Time for Sampling Is Now! Charting a New Course for Bayesian Deep Learning", "abstract": "The practical adoption of sampling-based inference (SAI) in Bayesian neural networks (BNNs) remains limited, partly due to persistent misconceptions about the feasibility and efficiency of sampling. This position paper argues that SAI has achieved computational parity with optimization-based methods and is at the verge of superseding such methods for effective and efficient inference in BNNs. This development should be in the interest of the whole community, promoting BNNs as a principled paradigm with its long-standing yet unfulfilled promise of providing principled uncertainty quantification for neural networks. SAI can even do more -- yielding superior prediction performance through model averaging, serving as the foundation for a plethora of possible downstream tasks, and providing crucial insights into the landscape of BNNs. In order to make such a change happen and unfold the potential of sampling, overcoming current misconceptions is a necessary first step. The next step is to realign research efforts toward addressing remaining challenges in SAI. In particular, the community must focus on two core problems: sufficient exploration of the posterior landscape and high-fidelity distillation of posterior samples for efficient downstream inference. By addressing conceptual and practical obstacles, we can unlock the full potential of SAI and establish it as a central tool in Bayesian deep learning.", "date": "2026-05-20", "categories": []}
{"id": "2511.21223", "title": "Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference", "abstract": "Variational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often rendering analytical treatment impossible and necessitating heavy reliance on approximations. Possibility theory, an imprecise probability framework, allows us to directly model epistemic uncertainty instead of relying on a subjective interpretation of probabilities. While this framework provides robustness and interpretability under sparse or imprecise information, adapting VI to the possibilistic setting requires rethinking core concepts such as divergences, which presuppose additivity. In this work, we develop a principled formulation for performing possibilistic VI by establishing a maxitive analogue of the classical Donsker-Varadhan formulation. The resulting framework enables us to derive a learning rule for possibilistic VI with exponential-family candidates and practical update rules for neural-network training, giving rise to a family of optimizers termed CBOpt. Finally, we demonstrate that CBOpt achieves competitive performance on both in-domain and out-of-domain image classification tasks.", "date": "2025-11-26", "categories": []}
{"id": "2602.06500", "title": "Can Microcanonical Langevin Dynamics Leverage Mini-Batch Gradient Noise?", "abstract": "Scaling inference methods such as Markov chain Monte Carlo to high-dimensional models remains a central challenge in Bayesian deep learning. A promising recent proposal, microcanonical Langevin Monte Carlo, has shown state-of-the-art performance across a wide range of problems. However, its reliance on full-dataset gradients makes it prohibitively expensive for large-scale problems. This paper addresses a fundamental question: Can microcanonical dynamics effectively leverage mini-batch gradient noise? We provide the first systematic study of this problem, establishing a novel continuous-time theoretical analysis of stochastic-gradient microcanonical dynamics. We reveal two critical failure modes: a theoretically derived bias due to anisotropic gradient noise and numerical instabilities in complex high-dimensional posteriors. To tackle these issues, we propose a principled gradient noise preconditioning scheme shown to significantly reduce this bias and develop a novel, energy-variance-based adaptive tuner that automates step size selection and dynamically informs numerical guardrails. The resulting algorithm is a robust and scalable microcanonical Monte Carlo sampler that achieves state-of-the-art performance on challenging high-dimensional inference tasks like Bayesian neural networks. Combined with recent ensemble techniques, our work unlocks a new class of stochastic microcanonical Langevin ensemble (SMILE) samplers for large-scale Bayesian inference.", "date": "2026-02-06", "categories": []}
{"id": "2605.20345", "title": "Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models", "abstract": "Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.", "date": "2026-05-19", "categories": []}
{"id": "2511.01959", "title": "Addressing prior dependence in hierarchical Bayesian modeling for PTA data analysis II: Noise and SGWB inference through parameter decorrelation", "abstract": "Pulsar Timing Arrays (PTA) provide a powerful framework to measure low-frequency gravitational waves, but accuracy and robustness of the results are challenged by complex noise processes that must be accurately modeled. Standard PTA analyses assign fixed uniform noise priors to each pulsar, an approach that can introduce systematic biases when combining the array. To overcome this limitation, we adopt a hierarchical Bayesian modeling strategy in which noise priors are parametrized by higher-level hyperparameters. To mitigate the sensitivity of the inferred parameters to the choice of noise hyperprior, we introduce a reparametrization of the hierarchical model based on the orthogonal projection of hyperparameters onto the physical parameter subspace. The transformation is implemented through Normalizing Flows (NFs), which provide an invertible, tractable representation and preserve shrinkage and inter-pulsar information pooling in the reparametrized model. We also employ i-nessai, a flow-guided nested sampler, to efficiently explore the resulting higher-dimensional parameter space. We apply our method to a minimal 3-pulsar case study, performing a simultaneous inference of noise and stochastic gravitational wave background (SGWB) parameters. Despite the limited dataset, the results consistently show that the reparametrized hierarchical treatment constrains the noise parameters more tightly and partially alleviates the red-noise-SGWB degeneracy, while the orthogonal reparametrization further enhances parameter independence without affecting the correlations intrinsic to the power-law modeling of the physical processes involved.", "date": "2025-11-03", "categories": []}
{"id": "2605.19610", "title": "Posterior Contraction of L\\'evy Adaptive B-spline Regression in Besov Spaces", "abstract": "We investigate the asymptotic properties of the L\\'evy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the L\\'evy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.", "date": "2026-05-19", "categories": []}
{"id": "2605.20028", "title": "Training-Free Bayesian Filtering with Generative Emulators", "abstract": "Bayesian filtering is a well-known problem that aims to estimate plausible states of a dynamical system from observations. Among existing approaches to solve this problem, particle filters are theoretically exact for non-linear dynamics and observations, but suffer from poor scalability in high dimensions. In this work, we show that diffusion-based emulators of dynamical systems can be used to implement, without additional training, an optimal variant of particle filters that has remained largely unexplored due to implementation challenges with classical numerical solvers. Experiments on nonlinear chaotic systems, including atmospheric dynamics, demonstrate that the proposed approach successfully scales particle filtering to high-dimensional settings.", "date": "2026-05-19", "categories": []}
{"id": "2405.14657", "title": "Anchor-Based Heteroscedastic Noise for Preferential Bayesian Optimization", "abstract": "Preferential Bayesian optimization (PBO) learns latent utilities from pairwise comparisons, but most existing methods assume homoscedastic comparison noise. This is inadequate in human-in-the-loop settings, where a user may compare some designs reliably and others only hesitantly. We propose a heteroscedastic noise model for PBO: before optimization, the user provides a small set of reliable examples, called anchors, and a kernel density estimator (KDE) turns these anchors into an input-dependent map of user uncertainty. We incorporate this map into preferential GP surrogates and derive risk-averse acquisition functions that trade off utility and ease of comparison. We further show that a risk-adjusted variant of the popular expected utility of the best option (EUBO) preserves the one-step Bayes-optimality guarantee up to an additive constant, and that under an idealized i.i.d. anchor model the KDE estimator enjoys standard consistency and concentration rates. Experiments on synthetic problems and human-preference datasets show improved risk-adjusted performance and clarify how anchor placement affects the method.", "date": "2024-05-23", "categories": []}
{"id": "2502.02463", "title": "Distribution Transformers: Fast Approximate Bayesian Inference With On-The-Fly Prior Adaptation", "abstract": "While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer -- a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset -- thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times-from minutes to milliseconds-while achieving log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.", "date": "2025-02-04", "categories": []}
{"id": "2601.20888", "title": "Latent-IMH: Efficient Bayesian Inference for Inverse Problems with Approximate Operators", "abstract": "We study sampling from posterior distributions in Bayesian linear inverse problems where $A$, the parameters to observables operator, is computationally expensive. In many applications, $A$ can be factored in a manner that facilitates the construction of a cost-effective approximation $\\tilde{A}$. In this framework, we introduce Latent-IMH, a sampling method based on the Metropolis-Hastings independence (IMH) sampler. Latent-IMH first generates intermediate latent variables using the approximate $\\tilde{A}$, and then refines them using the exact $A$. Its primary benefit is that it shifts the computational cost to an offline phase. We theoretically analyze the performance of Latent-IMH using KL divergence and mixing time bounds. Using numerical experiments on several model problems, we show that, under reasonable assumptions, it outperforms state-of-the-art methods such as the No-U-Turn sampler (NUTS) in computational efficiency. In some cases, Latent-IMH can be orders of magnitude faster than existing schemes.", "date": "2026-01-28", "categories": []}
{"id": "2602.21426", "title": "Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators", "abstract": "We consider the problem of sampling from a posterior distribution arising in Bayesian inverse problems in science, engineering, and imaging. Our method belongs to the family of independence Metropolis-Hastings (IMH) sampling algorithms, which are common in Bayesian inference. Relying on the existence of an approximate posterior distribution that is cheaper to sample from but may have significant bias, we introduce Proximal-IMH, a scheme that removes this bias by correcting samples from the approximate posterior through an auxiliary optimization problem. This yields a local adjustment that trades off adherence to the exact model against stability around the approximate reference point. For idealized settings, we prove that the proximal correction tightens the match between approximate and exact posteriors, thereby improving acceptance rates and mixing. The method applies to both linear and nonlinear input-output operators and is particularly suitable for inverse problems where exact posterior sampling is too expensive. We present numerical experiments including multimodal and data-driven priors with nonlinear input-output operators. The results show that Proximal-IMH reliably outperforms existing IMH variants.", "date": "2026-02-24", "categories": []}
{"id": "2605.15407", "title": "Amortized Energy-Based Bayesian Inference", "abstract": "We consider amortized Bayesian inference for nonlinear inverse problems in settings where only samples from the joint distribution of parameters and observations are available. Classical methods such as Markov chain Monte Carlo require solving a new inference problem for each observation, which can be computationally prohibitive when inference must be repeated many times. We propose a transport-based approach that learns an observation-dependent map pushing forward a reference measure to approximate the posterior distribution. The map is trained by minimizing an averaged energy-distance objective between the true posterior and the learned pushforward. This formulation is likelihood-free, requiring only joint samples, and avoids density evaluation, invertibility constraints, and Jacobian determinant computations. For function-space inverse problems with Gaussian priors, we parameterize the transport map as the identity plus a perturbation in the Cameron-Martin space of the prior, preserving absolute continuity with respect to the prior. In infinite-dimensional settings, the map is represented using neural operators. We illustrate the method on a finite-dimensional nonlinear inverse problem and two PDE-constrained inverse problems arising in porous medium flow and seismic inversion. The results show that the learned transport captures posterior structure, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.", "date": "2026-05-14", "categories": []}
{"id": "2605.15459", "title": "Don't Stop Me Yet: Sampling Loss Minima via Dissipative Riemannian Mechanics", "abstract": "The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DiMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.", "date": "2026-05-14", "categories": []}
{"id": "2605.15488", "title": "SurvivalPFN: Amortizing Survival Prediction via In-Context Bayesian Inference", "abstract": "Survival analysis provides a powerful statistical framework for modeling time-to-event outcomes in the presence of censoring. However, selecting an appropriate estimator from the many specialized survival approaches often requires substantial methodological and domain expertise. We introduce SurvivalPFN, a prior-data fitted network that amortizes Bayesian inference for censored observations through in-context learning. SurvivalPFN is pretrained on a diverse family of synthetic, identifiable, and right-censored data-generating processes, enabling it to amortize survival analysis in a single forward pass during inference. As a result, the model adapts to the effective complexity of each dataset without task-specific training or hyperparameter tuning, avoids restrictive parametric assumptions, and produces calibrated survival distributions. In a large-scale benchmark spanning 61 datasets, 21 methods, and 5 evaluation metrics, SurvivalPFN achieves strong predictive performance and often improves upon established survival models. These results suggest that SurvivalPFN offers a principled and practical foundation model for survival analysis, with potential applications in high-impact domains such as healthcare, finance, and engineering (https://github.com/rgklab/SurvivalPFN).", "date": "2026-05-15", "categories": []}
{"id": "2510.25240", "title": "Generative Bayesian Optimization: Generative Models as Acquisition Functions", "abstract": "We present a general strategy for turning generative models into candidate solution samplers for batch Bayesian optimization (BO). The use of generative models for BO enables large batch scaling as generative sampling, optimization of non-continuous design spaces, and high-dimensional and combinatorial design. Inspired by the success of direct preference optimization (DPO), we show that one can train a generative model with noisy, simple utility values directly computed from observations to then form proposal distributions whose densities are proportional to the expected utility, i.e., BO's acquisition function values. Furthermore, this approach is generalizable beyond preference-based feedback to general types of reward signals and loss functions. This perspective avoids the construction of surrogate (regression or classification) models, common in previous methods that have used generative models for black-box optimization. Theoretically, we show that the generative models within the BO process follow a sequence of distributions which asymptotically approximate an optimal target under certain conditions. We also evaluate the performance through experiments on challenging optimization problems involving large batches in high dimensions.", "date": "2025-10-29", "categories": []}
{"id": "2605.07060", "title": "Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks", "abstract": "Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.", "date": "2026-05-08", "categories": []}
{"id": "2605.14142", "title": "To discretize continually: Mean shift interacting particle systems for Bayesian inference", "abstract": "Integration against a probability distribution given its unnormalized density is a central task in Bayesian inference and other fields. We introduce new methods for approximating such expectations with a small set of weighted samples -- i.e., a quadrature rule -- constructed via an interacting particle system that minimizes maximum mean discrepancy (MMD) to the target distribution. These methods extend the classical mean shift algorithm, as well as recent algorithms for optimal quantization of empirical distributions, to the case of continuous distributions. Crucially, our approach creates dynamics for MMD minimization that are invariant to the unknown normalizing constant; they also admit both gradient-free and gradient-informed implementations. The resulting mean shift interacting particle systems converge quickly, capture anisotropy and multi-modality, avoid mode collapse, and scale to high dimensions. We demonstrate their performance on a wide range of benchmark sampling problems, including multi-modal mixtures, Bayesian hierarchical models, PDE-constrained inverse problems, and beyond.", "date": "2026-05-13", "categories": []}
{"id": "2605.14146", "title": "bde: A Python Package for Bayesian Deep Ensembles via MILE", "abstract": "bde is a user-friendly Python package for Bayesian Deep Ensembles with a particular focus on tabular data. Built on an efficient JAX implementation of the sampling-based inference method Microcanonical Langevin Ensembles (MILE), it provides scikit-learn compatible estimators for fast training, efficient Markov Chain Monte Carlo sampling, and uncertainty quantification in both regression and classification tasks.", "date": "2026-05-13", "categories": []}
{"id": "2507.22095", "title": "Posterior Bayesian Neural Networks with Dependent Weights", "abstract": "We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in [26], to address limitations of the standard Gaussian prior. It has been proved in [26] that, as the number of nodes in the hidden layers grows large, according to a sequential and ordered limit, the law of the output converges weakly to a Gaussian mixture. In this paper, we study the neural network through the lens of the posterior distribution with a Gaussian likelihood. If the random covariance matrix of the infinite-width limit is positive definite under the prior, we identify the posterior distribution of the output in the wide-width limit according to a sequential regime. Remarkably, we provide mild sufficient conditions to ensure the aforementioned invertibility of the random covariance matrix under the prior, thereby extending the results in [8]. Among our results, we present sufficient conditions on some model parameters (the activation function and the associated L\\'evy measures) which ensure that the sequential limits are independent of the order. We illustrate our findings with examples and numerical simulations.", "date": "2025-07-29", "categories": []}
{"id": "2509.19929", "title": "Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later", "abstract": "Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture-agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynolds-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.", "date": "2025-09-24", "categories": []}
{"id": "2602.23089", "title": "Physics-informed neural particle flow for the Bayesian update step", "abstract": "The Bayesian update step poses significant computational challenges in high-dimensional nonlinear estimation. While log-homotopy particle flow filters offer an alternative to stochastic sampling, existing formulations usually yield stiff differential equations. Conversely, existing deep learning approximations typically treat the update as a black-box task or rely on asymptotic relaxation, neglecting the exact geometric structure of the finite-horizon probability transport. In this work, we propose a physics-informed neural particle flow, which is an amortized inference framework. To construct the flow, we couple the log-homotopy trajectory of the prior to posterior density function with the continuity equation describing the density evolution. This derivation yields a governing partial differential equation (PDE), referred to as the master PDE. By embedding this PDE as a physical constraint into the loss function, we train a neural network to approximate the transport velocity field. This approach enables purely unsupervised training, eliminating the need for ground-truth posterior samples. We demonstrate that the neural parameterization acts as an implicit regularizer, mitigating the numerical stiffness inherent to analytic flows and reducing online computational complexity. Experimental validation on multimodal benchmarks and a challenging nonlinear scenario confirms better mode coverage and robustness compared to state-of-the-art baselines.", "date": "2026-02-26", "categories": []}
{"id": "2605.12843", "title": "Bayesian Model Merging", "abstract": "Model merging aims to combine multiple task-specific expert models into a single model without joint retraining, offering a practical alternative to multi-task learning when data access or computational budget is limited. Existing methods, however, face two key limitations: (1) they overlook the valuable inductive bias of strong anchor models and estimate the merged weights from scratch, and (2) they rely on a shared hyperparameter setting across different modules of the network, lacking a global optimization strategy. This paper introduces Bayesian Model Merging (BMM), a plug-and-play bi-level optimization framework, where the inner level formulates the model merging as an activation-based Bayesian regression under a strong prior induced by an anchor model, yielding an efficient closed-form solution; and the outer level leverages a Bayesian optimization procedure to search module-specific hyperparameters globally based on a small validation set. Furthermore, we reveal a key alignment between activation statistics and task vectors, enabling us to derive a data-free variant of BMM that estimates the Gram matrix for regression without any auxiliary data. Across extensive benchmarks, including up to 20-task merging in vision and 5-task merging in language, BMM consistently outperforms all plug-and-play anchor baselines (e.g., TA, WUDI-Merging, and TSV). In particular, on the ViT-L/14 benchmark for 8-task merging, a single merged model reaches 95.1, closely matching the average performance of eight task-specific experts (95.8).", "date": "2026-05-13", "categories": []}
{"id": "2605.13088", "title": "Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes", "abstract": "Dynamical modelling is central to many scientific domains, including pharmacometrics, systems biology, physiology, and epidemiology. In these settings, heterogeneity is often intrinsic: different subjects or units follow related but distinct continuous-time dynamics. Classical nonlinear mixed-effects Ordinary Differential Equation (ODE) models address this by combining population-level structure with subject-specific effects, but they rely on a parametric vector field and are therefore vulnerable to structural misspecification and unmodelled mechanisms. This motivates nonparametric approaches that can retain principled uncertainty quantification, yet existing nonparametric ODE methods typically assume a single shared dynamical system rather than an explicit mixed-effect hierarchy over subject-specific dynamics. We propose MEGPODE, a Bayesian nonparametric mixed-effect ODE model in which each subject's vector field is decomposed into a shared population component and a subject-specific deviation, both endowed with Gaussian process (GP) priors. To avoid repeated ODE solves per subject during training, we combine state-space GP trajectory priors with virtual collocation observations, yielding Kalman-smoothing trajectory updates and closed-form regressions for the vector fields. Across controlled heterogeneous ODE benchmarks spanning oscillatory, biomedical systems, MEGPODE improves population-field recovery and subject-level trajectory prediction relative to strong baselines.", "date": "2026-05-13", "categories": []}
{"id": "2411.00471", "title": "Dirichlet process mixtures of block $g$ priors for model selection and prediction in linear models", "abstract": "This paper introduces Dirichlet process mixtures of block $g$ priors for model selection and prediction in linear models. These priors are extensions of traditional mixtures of $g$ priors that allow for differential shrinkage for various (data-selected) blocks of parameters while fully accounting for the predictors' correlation structure, providing a bridge between the literatures on model selection and continuous shrinkage priors. We show that Dirichlet process mixtures of block $g$ priors are consistent in various senses and, in particular, that they avoid the conditional Lindley ``paradox'' highlighted by Som et al. (2016). Further, we develop a Markov chain Monte Carlo algorithm for posterior inference that requires only minimal ad-hoc tuning. Finally, we investigate the empirical performance of the prior in various real and simulated datasets. In the presence of a small number of very large effects, Dirichlet process mixtures of block $g$ priors lead to higher power for detecting smaller but significant effects without only a minimal increase in the number of false discoveries.", "date": "2024-11-01", "categories": []}
{"id": "2412.11875", "title": "Bayesian Surrogate Training on Multiple Data Sources: A Hybrid Modeling Strategy", "abstract": "Surrogate models are often used as computationally efficient approximations to complex simulation models, enabling tasks such as solving inverse problems, sensitivity analysis, and probabilistic forward predictions, which would otherwise be computationally infeasible. During training, surrogate parameters are fitted such that the surrogate reproduces the simulation model's outputs as closely as possible. However, the simulation model itself is merely a simplification of the real-world system, often missing relevant processes or suffering from misspecifications e.g., in inputs or boundary conditions. Hints about these might be captured in real-world measurement data, and yet, we typically ignore those hints during surrogate building. In this paper, we propose two novel probabilistic approaches to integrate simulation data and real-world measurement data during surrogate training. The first method trains separate surrogate models for each data source and combines their predictive distributions, while the second incorporates both data sources by training a single surrogate. Both hybrid modeling approaches employ a novel weighting strategy for combining heterogeneous data sources during surrogate training, which operates independently of the chosen surrogate family. We show the conceptual differences and benefits of the two approaches through both synthetic and real-world case studies. The results demonstrate the potential of these methods to improve predictive accuracy, predictive coverage, and to diagnose problems in the underlying simulation model. These insights can improve system understanding and future model development.", "date": "2024-12-16", "categories": []}
{"id": "2509.21711", "title": "Multi-modal Bayesian Neural Network Surrogates with Conjugate Last-Layer Estimation", "abstract": "As data collection and simulation capabilities advance, multi-modal learning, the task of learning from multiple modalities and sources of data, is becoming an increasingly important area of research. Surrogate models that learn from data of multiple auxiliary modalities to support the modeling of a highly expensive quantity of interest have the potential to aid outer loop applications such as optimization, inverse problems, or sensitivity analyses when multi-modal data are available. We develop two multi-modal Bayesian neural network surrogate models and leverage conditionally conjugate distributions in the last layer to estimate model parameters using stochastic variational inference (SVI). We provide a method to perform this conjugate SVI estimation in the presence of partially missing observations. We demonstrate improved prediction accuracy and uncertainty quantification compared to uni-modal surrogate models for both scalar and time series data.", "date": "2025-09-26", "categories": []}
{"id": "2510.09333", "title": "Efficient Bayesian Inference from Noisy Pairwise Comparisons", "abstract": "Evaluating generative models is challenging because standard metrics often fail to reflect human preferences. Human evaluations are more reliable but costly and noisy, as participants vary in expertise, attention, and diligence. Pairwise comparisons improve consistency, yet aggregating them into overall quality scores requires careful modeling. Bradley-Terry-based methods update item scores from comparisons, but existing approaches either ignore rater variability or lack convergence guarantees, limiting robustness and interpretability. We introduce BBQ, a Bayesian Bradley-Terry variant that explicitly models rater quality, downweighting or removing unreliable participants, and provides guaranteed monotonic likelihood convergence through an Expectation-Maximization algorithm. Empirical results show that BBQ provides efficient inference, well-calibrated uncertainty estimates, and more robust, interpretable rankings compared to baseline Bradley-Terry models, even with noisy or crowdsourced raters. This framework enables more reliable and cost-effective human evaluation of generative models.", "date": "2025-10-10", "categories": []}
{"id": "2603.14094", "title": "Maximin Robust Bayesian Experimental Design", "abstract": "We address the brittleness of Bayesian experimental design under model misspecification by formulating the problem as a max--min game between the experimenter and an adversarial nature subject to information-theoretic constraints. We demonstrate that this approach yields a robust objective governed by Sibson's $\\alpha$-mutual information (MI), which identifies the $\\alpha$-tilted posterior as the robust belief update and establishes the R\\'enyi divergence as the appropriate measure of conditional information gain. To mitigate the bias and variance of nested Monte Carlo estimators needed to estimate Sibson's $\\alpha$-MI, we adopt a PAC-Bayes framework to search over stochastic design policies, yielding rigorous high-probability lower bounds on the robust expected information gain that explicitly control finite-sample error.", "date": "2026-03-14", "categories": []}
{"id": "2605.08446", "title": "Direct Bethe Free Energy Minimization for Bayesian Neural Network", "abstract": "We propose training Bayesian neural networks by directly minimizing the Bethe free energy rather than maximizing a variational lower bound. On tree-structured factor graphs the Bethe free energy is exact; deterministic layers drop out of the objective and are trained by standard backpropagation, so the framework accommodates any mixture of probabilistic and deterministic subgraphs without modification. Restricting the weight posterior to a last-layer Gaussian yields analytically tractable losses: for a Gaussian likelihood the Bethe loss equals the exact marginal likelihood, and for a probit likelihood it reduces to a closed form via the probit-Gaussian convolution. Both objectives sit strictly between MAP and the ELBO ($L_\\text{MAP} \\leq L_\\text{Bethe} \\leq L_\\text{ELBO}$), removing the structural Jensen gap that no choice of variational family can close. The Z-consistent prior formulation makes the prior precision a differentiable parameter, enabling empirical Bayes - joint optimization of weights, covariance, and hyperparameters - in a single gradient pass, with no cross-validation or outer loop. All variants admit a closed-form predictive at MAP-equivalent inference cost, in contrast to ensemble and sampling-based methods. On 8 UCI regression and 12 UCI classification benchmarks evaluated under a single shared hyperparameter regime, Bethe is competitive with standard reference methods at single-pass cost. Independently, joint single-pass empirical Bayes matches grid-search cross-validation of the prior precision on essentially all dataset-variant combinations, eliminating the outer hyperparameter loop without measurable cost. Isolated optimization gaps on a few datasets reflect numerical rather than principled limitations of the framework.", "date": "2026-05-08", "categories": []}
{"id": "2605.12079", "title": "Elicitation-Augmented Bayesian Optimization", "abstract": "Human-in-the-loop Bayesian optimization (HITL BO) methods utilize human expertise to improve the sample-efficiency of BO. Most HITL BO methods assume that a domain expert can quantify their knowledge, for instance by pinpointing query locations or specifying their prior beliefs about the location of the maximum as a probability distribution. However, since human expertise is often tacit and cannot be explicitly quantified, we consider a setting where domain knowledge of an expert is elicited via pairwise comparisons of designs. We interpret the expert's pairwise judgements as noisy evidence about the values of the observable objective function and develop a principled method for combining the information obtained via direct observations and pairwise queries. Specifically, we derive a cost-aware value-of-information acquisition function that balances direct observations against pairwise queries. The proposed method approaches the convex hull of the trajectories of the individual information sources: when pairwise queries are cheap it substantially improves sample-efficiency over observation-only BO, and when pairwise queries are costly or noisy, it recovers the performance of standard BO by relying on direct observations alone.", "date": "2026-05-12", "categories": []}
{"id": "2510.09877", "title": "Batch Bayesian Active Learning with Partial Batch Label Sampling", "abstract": "Over the past couple of decades, many active learning acquisition functions have been proposed, leaving practitioners with an unclear choice of which to use. Bayesian-based active learning offers principled objectives with explainable intuition, including Expected Error Reduction (EER), Expected Predictive Information Gain (EPIG), and Bayesian Active Learning by Disagreements (BALD). A key challenge of such methods is the difficult scaling to large batch sizes, leading to either computational challenges (BatchBALD) or dramatic performance drops (top-$B$ selection). Here, using a particular formulation of Bayesian Decision Theory, we derive Partial Batch Label Sampling (ParBaLS) for the EPIG algorithm. We show experimentally for several datasets that ParBaLS EPIG gives superior performance for a fixed budget and Bayesian Logistic Regression on embeddings from large pre-trained models. Our code is available at https://github.com/ADDAPT-ML/ParBaLS.", "date": "2025-10-10", "categories": []}
{"id": "2601.23252", "title": "Nested Slice Sampling: Vectorized Nested Sampling for GPU-Accelerated Inference", "abstract": "Model comparison and calibrated uncertainty quantification often require integrating over parameters, but scalable inference can be challenging for complex, multimodal targets. Nested Sampling is a robust alternative to standard MCMC, yet its typically sequential structure and hard constraints make efficient accelerator implementations difficult. This paper introduces Nested Slice Sampling (NSS), a GPU-friendly, vectorized formulation of Nested Sampling that uses Hit-and-Run Slice Sampling for constrained updates. A tuning analysis yields a simple near-optimal rule for setting the slice width, improving high-dimensional behavior and making per-step compute more predictable for parallel execution. Experiments on challenging synthetic targets, high dimensional Bayesian inference, and Gaussian process hyperparameter marginalization show that NSS maintains accurate evidence estimates and high-quality posterior samples, and is particularly robust on difficult multimodal problems where current state-of-the-art methods such as tempered SMC baselines can struggle. An open-source implementation is released to facilitate adoption and reproducibility.", "date": "2026-01-30", "categories": []}
{"id": "2603.14681", "title": "Generalized Hierarchical Bayesian Segmentation with Irregular Designs, Multi-Sequence Hierarchies, and Grouped/Latent-Group Designs", "abstract": "Bayesian change-point and segmentation models provide uncertainty-aware piecewise-constant representations of ordered data, but exact inference is often limited to narrow likelihood classes, single sequences, or index-uniform designs. We present \\texttt{BayesBreak}, a modular offline Bayesian segmentation framework that separates local block scoring from global inference: each candidate block supplies a marginal likelihood and any needed moment numerators, while a dynamic program combines these scores to compute posteriors over segment counts, boundaries, and latent signals. For weighted exponential-family likelihoods with conjugate priors, block evidences and posterior moments are available in closed form from cumulative sufficient statistics, enabling exact sum-product inference for $p(y\\mid k)$, $p(k\\mid y)$, boundary marginals, and Bayes regression curves. We distinguish these summaries from the \\emph{joint} MAP segmentation, recovered by a separate max-sum recursion.\n  BayesBreak supports design-aware partition priors for irregular observations, exact pooling across replicates with shared boundaries, and latent-template mixtures with exact EM updates. For non-conjugate GLM blocks, the same DP layer can use deterministic local approximations such as Laplace, variational methods, EP, or quadrature. We prove a posterior-odds stability bound: uniform per-block log-evidence error $\\varepsilon$ perturbs $k$-odds and boundary-odds by at most $(k+k')\\varepsilon$ and $2k\\varepsilon$. Validation includes synthetic recovery, calibration, and scaling experiments, plus four real-data illustrations: well-log geology, array-CGH copy number, equity-return volatility, and CpG-atlas methylation.", "date": "2026-03-16", "categories": []}
{"id": "2605.08110", "title": "BaLoRA: Bayesian Low-Rank Adaptation of Large Scale Models", "abstract": "Low-Rank Adaptation (LoRA) has become the standard for fine-tuning large pre-trained models at reduced computational cost. However, its low-rank point-estimate updates limit expressiveness, leave a persistent gap relative to full fine-tuning accuracy, and provide no built-in uncertainty quantification, limiting its applicability in settings where reliability matters as much as accuracy. We introduce BaLoRA, a Bayesian extension of LoRA with a novel input-adaptive Bayesian parameterization of LoRA matrices that adds minimal parameters and compute. Surprisingly, not only does the Bayesian extension yield well-calibrated uncertainty estimates, but the adaptive noise injection underlying our approach also significantly improves prediction accuracy, narrowing the gap with full fine-tuning across both natural language reasoning and vision tasks. When applied to band gap prediction in metal-organic frameworks, BaLoRA produces zero-shot test-time uncertainty estimates that correlate more strongly with model error than a trained ensemble of LoRA models, and improve monotonically with compute without sacrificing accuracy.", "date": "2026-04-27", "categories": []}
{"id": "2605.09075", "title": "Optimality of Sub-network Laplace Approximations: New Results and Methods", "abstract": "Although the Laplace approximation offers a simple route to uncertainty quantification in deep neural networks, its reliance on inverting large Hessian matrices has motivated a range of computationally feasible low-dimensional or sparse approximations. A prominent class of such methods - sub-network Laplace approximations, constructs surrogates by restricting attention to a small subset of parameters. Existing approaches in this family typically rely on diagonal, layer-wise, or other architectural heuristics for subset selection, which ignore cross-parameter interactions and lack formal optimality guarantees. In this paper, we provide a rigorous theoretical analysis of the sub-network Laplace paradigm. We prove that all sub-network Laplace methods systematically underestimate the predictive variance of the full Laplace posterior, and that this bias decreases monotonically as the retained sub-matrix expands. Leveraging this insight, we propose two principled, analytically grounded sub-network Hessian approximations: \\textit{Gradient-Laplace} selects parameters with the largest average squared gradients of the model output with respect to the parameters over a reference dataset; while \\textit{Greedy-Laplace} iteratively refines this selection by accounting for off-diagonal interactions in the precision matrix. We establish theoretical guarantees characterizing their optimality properties and show that Gradient-Laplace provably outperforms existing heuristic approaches. Extensive numerical studies across diverse settings indicate that these methods perform strongly relative to existing benchmarks.", "date": "2026-05-09", "categories": []}
{"id": "2605.09509", "title": "Empirical Bayes 1-bit matrix completion", "abstract": "The problem of predicting unobserved entries in a binary matrix, known as 1-bit matrix completion, has found diverse applications in fields such as recommendation systems. In this study, we develop an empirical Bayes method for 1-bit matrix completion motivated by the Efron--Morris estimator, a matrix generalization of the James--Stein estimator that shrinks singular values toward zero. The proposed method exploits the underlying low-rank structure of binary matrices, drawing parallels with multidimensional item response theory. Simulation studies and real-data applications demonstrate that the proposed method achieves a superior balance of predictive accuracy, calibration reliability (uncertainty quantification), and computational efficiency compared to existing methods.", "date": "2026-05-10", "categories": []}
{"id": "2605.09834", "title": "Supercharging Bayesian Inference with Reliable AI-Informed Priors", "abstract": "Modern predictive systems encode beliefs that can act as useful prior information for statistical inference in data-limited settings. Using them for prior construction introduces a tradeoff: an informative prior built from a predictive model can sharpen inference from limited data, but also risks propagating error from the model into the posterior. We propose a framework for AI-informed prior elicitation that mitigates this tension by rectifying the AI-induced law that generates synthetic data before using it to inform a prior. The rectified law can be embedded into synthetic data-driven prior elicitation techniques, including as a base measure in a Dirichlet process (DP) prior on the data-generating process. We refer to the resulting prior and corresponding posterior as the rectified AI prior and rectified AI posterior. We establish Gaussian asymptotics for the rectified AI posterior under non-vanishing prior strength and derive a first-order expression for its centering bias. Our rectified AI priors substantially reduce bias compared to standard approaches, improve the coverage of credible intervals, and make AI-powered prior information more reliable. We additionally apply the rectified AI prior to a real skin disease classification task and show that it can meaningfully boost predictive performance.", "date": "2026-05-11", "categories": []}
{"id": "2605.09839", "title": "Free Energy Manifold: Score-Based Inference for Hybrid Bayesian Networks", "abstract": "We introduce the Free Energy Manifold (FEM), a score-trained conditional energy model specialized for inference in hybrid Bayesian networks with discrete and continuous variables. FEM represents each conditional factor as an energy landscape over learned discrete-parent embeddings and continuous observations, enabling posterior evaluation, generative sampling, and compositional inference across multiple continuous leaves by energy addition under conditional independence.\n  A central finding is the mode-bridge artifact: standard conditional energy models can create low-energy ridges between separated modes of the same class, producing overconfident posteriors at off-data interior points. We analyze this failure and propose valley regularization, an off-data calibration term that restores near-uniform posteriors in such regions while preserving in-data fit.\n  Across synthetic multimodal hybrid-BN benchmarks, FEM substantially reduces KL divergence relative to classical baselines and a vanilla conditional EBM, including large gains at mode-bridge midpoint queries and in multi-leaf evidence composition. We also evaluate high-cardinality discrete-parent settings and a UCI Breast Cancer sanity check, showing that FEM is most useful when multimodal or compositional Bayesian-network inference is required, while discriminative classifiers remain preferable for closed-world classification tasks.", "date": "2026-05-11", "categories": []}
{"id": "2605.10934", "title": "Variational Inference for L\\'evy Process-Driven SDEs via Neural Tilting", "abstract": "Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While L\\'evy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for L\\'evy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in L\\'evy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the L\\'evy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.", "date": "2026-05-11", "categories": []}
{"id": "2512.20974", "title": "Generalised Linear Models in Deep Bayesian RL with Learnable Basis Functions", "abstract": "Bayesian Reinforcement Learning (BRL), a subclass of Meta-Reinforcement Learning (Meta-RL), provides a principled framework for generalisation by explicitly incorporating Bayesian task parameters into transition and reward models. However, classical BRL methods assume known forms of transition and reward models. While recent deep BRL methods incorporate model learning to address this, applying neural networks directly to joint data and task parameters necessitates variational inference. This often yields indistinct task representations, compromising the resulting BRL policies. To overcome these limitations, we introduce Generalised Linear Models in Deep Bayesian RL with Learnable Basis Functions (GLiBRL). Our approach features fully tractable Bayesian inference over task parameters and model noise, alongside exact marginal likelihood evaluation for learning transition and reward models. The permutation-invariant nature of exact Bayesian inference in GLiBRL enables seamless integration with both on-policy and off-policy RL algorithms. We further show that GLiBRL admits a closed-form relationship between the $\\mathcal{L}_2$ distance of its task representations and empirical kernel-based correspondence between task samples, which is to our knowledge the first such structural result for online deep BRL. GLiBRL is compared against representative and recent Meta-RL methods, and improves state-of-the-art performance on both MuJoCo and MetaWorld benchmarks by up to 1.8$\\times$.", "date": "2025-12-24", "categories": []}
{"id": "2604.11995", "title": "Loss-Driven Bayesian Active Learning", "abstract": "The central goal of active learning is to gather data that maximises downstream predictive performance, but popular approaches have limited flexibility in customising this data acquisition to different downstream problems and losses. We propose a rigorous loss-driven approach to Bayesian active learning that allows data acquisition to directly target the loss associated with a given decision problem. In particular, we show how any loss can be used to derive a unique objective for optimal data acquisition. Critically, we then show that any loss taking the form of a weighted Bregman divergence permits analytic computation of a central component of its corresponding objective, making the approach applicable in practice. In regression and classification experiments with a range of different losses, we find our approach reduces test losses relative to existing techniques.", "date": "2026-04-13", "categories": []}
{"id": "2605.07706", "title": "Bayesian Fine-tuning in Projected Subspaces", "abstract": "Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning of large models by decomposing weight updates into low-rank matrices, significantly reducing storage and computational overhead. While effective, standard LoRA lacks mechanisms for uncertainty quantification, leading to overconfident and poorly calibrated models. Bayesian variants of LoRA address this limitation, but at the cost of a significantly increased number of trainable parameters, partially offsetting the original efficiency gains. Additionally, these models are harder to train and may suffer from unstable convergence. In this work, we propose a novel framework for parameter-efficient Bayesian fine-tuning, demonstrating that effective uncertainty quantification can be achieved in very low-dimensional parameter spaces. The proposed method achieves strong performance with improved calibration and generalization while maintaining computational efficiency. Our empirical findings show that, with the appropriate projection of the weight space uncertainty can be effectively modeled in a low-dimensional space, and weight covariances exhibit low ranks.", "date": "2026-05-08", "categories": []}
{"id": "2605.07765", "title": "Pre-trained Tabular Foundation Models as Versatile Summary Networks for Neural Posterior Estimation", "abstract": "In this work, we study TabPFN as a training-free, modular summary network for simulation-based Bayesian inference (SBI). Tabular foundation models such as TabPFN are pretrained on broad families of synthetic tabular data-generating processes and adapt at test time through in-context learning, making them natural candidates for SBI, where posterior estimation often depends on learning informative summaries of simulated observations. We propose PFN-NPE: a general recipe that uses a pretrained TabPFN encoder as a fixed summary network for simulator outputs, then pairs the resulting summaries with a downstream inference head chosen for the problem. With normalizing flows as the default inference head, PFN-NPE matches established posterior approximation methods and sometimes outperforms them. More importantly, diagnostic probes show that the TabPFN-derived summaries often preserve useful posterior location and marginal information. These analyses also reveal a limitation in that TabPFN-derived summaries may struggle to represent the joint posterior structure even when the marginals are well recovered. Still, our experiments show that TabPFN can serve as an effective summary network across a diverse set of SBI settings, with the inference network left modular and task-dependent.", "date": "2026-05-08", "categories": []}
{"id": "2605.07980", "title": "Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning", "abstract": "These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $\\phi$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $\\phi$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $n\\beta$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape.", "date": "2026-05-08", "categories": []}
{"id": "2605.06612", "title": "Online Bayesian Calibration under Gradual and Abrupt System Changes", "abstract": "Bayesian model calibration is central to digital twins and computer experiments, as it aligns model outputs with field observations by estimating calibration parameters and correcting systematic model bias. Classical Bayesian calibration introduces latent parameters and a discrepancy function to model bias, but suffers from parameter--discrepancy confounding and is typically formulated as an offline procedure under a stationary data-generating assumption. These limitations are restrictive in modern digital twin applications, where systems evolve over time and may exhibit gradual drift and abrupt regime shifts. While data assimilation methods enable sequential updates, they generally do not explicitly model systematic bias and are less effective under abrupt changes. We propose Bayesian Recursive Projected Calibration (BRPC), an online Bayesian calibration framework for streaming data under simulator mismatch and nonstationarity. BRPC extends projected calibration to the online setting by separating a discrepancy-free particle update for calibration parameters from a conditional Gaussian process update for discrepancy, preserving identifiability while enabling bias-aware adaptation under gradual system evolution. To handle abrupt changes, BRPC is integrated with restart mechanisms that detect regime shifts and reset the calibration process. We establish theoretical guarantees for both components, including tracking performance under gradual evolution and false-alarm and detection behavior for restart mechanisms. Empirical studies on synthetic and plant-simulation benchmarks show that BRPC improves calibration accuracy under gradual changes, while restart-augmented BRPC further improves robustness and predictive performance under abrupt regime shifts compared to sliding-window Bayesian calibration and data assimilation baselines.", "date": "2026-05-07", "categories": []}
{"id": "2601.02241", "title": "From Mice to Trains: Amortized Bayesian Inference on Graph Data", "abstract": "Graphs arise across diverse domains, from biology and chemistry to social and information networks, as well as in transportation and logistics. Inference on graph-structured data requires methods that are permutation-invariant, scalable across varying sizes and sparsities, and capable of capturing complex long-range dependencies, making posterior estimation on graph parameters particularly challenging. Amortized Bayesian Inference (ABI) is a simulation-based framework that employs generative neural networks to enable fast, likelihood-free posterior inference. We adapt ABI to graph data to address these challenges to perform inference on node-, edge-, and graph-level parameters. Our approach couples permutation-invariant graph encoders with flexible neural posterior estimators in a two-module pipeline: a summary network maps attributed graphs to fixed-length representations, and an inference network approximates the posterior over parameters. In this setting, several neural architectures can serve as the summary network. In this work we evaluate multiple architectures and assess their performance on controlled synthetic settings and two real-world domains - biology and logistics - in terms of recovery and calibration.", "date": "2026-01-05", "categories": []}
{"id": "2602.00387", "title": "Singular Bayesian Neural Networks", "abstract": "Bayesian neural networks promise calibrated uncertainty but require $O(mn)$ parameters for standard mean-field Gaussian posteriors. We argue this cost is often unnecessary, particularly when weight matrices exhibit fast singular value decay. By parameterizing weights as $W = AB^{\\top}$ with $A \\in \\mathbb{R}^{m \\times r}$, $B \\in \\mathbb{R}^{n \\times r}$, we induce a posterior that is \\emph{singular} with respect to the Lebesgue measure, concentrating on the rank-$r$ manifold. This singularity captures structured weight correlations through shared latent factors, geometrically distinct from mean-field's independence assumption. We derive PAC-Bayes generalization bounds whose complexity term scales as $\\sqrt{r(m+n)}$ instead of $\\sqrt{m n}$, and prove loss bounds that decompose the error into optimization and rank-induced bias using the Eckart-Young-Mirsky theorem. We further adapt recent Gaussian complexity bounds for low-rank deterministic networks to Bayesian predictive means. Empirically, across MLPs, LSTMs, and Transformers on standard benchmarks, our method achieves competitive predictive performance while using up to $33\\times$ fewer parameters than 5-member Deep Ensembles. It substantially improves OOD detection and often improves calibration relative to mean-field and perturbation baselines, while Deep Ensembles can still be stronger on in-distribution likelihood-based metrics.", "date": "2026-01-30", "categories": []}
{"id": "2604.06438", "title": "Cost-sensitive retraining via posterior learning debt", "abstract": "Deployed prediction systems are often retrained on fixed calendars, even when model staleness and retraining burden vary over time. This short communication formulates retraining for Bayesian prediction systems as a cost-sensitive predictive-regret decision. The central monitoring state is posterior learning debt, defined as the Kullback--Leibler divergence from a reference shadow posterior to the deployed frozen posterior. In the decision layer, a retraining cost is compared with the expected one-period predictive regret of waiting. A continuous-severity version retrains when calibrated expected regret exceeds the retraining cost, while the familiar two-state excess-loss rule is a special case. The empirical study is an exact-state proof-of-concept in a synthetic conjugate simulation with warm-started deployed and shadow normal-inverse-gamma posteriors, separate update, monitoring, and evaluation batches, lagged deployment actions, expanded baseline grids, and score-unit sensitivity. Under the primary 75th-percentile score-unit scaling, an age-adjusted debt-threshold policy improves on tuned calendar retraining in all 72 non-stable scenario cells and on tuned CUSUM in 58 of 72 cells, with mean relative objectives 0.677 and 0.975, respectively. Debt-utility and hybrid-utility policies also improve strongly over tuned calendar retraining, but they do not dominate tuned CUSUM. Median and mean score-unit sensitivities show the same main calendar result, while the CUSUM comparison remains policy-dependent. The contribution is a transparent decision layer for deployed Bayesian prediction systems, not a universal replacement for drift detection.", "date": "2026-04-07", "categories": []}
{"id": "2605.00966", "title": "Robust volatility updates for Hierarchical Gaussian Filtering", "abstract": "Hierarchical Gaussian Filtering (HGF) networks allow for efficient updating of posterior distributions (beliefs) about hidden states of an agent's environment. HGF parent nodes can target the mean or variance of their children. New information entering at input nodes leads to a cascade of belief updates across the network according to one-step update equations for each node's mean and precision (inverse variance). However, the original form of the update equations for variance-targeting parents(volatility coupling) can in some regions of parameter space lead to negative posterior precision, a logical impossibility which causes the updating algorithm to terminate with an error. In this report, we introduce a modified quadratic approximation to the variational energy of volatility-coupled nodes that avoids negative posterior precision. The key idea is to interpolate between two quadratic expansions of the variational energy: one at the prior prediction and one at a second mode whose location is obtained in closed form via the Lambert W function. The resulting update equations are robust across the entire parameter space and faithfully track the variational posterior even for large prediction errors.", "date": "2026-05-01", "categories": []}
{"id": "2605.01676", "title": "Missingness-aware Data Imputation via AI-powered Bayesian Generative Modeling", "abstract": "Missing data imputation remains a fundamental challenge in modern data science, especially when uncertainty quantification is essential. In this work, we propose MissBGM, an AI-powered missing data imputation method via Bayesian generative modeling that bridges the expressive flexibility of neural networks with the statistical rigor of Bayesian inference. Unlike existing methods that often focus on point estimates or treat the missingness mechanism implicitly, MissBGM explicitly and jointly models the data-generating and missingness mechanisms, providing principled posterior uncertainty over imputations rather than a single point estimate. We develop a stochastic optimization framework with alternating updates among missing values, model parameters, and latent variables until convergence. Our theoretical analysis shows that estimates of missing values from MissBGM converge consistently under mild assumptions. Empirically, we demonstrate that MissBGM achieves superior performance over traditional imputers and recent neural network-based methods across extensive experimental settings. These results establish MissBGM as a principled and scalable solution for modern missing data imputation. The code for MissBGM is open sourced at https://github.com/liuq-lab/MissBGM.", "date": "2026-05-03", "categories": []}
{"id": "2402.14532", "title": "A Framework for Variational Inference of Lightweight Bayesian Neural Networks with Heteroscedastic Uncertainties", "abstract": "Obtaining heteroscedastic predictive uncertainties from a Bayesian Neural Network (BNN) is vital to many applications. Often, heteroscedastic aleatoric uncertainties are learned as outputs of the BNN in addition to the predictive means, however doing so may necessitate adding more learnable parameters to the network. In this work, we demonstrate that both the heteroscedastic aleatoric and epistemic variance can be embedded into the variances of learned BNN parameters, improving predictive performance for lightweight networks. By complementing this approach with a moment propagation approach to inference, we introduce a relatively simple framework for sampling-free variational inference suitable for lightweight BNNs.", "date": "2024-02-22", "categories": []}
{"id": "2603.10252", "title": "Bayesian Hierarchical Models and the Maximum Entropy Principle", "abstract": "Bayesian hierarchical models are frequently used in practical data analysis contexts. One interpretation of these models is that they provide an indirect way of assigning a prior for unknown parameters, through the introduction of hyperparameters. The resulting marginal prior for the parameters (integrating over the hyperparameters) is usually dependent, so that learning one parameter provides some information about the others. In this contribution, I will demonstrate that, when the prior given the hyperparameters is a canonical distribution (a maximum entropy distribution with moment constraints), the dependent marginal prior also has a maximum entropy property, with a different constraint. This constraint is on the marginal distribution of some function of the unknown quantities. The results shed light on what information is actually being assumed when we assign a hierarchical model.", "date": "2026-03-10", "categories": []}
{"id": "2604.26673", "title": "Laplace Approximation for Bayesian Tensor Network Kernel Machines", "abstract": "Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.", "date": "2026-04-29", "categories": []}
{"id": "2604.24911", "title": "Learning with Embedded Linear Equality Constraints via Variational Bayesian Inference", "abstract": "Machine Learning is becoming more prevalent in science and engineering, but many approaches do not provide meaningful uncertainty estimates and predictions may also violate known physical knowledge. We propose a Bayesian framework to embed linear relationships across inputs and outputs into the learning process, whilst characterizing full predictive uncertainty over both the model parameters and the domain knowledge. We evaluated our method on learning the single particle battery model subject to voltage and energy balances, showing its ability to provide reduced credible intervals and constraint violations compared to standard Bayesian neural networks based on variational inference.", "date": "2026-04-27", "categories": []}
{"id": "2504.09499", "title": "Decoding the mechanisms of the Hattrick football manager game using Bayesian network structure learning", "abstract": "Hattrick is a free web-based probabilistic football manager game with over 200,000 users competing for titles at national and international levels. Launched in Sweden in 1997 as part of an MSc project, the game's slow-paced design has fostered a loyal community, with users remaining active for decades. Hattrick's game-engine mechanics are partially hidden, and users have attempted to decode them with incremental success over the years. Rule-based, statistical and machine learning models have been developed to aid this effort and are widely used by the community, but have not been formally evaluated in the scientific literature. This study is the first to explore Hattrick using structure learning techniques and Bayesian networks, integrating expert knowledge with data to develop models that simulate and explain the game-engine. We assess the effectiveness of structure learning algorithms in relation to knowledge-based structures, and publicly share a fully specified Bayesian network model that matches the performance of top models used by the Hattrick community. We further demonstrate how analysis extends beyond prediction by providing a visual representation of dependencies between features, and using the optimal model for in-game decision-making. To support future research, we make all data, graphical structures, and models publicly available online.", "date": "2025-04-13", "categories": []}
{"id": "2604.23102", "title": "Unstable Rankings in Bayesian Deep Learning Evaluation", "abstract": "Standard evaluations of Bayesian deep learning methods assume that metric estimates are reliable, but we show this assumption fails under data scarcity. Method rankings are not only unreliable at small $n$, but also dataset-dependent in ways that point estimates cannot reveal: the same method comparison yields $P(\\mathrm{MCD} \\prec \\mathrm{Ensemble}) = 1.000$ at $n = 50$ on one dataset and remains below $0.95$ even at $n = 500$ on another. Across the datasets we consider, no universal sample size threshold exists, which is precisely why dataset-specific posterior inference is necessary. To address this, we use a Bayesian hierarchical model with method-specific variances to treat evaluation metrics as random variables across data realizations, and we use a predictive Minimum Detectable Difference curve to assess whether an observed gap would be detectable at a given training size. Across six Bayesian deep learning methods and five regression datasets, our results show that uncertainty-aware evaluation is necessary in low-data settings, because current evidence for method superiority and predictive detectability at the same training size can diverge substantially. Our framework provides practitioners with principled tools to determine whether their evaluation data is sufficient before drawing conclusions about method superiority.", "date": "2026-04-25", "categories": []}
{"id": "2604.23114", "title": "A Tale of Two Variances: When Single-Seed Benchmarks Fail in Bayesian Deep Learning", "abstract": "In limited-data settings, a single endpoint mean of an evaluation metric such as the Continuous Ranked Probability Score (CRPS) is itself a random variable, yet it is routinely reported as if it were a stable property of the method. We study when this practice fails. Using 50 independent repetitions across six regression datasets, we show that CRPS variance trajectories differ substantially across methods and are not always well described by a smooth power-law decay. Methods with a learned heteroscedastic variance head, namely MAP and Deep Ensembles, can develop pronounced, reproducible variance peaks at intermediate training sizes on real datasets, whereas MC Dropout and Bayes by Backprop typically show smooth variance contraction. These peaks have direct practical consequences: at the variance peak on Seoul Bike, the relative RMSE of a single-seed MAP estimate reaches 93.6\\%, and the probability of falling within \\(\\pm 10\\%\\) of the repeated-run mean drops to 5.9\\%. We show that local CRPS variance provides a direct signal of single-seed estimation error, with Spearman correlations above 0.96 on every real dataset. Power-law fit quality and monotonicity together provide compact method-level summaries of trajectory regularity. Finally, replacing the standard heteroscedastic objective with \\(\\beta\\)-NLL substantially reduces the irregular behavior, consistent with the view that the heteroscedastic training objective contributes to the instability. Practitioners should report trajectory summaries alongside endpoint means and concentrate repeated evaluation in high-variance regions.", "date": "2026-04-25", "categories": []}
{"id": "2604.07169", "title": "FLUID: Flow-based Unified Inference for Dynamics", "abstract": "Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. In this work, we propose FLUID, a flow-based unified amortized inference framework for filtering and smoothing dynamics. The core idea is to encode each observation history into a fixed-dimensional summary statistic and use this shared representation to learn both a forward flow for the filtering distribution and a backward flow for the backward transition kernel. Specifically, a recurrent encoder maps each observation history to a fixed-dimensional summary statistic whose dimension does not depend on the length of the time series. Conditioned on this shared summary statistic, the forward flow approximates the filtering distribution, while the backward flow approximates the backward transition kernel. The smoothing distribution over an entire trajectory is then recovered by combining the terminal filtering distribution with the learned backward flow through the standard backward recursion. By learning the underlying temporal evolution structure, FLUID also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, FLUID induces an implicit regularization across latent state trajectories and improves trajectory-level smoothing. In addition, we develop a flow-based particle filtering variant that provides an alternative filtering procedure and enables ESS-based diagnostics when explicit model factors are available. Numerical experiments demonstrate that FLUID provides accurate approximations of both filtering distributions and smoothing paths.", "date": "2026-04-08", "categories": []}
{"id": "2603.15055", "title": "Spatio-temporal probabilistic forecast using MMAF-guided learning", "abstract": "We present a theory-guided generalized Bayesian methodology for spatio-temporal raster data, which we use to train an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights. The methodology incorporates the dependence and causal structure of a spatio-temporal Ornstein-Uhlenbeck process into training and inference by enforcing constraints on the design of the data embedding and the related optimization routine. In inference mode, the networks are employed to generate causal ensemble forecasts by applying different initial conditions at different horizons. We call this workflow MMAF-guided learning. Experiments conducted on both synthetic and real data demonstrate that our forecasts remain calibrated across multiple time horizons. Moreover, we show that on such data, shallow feed-forward architectures can achieve performance comparable to, and in some cases better than, convolutional or diffusion deep learning architectures used in probabilistic forecasting tasks.", "date": "2026-03-16", "categories": []}
{"id": "2604.20551", "title": "On Bayesian Softmax-Gated Mixture-of-Experts Models", "abstract": "Mixture-of-experts models provide a flexible framework for learning complex probabilistic input-output relationships by combining multiple expert models through an input-dependent gating mechanism. These models have become increasingly prominent in modern machine learning, yet their theoretical properties in the Bayesian framework remain largely unexplored. In this paper, we study Bayesian mixture-of-experts models, focusing on the ubiquitous softmax-based gating mechanism. Specifically, we investigate the asymptotic behavior of the posterior distribution for three fundamental statistical tasks: density estimation, parameter estimation, and model selection. First, we establish posterior contraction rates for density estimation, both in the regimes with a fixed, known number of experts and with a random learnable number of experts. We then analyze parameter estimation and derive convergence guarantees based on tailored Voronoi-type losses, which account for the complex identifiability structure of mixture-of-experts models. Finally, we propose and analyze two complementary strategies for selecting the number of experts. Taken together, these results provide one of the first systematic theoretical analyses of Bayesian mixture-of-experts models with softmax gating, and yield several theory-grounded insights for practical model design.", "date": "2026-04-22", "categories": []}
{"id": "2510.13763", "title": "PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inference", "abstract": "Amortized simulator-based inference offers a powerful framework for tackling Bayesian inference in computational fields such as engineering or neuroscience, increasingly leveraging modern generative methods like diffusion models to map observed data to model parameters or future predictions. These approaches yield posterior or posterior-predictive samples for new datasets without requiring further simulator calls after training on simulated parameter-data pairs. However, their applicability is often limited by the prior distribution(s) used to generate model parameters during this training phase. To overcome this constraint, we introduce PriorGuide, a technique specifically designed for diffusion-based amortized inference methods. PriorGuide leverages a novel guidance approximation that enables flexible adaptation of the trained diffusion model to new priors at test time, crucially without costly retraining. This allows users to readily incorporate updated information or expert knowledge post-training, enhancing the versatility of pre-trained inference models.", "date": "2025-10-15", "categories": []}
{"id": "2604.18089", "title": "Towards E-Value Based Stopping Rules for Bayesian Deep Ensembles", "abstract": "Bayesian Deep Ensembles (BDEs) represent a powerful approach for uncertainty quantification in deep learning, combining the robustness of Deep Ensembles (DEs) with flexible multi-chain MCMC. While DEs are affordable in most deep learning settings, (long) sampling of Bayesian neural networks can be prohibitively costly. Yet, adding sampling after optimizing the DEs has been shown to yield significant improvements. This leaves a critical practical question: How long should the sequential sampling process continue to yield significant improvements over the initial optimized DE baseline? To tackle this question, we propose a stopping rule based on E-values. We formulate the ensemble construction as a sequential anytime-valid hypothesis test, providing a principled way to decide whether or not to reject the null hypothesis that MCMC offers no improvement over a strong baseline, to early stop the sampling. Empirically, we study this approach for diverse settings. Our results demonstrate the efficacy of our approach and reveal that only a fraction of the full-chain budget is often required.", "date": "2026-04-20", "categories": []}
{"id": "2601.21003", "title": "Bayesian-LoRA: Probabilistic Low-Rank Adaptation of Large Language Models", "abstract": "Large Language Models usually put more emphasis on accuracy and therefore, will guess even when not certain about the prediction, which is especially severe when fine-tuned on small datasets due to the inherent tendency toward miscalibration. In this work, we introduce Bayesian-LoRA, which reformulates the deterministic LoRA update as a probabilistic low-rank representation inspired by Sparse Gaussian Processes. We identify a structural isomorphism between LoRA's factorization and Kronecker-factored SGP posteriors, and show that LoRA emerges as a limiting case when posterior uncertainty collapses. We conduct extensive experiments on various LLM architectures across commonsense reasoning benchmarks. With only approximately 0.42M additional parameters and ${\\approx}1.2{\\times}$ training cost relative to standard LoRA, Bayesian-LoRA significantly improves calibration across models up to 30B, achieving up to 84% ECE reduction and 76% NLL reduction while maintaining competitive accuracy for both in-distribution and out-of-distribution (OoD) evaluations.", "date": "2026-01-28", "categories": []}
{"id": "2603.18640", "title": "A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Sufficient Convergence Conditions and Mixing Time Analysis under Gaussian Targets", "abstract": "The No-U-Turn Sampler (NUTS) is the computational workhorse of modern Bayesian software libraries, yet its qualitative and quantitative convergence guarantees were established only recently. A significant gap remains in the theoretical comparison of its two main variants: NUTS-mul and NUTS-BPS, which use multinomial sampling and biased progressive sampling, respectively, for index selection. In this paper, we address this gap in three contributions. First, we derive the first necessary conditions for geometric ergodicity for both variants. Second, we establish the first sufficient conditions for geometric ergodicity and ergodicity for NUTS-mul. Third, we obtain the first mixing time result for NUTS-BPS on a standard Gaussian distribution. Our results show that NUTS-mul and NUTS-BPS exhibit nearly identical qualitative behavior, with geometric ergodicity depending on the tail properties of the target distribution. However, they differ quantitatively in their convergence rates. More precisely, when initialized in the typical set of the canonical Gaussian measure, the mixing times of both NUTS-mul and NUTS-BPS scale as $O(d^{1/4})$ up to logarithmic factors, where $d$ denotes the dimension. Nevertheless, the associated constants are strictly smaller for NUTS-BPS.", "date": "2026-03-19", "categories": []}
{"id": "2604.12519", "title": "Instantiating Bayesian CVaR lower bounds in Interactive Decision Making Problems", "abstract": "Recent work established a generalized-Fano framework for lower bounding prior-predictive (Bayesian) CVaR in interactive statistical decision making. In this paper, we show how to instantiate that framework in concrete interactive problems and derive explicit Bayesian CVaR lower bounds from its abstract corollaries. Our approach compares a hard model with a reference model using squared Hellinger distance, and combines a lower bound on a reference hinge term with a bound on the distinguishability of the two models. We apply this approach to canonical examples, including Gaussian bandits, and obtain explicit bounds that make the dependence on key problem parameters transparent. These results show how the generalized-Fano Bayesian CVaR framework can be used as a practical lower-bound tool for interactive learning and risk-sensitive decision making.", "date": "2026-04-14", "categories": []}
{"id": "2604.12719", "title": "Monte Carlo Stochastic Depth for Uncertainty Estimation in Deep Learning", "abstract": "The deployment of deep neural networks in safety-critical systems necessitates reliable and efficient uncertainty quantification (UQ). A practical and widespread strategy for UQ is repurposing stochastic regularizers as scalable approximate Bayesian inference methods, such as Monte Carlo Dropout (MCD) and MC-DropBlock (MCDB). However, this paradigm remains under-explored for Stochastic Depth (SD), a regularizer integral to the residual-based backbones of most modern architectures. While prior work demonstrated its empirical promise for segmentation, a formal theoretical connection to Bayesian variational inference and a benchmark on complex, multi-task problems like object detection are missing. In this paper, we first provide theoretical insights connecting Monte Carlo Stochastic Depth (MCSD) to principled approximate variational inference. We then present the first comprehensive empirical benchmark of MCSD against MCD and MCDB on state-of-the-art detectors (YOLO, RT-DETR) using the COCO and COCO-O datasets. Our results position MCSD as a robust and computationally efficient method that achieves highly competitive predictive accuracy (mAP), notably yielding slight improvements in calibration (ECE) and uncertainty ranking (AUARC) compared to MCD. We thus establish MCSD as a theoretically-grounded and empirically-validated tool for efficient Bayesian approximation in modern deep learning.", "date": "2026-04-14", "categories": []}
{"id": "2602.17375", "title": "MDP Planning as Policy Inference", "abstract": "We cast episodic Markov decision process (MDP) planning as Bayesian inference over policies. A policy is treated as the latent variable and is assigned an unnormalized probability of optimality that is monotone in its expected return, yielding a posterior distribution whose modes coincide with return-maximizing solutions while posterior dispersion represents uncertainty over optimal behavior. To approximate this posterior in discrete domains, we adapt variational sequential Monte Carlo (VSMC) to inference over deterministic policies under stochastic dynamics, introducing a sweep that enforces policy consistency across revisited states and couples transition randomness across particles to avoid confounding from simulator noise. Acting is performed by posterior predictive sampling, which induces a stochastic control policy through a Thompson-sampling interpretation rather than entropy regularization. Across grid worlds, Blackjack, Triangle Tireworld, and Academic Advising, we analyze the structure of inferred policy distributions and compare the resulting behavior to discrete Soft Actor-Critic, highlighting qualitative and statistical differences that arise from policy-level uncertainty.", "date": "2026-02-19", "categories": []}
{"id": "2502.02345", "title": "Low Rank Based Subspace Inference for the Laplace Approximation of Bayesian Neural Networks", "abstract": "Subspace inference for neural networks assumes that a subspace of their parameter space suffices to produce a reliable uncertainty quantification. In this work, we underpin the validity of this assumption by using low rank techniques. We derive an expression for a subspace model to a Bayesian inference scenario based on the Laplace approximation that is, in a certain sense, optimal given a specific dataset. We empirically show that a Laplace approximation constructed with a dimensionally reduced covariance matrix closely matches the full Laplace approximation obtained using the exact covariance matrix. Where feasible, this subspace model can serve as a baseline for benchmarking the performance of subspace models. In addition, we provide a scalable approximation of this subspace construction that is usable in practice and compare it to existing subspace models from the literature. In general, our approximation scheme outperforms previous work. Furthermore, we present a metric to qualitatively compare the approximation quality of different subspace models even if the exact Laplace approximation is unknown.", "date": "2025-02-04", "categories": []}
{"id": "2505.08518", "title": "SPP-SBL: Space-Power Prior Sparse Bayesian Learning for Block Sparse Recovery", "abstract": "The recovery of block-sparse signals with unknown structural patterns remains a fundamental challenge in structured sparse signal reconstruction. By proposing a variance transformation framework, this paper unifies existing pattern-based block sparse Bayesian learning methods, and introduces a novel space power prior based on undirected graph models to adaptively capture the unknown patterns of block-sparse signals. By combining the EM algorithm with high-order equation root-solving, we develop a new structured sparse Bayesian learning method, SPP-SBL, which effectively addresses the open problem of space coupling parameter estimation in pattern-based methods. We further demonstrate that learning the relative values of space coupling parameters is key to capturing unknown block-sparse patterns and improving recovery accuracy. Experiments validate that SPP-SBL successfully recovers various challenging structured sparse signals (e.g., chain-structured signals and multi-pattern sparse signals) and real-world multi-modal structured sparse signals (images, audio), showing significant advantages in recovery accuracy across multiple metrics.", "date": "2025-05-13", "categories": []}
{"id": "2511.11666", "title": "Adaptive Stepsizing for Stochastic Gradient Langevin Dynamics in Bayesian Neural Networks", "abstract": "Bayesian neural networks (BNNs) require scalable sampling algorithms to approximate posterior distributions over parameters. Existing stochastic gradient Markov Chain Monte Carlo (SGMCMC) methods are highly sensitive to the choice of stepsize and adaptive variants such as pSGLD typically fail to sample the correct invariant measure without addition of a costly divergence correction term. In this work, we build on the recently proposed `SamAdams' framework for timestep adaptation (Leimkuhler, Lohmann, and Whalley 2025), introducing an adaptive scheme: SA-SGLD, which employs time rescaling to modulate the stepsize according to a monitored quantity (typically the local gradient norm). SA-SGLD can automatically shrink stepsizes in regions of high curvature and expand them in flatter regions, improving both stability and mixing without introducing bias. We show that our method can achieve more accurate posterior sampling than SGLD on high-curvature 2D toy examples and in image classification with BNNs using sharp priors.", "date": "2025-11-11", "categories": []}
{"id": "2302.08724", "title": "Piecewise Deterministic Markov Processes for Bayesian Neural Networks", "abstract": "Inference on modern Bayesian Neural Networks (BNNs) often relies on a variational inference treatment, imposing violated assumptions of independence and the form of the posterior. Traditional MCMC approaches avoid these assumptions at the cost of increased computation due to its incompatibility to subsampling of the likelihood. New Piecewise Deterministic Markov Process (PDMP) samplers permit subsampling, though introduce a model specific inhomogenous Poisson Process (IPPs) which is difficult to sample from. This work introduces a new generic and adaptive thinning scheme for sampling from these IPPs, and demonstrates how this approach can accelerate the application of PDMPs for inference in BNNs. Experimentation illustrates how inference with these methods is computationally feasible, can improve predictive accuracy, MCMC mixing performance, and provide informative uncertainty measurements when compared against other approximate inference schemes.", "date": "2023-02-17", "categories": []}
{"id": "2604.03981", "title": "Multirate Stein Variational Gradient Descent for Efficient Bayesian Sampling", "abstract": "Many particle-based Bayesian inference methods use a single global step size for all parts of the update. In Stein variational gradient descent (SVGD), however, each update combines two qualitatively different effects: attraction toward high-posterior regions and repulsion that preserves particle diversity. These effects can evolve at different rates, especially in high-dimensional, anisotropic, or hierarchical posteriors, so one step size can be unstable in some regions and inefficient in others. We derive a multirate version of SVGD that updates these components on different time scales. The framework yields practical algorithms, including a symmetric split method, a fixed multirate method (MR-SVGD), and an adaptive multirate method (Adapt-MR-SVGD) with local error control. We evaluate the methods in a broad and rigorous benchmark suite covering six problem families: a 50D Gaussian target, multiple 2D synthetic targets, UCI Bayesian logistic regression, multimodal Gaussian mixtures, Bayesian neural networks, and large-scale hierarchical logistic regression. Evaluation includes posterior-matching metrics, predictive performance, calibration quality, mixing, and explicit computational cost accounting. Across these six benchmark families, multirate SVGD variants improve robustness and quality-cost tradeoffs relative to vanilla SVGD. The strongest gains appear on stiff hierarchical, strongly anisotropic, and multimodal targets, where adaptive multirate SVGD is usually the strongest variant and fixed multirate SVGD provides a simpler robust alternative at lower cost.", "date": "2026-04-05", "categories": []}
{"id": "2604.04673", "title": "Minimaxity and Admissibility of Bayesian Neural Networks", "abstract": "Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.", "date": "2026-04-06", "categories": []}
{"id": "2604.04736", "title": "Sampling Parallelism for Fast and Efficient Bayesian Learning", "abstract": "Machine learning models, and deep neural networks in particular, are increasingly deployed in risk-sensitive domains such as healthcare, environmental forecasting, and finance, where reliable quantification of predictive uncertainty is essential. However, many uncertainty quantification (UQ) methods remain difficult to apply due to their substantial computational cost. Sampling-based Bayesian learning approaches, such as Bayesian neural networks (BNNs), are particularly expensive since drawing and evaluating multiple parameter samples rapidly exhausts memory and compute resources. These constraints have limited the accessibility and exploration of Bayesian techniques thus far. To address these challenges, we introduce sampling parallelism, a simple yet powerful parallelization strategy that targets the primary bottleneck of sampling-based Bayesian learning: the samples themselves. By distributing sample evaluations across multiple GPUs, our method reduces memory pressure and training time without requiring architectural changes or extensive hyperparameter tuning. We detail the methodology and evaluate its performance on a few example tasks and architectures, comparing against distributed data parallelism (DDP) as a baseline. We further demonstrate that sampling parallelism is complementary to existing strategies by implementing a hybrid approach that combines sample and data parallelism. Our experiments show near-perfect scaling when the sample number is scaled proportionally to the computational resources, confirming that sample evaluations parallelize cleanly. Although DDP achieves better raw speedups under scaling with constant workload, sampling parallelism has a notable advantage: by applying independent stochastic augmentations to the same batch on each GPU, it increases augmentation diversity and thus reduces the number of epochs required for convergence.", "date": "2026-04-06", "categories": []}
{"id": "2511.13394", "title": "Fast and Robust Simulation-Based Inference With Optimization Monte Carlo", "abstract": "Bayesian parameter inference for complex stochastic simulators is challenging due to intractable likelihood functions. Existing simulation-based inference methods often require large number of simulations and become costly to use in high-dimensional parameter spaces or in problems with partially uninformative outputs. We propose a new method for differentiable simulators that delivers accurate posterior inference with substantially reduced runtimes. Building on the Optimization Monte Carlo framework, our approach reformulates inference for stochastic simulators in terms of deterministic optimization problems. Gradient-based methods are then applied to efficiently navigate toward high-density posterior regions and avoid wasteful simulations in low-probability areas. A JAX-based implementation further enhances the performance through vectorization of key method components. Extensive experiments, including high-dimensional parameter spaces, uninformative outputs, multiple observations and multimodal posteriors show that our method consistently matches, and often exceeds, the accuracy of state-of-the-art approaches, while reducing the runtime by a substantial margin.", "date": "2025-11-17", "categories": []}
{"id": "2604.01021", "title": "Transfer learning for nonparametric Bayesian networks", "abstract": "This paper introduces two transfer learning methodologies for estimating nonparametric Bayesian networks under scarce data. We propose two algorithms, a constraint-based structure learning method, called PC-stable-transfer learning (PCS-TL), and a score-based method, called hill climbing transfer learning (HC-TL). We also define particular metrics to tackle the negative transfer problem in each of them, a situation in which transfer learning has a negative impact on the model's performance. Then, for the parameters, we propose a log-linear pooling approach. For the evaluation, we learn kernel density estimation Bayesian networks, a type of nonparametric Bayesian network, and compare their transfer learning performance with the models alone. To do so, we sample data from small, medium and large-sized synthetic networks and datasets from the UCI Machine Learning repository. Then, we add noise and modifications to these datasets to test their ability to avoid negative transfer. To conclude, we perform a Friedman test with a Bergmann-Hommel post-hoc analysis to show statistical proof of the enhanced experimental behavior of our methods. Thus, PCS-TL and HC-TL demonstrate to be reliable algorithms for improving the learning performance of a nonparametric Bayesian network with scarce data, which in real industrial environments implies a reduction in the required time to deploy the network.", "date": "2026-04-01", "categories": []}
{"id": "2604.02513", "title": "Sparse Bayesian Learning Algorithms Revisited: From Learning Majorizers to Structured Algorithmic Learning using Neural Networks", "abstract": "Sparse Bayesian Learning is one of the most popular sparse signal recovery methods, and various algorithms exist under the SBL paradigm. However, given a performance metric and a sparse recovery problem, it is difficult to know a-priori the best algorithm to choose. This difficulty is in part due to a lack of a unified framework to derive SBL algorithms. We address this issue by first showing that the most popular SBL algorithms can be derived using the majorization-minimization (MM) principle, providing hitherto unknown convergence guarantees to this class of SBL methods. Moreover, we show that the two most popular SBL update rules not only fall under the MM framework but are both valid descent steps for a common majorizer, revealing a deeper analytical compatibility between these algorithms. Using this insight and properties from MM theory we expand the class of SBL algorithms, and address finding the best SBL algorithm via data within the MM framework. Second, we go beyond the MM framework by introducing the powerful modeling capabilities of deep learning to further expand the class of SBL algorithms, aiming to learn a superior SBL update rule from data. We propose a novel deep learning architecture that can outperform the classical MM based ones across different sparse recovery problems. Our architecture's complexity does not scale with the measurement matrix dimension, hence providing a unique opportunity to test generalization capability across different matrices. For parameterized dictionaries, this invariance allows us to train and test the model across different parameter ranges. We also showcase our model's ability to learn a functional mapping by its zero-shot performance on unseen measurement matrices. Finally, we test our model's performance across different numbers of snapshots, signal-to-noise ratios, and sparsity levels.", "date": "2026-04-02", "categories": []}
{"id": "2504.10807", "title": "Power-scaled Bayesian Inference with Score-based Generative Models", "abstract": "We propose a score-based generative algorithm for sampling from power-scaled priors and likelihoods within the Bayesian inference framework. Our algorithm enables flexible control over prior-likelihood influence without requiring retraining for different power-scaling configurations. Specifically, we focus on synthesizing seismic velocity models conditioned on imaged seismic. Our method enables sensitivity analysis by sampling from intermediate power posteriors, allowing us to assess the relative influence of the prior and likelihood on samples of the posterior distribution. Through a comprehensive set of experiments, we evaluate the effects of varying the power parameter in different settings: applying it solely to the prior, to the likelihood of a Bayesian formulation, and to both simultaneously. The results show that increasing the power of the likelihood up to a certain threshold improves the fidelity of posterior samples to the conditioning data (e.g., seismic images), while decreasing the prior power promotes greater structural diversity among samples. Moreover, we find that moderate scaling of the likelihood leads to a reduced shot data residual, confirming its utility in posterior refinement.", "date": "2025-04-15", "categories": []}
{"id": "2510.09534", "title": "Conditional Flow Matching for Bayesian Posterior Inference", "abstract": "We propose a generative multivariate posterior sampler via flow matching. It offers a simple training objective, and does not require access to likelihood evaluation. The method learns a dynamic, block-triangular velocity field in the joint space of data and parameters, which results in a deterministic transport map from a source distribution to the desired posterior. The inverse map, named vector rank, is accessible by reversibly integrating the velocity over time. It is advantageous to leverage the dynamic design: proper constraints on the velocity yield a monotone map, which leads to a conditional Brenier map, enabling a fast and simultaneous generation of Bayesian credible sets whose contours correspond to level sets of Monge-Kantorovich data depth. Our approach is computationally lighter compared to GAN-based and diffusion-based counterparts, and is capable of capturing complex posterior structures. Finally, frequentist theoretical guarantee on the consistency of the recovered posterior distribution, and of the corresponding Bayesian credible sets, is provided.", "date": "2025-10-10", "categories": []}
{"id": "2511.12749", "title": "Taxonomy-Conditioned Hierarchical Bayesian TSB Models for Heterogeneous Intermittent Demand Forecasting", "abstract": "Intermittent demand forecasting poses unique challenges due to sparse observations, cold-start items, and obsolescence. Classical models such as Croston, SBA, and the Teunter--Syntetos--Babai (TSB) method provide simple heuristics but lack a principled generative foundation. We introduce TSB-HB, a hierarchical Bayesian extension of TSB. Demand occurrence is modeled with a Beta--Binomial distribution, while nonzero demand sizes follow a Log-Normal distribution. Crucially, hierarchical priors enable partial pooling across items, stabilizing estimates for sparse or cold-start series while preserving heterogeneity. This framework provides a coherent generative reinterpretation of the classical TSB structure. On the UCI Online Retail dataset, TSB-HB achieves the lowest RMSE and RMSSE among all baselines, while remaining competitive in MAE. On a 5,000-series M5 sample, it improves MAE and RMSE over classical intermittent baselines. Under the calibrated probabilistic configuration, TSB-HB yields competitive pinball loss and a favorable sharpness--calibration tradeoff among the parametric baselines reported in the main text.", "date": "2025-11-16", "categories": []}
{"id": "2509.03317", "title": "Bayesian Additive Regression Trees for functional ANOVA model", "abstract": "Bayesian Additive Regression Trees (BART) is a powerful statistical model that leverages the strengths of Bayesian inference and regression trees. It has received significant attention for capturing complex non-linear relationships and interactions among predictors. However, the accuracy of BART often comes at the cost of interpretability. To address this limitation, we propose ANOVA Bayesian Additive Regression Trees (ANOVA-BART), a novel extension of BART based on the functional ANOVA decomposition, which is used to decompose the variability of a function into different interactions, each representing the contribution of a different set of covariates or factors. Our proposed ANOVA-BART enhances interpretability, preserves and extends the theoretical guarantees of BART, and achieves comparable prediction performance. Specifically, we establish that the posterior concentration rate of ANOVA-BART is nearly minimax optimal, and further provides the same convergence rates for each interaction that are not available for BART. Moreover, comprehensive experiments confirm that ANOVA-BART is comparable to BART in both accuracy and uncertainty quantification, while also demonstrating its effectiveness in component selection. These results suggest that ANOVA-BART offers a compelling alternative to BART by balancing predictive accuracy, interpretability, and theoretical consistency.", "date": "2025-09-03", "categories": []}
{"id": "2307.07753", "title": "Learning Expressive Priors for Generalization and Uncertainty Estimation in Neural Networks", "abstract": "In this work, we propose a novel prior learning method for advancing generalization and uncertainty estimation in deep neural networks. The key idea is to exploit scalable and structured posteriors of neural networks as informative priors with generalization guarantees. Our learned priors provide expressive probabilistic representations at large scale, like Bayesian counterparts of pre-trained models on ImageNet, and further produce non-vacuous generalization bounds. We also extend this idea to a continual learning framework, where the favorable properties of our priors are desirable. Major enablers are our technical contributions: (1) the sums-of-Kronecker-product computations, and (2) the derivations and optimizations of tractable objectives that lead to improved generalization bounds. Empirically, we exhaustively show the effectiveness of this method for uncertainty estimation and generalization.", "date": "2023-07-15", "categories": []}
{"id": "2510.27643", "title": "Bayesian Optimization on Networks", "abstract": "This paper studies optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Mat\\'ern Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Mat\\'ern prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.", "date": "2025-10-31", "categories": []}
{"id": "2511.17339", "title": "ReBaPL: Repulsive Bayesian Prompt Learning", "abstract": "Prompt learning has emerged as an effective technique for fine-tuning large-scale foundation models for downstream tasks. However, conventional prompt learning methods are prone to overfitting and can struggle with out-of-distribution generalization. To address these limitations, Bayesian prompt learning has been proposed, which frames prompt optimization as a Bayesian inference problem to enhance robustness. This paper introduces Repulsive Bayesian Prompt Learning (ReBaPL), a novel method for Bayesian prompt learning, designed to efficiently explore the complex and often multimodal posterior landscape of prompts. Our method integrates a cyclical step-size schedule with a stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm, enabling alternating phases of exploration to discover new modes, and exploitation to refine existing modes. Furthermore, we introduce a repulsive force derived from a potential function over probability metrics (including Maximum Mean Discrepancy and Wasserstein distance) computed on the distributions of representations produced by different prompts. This representation-space repulsion diversifies exploration and prevents premature collapse to a single mode. Our approach allows for a more comprehensive characterization of the prompt posterior distribution, leading to improved generalization. In contrast to prior Bayesian prompt learning methods, our method provides a modular plug-and-play Bayesian extension of any existing prompt learning method based on maximum likelihood estimation. We demonstrate the efficacy of ReBaPL on several benchmark datasets, showing superior performance over state-of-the-art prompt learning methods.", "date": "2025-11-21", "categories": []}
{"id": "2603.26644", "title": "Automatic Laplace Collapsed Sampling: Scalable Marginalisation of Latent Parameters via Automatic Differentiation", "abstract": "We present Automatic Laplace Collapsed Sampling (ALCS), a general framework for marginalising latent parameters in Bayesian models using automatic differentiation, which we combine with nested sampling to explore the hyperparameter space in a robust and efficient manner. At each nested sampling likelihood evaluation, ALCS collapses the high-dimensional latent variables $z$ to a scalar contribution via maximum a posteriori (MAP) optimisation and a Laplace approximation, both computed using autodiff. This reduces the effective dimension from $d_\\theta + d_z$ to just $d_\\theta$, making Bayesian evidence computation tractable for high-dimensional settings without hand-derived gradients or Hessians, and with minimal model-specific engineering. The MAP optimisation and Hessian evaluation are parallelised across live points on GPU-hardware, making the method practical at scale. We also show that automatic differentiation enables local approximations beyond Laplace to parametric families such as the Student-$t$, which improves evidence estimates for heavy-tailed latents. We validate ALCS on a suite of benchmarks spanning hierarchical, time-series, and discrete-likelihood models and establish where the Gaussian approximation holds. This enables a post-hoc ESS diagnostic that localises failures across hyperparameter space without expensive joint sampling.", "date": "2026-03-27", "categories": []}
{"id": "2601.07325", "title": "Robust Bayesian Inference via Variational Approximations of Generalized Rho-Posteriors", "abstract": "We introduce the $\\widetilde{\\rho}$-posterior, a modified version of the $\\rho$-posterior, obtained by replacing the supremum over competitor parameters with a softmax aggregation. This modification allows a PAC-Bayesian analysis of the $\\widetilde{\\rho}$-posterior. This yields finite-sample oracle inequalities with explicit convergence rates that inherit the key robustness properties of the original framework, in particular, graceful degradation under model misspecification and data contamination. Crucially, the PAC-Bayesian oracle inequalities extend to variational approximations of the $\\widetilde{\\rho}$-posterior, providing theoretical guarantees for tractable inference. Numerical experiments on exponential families, regression, and real-world datasets confirm that the resulting variational procedures achieve robustness competitive with theoretical predictions at computational cost comparable to standard variational Bayes.", "date": "2026-01-12", "categories": []}
{"id": "2603.25024", "title": "Improving Infinitely Deep Bayesian Neural Networks with Nesterov's Accelerated Gradient Method", "abstract": "As a representative continuous-depth neural network approach, stochastic differential equation (SDE)-based Bayesian neural networks (BNNs) have attracted considerable attention due to their solid theoretical foundations and strong potential for real-world applications. However, their reliance on numerical SDE solvers inevitably incurs a large number of function evaluations (NFEs), resulting in high computational cost and occasional convergence instability. To address these challenges, we propose a Nesterov-accelerated gradient (NAG) enhanced SDE-BNN model. By integrating NAG into the SDE-BNN framework along with an NFE-dependent residual skip connection, our method accelerates convergence and substantially reduces NFEs during both training and testing. Extensive empirical results show that our model consistently outperforms conventional SDE-BNNs across various tasks, including image classification and sequence modeling, achieving lower NFEs and improved predictive accuracy.", "date": "2026-03-26", "categories": []}
{"id": "2402.08151", "title": "Perturbative adaptive importance sampling for Bayesian LOO cross-validation", "abstract": "Importance sampling (IS) is an efficient stand-in for model refitting in performing (LOO) cross-validation (CV) on a Bayesian model. IS inverts the Bayesian update for a single observation by reweighting posterior samples. The so-called importance weights have high variance -- we resolve this issue through adaptation by transformation. We observe that removing a single observation perturbs the posterior by $\\mathcal{O}(1/n)$, motivating bijective transformations of the form $T(\\theta)=\\theta + h Q(\\theta)$ for $0<h\\ll 1.$ We introduce several such transformations: partial moment matching, which generalizes prior work on affine moment-matching with a tunable step size; log-likelihood descent, which partially invert the Bayesian update for an observation; and gradient flow steps that minimize the KL divergence or IS variance. The gradient flow and likelihood descent transformations require Jacobian determinants, which are available via auto-differentiation; we additionally derive closed-form expressions for logistic regression and shallow ReLU networks. We tested the methodology on classification ($n\\ll p$), count regression (Poisson and zero-inflated negative binomial), and survival analysis problems, finding that no single transformation dominates but their combination nearly eliminates the need to refit.", "date": "2024-02-13", "categories": []}
{"id": "2310.09335", "title": "The surrogate Gibbs-posterior of a corrected stochastic MALA: Towards uncertainty quantification for neural networks", "abstract": "MALA is a popular gradient-based Markov chain Monte Carlo method to access the Gibbs-posterior distribution. Stochastic MALA (sMALA) scales to large data sets, but changes the target distribution from the Gibbs-posterior to a surrogate posterior which only exploits a reduced sample size. We introduce a corrected stochastic MALA (csMALA) with a simple correction term for which distance between the resulting surrogate posterior and the original Gibbs-posterior decreases in the full sample size while retaining scalability. In a nonparametric regression model, we prove a PAC-Bayes oracle inequality for the surrogate posterior. Uncertainties can be quantified by sampling from the surrogate posterior. Focusing on Bayesian neural networks, we analyze the diameter and coverage of credible balls for shallow neural networks and we show optimal contraction rates for deep neural networks. Our credibility result is independent of the correction and can also be applied to the standard Gibbs-posterior. A simulation study in a high-dimensional parameter space demonstrates that an estimator drawn from csMALA based on its surrogate Gibbs-posterior indeed exhibits these advantages in practice.", "date": "2023-10-13", "categories": []}
{"id": "2602.07098", "title": "BayesFlow 2: Multi-Backend Amortized Bayesian Inference in Python", "abstract": "Modern Bayesian inference involves a mixture of computational methods for estimating, validating, and drawing conclusions from probabilistic models as part of principled workflows. An overarching motif of many Bayesian methods is that they are relatively slow, which often becomes prohibitive when fitting complex models to large data sets. Amortized Bayesian inference (ABI) offers a path to solving the computational challenges of Bayes. ABI trains neural networks on model simulations, rewarding users with rapid inference of any model-implied quantity, such as point estimates, likelihoods, or full posterior distributions. In this work, we present the Python library BayesFlow, Version 2.0, for general-purpose ABI. Along with direct posterior, likelihood, and ratio estimation, the software includes support for multiple popular deep learning backends, a rich collection of generative networks for sampling and density estimation, complete customization and high-level interfaces, as well as new capabilities for hyperparameter optimization, design optimization, and hierarchical modeling. Using a case study on dynamical system parameter estimation, combined with comparisons to similar software, we show that our streamlined, user-friendly workflow has strong potential to support broad adoption.", "date": "2026-02-06", "categories": []}
{"id": "2603.22030", "title": "On the Interplay of Priors and Overparametrization in Bayesian Neural Network Posteriors", "abstract": "Bayesian neural network (BNN) posteriors are often considered impractical for inference, as symmetries fragment them, non-identifiabilities inflate dimensionality, and weight-space priors are seen as meaningless. In this work, we study how overparametrization and priors together reshape BNN posteriors and derive implications allowing us to better understand their interplay. We show that redundancy introduces three key phenomena that fundamentally reshape the posterior geometry: balancedness, weight reallocation on equal-probability manifolds, and prior conformity. We validate our findings through extensive experiments with posterior sampling budgets that far exceed those of earlier works, and demonstrate how overparametrization induces structured, prior-aligned weight posterior distributions.", "date": "2026-03-23", "categories": []}
{"id": "2409.19435", "title": "Simulation-based Inference with the Python Package sbijax", "abstract": "Neural simulation-based inference (SBI) describes an emerging family of methods for Bayesian inference with intractable likelihood functions that use neural networks as surrogate models. Here we introduce sbijax, a Python package that implements a wide variety of state-of-the-art methods in neural simulation-based inference using a user-friendly programming interface. sbijax offers high-level functionality to quickly construct SBI estimators, and compute and visualize posterior distributions with only a few lines of code. In addition, the package provides functionality for conventional approximate Bayesian computation, to compute model diagnostics, and to automatically estimate summary statistics. By virtue of being entirely written in JAX, sbijax is extremely computationally efficient, allowing rapid training of neural networks and executing code automatically in parallel on both CPU and GPU.", "date": "2024-09-28", "categories": []}
{"id": "2603.19139", "title": "Hierarchical Latent Structure Learning through Online Inference", "abstract": "Learning systems must balance generalization across experiences with discrimination of task-relevant details. Effective learning therefore requires representations that support both. Online latent-cause models support incremental inference but assume flat partitions, whereas hierarchical Bayesian models capture multilevel structure but typically require offline inference. We introduce the Hierarchical Online Learning of Multiscale Experience Structure (HOLMES) model, a computational framework for hierarchical latent structure learning through online inference. HOLMES combines a variation on the nested Chinese Restaurant Process prior with sequential Monte Carlo inference to perform tractable trial-by-trial inference over hierarchical latent representations without explicit supervision over the latent structure. In simulations, HOLMES matched the predictive performance of flat models while learning more compact representations that supported one-shot transfer to higher-level latent categories. In a context-dependent task with nested temporal structure, HOLMES also improved outcome prediction relative to flat models. These results provide a tractable computational framework for discovering hierarchical structure in sequential data.", "date": "2026-03-19", "categories": []}
{"id": "2603.17063", "title": "Transformers are Bayesian Networks", "abstract": "Transformers are the dominant architecture in AI, yet why they work remains poorly understood. This paper offers a precise answer: a transformer is a Bayesian network. We establish this in five ways.\n  First, we prove that every sigmoid transformer with any weights implements weighted loopy belief propagation on its implicit factor graph. One layer is one round of BP. This holds for any weights -- trained, random, or constructed. Formally verified against standard mathematical axioms.\n  Second, we give a constructive proof that a transformer can implement exact belief propagation on any declared knowledge base. On knowledge bases without circular dependencies this yields provably correct probability estimates at every node. Formally verified against standard mathematical axioms.\n  Third, we prove uniqueness: a sigmoid transformer that produces exact posteriors necessarily has BP weights. There is no other path through the sigmoid architecture to exact posteriors. Formally verified against standard mathematical axioms.\n  Fourth, we delineate the AND/OR boolean structure of the transformer layer: attention is AND, the FFN is OR, and their strict alternation is Pearl's gather/update algorithm exactly.\n  Fifth, we confirm all formal results experimentally, corroborating the Bayesian network characterization in practice. We also establish the practical viability of loopy belief propagation despite the current lack of a theoretical convergence guarantee.\n  We further prove that verifiable inference requires a finite concept space. Any finite verification procedure can distinguish at most finitely many concepts. Without grounding, correctness is not defined. Hallucination is not a bug that scaling can fix. It is the structural consequence of operating without concepts. Formally verified against standard mathematical axioms.", "date": "2026-03-17", "categories": []}
{"id": "2503.17025", "title": "A Guide to Bayesian Networks Software Packages for Structure and Parameter Learning -- 2025 Edition", "abstract": "A representation of the cause-effect mechanism is needed to enable artificial intelligence to represent how the world works. Bayesian Networks (BNs) have proven to be an effective and versatile tool for this task. BNs require constructing a structure of dependencies among variables and learning the parameters that govern these relationships. These tasks, referred to as structural learning and parameter learning, are actively investigated by the research community, with several algorithms proposed and no single method having established itself as standard. A wide range of software, tools, and packages have been developed for BNs analysis and made available to academic researchers and industry practitioners. As a consequence of having no one-size-fits-all solution, moving the first practical steps and getting oriented into this field is proving to be challenging to outsiders and beginners. In this paper, we review the most relevant tools and software for BNs structural and parameter learning to date, providing our subjective recommendations directed to an audience of beginners. In addition, we provide an extensive easy-to-consult overview table summarizing all software packages and their main features. By improving the reader understanding of which available software might best suit their needs, we improve accessibility to the field and make it easier for beginners to take their first step into it.", "date": "2025-03-21", "categories": []}
{"id": "2510.24318", "title": "Transformers can do Bayesian Clustering", "abstract": "Bayesian clustering accounts for uncertainty but is computationally demanding at scale. Furthermore, real-world datasets often contain missing values, and simple imputation ignores the associated uncertainty, resulting in suboptimal results. We present Cluster-PFN, a Transformer-based model that extends Prior-Data Fitted Networks (PFNs) to unsupervised Bayesian clustering. Trained entirely on synthetic datasets generated from a finite Gaussian Mixture Model (GMM) prior, Cluster-PFN learns to estimate the posterior distribution over both the number of clusters and the cluster assignments. Our method estimates the number of clusters more accurately than handcrafted model selection procedures such as AIC, BIC and Variational Inference (VI), and achieves clustering quality competitive with VI while being orders of magnitude faster. Cluster-PFN can be trained on complex priors that include missing data, outperforming imputation-based baselines on real-world genomic datasets, at high missingness. These results show that the Cluster-PFN can provide scalable and flexible Bayesian clustering.", "date": "2025-10-28", "categories": []}
{"id": "2501.10229", "title": "Amortized Bayesian Mixture Models", "abstract": "Finite mixtures are a broad class of models useful in scenarios where observed data is generated by multiple distinct processes but without explicit information about the responsible process for each data point. Estimating Bayesian mixture models is computationally challenging due to issues such as high-dimensional posterior inference and label switching. Furthermore, traditional methods such as MCMC are applicable only if the likelihoods for each mixture component are analytically tractable.\n  Amortized Bayesian Inference (ABI) is a simulation-based framework for estimating Bayesian models using generative neural networks. This allows the fitting of models without explicit likelihoods, and provides fast inference. ABI is therefore an attractive framework for estimating mixture models. This paper introduces a novel extension of ABI tailored to mixture models. We factorize the posterior into a distribution of the parameters and a distribution of (categorical) mixture indicators, which allows us to use a combination of generative neural networks for parameter inference, and classification networks for mixture membership identification. The proposed framework accommodates both independent and dependent mixture models, enabling filtering and smoothing. We validate and demonstrate our approach through synthetic and real-world datasets.", "date": "2025-01-17", "categories": []}
{"id": "2505.20235", "title": "Variational Deep Learning via Implicit Regularization", "abstract": "Modern deep learning models generalize remarkably well in-distribution, despite being overparametrized and trained with little to no explicit regularization. Instead, current theory credits implicit regularization imposed by the choice of architecture, hyperparameters, and optimization procedure. However, deep neural networks can be surprisingly non-robust, resulting in overconfident predictions and poor out-of-distribution generalization. Bayesian deep learning addresses this via model averaging, but typically requires significant computational resources as well as carefully elicited priors to avoid overriding the benefits of implicit regularization. Instead, in this work, we propose to regularize variational neural networks solely by relying on the implicit bias of (stochastic) gradient descent. We theoretically characterize this inductive bias in overparametrized linear models as generalized variational inference and demonstrate the importance of the choice of parametrization. Empirically, our approach demonstrates strong in- and out-of-distribution performance without additional hyperparameter tuning and with minimal computational overhead.", "date": "2025-05-26", "categories": []}
{"id": "2311.02766", "title": "Riemannian Laplace Approximation with the Fisher Metric", "abstract": "Laplace's method approximates a target density with a Gaussian distribution at its mode. It is computationally efficient and asymptotically exact for Bayesian inference due to the Bernstein-von Mises theorem, but for complex targets and finite-data posteriors it is often too crude an approximation. A recent generalization of the Laplace Approximation transforms the Gaussian approximation according to a chosen Riemannian geometry providing a richer approximation family, while still retaining computational efficiency. However, as shown here, its properties depend heavily on the chosen metric, indeed the metric adopted in previous work results in approximations that are overly narrow as well as being biased even at the limit of infinite data. We correct this shortcoming by developing the approximation family further, deriving two alternative variants that are exact at the limit of infinite data, extending the theoretical analysis of the method, and demonstrating practical improvements in a range of experiments.", "date": "2023-11-05", "categories": []}
{"id": "2601.05355", "title": "An AI-powered Bayesian Generative Modeling Approach for Arbitrary Conditional Inference", "abstract": "Modern data analysis increasingly requires flexible conditional inference P(X_B | X_A) where (X_A, X_B) is an arbitrary partition of observed variable X. Existing approaches are either restricted to a fixed conditioning structure or depend strongly on the distribution of conditioning masks during training. To address these limitations, we introduce Bayesian generative modeling (BGM), a unified framework for arbitrary conditional inference. BGM learns a generative model of X via a stochastic iterative Bayesian updating algorithm in which model parameters and latent variables are updated until convergence. Once trained, any conditional distribution can be obtained without retraining. Empirically, BGM achieves superior predictive performance with posterior predictive intervals, demonstrating that a single learned model can serve as a universal engine for conditional prediction with principled uncertainty quantification. We provide theoretical guarantees for convergence of the stochastic iterative algorithm, statistical consistency, and conditional risk bounds. The proposed BGM framework leverages modern AI to capture complex relationships among variables while adhering to Bayesian principles, offering a promising approach for a wide range of applications in modern data science. Code for BGM is available at https://github.com/liuq-lab/bayesgm. Document of BGM is available at https://bayesgm.readthedocs.io.", "date": "2026-01-08", "categories": []}
{"id": "2408.06710", "title": "Variational Learning of Gaussian Process Latent Variable Models through Stochastic Gradient Annealed Importance Sampling", "abstract": "Gaussian Process Latent Variable Models (GPLVMs) have become increasingly popular for unsupervised tasks such as dimensionality reduction and missing data recovery due to their flexibility and non-linear nature. An importance-weighted version of the Bayesian GPLVMs has been proposed to obtain a tighter variational bound. However, this version of the approach is primarily limited to analyzing simple data structures, as the generation of an effective proposal distribution can become quite challenging in high-dimensional spaces or with complex data sets. In this work, we propose an Annealed Importance Sampling (AIS) approach to address these issues. By transforming the posterior into a sequence of intermediate distributions using annealing, we combine the strengths of Sequential Monte Carlo samplers and VI to explore a wider range of posterior distributions and gradually approach the target distribution. We further propose an efficient algorithm by reparameterizing all variables in the evidence lower bound (ELBO). Experimental results on both toy and image datasets demonstrate that our method outperforms state-of-the-art methods in terms of tighter variational bounds, higher log-likelihoods, and more robust convergence.", "date": "2024-08-13", "categories": []}
{"id": "2510.23745", "title": "Bayesian neural networks with interpretable priors from Mercer kernels", "abstract": "Quantifying the uncertainty in the output of a neural network is essential for deployment in scientific or engineering applications where decisions must be made under limited or noisy data. Bayesian neural networks (BNNs) provide a framework for this purpose by constructing a Bayesian posterior distribution over the network parameters. However, the prior, which is of key importance in any Bayesian setting, is rarely meaningful for BNNs. This is because the complexity of the input-to-output map of a BNN makes it difficult to understand how certain distributions enforce any interpretable constraint on the output space of the network. Gaussian processes (GPs), on the other hand, are often preferred in uncertainty quantification tasks due to their interpretability. The drawback is that GPs are limited to small datasets without advanced techniques, which often rely on the covariance kernel having a specific structure. To address these challenges, we introduce a new class of priors for BNNs, called Mercer priors, such that the resulting BNN has samples which approximate that of a specified GP. The method works by defining a prior directly over the network parameters from the Mercer representation of the covariance kernel, and does not rely on the network having a specific structure. In doing so, we can exploit the scalability of BNNs in a meaningful Bayesian way.", "date": "2025-10-27", "categories": []}
{"id": "2501.13483", "title": "Robust Amortized Bayesian Inference with Self-Consistency Losses on Unlabeled Data", "abstract": "Amortized Bayesian inference (ABI) with neural networks can solve probabilistic inverse problems orders of magnitude faster than classical methods. However, ABI is not yet sufficiently robust for widespread and safe application. When performing inference on observations outside the scope of the simulated training data, posterior approximations are likely to become highly biased, which cannot be corrected by additional simulations due to the bad pre-asymptotic behavior of current neural posterior estimators. In this paper, we propose a semi-supervised approach that enables training not only on labeled simulated data generated from the model, but also on \\textit{unlabeled} data originating from any source, including real data. To achieve this, we leverage Bayesian self-consistency properties that can be transformed into strictly proper losses that do not require knowledge of ground-truth parameters. We test our approach on several real-world case studies, including applications to high-dimensional time-series and image data. Our results show that semi-supervised learning with unlabeled data drastically improves the robustness of ABI in the out-of-simulation regime. Notably, inference remains accurate even when evaluated on observations far away from the labeled and unlabeled data seen during training.", "date": "2025-01-23", "categories": []}
{"id": "2501.04272", "title": "On weight and variance uncertainty in neural networks for regression tasks", "abstract": "We investigate the problem of weight uncertainty originally proposed by [Blundell et al. (2015). Weight uncertainty in neural networks. In International conference on machine learning, 1613-1622, PMLR.] in the context of neural networks designed for regression tasks, and we extend their framework by incorporating variance uncertainty into the model. Our analysis demonstrates that explicitly modeling uncertainty in the variance parameter can significantly enhance the predictive performance of Bayesian neural networks. By considering a full posterior distribution over the variance, the model achieves improved generalization compared to approaches that treat variance as fixed or deterministic. We evaluate the generalization capability of our proposed approach through a function approximation example and further validate it on the riboflavin genetic dataset. Our exploration encompasses both fully connected dense networks and dropout neural networks, employing Gaussian and spike-and-slab priors respectively for the network weights, providing a comprehensive assessment of how variance uncertainty affects model performance across different architectural choices.", "date": "2025-01-08", "categories": []}
{"id": "2602.00640", "title": "Combinatorial Bandit Bayesian Optimization for Tensor Outputs", "abstract": "Bayesian optimization (BO) has been widely used to optimize expensive and black-box functions across various domains. However, existing BO methods have not addressed tensor-output functions. To fill this gap, we propose a novel tensor-output BO framework. Specifically, we first introduce a tensor-output Gaussian process (TOGP) with two classes of tensor-output kernels as a surrogate model of the tensor-output function, which can effectively capture the structural dependencies within the tensor. Based on it, we develop an upper confidence bound (UCB) acquisition function to select query points. Furthermore, we introduce a more practical and challenging problem setting, termed combinatorial bandit Bayesian optimization (CBBO), where only a subset of the tensor outputs can be selected to contribute to the objective. To tackle this, we propose a tensor-output CBBO method, which extends TOGP to handle partially observed tensor outputs, and accordingly design a novel combinatorial multi-arm bandit-UCB2 (CMAB-UCB2) criterion to sequentially select both the query points and the output subset. We establish theoretical regret bounds for both methods, guaranteeing sublinear regret. Extensive experiments on synthetic and real-world datasets demonstrate the superiority of our methods.", "date": "2026-01-31", "categories": []}
{"id": "2603.00393", "title": "Dual-space posterior sampling for Bayesian inference in constrained inverse problems", "abstract": "Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy and incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion, which estimates Earth's subsurface properties by fitting seismic measurements subject to the wave equation, where ill-conditioning is inherent to noisy, band-limited, finite-aperture data and shadow zones. Casting the inverse problem into a Bayesian framework allows for a more comprehensive description of its solution, where instead of a single estimate, the posterior distribution characterizes non-uniqueness and can be sampled to quantify uncertainty. However, no clear procedure exists for translating hard physical constraints, such as the wave equation, into prior distributions amenable to existing sampling techniques. To address this, we perform posterior sampling in the dual space using an augmented Lagrangian formulation, which translates hard constraints into penalties amenable to sampling algorithms while ensuring their exact satisfaction. We achieve this by seamlessly integrating the alternating direction method of multipliers (ADMM) with Stein variational gradient descent (SVGD) -- a particle-based sampler -- where the constraint is relaxed at each iteration and multiplier updates progressively enforce satisfaction. This enables constrained posterior sampling while inheriting the favorable conditioning properties of dual-space solvers, where partial constraint relaxation allows productive updates even when the current model is far from the true solution. We validate the method on a stylized Rosenbrock conditional inference problem and on frequency-domain full waveform inversion for a Gaussian anomaly model and the Marmousi~II benchmark, demonstrating well-calibrated uncertainty estimates and posterior contraction with increasing data coverage.", "date": "2026-02-28", "categories": []}
{"id": "2603.00888", "title": "Probabilistic Learning and Generation in Deep Sequence Models", "abstract": "Despite exceptional predictive performance of Deep sequence models (DSMs), the main concern of their deployment centers around the lack of uncertainty awareness. In contrast, probabilistic models quantify the uncertainty associated with unobserved variables with rules of probability. Notably, Bayesian methods leverage Bayes' rule to express our belief of unobserved variables in a principled way. Since exact Bayesian inference is computationally infeasible at scale, approximate inference is required in practice. Two major bottlenecks of Bayesian methods, especially when applied in deep neural networks, are prior specification and approximation quality. In Chapter 3 & 4, we investigate how the architectures of DSMs themselves can be informative for the design of priors or approximations in probabilistic models. We first develop an approximate Bayesian inference method tailored to the Transformer based on the similarity between attention and sparse Gaussian process. Next, we exploit the long-range memory preservation capability of HiPPOs (High-order Polynomial Projection Operators) to construct an interdomain inducing point for Gaussian process, which successfully memorizes the history in online learning. In addition to the progress of DSMs in predictive tasks, sequential generative models consisting of a sequence of latent variables are popularized in the domain of deep generative models. Inspired by the explicit self-supervised signals for these latent variables in diffusion models, in Chapter 5, we explore the possibility of improving other generative models with self-supervision for their sequential latent states, and investigate desired probabilistic structures over them. Overall, this thesis leverages inductive biases in DSMs to design probabilistic inference or structure, which bridges the gap between DSMs and probabilistic models, leading to mutually reinforced improvement.", "date": "2026-03-01", "categories": []}
{"id": "2602.23672", "title": "General Bayesian Policy Learning", "abstract": "This study proposes the General Bayes framework for policy learning. We consider decision problems in which a decision-maker chooses an action from an action set to maximize its expected welfare. Typical examples include treatment choice and portfolio selection. In such problems, the statistical target is a decision rule, and the prediction of each outcome $Y(a)$ is not necessarily of primary interest. We formulate this policy learning problem by loss-based Bayesian updating. Our main technical device is a squared-loss surrogate for welfare maximization. We show that maximizing empirical welfare over a policy class is equivalent to minimizing a scaled squared error in the outcome difference, up to a quadratic regularization controlled by a tuning parameter $\\zeta>0$. This rewriting yields a General Bayes posterior over decision rules that admits a Gaussian pseudo-likelihood interpretation. We clarify two Bayesian interpretations of the resulting generalized posterior, a working Gaussian view and a decision-theoretic loss-based view. As one implementation example, we introduce neural networks with tanh-squashed outputs. Finally, we provide theoretical guarantees in a PAC-Bayes style.", "date": "2026-02-27", "categories": []}
{"id": "2602.22428", "title": "Calibrated Test-Time Guidance for Bayesian Inference", "abstract": "Test-time guidance is a widely used mechanism for steering pretrained diffusion models toward outcomes specified by a reward function. Existing approaches, however, focus on maximizing reward rather than sampling from the true Bayesian posterior, leading to miscalibrated inference. In this work, we show that common test-time guidance methods do not recover the correct posterior distribution and identify the structural approximations responsible for this failure. We then propose consistent alternative estimators that enable calibrated sampling from the Bayesian posterior. We significantly outperform previous methods on a set of Bayesian inference tasks, and match state-of-the-art in black hole image reconstruction.", "date": "2026-02-25", "categories": []}
{"id": "2602.22492", "title": "From Shallow Bayesian Neural Networks to Gaussian Processes: General Convergence, Identifiability and Scalable Inference", "abstract": "In this work, we study scaling limits of shallow Bayesian neural networks (BNNs) via their connection to Gaussian processes (GPs), with an emphasis on statistical modeling, identifiability, and scalable inference. We first establish a general convergence result from BNNs to GPs by relaxing assumptions used in prior formulations, and we compare alternative parameterizations of the limiting GP model. Building on this theory, we propose a new covariance function defined as a convex mixture of components induced by four widely used activation functions, and we characterize key properties including positive definiteness and both strict and practical identifiability under different input designs. For computation, we develop a scalable maximum a posterior (MAP) training and prediction procedure using a Nystr\\\"om approximation, and we show how the Nystr\\\"om rank and anchor selection control the cost-accuracy trade-off. Experiments on controlled simulations and real-world tabular datasets demonstrate stable hyperparameter estimates and competitive predictive performance at realistic computational cost.", "date": "2026-02-26", "categories": []}
{"id": "2602.22884", "title": "Unsupervised Continual Learning for Amortized Bayesian Inference", "abstract": "Amortized Bayesian Inference (ABI) enables efficient posterior estimation using generative neural networks trained on simulated data, but often suffers from performance degradation under model misspecification. While self-consistency (SC) training on unlabeled empirical data can enhance network robustness, current approaches are limited to static, single-task settings and fail to handle sequentially arriving data or distribution shifts. We propose a continual learning framework for ABI that decouples simulation-based pre-training from unsupervised sequential SC fine-tuning on real-world data. To address the challenge of catastrophic forgetting, we introduce two adaptation strategies: (1) SC with episodic replay, utilizing a memory buffer of past observations, and (2) SC with elastic weight consolidation, which regularizes updates to preserve task-critical parameters. Across three diverse case studies, our methods significantly mitigate forgetting and yield posterior estimates that outperform standard simulation-based training, achieving estimates closer to MCMC reference, providing a viable path for trustworthy ABI across a range of different tasks.", "date": "2026-02-26", "categories": []}
{"id": "2602.22925", "title": "Beyond NNGP: Large Deviations and Feature Learning in Bayesian Neural Networks", "abstract": "We study wide Bayesian neural networks focusing on the rare but statistically dominant fluctuations that govern posterior concentration, beyond Gaussian-process limits. Large-deviation theory provides explicit variational objectives-rate functions-on predictors, providing an emerging notion of complexity and feature learning directly at the functional level. We show that the posterior output rate function is obtained by a joint optimization over predictors and internal kernels, in contrast with fixed-kernel (NNGP) theory. Numerical experiments demonstrate that the resulting predictions accurately describe finite-width behavior for moderately sized networks, capturing non-Gaussian tails, posterior deformation, and data-dependent kernel selection effects.", "date": "2026-02-26", "categories": []}
{"id": "2602.21357", "title": "Conditional neural control variates for variance reduction in Bayesian inverse problems", "abstract": "Bayesian inference for inverse problems involves computing expectations under posterior distributions -- e.g., posterior means, variances, or predictive quantities -- typically via Monte Carlo (MC) estimation. When the quantity of interest varies significantly under the posterior, accurate estimates demand many samples -- a cost often prohibitive for partial differential equation-constrained problems. To address this challenge, we introduce conditional neural control variates, a modular method that learns amortized control variates from joint model-data samples to reduce the variance of MC estimators. To scale to high-dimensional problems, we leverage Stein's identity to design an architecture based on an ensemble of hierarchical coupling layers with tractable Jacobian trace computation. Training requires: (i) samples from the joint distribution of unknown parameters and observed data; and (ii) the posterior score function, which can be computed from physics-based likelihood evaluations, neural operator surrogates, or learned generative models such as conditional normalizing flows. Once trained, the control variates generalize across observations without retraining. We validate our approach on stylized and partial differential equation-constrained Darcy flow inverse problems, demonstrating substantial variance reduction, even when the analytical score is replaced by a learned surrogate.", "date": "2026-02-24", "categories": []}
{"id": "2602.22015", "title": "Function-Space Empirical Bayes Regularisation with Student's t Priors", "abstract": "Bayesian deep learning (BDL) has emerged as a principled approach to produce reliable uncertainty estimates by integrating deep neural networks with Bayesian inference, and the selection of informative prior distributions remains a significant challenge. Various function-space variational inference (FSVI) regularisation methods have been presented, assigning meaningful priors over model predictions. However, these methods typically rely on a Gaussian prior, which fails to capture the heavy-tailed statistical characteristics inherent in neural network outputs. By contrast, this work proposes a novel function-space empirical Bayes regularisation framework -- termed ST-FS-EB -- which employs heavy-tailed Student's $t$ priors in both parameter and function spaces. Also, we approximate the posterior distribution through variational inference (VI), inducing an evidence lower bound (ELBO) objective based on Monte Carlo (MC) dropout. Furthermore, the proposed method is evaluated against various VI-based BDL baselines, and the results demonstrate its robust performance in in-distribution prediction, out-of-distribution (OOD) detection and handling distribution shifts.", "date": "2026-02-25", "categories": []}
{"id": "2602.20611", "title": "Amortized Bayesian inference for actigraph time sheet data from mobile devices", "abstract": "Mobile data technologies use ``actigraphs'' to furnish information on health variables as a function of a subject's movement. The advent of wearable devices and related technologies has propelled the creation of health databases consisting of human movement data to conduct research on mobility patterns and health outcomes. Statistical methods for analyzing high-resolution actigraph data depend on the specific inferential context, but the advent of Artificial Intelligence (AI) frameworks require that the methods be congruent to transfer learning and amortization. This article devises amortized Bayesian inference for actigraph time sheets. We pursue a Bayesian approach to ensure full propagation of uncertainty and its quantification using a hierarchical dynamic linear model. We build our analysis around actigraph data from the Physical Activity through Sustainable Transport Approaches in Los Angeles (PASTA-LA) study conducted by the Fielding School of Public Health in the University of California, Los Angeles. Apart from achieving probabilistic imputation of actigraph time sheets, we are also able to statistically learn about the time-varying impact of explanatory variables on the magnitude of acceleration (MAG) for a cohort of subjects.", "date": "2026-02-24", "categories": []}
{"id": "2602.18825", "title": "Bayesian Lottery Ticket Hypothesis", "abstract": "Bayesian neural networks (BNNs) are a useful tool for uncertainty quantification, but require substantially more computational resources than conventional neural networks. For non-Bayesian networks, the Lottery Ticket Hypothesis (LTH) posits the existence of sparse subnetworks that can train to the same or even surpassing accuracy as the original dense network. Such sparse networks can lower the demand for computational resources at inference, and during training. The existence of the LTH and corresponding sparse subnetworks in BNNs could motivate the development of sparse training algorithms and provide valuable insights into the underlying training process. Towards this end, we translate the LTH experiments to a Bayesian setting using common computer vision models. We investigate the defining characteristics of Bayesian lottery tickets, and extend our study towards a transplantation method connecting BNNs with deterministic Lottery Tickets. We generally find that the LTH holds in BNNs, and winning tickets of matching and surpassing accuracy are present independent of model size, with degradation at very high sparsities. However, the pruning strategy should rely primarily on magnitude, secondly on standard deviation. Furthermore, our results demonstrate that models rely on mask structure and weight initialization to varying degrees.", "date": "2026-02-21", "categories": []}
{"id": "2602.19610", "title": "Variational Inference for Bayesian MIDAS Regression", "abstract": "We develop a Coordinate Ascent Variational Inference (CAVI) algorithm for Bayesian Mixed Data Sampling (MIDAS) regression with linear weight parameterizations. The model separates impact coeffcients from weighting function parameters through a normalization constraint, creating a bilinear structure that renders generic Hamiltonian Monte Carlo samplers unreliable while preserving conditional conjugacy exploitable by CAVI. Each variational update admits a closed-form solution: Gaussian for regression coefficients and weight parameters, Inverse-Gamma for the error variance. The algorithm propagates uncertainty across blocks through second moments, distinguishing it from naive plug-in approximations. In a Monte Carlo study spanning 21 data-generating configurations with up to 50 predictors, CAVI produces posterior means nearly identical to a block Gibbs sampler benchmark while achieving speedups of 107x to 1,772x (Table 9). Generic automatic differentiation VI (ADVI), by contrast, produces bias 714 times larger while being orders of magnitude slower, confirming the value of model-specific derivations. Weight function parameters maintain excellent calibration (coverage above 92%) across all configurations. Impact coefficient credible intervals exhibit the underdispersion characteristic of mean-field approximations, with coverage declining from 89% to 55% as the number of predictors grows a documented trade-off between speed and interval calibration that structured variational methods can address. An empirical application to realized volatility forecasting on S&P 500 daily returns cofirms that CAVI and Gibbs sampling yield virtually identical point forecasts, with CAVI completing each monthly estimation in under 10 milliseconds.", "date": "2026-02-23", "categories": []}
{"id": "2602.19859", "title": "Dirichlet Scale Mixture Priors for Bayesian Neural Networks", "abstract": "Neural networks are the cornerstone of modern machine learning, yet can be difficult to interpret, give overconfident predictions and are vulnerable to adversarial attacks. Bayesian neural networks (BNNs) provide some alleviation of these limitations, but have problems of their own. The key step of specifying prior distributions in BNNs is no trivial task, yet is often skipped out of convenience. In this work, we propose a new class of prior distributions for BNNs, the Dirichlet scale mixture (DSM) prior, that addresses current limitations in Bayesian neural networks through structured, sparsity-inducing shrinkage. Theoretically, we derive general dependence structures and shrinkage results for DSM priors and show how they manifest under the geometry induced by neural networks. In experiments on simulated and real world data we find that the DSM priors encourages sparse networks through implicit feature selection, show robustness under adversarial attacks and deliver competitive predictive performance with substantially fewer effective parameters. In particular, their advantages appear most pronounced in correlated, moderately small data regimes, and are more amenable to weight pruning. Moreover, by adopting heavy-tailed shrinkage mechanisms, our approach aligns with recent findings that such priors can mitigate the cold posterior effect, offering a principled alternative to the commonly used Gaussian priors.", "date": "2026-02-23", "categories": []}
{"id": "2602.17809", "title": "Calibrated Adaptation: Bayesian Stiefel Manifold Priors for Reliable Parameter-Efficient Fine-Tuning", "abstract": "Parameter-efficient fine-tuning methods such as LoRA enable practical adaptation of large language models but provide no principled uncertainty estimates, leading to poorly calibrated predictions and unreliable behavior under domain shift. We introduce Stiefel-Bayes Adapters (SBA), a Bayesian PEFT framework that places a Matrix Langevin prior over orthonormal adapter factors on the Stiefel manifold $\\St$ and performs approximate posterior inference via tangent space Laplace approximation with geodesic retraction. Unlike Gaussian priors in flat space projected onto orthogonality constraints, our prior on the manifold naturally encodes the inductive bias that adapter subspaces should be well conditioned and orthogonal, while the posterior provides calibrated predictive uncertainty without recalibration. We prove formally that the tangent space approximation strictly avoids the structural variance inflation inherent in projecting from ambient space, establishing a rigorous theoretical advantage for intrinsic manifold inference. Across GLUE and SuperGLUE benchmarks on RoBERTa-large, LLaMA-2-7B, LLaMA-2-13B, Mistral-7B, and Qwen2.5-7B, domain shift evaluations, selective prediction protocols, and an abstractive summarization task, SBA achieves task performance comparable to LoRA and DoRA while reducing Expected Calibration Error by 18 to 34\\% over deterministic baselines, improving selective prediction AUROC by 12 to 25\\% under domain shift, and outperforming deep ensembles of five LoRA models on OOD detection at a fraction of the parameter cost. Our results demonstrate that where you place uncertainty, on the right geometric structure, matters more than simply adding any Bayesian treatment to adapters.", "date": "2026-02-19", "categories": []}
{"id": "2602.18250", "title": "Variational Distributional Neuron", "abstract": "We propose a proof of concept for a variational distributional neuron: a compute unit formulated as a VAE brick, explicitly carrying a prior, an amortized posterior and a local ELBO. The unit is no longer a deterministic scalar but a distribution: computing is no longer about propagating values, but about contracting a continuous space of possibilities under constraints. Each neuron parameterizes a posterior, propagates a reparameterized sample and is regularized by the KL term of a local ELBO - hence, the activation is distributional. This \"contraction\" becomes testable through local constraints and can be monitored via internal measures. The amount of contextual information carried by the unit, as well as the temporal persistence of this information, are locally tuned by distinct constraints. This proposal addresses a structural tension: in sequential generation, causality is predominantly organized in the symbolic space and, even when latents exist, they often remain auxiliary, while the effective dynamics are carried by a largely deterministic decoder. In parallel, probabilistic latent models capture factors of variation and uncertainty, but that uncertainty typically remains borne by global or parametric mechanisms, while units continue to propagate scalars - hence the pivot question: if uncertainty is intrinsic to computation, why does the compute unit not carry it explicitly? We therefore draw two axes: (i) the composition of probabilistic constraints, which must be made stable, interpretable and controllable; and (ii) granularity: if inference is a negotiation of distributions under constraints, should the primitive unit remain deterministic or become distributional? We analyze \"collapse\" modes and the conditions for a \"living neuron\", then extend the contribution over time via autoregressive priors over the latent, per unit.", "date": "2026-02-20", "categories": []}
{"id": "2409.04332", "title": "Amortized Bayesian Workflow", "abstract": "Bayesian inference often faces a trade-off between computational speed and sampling accuracy. We propose an adaptive workflow that integrates rapid amortized inference with gold-standard MCMC techniques to achieve a favorable combination of both speed and accuracy when performing inference on many observed datasets. Our approach uses principled diagnostics to guide the choice of inference method for each dataset, moving along the Pareto front from fast amortized sampling via generative neural networks to slower but guaranteed-accurate MCMC when needed. By reusing computations across steps, our workflow synergizes amortized and MCMC-based inference. We demonstrate the effectiveness of this integrated approach on several synthetic and real-world problems with tens of thousands of datasets, showing efficiency gains while maintaining high posterior quality.", "date": "2024-09-06", "categories": []}
{"id": "2411.16370", "title": "A Review of Bayesian Uncertainty Quantification in Deep Probabilistic Image Segmentation", "abstract": "Advances in architectural design, data availability, and compute have driven remarkable progress in semantic segmentation. Yet, these models often rely on relaxed Bayesian assumptions, omitting critical uncertainty information needed for robust decision-making. Despite growing interest in probabilistic segmentation to address point-estimate limitations, the research landscape remains fragmented. In response, this review synthesizes foundational concepts in uncertainty modeling, analyzing how feature- and parameter-distribution modeling impact four key segmentation tasks: Observer Variability, Active Learning, Model Introspection, and Model Generalization. Our work establishes a common framework by standardizing theory, notation, and terminology, thereby bridging the gap between method developers, task specialists, and applied researchers. We then discuss critical challenges, including the nuanced distinction between uncertainty types, strong assumptions in spatial aggregation, the lack of standardized benchmarks, and pitfalls in current quantification methods. We identify promising avenues for future research, such as uncertainty-aware active learning, data-driven benchmarks, transformer-based models, and novel techniques to move from simple segmentation problems to uncertainty in holistic scene understanding. Based on our analysis, we offer practical guidelines for researchers on method selection, evaluation, reproducibility, and meaningful uncertainty estimation. Ultimately, our goal is to facilitate the development of more reliable, efficient, and interpretable segmentation models that can be confidently deployed in real-world applications.", "date": "2024-11-25", "categories": []}
{"id": "2602.15925", "title": "Robust Stochastic Gradient Posterior Sampling with Lattice Based Discretisation", "abstract": "Stochastic-gradient MCMC methods enable scalable Bayesian posterior sampling but often suffer from sensitivity to minibatch size and gradient noise. To address this, we propose Stochastic Gradient Lattice Random Walk (SGLRW), an extension of the Lattice Random Walk discretization. Unlike conventional Stochastic Gradient Langevin Dynamics (SGLD), SGLRW introduces stochastic noise only through the off-diagonal elements of the update covariance; this yields greater robustness to minibatch size while retaining asymptotic correctness. Furthermore, as comparison we analyze a natural analogue of SGLD utilizing gradient clipping. Experimental validation on Bayesian regression and classification demonstrates that SGLRW remains stable in regimes where SGLD fails, including in the presence of heavy-tailed gradient noise, and matches or improves predictive performance.", "date": "2026-02-17", "categories": []}
{"id": "2602.16218", "title": "Bayesian Quadrature: Gaussian Processes for Integration", "abstract": "Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.", "date": "2026-02-18", "categories": []}
{"id": "2602.15136", "title": "Universal priors: solving empirical Bayes via Bayesian inference and pretraining", "abstract": "We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of $\\widetilde{O}(\\frac{1}{n})$ uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.", "date": "2026-02-16", "categories": []}
{"id": "2509.19078", "title": "Diffusion Bridge Variational Inference for Deep Gaussian Processes", "abstract": "Deep Gaussian processes (DGPs) enable expressive hierarchical Bayesian modeling but pose substantial challenges for posterior inference, especially over inducing variables. Denoising diffusion variational inference (DDVI) addresses this by modeling the posterior as a time-reversed diffusion from a simple Gaussian prior. However, DDVI's fixed unconditional starting distribution remains far from the complex true posterior, resulting in inefficient inference trajectories and slow convergence. In this work, we propose Diffusion Bridge Variational Inference (DBVI), a principled extension of DDVI that initiates the reverse diffusion from a learnable, data-dependent initial distribution. This initialization is parameterized via an amortized neural network and progressively adapted using gradients from the ELBO objective, reducing the posterior gap and improving sample efficiency. To enable scalable amortization, we design the network to operate on the inducing inputs, which serve as structured, low-dimensional summaries of the dataset and naturally align with the inducing variables' shape. DBVI retains the mathematical elegance of DDVI, including Girsanov-based ELBOs and reverse-time SDEs,while reinterpreting the prior via a Doob-bridged diffusion process. We derive a tractable training objective under this formulation and implement DBVI for scalable inference in large-scale DGPs. Across regression, classification, and image reconstruction tasks, DBVI consistently outperforms DDVI and other variational baselines in predictive accuracy, convergence speed, and posterior quality.", "date": "2025-09-23", "categories": []}
{"id": "2602.11711", "title": "Estimation of instrument and noise parameters for inverse problem based on prior diffusion model", "abstract": "This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is well known to be thorny, and a recent paper proposes a notably simple and effective solution. Consequently, it offers an remarkable additional flexibility when it comes to estimating observation parameters. The proposed strategy enables us to define an optimal estimator for both the observation parameters and the image of interest. Furthermore, the strategy provides a means of quantifying uncertainty. In addition, MCMC algorithms allow for the efficient computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainties quantification.", "date": "2026-02-12", "categories": []}
{"id": "2602.12082", "title": "Empirical Gaussian Processes", "abstract": "Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of standard functions, a process that requires expert knowledge, results in limited adaptivity to data, and imposes strong assumptions on the hypothesis space. We study Empirical GPs, a principled framework for constructing flexible, data-driven GP priors that overcome these limitations. Rather than relying on standard parametric kernels, we estimate the mean and covariance functions empirically from a corpus of historical observations, enabling the prior to reflect rich, non-trivial covariance structures present in the data. Theoretically, we show that the resulting model converges to the GP that is closest (in KL-divergence sense) to the real data generating process. Practically, we formulate the problem of learning the GP prior from independent datasets as likelihood estimation and derive an Expectation-Maximization algorithm with closed-form updates, allowing the model handle heterogeneous observation locations across datasets. We demonstrate that Empirical GPs achieve competitive performance on learning curve extrapolation and time series forecasting benchmarks.", "date": "2026-02-12", "categories": []}
{"id": "2502.03366", "title": "Rethinking Approximate Gaussian Inference in Classification", "abstract": "In classification tasks, softmax functions are ubiquitously used as output activations to produce predictive probabilities. Such outputs only capture aleatoric uncertainty. To capture epistemic uncertainty, approximate Gaussian inference methods have been proposed. We develop a common formalism to describe such methods, which we view as outputting Gaussian distributions over the logit space. Predictives are then obtained as the expectations of the Gaussian distributions pushed forward through the softmax. However, such softmax Gaussian integrals cannot be solved analytically, and Monte Carlo (MC) approximations can be costly and noisy. We propose to replace the softmax activation by element-wise normCDF or sigmoid, which allows for the accurate sampling-free approximation of predictives. This also enables the approximation of the Gaussian pushforwards by Dirichlet distributions with moment matching. This approach entirely eliminates the runtime and memory overhead associated with MC sampling. We evaluate it combined with several approximate Gaussian inference methods (Laplace, HET, SNGP) on large- and small-scale datasets (ImageNet, CIFAR-100, CIFAR-10), demonstrating improved uncertainty quantification capabilities compared to softmax MC sampling. Our code is available at https://github.com/bmucsanyi/probit.", "date": "2025-02-05", "categories": []}
{"id": "2602.10792", "title": "Bayesian Signal Component Decomposition via Diffusion-within-Gibbs Sampling", "abstract": "In signal processing, the data collected from sensing devices is often a noisy linear superposition of multiple components, and the estimation of components of interest constitutes a crucial pre-processing step. In this work, we develop a Bayesian framework for signal component decomposition, which combines Gibbs sampling with plug-and-play (PnP) diffusion priors to draw component samples from the posterior distribution. Unlike many existing methods, our framework supports incorporating model-driven and data-driven prior knowledge into the diffusion prior in a unified manner. Moreover, the proposed posterior sampler allows component priors to be learned separately and flexibly combined without retraining. Under suitable assumptions, the proposed DiG sampler provably produces samples from the posterior distribution. We also show that DiG can be interpreted as an extension of a class of recently proposed diffusion-based samplers, and that, for suitable classes of sensing operators, DiG better exploits the structure of the measurement model. Numerical experiments demonstrate the superior performance of our method over existing approaches.", "date": "2026-02-11", "categories": []}
{"id": "2406.19958", "title": "On the Computational Efficiency of Bayesian Additive Regression Trees: An Asymptotic Analysis", "abstract": "Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression model that is commonly used in causal inference and beyond. Its strong predictive performance is supported by well-developed estimation theory, comprising guarantees that its posterior distribution concentrates around the true regression function at optimal rates under various data generative settings and for appropriate prior choices. However, the computational properties of the widely-used BART sampler proposed by Chipman et al. (2010) are yet to be well-understood. In this paper, we perform an asymptotic analysis of a slightly modified version of the default BART sampler when fitted to data-generating processes with discrete covariates. We show that the sampler's time to convergence, evaluated in terms of the hitting time of a high posterior density set, increases with the number of training samples, due to the multi-modal nature of the target posterior. On the other hand, we show that this trend can be dampened by simple changes, such as increasing the number of trees in the ensemble or raising the temperature of the sampler. These results provide a nuanced picture on the computational efficiency of the BART sampler in the presence of large amounts of training data while suggesting strategies to improve the sampler. We complement our theoretical analysis with a simulation study focusing on the default BART sampler. We observe that the increasing trend of convergence time against number training samples holds for the default BART sampler and is robust to changes in sampler initialization, number of burn-in iterations, feature selection prior, and discretization strategy. On the other hand, increasing the number of trees or raising the temperature sharply dampens this trend, as indicated by our theory.", "date": "2024-06-28", "categories": []}
{"id": "2602.07411", "title": "Nonparametric Bayesian Optimization for General Rewards", "abstract": "This work focuses on Bayesian optimization (BO) under reward model uncertainty. We propose the first BO algorithm that achieves no-regret guarantee in a general reward setting, requiring only Lipschitz continuity of the objective function and accommodating a broad class of measurement noise. The core of our approach is a novel surrogate model, termed as infinite Gaussian process ($\\infty$-GP). It is a Bayesian nonparametric model that places a prior on the space of reward distributions, enabling it to represent a substantially broader class of reward models than classical Gaussian process (GP). The $\\infty$-GP is used in combination with Thompson Sampling (TS) to enable effective exploration and exploitation. Correspondingly, we develop a new TS regret analysis framework for general rewards, which relates the regret to the total variation distance between the surrogate model and the true reward distribution. Furthermore, with a truncated Gibbs sampling procedure, our method is computationally scalable, incurring minimal additional memory and computational complexities compared to classical GP. Empirical results demonstrate state-of-the-art performance, particularly in settings with non-stationary, heavy-tailed, or other ill-conditioned rewards.", "date": "2026-02-07", "categories": []}
{"id": "2602.08782", "title": "Amortising Inference and Meta-Learning Priors in Neural Networks", "abstract": "One of the core facets of Bayesianism is in the updating of prior beliefs in light of new evidence$\\text{ -- }$so how can we maintain a Bayesian approach if we have no prior beliefs in the first place? This is one of the central challenges in the field of Bayesian deep learning, where it is not clear how to represent beliefs about a prediction task by prior distributions over model parameters. Bridging the fields of Bayesian deep learning and probabilistic meta-learning, we introduce a way to $\\textit{learn}$ a weights prior from a collection of datasets by introducing a way to perform per-dataset amortised variational inference. The model we develop can be viewed as a neural process whose latent variable is the set of weights of a BNN and whose decoder is the neural network parameterised by a sample of the latent variable itself. This unique model allows us to study the behaviour of Bayesian neural networks under well-specified priors, use Bayesian neural networks as flexible generative models, and perform desirable but previously elusive feats in neural processes such as within-task minibatching or meta-learning under extreme data-starvation.", "date": "2026-02-09", "categories": []}
{"id": "2512.12742", "title": "A Novel Framework Using Variational Inference with Normalizing Flows to Train Transport Reversible Jump Proposals", "abstract": "We propose a unified framework that employs variational inference (VI) with (conditional) normalizing flows (NFs) to train both between-model and within-model proposals for reversible jump Markov chain Monte Carlo, enabling efficient trans-dimensional Bayesian inference. In contrast to the transport reversible jump (TRJ) of Davies et al. (2023), which optimizes forward KL divergence using pilot samples from the complex target distribution, our approach minimizes the reverse KL divergence, requiring only samples from a simple base distribution and largely reducing computational cost. Especially, we develop a novel trans-dimensional VI method with conditional NFs to fit the conditional transport proposal of Davies et al. (2023). We use RealNVP flows to learn the model-specific transport maps used for constructing proposals so that the calculation is parallelizable. Our framework also provides accurate estimates of marginal likelihoods, which may facilitate efficient model comparison and help design rejection-free proposals. Extensive numerical studies demonstrate that the TRJ method trained under our framework achieves faster mixing compared to existing baselines.", "date": "2025-12-14", "categories": []}
{"id": "2601.21835", "title": "Scalable Linearized Laplace Approximation via Surrogate Neural Kernel", "abstract": "We introduce a scalable method to approximate the kernel of the Linearized Laplace Approximation (LLA). For this, we use a surrogate deep neural network (DNN) that learns a compact feature representation whose inner product replicates the Neural Tangent Kernel (NTK). This avoids the need to compute large Jacobians. Training relies solely on efficient Jacobian-vector products, allowing to compute predictive uncertainty on large-scale pre-trained DNNs. Experimental results show similar or improved uncertainty estimation and calibration compared to existing LLA approximations. Notwithstanding, biasing the learned kernel significantly enhances out-of-distribution detection. This remarks the benefits of the proposed method for finding better kernels than the NTK in the context of LLA to compute prediction uncertainty given a pre-trained DNN.", "date": "2026-01-29", "categories": []}
{"id": "2602.03119", "title": "Function-Space Empirical Bayes Regularisation with Large Vision-Language Model Priors", "abstract": "Bayesian deep learning (BDL) provides a principled framework for reliable uncertainty quantification by combining deep neural networks with Bayesian inference. A central challenge in BDL lies in the design of informative prior distributions that scale effectively to high-dimensional data. Recent functional variational inference (VI) approaches address this issue by imposing priors directly in function space; however, most existing methods rely on Gaussian process (GP) priors, whose expressiveness and generalisation capabilities become limited in high-dimensional regimes. In this work, we propose VLM-FS-EB, a novel function-space empirical Bayes regularisation framework, leveraging large vision-language models (VLMs) to generates semantically meaningful context points. These synthetic samples are then used VLMs for embeddings to construct expressive functional priors. Furthermore, the proposed method is evaluated against various baselines, and experimental results demonstrate that our method consistently improves predictive performance and yields more reliable uncertainty estimates, particularly in out-of-distribution (OOD) detection tasks and data-scarce regimes.", "date": "2026-02-03", "categories": []}
{"id": "2602.03394", "title": "Improving the Linearized Laplace Approximation via Quadratic Approximations", "abstract": "Deep neural networks (DNNs) often produce overconfident out-of-distribution predictions, motivating Bayesian uncertainty quantification. The Linearized Laplace Approximation (LLA) achieves this by linearizing the DNN and applying Laplace inference to the resulting model. Importantly, the linear model is also used for prediction. We argue this linearization in the posterior may degrade fidelity to the true Laplace approximation. To alleviate this problem, without increasing significantly the computational cost, we propose the Quadratic Laplace Approximation (QLA). QLA approximates each second order factor in the approximate Laplace log-posterior using a rank-one factor obtained via efficient power iterations. QLA is expected to yield a posterior precision closer to that of the full Laplace without forming the full Hessian, which is typically intractable. For prediction, QLA also uses the linearized model. Empirically, QLA yields modest yet consistent uncertainty estimation improvements over LLA on five regression datasets.", "date": "2026-02-03", "categories": []}
{"id": "2410.06329", "title": "Joint Bayesian Parameter and Model Order Estimation for Low-Rank Probability Mass Tensors", "abstract": "Obtaining a reliable estimate of the joint probability mass function (PMF) of a set of random variables from observed data is a significant objective in statistical signal processing and machine learning. Modelling the joint PMF as a tensor that admits a low-rank canonical polyadic decomposition (CPD) has enabled the development of efficient PMF estimation algorithms. However, these algorithms require the rank (model order) of the tensor to be specified beforehand. In real-world applications, the true rank is unknown. Therefore, an appropriate rank is usually selected from a candidate set either by observing validation errors or by computing various likelihood-based information criteria, a procedure that could be costly in terms of computational time or hardware resources, or could result in mismatched models which affect the model accuracy. This paper presents a novel Bayesian framework for estimating the low-rank components of a joint PMF tensor and simultaneously inferring its rank from the observed data. We specify a Bayesian PMF estimation model and employ appropriate prior distributions for the model parameters, allowing the rank to be inferred without cross-validation.We then derive a deterministic solution based on variational inference (VI) to approximate the posterior distributions of various model parameters. Numerical experiments involving both synthetic data and real classification and item recommendation data illustrate the advantages of our VI-based method in terms of estimation accuracy, automatic rank detection, and computational efficiency.", "date": "2024-10-08", "categories": []}
{"id": "2602.01898", "title": "Observation-dependent Bayesian active learning via input-warped Gaussian processes", "abstract": "Bayesian active learning relies on the precise quantification of predictive uncertainty to explore unknown function landscapes. While Gaussian process surrogates are the standard for such tasks, an underappreciated fact is that their posterior variance depends on the observed outputs only through the hyperparameters, rendering exploration largely insensitive to the actual measurements. We propose to inject observation-dependent feedback by warping the input space with a learned, monotone reparameterization. This mechanism allows the design policy to expand or compress regions of the input space in response to observed variability, thereby shaping the behavior of variance-based acquisition functions. We demonstrate that while such warps can be trained via marginal likelihood, a novel self-supervised objective yields substantially better performance. Our approach improves sample efficiency across a range of active learning benchmarks, particularly in regimes where non-stationarity challenges traditional methods.", "date": "2026-02-02", "categories": []}
{"id": "2404.09113", "title": "Extending Mean-Field Variational Inference via Entropic Regularization: Theory and Computation", "abstract": "Variational inference (VI) has emerged as a popular method for approximate inference for high-dimensional Bayesian models. In this paper, we propose a novel VI method that extends the naive mean field via entropic regularization, referred to as $\\Xi$-variational inference ($\\Xi$-VI). $\\Xi$-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm. We show that $\\Xi$-variational posteriors effectively recover the true posterior dependency, where the dependence is downweighted by the regularization parameter. We analyze the role of dimensionality of the parameter space on the accuracy of $\\Xi$-variational approximation and how it affects computational considerations, providing a rough characterization of the statistical-computational trade-off in $\\Xi$-VI. We also investigate the frequentist properties of $\\Xi$-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability. We provide sufficient criteria for achieving polynomial-time approximate inference using the method. Finally, we demonstrate the practical advantage of $\\Xi$-VI over mean-field variational inference on simulated and real data.", "date": "2024-04-14", "categories": []}
{"id": "2405.01761", "title": "Multivariate Bayesian Last Layer for Regression with Uncertainty Quantification and Decomposition", "abstract": "We present new Bayesian Last Layer neural network models in the setting of multivariate regression under heteroscedastic noise, and propose EM algorithms for parameter learning. Bayesian modeling of a neural network's final layer has the attractive property of uncertainty quantification with a single forward pass. The proposed framework is capable of disentangling the aleatoric and epistemic uncertainty, and can be used to enhance a canonically trained deep neural network with uncertainty-aware capabilities.", "date": "2024-05-02", "categories": []}
{"id": "2601.22441", "title": "Simulation-based Bayesian inference with ameliorative learned summary statistics -- Part I", "abstract": "This paper, which is Part 1 of a two-part paper series, considers a simulation-based inference with learned summary statistics, in which such a learned summary statistic serves as an empirical-likelihood with ameliorative effects in the Bayesian setting, when the exact likelihood function associated with the observation data and the simulation model is difficult to obtain in a closed form or computationally intractable. In particular, a transformation technique which leverages the Cressie-Read discrepancy criterion under moment restrictions is used for summarizing the learned statistics between the observation data and the simulation outputs, while preserving the statistical power of the inference. Here, such a transformation of data-to-learned summary statistics also allows the simulation outputs to be conditioned on the observation data, so that the inference task can be performed over certain sample sets of the observation data that are considered as an empirical relevance or believed to be particular importance. Moreover, the simulation-based inference framework discussed in this paper can be extended further, and thus handling weakly dependent observation data. Finally, we remark that such an inference framework is suitable for implementation in distributed computing, i.e., computational tasks involving both the data-to-learned summary statistics and the Bayesian inferencing problem can be posed as a unified distributed inference problem that will exploit distributed optimization and MCMC algorithms for supporting large datasets associated with complex simulation models.", "date": "2026-01-30", "categories": []}
{"id": "2601.22539", "title": "Neural-Inspired Posterior Approximation (NIPA)", "abstract": "Humans learn efficiently from their environment by engaging multiple interacting neural systems that support distinct yet complementary forms of control, including model-based (goal-directed) planning, model-free (habitual) responding, and episodic memory-based learning. Model-based mechanisms compute prospective action values using an internal model of the environment, supporting flexible but computationally costly planning; model-free mechanisms cache value estimates and build heuristics that enable fast, efficient habitual responding; and memory-based mechanisms allow rapid adaptation from individual experience. In this work, we aim to elucidate the computational principles underlying this biological efficiency and translate them into a sampling algorithm for scalable Bayesian inference through effective exploration of the posterior distribution. More specifically, our proposed algorithm comprises three components: a model-based module that uses the target distribution for guided but computationally slow sampling; a model-free module that uses previous samples to learn patterns in the parameter space, enabling fast, reflexive sampling without directly evaluating the expensive target distribution; and an episodic-control module that supports rapid sampling by recalling specific past events (i.e., samples). We show that this approach advances Bayesian methods and facilitates their application to large-scale statistical machine learning problems. In particular, we apply our proposed framework to Bayesian deep learning, with an emphasis on proper and principled uncertainty quantification.", "date": "2026-01-30", "categories": []}
{"id": "2601.22765", "title": "Bayesian Matrix Completion Under Geometric Constraints", "abstract": "The completion of a Euclidean distance matrix (EDM) from sparse and noisy observations is a fundamental challenge in signal processing, with applications in sensor network localization, acoustic room reconstruction, molecular conformation, and manifold learning. Traditional approaches, such as rank-constrained optimization and semidefinite programming, enforce geometric constraints but often struggle under sparse or noisy conditions. This paper introduces a hierarchical Bayesian framework that places structured priors directly on the latent point set generating the EDM, naturally embedding geometric constraints. By incorporating a hierarchical prior on latent point set, the model enables automatic regularization and robust noise handling. Posterior inference is performed using a Metropolis-Hastings within Gibbs sampler to handle coupled latent point posterior. Experiments on synthetic data demonstrate improved reconstruction accuracy compared to deterministic baselines in sparse regimes.", "date": "2026-01-30", "categories": []}
{"id": "2601.22899", "title": "Uncertainty-Aware Extrapolation in Bayesian Oblique Trees", "abstract": "Decision trees are widely used due to their interpretability and efficiency, but they struggle in regression tasks that require reliable extrapolation and well-calibrated uncertainty. Piecewise-constant leaf predictions are bounded by the training targets and often become overconfident under distribution shift. We propose a single-tree Bayesian model that extends VSPYCT by equipping each leaf with a GP predictor. Bayesian oblique splits provide uncertainty-aware partitioning of the input space, while GP leaves model local functional behaviour and enable principled extrapolation beyond the observed target range. We present an efficient inference and prediction scheme that combines posterior sampling of split parameters with \\gls{gp} posterior predictions, and a gating mechanism that activates GP-based extrapolation when inputs fall outside the training support of a leaf. Experiments on benchmark regression tasks show improvements in the predictive performance compared to standard variational oblique trees, and substantial performance gains in extrapolation scenarios.", "date": "2026-01-30", "categories": []}
{"id": "2504.02627", "title": "Incorporating the ChEES Criterion into Sequential Monte Carlo Samplers", "abstract": "Markov chain Monte Carlo (MCMC) methods are a powerful but computationally expensive way of performing non-parametric Bayesian inference. MCMC proposals which utilise gradients, such as Hamiltonian Monte Carlo (HMC), can better explore the parameter space of interest if the additional hyper-parameters are chosen well. The No-U-Turn Sampler (NUTS) is a variant of HMC which is extremely effective at selecting these hyper-parameters but is slow to run and is not suited to GPU architectures. An alternative to NUTS, Change in the Estimator of the Expected Square HMC (ChEES-HMC) was shown not only to run faster than NUTS on GPU but also sample from posteriors more efficiently. Sequential Monte Carlo (SMC) samplers are another sampling method which instead output weighted samples from the posterior. They are very amenable to parallelisation and therefore being run on GPUs while having additional flexibility in their choice of proposal over MCMC. We incorporate (ChEEs-HMC) as a proposal into SMC samplers and demonstrate competitive but faster performance than NUTS on a number of tasks.", "date": "2025-04-03", "categories": []}
{"id": "2505.03797", "title": "Utilising Gradient-Based Proposals Within Sequential Monte Carlo Samplers for Training of Partial Bayesian Neural Networks", "abstract": "Partial Bayesian neural networks (pBNNs) have been shown to perform competitively with fully Bayesian neural networks while only having a subset of the parameters be stochastic. Using sequential Monte Carlo (SMC) samplers as the inference method for pBNNs gives a non-parametric probabilistic estimation of the stochastic parameters, and has shown improved performance over parametric methods. In this paper we introduce a new SMC-based training method for pBNNs by utilising a guided proposal and incorporating gradient-based Markov kernels, which gives us better scalability on high dimensional problems. We show that our new method outperforms the state-of-the-art in terms of predictive performance and optimal loss. We also show that pBNNs scale well with larger batch sizes, resulting in significantly reduced training times and often better performance.", "date": "2025-05-01", "categories": []}
{"id": "2507.14057", "title": "Step-DAD: Semi-Amortized Policy-Based Bayesian Experimental Design", "abstract": "We develop a semi-amortized, policy-based, approach to Bayesian experimental design (BED) called Stepwise Deep Adaptive Design (Step-DAD). Like existing, fully amortized, policy-based BED approaches, Step-DAD trains a design policy upfront before the experiment. However, rather than keeping this policy fixed, Step-DAD periodically updates it as data is gathered, refining it to the particular experimental instance. This test-time adaptation improves both the flexibility and the robustness of the design strategy compared with existing approaches. Empirically, Step-DAD consistently demonstrates superior decision-making and robustness compared with current state-of-the-art BED methods.", "date": "2025-07-18", "categories": []}
{"id": "2511.04403", "title": "Online Bayesian Experimental Design for Partially Observed Dynamical Systems", "abstract": "Bayesian experimental design (BED) provides a principled framework for optimizing data collection by choosing experiments that are maximally informative about unknown parameters. However, existing methods cannot deal with the joint challenge of (a) partially observable dynamical systems, where only noisy and incomplete observations are available, and (b) fully online inference, which updates posterior distributions and selects designs sequentially in a computationally efficient manner. Under partial observability, dynamical systems are naturally modeled as state-space models (SSMs), where latent states mediate the link between parameters and data, making the likelihood -- and thus information-theoretic objectives like the expected information gain (EIG) -- intractable. We address these challenges by deriving new estimators of the EIG and its gradient that explicitly marginalize latent states, enabling scalable stochastic optimization in nonlinear SSMs. Our approach leverages nested particle filters for efficient online state-parameter inference with convergence guarantees. Applications to realistic models, such as the susceptible-infectious-recovered (SIR) and a moving source location task, show that our framework successfully handles both partial observability and online inference.", "date": "2025-11-06", "categories": []}
{"id": "2601.21983", "title": "Investigating Batch Inference in a Sequential Monte Carlo Framework for Neural Networks", "abstract": "Bayesian inference allows us to define a posterior distribution over the weights of a generic neural network (NN). Exact posteriors are usually intractable, in which case approximations can be employed. One such approximation - variational inference - is computationally efficient when using mini-batch stochastic gradient descent as subsets of the data are used for likelihood and gradient evaluations, though the approach relies on the selection of a variational distribution which sufficiently matches the form of the posterior. Particle-based methods such as Markov chain Monte Carlo and Sequential Monte Carlo (SMC) do not assume a parametric family for the posterior by typically require higher computational cost. These sampling methods typically use the full-batch of data for likelihood and gradient evaluations, which contributes to this computational expense. We explore several methods of gradually introducing more mini-batches of data (data annealing) into likelihood and gradient evaluations of an SMC sampler. We find that we can achieve up to $6\\times$ faster training with minimal loss in accuracy on benchmark image classification problems using NNs.", "date": "2026-01-29", "categories": []}
{"id": "2601.20043", "title": "Regime-Adaptive Bayesian Optimization via Dirichlet Process Mixtures of Gaussian Processes", "abstract": "Standard Bayesian Optimization (BO) assumes uniform smoothness across the search space an assumption violated in multi-regime problems such as molecular conformation search through distinct energy basins or drug discovery across heterogeneous molecular scaffolds. A single GP either oversmooths sharp transitions or hallucinates noise in smooth regions, yielding miscalibrated uncertainty. We propose RAMBO, a Dirichlet Process Mixture of Gaussian Processes that automatically discovers latent regimes during optimization, each modeled by an independent GP with locally-optimized hyperparameters. We derive collapsed Gibbs sampling that analytically marginalizes latent functions for efficient inference, and introduce adaptive concentration parameter scheduling for coarse-to-fine regime discovery. Our acquisition functions decompose uncertainty into intra-regime and inter-regime components. Experiments on synthetic benchmarks and real-world applications, including molecular conformer optimization, virtual screening for drug discovery, and fusion reactor design, demonstrate consistent improvements over state-of-the-art baselines on multi-regime objectives.", "date": "2026-01-27", "categories": []}
{"id": "2505.08683", "title": "Uncertainty-Aware Surrogate-based Amortized Bayesian Inference for Computationally Expensive Models", "abstract": "Bayesian inference typically relies on a large number of model evaluations to estimate posterior distributions. Established methods like Markov Chain Monte Carlo (MCMC) and Amortized Bayesian Inference (ABI) can become computationally challenging. While ABI enables fast inference after training, generating sufficient training data still requires thousands of model simulations, which is infeasible for expensive models. Surrogate models offer a solution by providing approximate simulations at a lower computational cost, allowing the generation of large data sets for training. However, the introduced approximation errors and uncertainties can lead to overconfident posterior estimates. To address this, we propose Uncertainty-Aware Surrogate-based Amortized Bayesian Inference (UA-SABI) -- a framework that combines surrogate modeling and ABI while explicitly quantifying and propagating surrogate uncertainties through the inference pipeline. Our experiments show that this approach enables reliable, fast, and repeated Bayesian inference for computationally expensive models, even under tight time constraints.", "date": "2025-05-13", "categories": []}
{"id": "2601.16332", "title": "Efficient Gaussian process learning via subspace projections", "abstract": "We propose a novel training objective for GPs constructed using lower-dimensional linear projections of the data, referred to as \\emph{projected likelihood} (PL). We provide a closed-form expression for the information loss related to the PL and empirically show that it can be reduced with random projections on the unit sphere. We show the superiority of the PL, in terms of accuracy and computational efficiency, over the exact GP training and the variational free energy approach to sparse GPs over different optimisers, kernels and datasets of moderately large sizes.", "date": "2026-01-22", "categories": []}
{"id": "2601.19007", "title": "Accelerated training of Gaussian processes using banded square exponential covariances", "abstract": "We propose a novel approach to computationally efficient GP training based on the observation that square-exponential (SE) covariance matrices contain several off-diagonal entries extremely close to zero. We construct a principled procedure to eliminate those entries to produce a \\emph{banded}-matrix approximation to the original covariance, whose inverse and determinant can be computed at a reduced computational cost, thus contributing to an efficient approximation to the likelihood function. We provide a theoretical analysis of the proposed method to preserve the structure of the original covariance in the 1D setting with SE kernel, and validate its computational efficiency against the variational free energy approach to sparse GPs.", "date": "2026-01-26", "categories": []}
{"id": "2312.05153", "title": "Uncertainty Quantification and Propagation in Surrogate-based Bayesian Inference", "abstract": "Surrogate models are statistical or conceptual approximations for more complex simulation models. In this context, it is crucial to propagate the uncertainty induced by limited simulation budget and surrogate approximation error to predictions, inference, and subsequent decision-relevant quantities. However, quantifying and then propagating the uncertainty of surrogates is usually limited to special analytic cases or is otherwise computationally very expensive. In this paper, we propose a framework enabling a scalable, Bayesian approach to surrogate modeling with thorough uncertainty quantification, propagation, and validation. Specifically, we present three methods for Bayesian inference with surrogate models given measurement data. This is a task where the propagation of surrogate uncertainty is especially relevant, because failing to account for it may lead to biased and/or overconfident estimates of the parameters of interest. We showcase our approach in three detailed case studies for linear and nonlinear real-world modeling scenarios. Uncertainty propagation in surrogate models enables more reliable and safe approximation of expensive simulators and will therefore be useful in various fields of applications.", "date": "2023-12-08", "categories": []}
{"id": "2512.14308", "title": "Improving the Accuracy of Amortized Model Comparison with Self-Consistency", "abstract": "Amortized Bayesian inference (ABI) offers fast, scalable approximations to posterior densities by training neural surrogates on data simulated from the statistical model. However, ABI methods are highly sensitive to model misspecification: when observed data fall outside the training distribution (generative scope of the statistical models), neural surrogates can behave unpredictably. This makes it a challenge in a model comparison setting, where multiple statistical models are considered, of which at least some are misspecified. Recent work on self-consistency (SC) provides a promising remedy to this issue, accessible even for empirical data (without ground-truth labels). In this work, we investigate how SC can improve amortized model comparison conceptualized in four different ways. Across two synthetic and two real-world case studies, we find that approaches for model comparison that estimate marginal likelihoods through approximate parameter posteriors consistently outperform methods that directly approximate model evidence or posterior model probabilities. SC training improves robustness when the likelihood is available, even under severe model misspecification. The benefits of SC for methods without access of analytic likelihoods are more limited and inconsistent. Our results suggest practical guidance for reliable amortized Bayesian model comparison: prefer parameter posterior-based methods and augment them with SC training on empirical datasets to mitigate extrapolation bias under model misspecification.", "date": "2025-12-16", "categories": []}
{"id": "2601.18683", "title": "Learned harmonic mean estimation of the marginal likelihood for multimodal posteriors with flow matching", "abstract": "The marginal likelihood, or Bayesian evidence, is a crucial quantity for Bayesian model comparison but its computation can be challenging for complex models, even in parameters space of moderate dimension. The learned harmonic mean estimator has been shown to provide accurate and robust estimates of the marginal likelihood simply using posterior samples. It is agnostic to the sampling strategy, meaning that the samples can be obtained using any method. This enables marginal likelihood calculation and model comparison with whatever sampling is most suitable for the task. However, the internal density estimators considered previously for the learned harmonic mean can struggle with highly multimodal posteriors. In this work we introduce flow matching-based continuous normalizing flows as a powerful architecture for the internal density estimation of the learned harmonic mean. We demonstrate the ability to handle challenging multimodal posteriors, including an example in 20 parameter dimensions, showcasing the method's ability to handle complex posteriors without the need for fine-tuning or heuristic modifications to the base distribution.", "date": "2026-01-26", "categories": []}
{"id": "2505.15638", "title": "Bayesian Ensembling: Insights from Online Optimization and Empirical Bayes", "abstract": "We revisit the classical problem of Bayesian ensembles and address the challenge of learning optimal combinations of Bayesian models in an online, continual learning setting. To this end, we reinterpret existing approaches such as Bayesian model averaging (BMA) and Bayesian stacking through a novel empirical Bayes lens, shedding new light on the limitations and pathologies of BMA. Further motivated by insights from online optimization, we propose Online Bayesian Stacking (OBS), a method that optimizes the log-score over predictive distributions to adaptively combine Bayesian models. A key contribution of our work is establishing a novel connection between OBS and portfolio selection, bridging Bayesian ensemble learning with a rich, well-studied theoretical framework that offers efficient algorithms and extensive regret analysis. We further clarify the relationship between OBS and online BMA, showing that they optimize related but distinct cost functions. Through theoretical analysis and empirical evaluation, we identify scenarios where OBS outperforms online BMA and provide principled methods and guidance on when practitioners should prefer one approach over the other.", "date": "2025-05-21", "categories": []}
{"id": "2601.16425", "title": "Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance", "abstract": "Designing experiments that systematically gather data from complex physical systems is central to accelerating scientific discovery. While Bayesian experimental design (BED) provides a principled, information-based framework that integrates experimental planning with probabilistic inference, the selection of utility functions in BED is a long-standing and active topic, where different criteria emphasize different notions of information. Although Kullback--Leibler (KL) divergence has been one of the most common choices, recent studies have proposed Wasserstein distance as an alternative. In this work, we first employ a toy example to illustrate an issue of Wasserstein distance - the value of Wasserstein distance of a fixed-shape posterior depends on the relative position of its main mass within the support and can exhibit false rewards unrelated to information gain, especially with a non-informative prior (e.g., uniform distribution). We then further provide a systematic comparison between these two criteria through a classical source inversion problem in the BED literature, revealing that the KL divergence tends to lead to faster convergence in the absence of model discrepancy, while Wasserstein metrics provide more robust sequential BED results if model discrepancy is non-negligible. These findings clarify the trade-offs between KL divergence and Wasserstein metrics for the utility function and provide guidelines for selecting suitable criteria in practical BED applications.", "date": "2026-01-23", "categories": []}
{"id": "2509.10132", "title": "Cost-Free Personalization via Information-Geometric Projection in Bayesian Federated Learning", "abstract": "Bayesian Federated Learning (BFL) combines uncertainty modeling with decentralized training, enabling the development of personalized and reliable models under data heterogeneity and privacy constraints. Existing approaches typically rely on Markov Chain Monte Carlo (MCMC) sampling or variational inference, often incorporating personalization mechanisms to better adapt to local data distributions. In this work, we propose an information-geometric projection framework for personalization in parametric BFL. By projecting the global model onto a neighborhood of the user's local model, our method enables a tunable trade-off between global generalization and local specialization. Under mild assumptions, we show that this projection step is equivalent to computing a barycenter on the statistical manifold, allowing us to derive closed-form solutions and achieve cost-free personalization. We apply the proposed approach to a variational learning setup using the Improved Variational Online Newton (IVON) optimizer and extend its application to general aggregation schemes in BFL. Empirical evaluations under heterogeneous data distributions confirm that our method effectively balances global and local performance with minimal computational overhead.", "date": "2025-09-12", "categories": []}
{"id": "2601.12442", "title": "Constraint-Aware Neurosymbolic Uncertainty Quantification with Bayesian Deep Learning for Scientific Discovery", "abstract": "Scientific Artificial Intelligence (AI) applications require models that deliver trustworthy uncertainty estimates while respecting domain constraints. Existing uncertainty quantification methods lack mechanisms to incorporate symbolic scientific knowledge, while neurosymbolic approaches operate deterministically without principled uncertainty modeling. We introduce the Constraint-Aware Neurosymbolic Uncertainty Framework (CANUF), unifying Bayesian deep learning with differentiable symbolic reasoning. The architecture comprises three components: automated constraint extraction from scientific literature, probabilistic neural backbone with variational inference, and differentiable constraint satisfaction layer ensuring physical consistency. Experiments on Materials Project (140,000+ materials), QM9 molecular properties, and climate benchmarks show CANUF reduces Expected Calibration Error by 34.7% versus Bayesian neural networks while maintaining 99.2% constraint satisfaction. Ablations reveal constraint-guided recalibration contributes 18.3% performance gain, with constraint extraction achieving 91.4% precision. CANUF provides the first end-to-end differentiable pipeline simultaneously addressing uncertainty quantification, constraint satisfaction, and interpretable explanations for scientific predictions.", "date": "2026-01-18", "categories": []}
{"id": "2412.18717", "title": "Variational Bayesian Inference for Tensor Robust Principal Component Analysis", "abstract": "Tensor Robust Principal Component Analysis (TRPCA) holds a crucial position in machine learning and computer vision. It aims to recover underlying low-rank structures and to characterize the sparse structures of noise. Current approaches often encounter difficulties in accurately capturing the low-rank properties of tensors and balancing the trade-off between low-rank and sparse components, especially in a mixed-noise scenario. To address these challenges, we introduce a Bayesian framework for TRPCA, which integrates a low-rank tensor nuclear norm prior and a generalized sparsity-inducing prior. By embedding the priors within the Bayesian framework, our method can automatically determine the optimal tensor nuclear norm and achieve a balance between the nuclear norm and sparse components. Furthermore, our method can be efficiently extended to the weighted tensor nuclear norm model. Experiments conducted on synthetic and real-world datasets demonstrate the effectiveness and superiority of our method compared to state-of-the-art approaches.", "date": "2024-12-25", "categories": []}
{"id": "2506.06087", "title": "Multilevel neural simulation-based inference", "abstract": "Neural simulation-based inference (SBI) is a popular set of methods for Bayesian inference when models are only available in the form of a simulator. These methods are widely used in the sciences and engineering, where writing down a likelihood can be significantly more challenging than constructing a simulator. However, the performance of neural SBI can suffer when simulators are computationally expensive, thereby limiting the number of simulations that can be performed. In this paper, we propose a novel approach to neural SBI which leverages multilevel Monte Carlo techniques for settings where several simulators of varying cost and fidelity are available. We demonstrate through both theoretical analysis and extensive experiments that our method can significantly enhance the accuracy of SBI methods given a fixed computational budget.", "date": "2025-06-06", "categories": []}
{"id": "2601.09043", "title": "Horseshoe Mixtures-of-Experts (HS-MoE)", "abstract": "Horseshoe mixtures-of-experts (HS-MoE) models provide a Bayesian framework for sparse expert selection in mixture-of-experts architectures. We combine the horseshoe prior's adaptive global-local shrinkage with input-dependent gating, yielding data-adaptive sparsity in expert usage. Our primary methodological contribution is a particle learning algorithm for sequential inference, in which the filter is propagated forward in time while tracking only sufficient statistics. We also discuss how HS-MoE relates to modern mixture-of-experts layers in large language models, which are deployed under extreme sparsity constraints (e.g., activating a small number of experts per token out of a large pool).", "date": "2026-01-14", "categories": []}
{"id": "2601.05371", "title": "The Kernel Manifold: A Geometric Approach to Gaussian Process Model Selection", "abstract": "Gaussian Process (GP) regression is a powerful nonparametric Bayesian framework, but its performance depends critically on the choice of covariance kernel. Selecting an appropriate kernel is therefore central to model quality, yet remains one of the most challenging and computationally expensive steps in probabilistic modeling. We present a Bayesian optimization framework built on kernel-of-kernels geometry, using expected divergence-based distances between GP priors to explore kernel space efficiently. A multidimensional scaling (MDS) embedding of this distance matrix maps a discrete kernel library into a continuous Euclidean manifold, enabling smooth BO. In this formulation, the input space comprises kernel compositions, the objective is the log marginal likelihood, and featurization is given by the MDS coordinates. When the divergence yields a valid metric, the embedding preserves geometry and produces a stable BO landscape. We demonstrate the approach on synthetic benchmarks, real-world time-series datasets, and an additive manufacturing case study predicting melt-pool geometry, achieving superior predictive accuracy and uncertainty calibration relative to baselines including Large Language Model (LLM)-guided search. This framework establishes a reusable probabilistic geometry for kernel search, with direct relevance to GP modeling and deep kernel learning.", "date": "2026-01-08", "categories": []}
{"id": "2601.07944", "title": "A Statistical Assessment of Amortized Inference Under Signal-to-Noise Variation and Distribution Shift", "abstract": "Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex and large-scale predictive problems. The recent success of deep neural networks and foundation models has now given rise to a new paradigm in statistical modeling, in which Bayesian inference can be amortized through large-scale learned predictors. In amortized inference, substantial computation is invested upfront to train a neural network that can subsequently produce approximate posterior or predictions at negligible marginal cost across a wide range of tasks. At deployment, amortized inference offers substantial computational savings compared with traditional Bayesian procedures, which generally require repeated likelihood evaluations or Monte Carlo simulations for predictions for each new dataset.\n  Despite the growing popularity of amortized inference, its statistical interpretation and its role within Bayesian inference remain poorly understood. This paper presents statistical perspectives on the working principles of several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examines how these architectures naturally support amortized Bayesian inference. We discuss how these models perform structured approximation and probabilistic reasoning in ways that yield controlled generalization error across a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Through simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference under varying signal-to-noise ratios and distributional shifts, highlighting both its strengths and its limitations.", "date": "2026-01-12", "categories": []}
{"id": "2601.07094", "title": "Robust Bayesian Optimization via Tempered Posteriors", "abstract": "Bayesian optimization (BO) iteratively fits a Gaussian process (GP) surrogate to accumulated evaluations and selects new queries via an acquisition function such as expected improvement (EI). In practice, BO often concentrates evaluations near the current incumbent, causing the surrogate to become overconfident and to understate predictive uncertainty in the region guiding subsequent decisions. We develop a robust GP-based BO via tempered posterior updates, which downweight the likelihood by a power $\\alpha \\in (0,1]$ to mitigate overconfidence under local misspecification. We establish cumulative regret bounds for tempered BO under a family of generalized improvement rules, including EI, and show that tempering yields strictly sharper worst-case regret guarantees than the standard posterior $(\\alpha=1)$, with the most favorable guarantees occurring near the classical EI choice.\n  Motivated by our theoretic findings, we propose a prequential procedure for selecting $\\alpha$ online: it decreases $\\alpha$ when realized prediction errors exceed model-implied uncertainty and returns $\\alpha$ toward one as calibration improves. Empirical results demonstrate that tempering provides a practical yet theoretically grounded tool for stabilizing BO surrogates under localized sampling.", "date": "2026-01-11", "categories": []}
{"id": "2601.07413", "title": "The Practicality of Normalizing Flow Test-Time Training in Bayesian Inference for Agent-Based Models", "abstract": "Agent-Based Models (ABMs) are gaining great popularity in economics and social science because of their strong flexibility to describe the realistic and heterogeneous decisions and interaction rules between individual agents. In this work, we investigate for the first time the practicality of test-time training (TTT) of deep models such as normalizing flows, in the parameters posterior estimations of ABMs. We propose several practical TTT strategies for fine-tuning the normalizing flow against distribution shifts. Our numerical study demonstrates that TTT schemes are remarkably effective, enabling real-time adjustment of flow-based inference for ABM parameters.", "date": "2026-01-12", "categories": []}
{"id": "2503.16222", "title": "Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems", "abstract": "This paper studies plug-and-play (PnP) Langevin sampling strategies for Bayesian inference in low-photon Poisson imaging problems, a challenging class of problems with significant applications in astronomy, medicine, and biology. PnP Langevin sampling offers a powerful framework for Bayesian image restoration, enabling accurate point estimation as well as advanced inference tasks, including uncertainty quantification and visualization analyses, and empirical Bayesian inference for automatic model parameter tuning. Herein, we leverage and adapt recent developments in this framework to tackle challenging imaging problems involving weakly informative Poisson data. Existing PnP Langevin algorithms are not well-suited for low-photon Poisson imaging due to high solution uncertainty and poor regularity properties, such as exploding gradients and non-negativity constraints. To address these challenges, we explore two strategies for extending Langevin PnP sampling to Poisson imaging models: (i) an accelerated PnP Langevin method that incorporates boundary reflections and a Poisson likelihood approximation and (ii) a mirror sampling algorithm that leverages a Riemannian geometry to handle the constraints and the poor regularity of the likelihood without approximations. The effectiveness of these approaches is evaluated and contrasted through extensive numerical experiments and comparisons with state-of-the-art methods. The source code accompanying this paper is available at https://github.com/freyyia/pnp-langevin-poisson.", "date": "2025-03-20", "categories": []}
{"id": "2601.05273", "title": "Bayesian Recovery for Probabilistic Coalition Structures", "abstract": "Probabilistic Coalition Structure Generation (PCSG) is NP-hard and can be recast as an $l_0$-type sparse recovery problem by representing coalition structures as sparse coefficient vectors over a coalition-incidence design. A natural question is whether standard sparse methods, such as $l_1$ relaxations and greedy pursuits, can reliably recover the optimal coalition structure in this setting. We show that the answer is negative in a PCSG-inspired regime where overlapping coalitions generate highly coherent, near-duplicate columns: the irrepresentable condition fails for the design, and $k$-step Orthogonal Matching Pursuit (OMP) exhibits a nonvanishing probability of irreversible mis-selection. In contrast, we prove that Sparse Bayesian Learning (SBL) with a Gaussian-Gamma hierarchy is support consistent under the same structural assumptions. The concave sparsity penalty induced by SBL suppresses spurious near-duplicates and recovers the true coalition support with probability tending to one. This establishes a rigorous separation between convex, greedy, and Bayesian sparse approaches for PCSG.", "date": "2025-12-25", "categories": []}
{"id": "2601.05586", "title": "Poisson Hyperplane Processes with Rectified Linear Units", "abstract": "Neural networks have shown state-of-the-art performances in various classification and regression tasks. Rectified linear units (ReLU) are often used as activation functions for the hidden layers in a neural network model. In this article, we establish the connection between the Poisson hyperplane processes (PHP) and two-layer ReLU neural networks. We show that the PHP with a Gaussian prior is an alternative probabilistic representation to a two-layer ReLU neural network. In addition, we show that a two-layer neural network constructed by PHP is scalable to large-scale problems via the decomposition propositions. Finally, we propose an annealed sequential Monte Carlo algorithm for Bayesian inference. Our numerical experiments demonstrate that our proposed method outperforms the classic two-layer ReLU neural network. The implementation of our proposed model is available at https://github.com/ShufeiGe/Pois_Relu.git.", "date": "2026-01-09", "categories": []}
{"id": "2503.02108", "title": "Correcting Mode Proportion Bias in Generalized Bayesian Inference via a Weighted Kernel Stein Discrepancy", "abstract": "Generalized Bayesian Inference (GBI) provides a flexible framework for updating prior distributions using various loss functions instead of the traditional likelihoods, thereby enhancing the model robustness to model misspecification. However, GBI often suffers the problem associated with intractable likelihoods. Kernelized Stein Discrepancy (KSD), as utilized in a recent study, addresses this challenge by relying only on the gradient of the log-likelihood. Despite this innovation, KSD-Bayes suffers from critical pathologies, including insensitivity to well-separated modes in multimodal posteriors. To address this limitation, we propose a weighted KSD method that retains computational efficiency while effectively capturing multimodal structures. Our method improves the GBI framework for handling intractable multimodal posteriors while maintaining key theoretical properties such as posterior consistency and asymptotic normality. Experimental results demonstrate that our method substantially improves mode sensitivity compared to standard KSD-Bayes, while retaining robust performance in unimodal settings and in the presence of outliers.", "date": "2025-03-03", "categories": []}
{"id": "2512.24381", "title": "Tubular Riemannian Laplace Approximations for Bayesian Neural Networks", "abstract": "Laplace approximations are among the simplest and most practical methods for approximate Bayesian inference in neural networks, yet their Euclidean formulation struggles with the highly anisotropic, curved loss surfaces and large symmetry groups that characterize modern deep models. Recent work has proposed Riemannian and geometric Gaussian approximations to adapt to this structure. Building on these ideas, we introduce the Tubular Riemannian Laplace (TRL) approximation. TRL explicitly models the posterior as a probabilistic tube that follows a low-loss valley induced by functional symmetries, using a Fisher/Gauss-Newton metric to separate prior-dominated tangential uncertainty from data-dominated transverse uncertainty. We interpret TRL as a scalable reparametrised Gaussian approximation that utilizes implicit curvature estimates to operate in high-dimensional parameter spaces. Our empirical evaluation on ResNet-18 (CIFAR-10 and CIFAR-100) demonstrates that TRL achieves excellent calibration, matching or exceeding the reliability of Deep Ensembles (in terms of ECE) while requiring only a fraction (1/5) of the training cost. TRL effectively bridges the gap between single-model efficiency and ensemble-grade reliability.", "date": "2025-12-30", "categories": []}
{"id": "2601.01238", "title": "Evidence Slopes and Effective Dimension in Singular Linear Models", "abstract": "Bayesian model selection commonly relies on Laplace approximation or the Bayesian Information Criterion (BIC), which assume that the effective model dimension equals the number of parameters. Singular learning theory replaces this assumption with the real log canonical threshold (RLCT), an effective dimension that can be strictly smaller in overparameterized or rank-deficient models.\n  We study linear-Gaussian rank models and linear subspace (dictionary) models in which the exact marginal likelihood is available in closed form and the RLCT is analytically tractable. In this setting, we show theoretically and empirically that the error of Laplace/BIC grows linearly with (d/2 minus lambda) times log n, where d is the ambient parameter dimension and lambda is the RLCT. An RLCT-aware correction recovers the correct evidence slope and is invariant to overcomplete reparameterizations that represent the same data subspace.\n  Our results provide a concrete finite-sample characterization of Laplace failure in singular models and demonstrate that evidence slopes can be used as a practical estimator of effective dimension in simple linear settings.", "date": "2026-01-03", "categories": []}
{"id": "2601.01442", "title": "Fast Gibbs Sampling on Bayesian Hidden Markov Model with Missing Observations", "abstract": "The Hidden Markov Model (HMM) is a widely-used statistical model for handling sequential data. However, the presence of missing observations in real-world datasets often complicates the application of the model. The EM algorithm and Gibbs samplers can be used to estimate the model, yet suffering from various problems including non-convexity, high computational complexity and slow mixing. In this paper, we propose a collapsed Gibbs sampler that efficiently samples from HMMs' posterior by integrating out both the missing observations and the corresponding latent states. The proposed sampler is fast due to its three advantages. First, it achieves an estimation accuracy that is comparable to existing methods. Second, it can produce a larger Effective Sample Size (ESS) per iteration, which can be justified theoretically and numerically. Third, when the number of missing entries is large, the sampler has a significant smaller computational complexity per iteration compared to other methods, thus is faster computationally. In summary, the proposed sampling algorithm is fast both computationally and theoretically and is particularly advantageous when there are a lot of missing entries. Finally, empirical evaluations based on numerical simulations and real data analysis demonstrate that the proposed algorithm consistently outperforms existing algorithms in terms of time complexity and sampling efficiency (measured in ESS).", "date": "2026-01-04", "categories": []}
{"id": "2412.20385", "title": "A Particle Algorithm for Mean-Field Variational Inference", "abstract": "Variational inference is a fast and scalable alternative to Markov chain Monte Carlo and has been widely applied to posterior inference tasks in statistics and machine learning. A traditional approach for implementing mean-field variational inference (MFVI) is coordinate ascent variational inference (CAVI), which relies crucially on parametric assumptions on complete conditionals. We introduce a novel particle-based algorithm for MFVI, named PArticle VI (PAVI), for nonparametric mean-field approximation. We obtain non-asymptotic error bounds for our algorithm. To our knowledge, this is the first end-to-end guarantee for particle-based MFVI.", "date": "2024-12-29", "categories": []}
{"id": "2512.23728", "title": "Spike-Timing-Dependent Plasticity for Bernoulli Message Passing", "abstract": "Bayesian inference provides a principled framework for understanding brain function, while neural activity in the brain is inherently spike-based. This paper bridges these two perspectives by designing spiking neural networks that simulate Bayesian inference through message passing for Bernoulli messages. To train the networks, we employ spike-timing-dependent plasticity, a biologically plausible mechanism for synaptic plasticity which is based on the Hebbian rule. Our results demonstrate that the network's performance closely matches the true numerical solution. We further demonstrate the versatility of our approach by implementing a factor graph example from coding theory, illustrating signal transmission over an unreliable channel.", "date": "2025-12-19", "categories": []}
{"id": "2512.24515", "title": "Improving the stability of the covariance-controlled adaptive Langevin thermostat for large-scale Bayesian sampling", "abstract": "Stochastic gradient Langevin dynamics and its variants approximate the likelihood of an entire dataset, via random (and typically much smaller) subsets, in the setting of Bayesian sampling. Due to the (often substantial) improvement of the computational efficiency, they have been widely used in large-scale machine learning applications. It has been demonstrated that the so-called covariance-controlled adaptive Langevin (CCAdL) thermostat, which incorporates an additional term involving the covariance matrix of the noisy force, outperforms popular alternative methods. A moving average is used in CCAdL to estimate the covariance matrix of the noisy force, in which case the covariance matrix will converge to a constant matrix in long-time limit. Moreover, it appears in our numerical experiments that the use of a moving average could reduce the stability of the numerical integrators, thereby limiting the largest usable stepsize. In this article, we propose a modified CCAdL (i.e., mCCAdL) thermostat that uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential to numerically approximate the exact solution to the subsystem involving the additional term proposed in CCAdL. We also propose a symmetric splitting method for mCCAdL, instead of an Euler-type discretisation used in the original CCAdL thermostat. We demonstrate in our numerical experiments that the newly proposed mCCAdL thermostat achieves a substantial improvement in the numerical stability over the original CCAdL thermostat, while significantly outperforming popular alternative stochastic gradient methods in terms of the numerical accuracy for large-scale machine learning applications.", "date": "2025-12-30", "categories": []}
{"id": "2512.22999", "title": "JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference", "abstract": "We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences. Inference networks are instantiated with diffusion-based posterior estimators that can approximate high-dimensional and multimodal posteriors at every experimental step. Across standard adaptive design benchmarks, JADAI achieves superior or competitive performance.", "date": "2025-12-28", "categories": []}
{"id": "2512.21486", "title": "When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning", "abstract": "Functional tensor decomposition can analyze multi-dimensional data with real-valued indices, paving the path for applications in machine learning and signal processing. A limitation of existing approaches is the assumption that the tensor rank-a critical parameter governing model complexity-is known. However, determining the optimal rank is a non-deterministic polynomial-time hard (NP-hard) task and there is a limited understanding regarding the expressive power of functional low-rank tensor models for continuous signals. We propose a rank-revealing functional Bayesian tensor completion (RR-FBTC) method. Modeling the latent functions through carefully designed multioutput Gaussian processes, RR-FBTC handles tensors with real-valued indices while enabling automatic tensor rank determination during the inference process. We establish the universal approximation property of the model for continuous multi-dimensional signals, demonstrating its expressive power in a concise format. To learn this model, we employ the variational inference framework and derive an efficient algorithm with closed-form updates. Experiments on both synthetic and real-world datasets demonstrate the effectiveness and superiority of the RR-FBTC over state-of-the-art approaches. The code is available at https://github.com/OceanSTARLab/RR-FBTC.", "date": "2025-12-25", "categories": []}
{"id": "2401.15502", "title": "Differentially private Bayesian tests", "abstract": "Differential privacy has emerged as an significant cornerstone in the realm of scientific hypothesis testing utilizing confidential data. In reporting scientific discoveries, Bayesian tests are widely adopted since they effectively circumnavigate the key criticisms of P-values, namely, lack of interpretability and inability to quantify evidence in support of the competing hypotheses. We present a novel differentially private Bayesian hypotheses testing framework that arise naturally under a principled data generative mechanism, inherently maintaining the interpretability of the resulting inferences. Furthermore, by focusing on differentially private Bayes factors based on widely used test statistics, we circumvent the need to model the complete data generative mechanism and ensure substantial computational benefits. We also provide a set of sufficient conditions to establish results on Bayes factor consistency under the proposed framework. The utility of the devised technology is showcased via several numerical experiments.", "date": "2024-01-27", "categories": []}
{"id": "2506.18283", "title": "Quantifying Uncertainty in the Presence of Distribution Shifts", "abstract": "Neural networks make accurate predictions but often fail to provide reliable uncertainty estimates, especially under covariate distribution shifts between training and testing. To address this problem, we propose a Bayesian framework for uncertainty estimation that explicitly accounts for covariate shifts. While conventional approaches rely on fixed priors, the key idea of our method is an adaptive prior, conditioned on both training and new covariates. This prior naturally increases uncertainty for inputs that lie far from the training distribution in regions where predictive performance is likely to degrade. To efficiently approximate the resulting posterior predictive distribution, we employ amortized variational inference. Finally, we construct synthetic environments by drawing small bootstrap samples from the training data, simulating a range of plausible covariate shift using only the original dataset. We evaluate our method on both synthetic and real-world data. It yields substantially improved uncertainty estimates under distribution shifts.", "date": "2025-06-23", "categories": []}
{"id": "2512.17677", "title": "Toward Ethical AI Through Bayesian Uncertainty in Neural Question Answering", "abstract": "We explore Bayesian reasoning as a means to quantify uncertainty in neural networks for question answering. Starting with a multilayer perceptron on the Iris dataset, we show how posterior inference conveys confidence in predictions. We then extend this to language models, applying Bayesian inference first to a frozen head and finally to LoRA-adapted transformers, evaluated on the CommonsenseQA benchmark. Rather than aiming for state-of-the-art accuracy, we compare Laplace approximations against maximum a posteriori (MAP) estimates to highlight uncertainty calibration and selective prediction. This allows models to abstain when confidence is low. An ``I don't know'' response not only improves interpretability but also illustrates how Bayesian methods can contribute to more responsible and ethical deployment of neural question-answering systems.", "date": "2025-12-19", "categories": []}
{"id": "2503.00213", "title": "An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation", "abstract": "Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to the latent field. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational problem associated with the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and the minimizer to the variational problem agree, thereby placing this PDE-based regularization within a fully Bayesian framework. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We then develop a finite-dimensional representation in function space and prove convergence of the projected prior and resulting posterior in Wasserstein distance. Finally, we connect the method to the important problem of identifying model-form error in applications, providing a diagnostic for model misspecification.", "date": "2025-02-28", "categories": []}
{"id": "2505.18077", "title": "Bayesian Deep Learning for Discrete Choice", "abstract": "Discrete choice models (DCMs) are used to analyze individual decision-making in contexts such as transportation choices, political elections, and consumer preferences. DCMs play a central role in applied econometrics by enabling inference on key economic variables, such as marginal rates of substitution, rather than focusing solely on predicting choices on new unlabeled data. However, while traditional DCMs offer high interpretability and support for point and interval estimation of economic quantities, these models often underperform in predictive tasks compared to deep learning (DL) models. Despite their predictive advantages, DL models remain largely underutilized in discrete choice due to concerns about their lack of interpretability, unstable parameter estimates, and the absence of established methods for uncertainty quantification. Here, we introduce a deep learning model architecture specifically designed to integrate with approximate Bayesian inference methods, such as Stochastic Gradient Langevin Dynamics (SGLD). Our proposed model collapses to behaviorally informed hypotheses when data is limited, mitigating overfitting and instability in underspecified settings while retaining the flexibility to capture complex nonlinear relationships when sufficient data is available. We demonstrate our approach using SGLD through a Monte Carlo simulation study, evaluating both predictive metrics--such as out-of-sample balanced accuracy--and inferential metrics--such as empirical coverage for marginal rates of substitution interval estimates. Additionally, we present results from two empirical case studies: one using revealed mode choice data in NYC, and the other based on the widely used Swiss train choice stated preference data.", "date": "2025-05-23", "categories": []}
{"id": "2512.16105", "title": "BayesSum: Bayesian Quadrature in Discrete Spaces", "abstract": "This paper addresses the challenging computational problem of estimating intractable expectations over discrete domains. Existing approaches, including Monte Carlo and Russian Roulette estimators, are consistent but often require a large number of samples to achieve accurate results. We propose a novel estimator, \\emph{BayesSum}, which is an extension of Bayesian quadrature to discrete domains. It is more sample efficient than alternatives due to its ability to make use of prior information about the integrand through a Gaussian process. We show this through theory, deriving a convergence rate significantly faster than Monte Carlo in a broad range of settings. We also demonstrate empirically that our proposed method does indeed require fewer samples on several synthetic settings as well as for parameter estimation for Conway-Maxwell-Poisson and Potts models.", "date": "2025-12-18", "categories": []}
{"id": "2512.12676", "title": "Robust Variational Bayes by Min-Max Median Aggregation", "abstract": "We propose a robust and scalable variational Bayes (VB) framework designed to effectively handle contamination and outliers in dataset. Our approach partitions the data into $m$ disjoint subsets and formulates a joint optimization problem based on robust aggregation principles. A key insight is that the full posterior distribution is equivalent to the minimizer of the mean Kullback-Leibler (KL) divergence from the $m$-powered local posterior distributions. To enhance robustness, we replace the mean KL divergence with a min-max median formulation. The min-max formulation not only ensures consistency between the KL minimizer and the Evidence Lower Bound (ELBO) maximizer but also facilitates the establishment of improved statistical rates for the mean of variational posterior. We observe a notable discrepancy in the $m$-powered marginal log likelihood function contingent on the presence of local latent variables. To address this, we treat these two scenarios separately to guarantee the consistency of the aggregated variational posterior. Specifically, when local latent variables are present, we introduce an aggregate-and-rescale strategy. Theoretically, we provide a non-asymptotic analysis of our proposed posterior, incorporating a refined analysis of Bernstein-von Mises (BvM) theorem to accommodate a diverging number of subsets $m$. Our findings indicate that the two-stage approach yields a smaller approximation error compared to directly aggregating the $m$-powered local posteriors. Furthermore, we establish a nearly optimal statistical rate for the mean of the proposed posterior, advancing existing theories related to min-max median estimators. The efficacy of our method is demonstrated through extensive simulation studies.", "date": "2025-12-14", "categories": []}
{"id": "2512.13111", "title": "From Overfitting to Reliability: Introducing the Hierarchical Approximate Bayesian Neural Network", "abstract": "In recent years, neural networks have revolutionized various domains, yet challenges such as hyperparameter tuning and overfitting remain significant hurdles. Bayesian neural networks offer a framework to address these challenges by incorporating uncertainty directly into the model, yielding more reliable predictions, particularly for out-of-distribution data. This paper presents Hierarchical Approximate Bayesian Neural Network, a novel approach that uses a Gaussian-inverse-Wishart distribution as a hyperprior of the network's weights to increase both the robustness and performance of the model. We provide analytical representations for the predictive distribution and weight posterior, which amount to the calculation of the parameters of Student's t-distributions in closed form with linear complexity with respect to the number of weights. Our method demonstrates robust performance, effectively addressing issues of overfitting and providing reliable uncertainty estimates, particularly for out-of-distribution tasks. Experimental results indicate that HABNN not only matches but often outperforms state-of-the-art models, suggesting a promising direction for future applications in safety-critical environments.", "date": "2025-12-15", "categories": []}
{"id": "2512.10602", "title": "Uncertainty-Preserving QBNNs: Multi-Level Quantization of SVI-Based Bayesian Neural Networks for Image Classification", "abstract": "Bayesian Neural Networks (BNNs) provide principled uncertainty quantification but suffer from substantial computational and memory overhead compared to deterministic networks. While quantization techniques have successfully reduced resource requirements in standard deep learning models, their application to probabilistic models remains largely unexplored. We introduce a systematic multi-level quantization framework for Stochastic Variational Inference based BNNs that distinguishes between three quantization strategies: Variational Parameter Quantization (VPQ), Sampled Parameter Quantization (SPQ), and Joint Quantization (JQ). Our logarithmic quantization for variance parameters, and specialized activation functions to preserve the distributional structure are essential for calibrated uncertainty estimation. Through comprehensive experiments on Dirty-MNIST, we demonstrate that BNNs can be quantized down to 4-bit precision while maintaining both classification accuracy and uncertainty disentanglement. At 4 bits, Joint Quantization achieves up to 8x memory reduction compared to floating-point implementations with minimal degradation in epistemic and aleatoric uncertainty estimation. These results enable deployment of BNNs on resource-constrained edge devices and provide design guidelines for future analog \"Bayesian Machines\" operating at inherently low precision.", "date": "2025-12-11", "categories": []}
{"id": "2506.14262", "title": "Knowledge Adaptation as Posterior Correction", "abstract": "Adaptation is the holy grail of intelligence, but even the best AI models lack the adaptability of toddlers. In spite of great progress, little is known about the mechanisms by which machines can learn to adapt as fast as humans and animals. Here, we cast adaptation as `correction' of old posteriors and show that a wide-variety of existing adaptation methods follow this very principle, including those used for continual learning, federated learning, unlearning, and model merging. In all these settings, more accurate posteriors often lead to smaller corrections and can enable faster adaptation. Posterior correction is derived by using the dual representation of the Bayesian Learning Rule of Khan and Rue (2023), where the interference between the old representation and new information is quantified by using the natural-gradient mismatch. We present many examples demonstrating how machines can learn to adapt quickly by using posterior correction.", "date": "2025-06-17", "categories": []}
{"id": "2512.08022", "title": "Provable Diffusion Posterior Sampling for Bayesian Inversion", "abstract": "This paper proposes a novel diffusion-based posterior sampling method within a plug-and-play (PnP) framework. Our approach constructs a probability transport from an easy-to-sample terminal distribution to the target posterior, using a warm-start strategy to initialize the particles. To approximate the posterior score, we develop a Monte Carlo estimator in which particles are generated using Langevin dynamics, avoiding the heuristic approximations commonly used in prior work. The score governing the Langevin dynamics is learned from data, enabling the model to capture rich structural features of the underlying prior distribution. On the theoretical side, we provide non-asymptotic error bounds, showing that the method converges even for complex, multi-modal target posterior distributions. These bounds explicitly quantify the errors arising from posterior score estimation, the warm-start initialization, and the posterior sampling procedure. Our analysis further clarifies how the prior score-matching error and the condition number of the Bayesian inverse problem influence overall performance. Finally, we present numerical experiments demonstrating the effectiveness of the proposed method across a range of inverse problems.", "date": "2025-12-08", "categories": []}
{"id": "2404.03764", "title": "Covariate-Elaborated Robust Partial Information Transfer with Conditional Spike-and-Slab Prior", "abstract": "The popularity of transfer learning stems from the fact that it can borrow information from useful auxiliary datasets. Existing statistical transfer learning methods usually adopt a global similarity measure between the source data and the target data, which may lead to inefficiency when only partial information is shared. In this paper, we propose a novel Bayesian transfer learning method named ``CONCERT'' to allow robust partial information transfer for high-dimensional data analysis. A conditional spike-and-slab prior is introduced in the joint distribution of target and source parameters for information transfer. By incorporating covariate-specific priors, we can characterize partial similarities and integrate source information collaboratively to improve the performance on the target. In contrast to existing work, the CONCERT is a one-step procedure which achieves variable selection and information transfer simultaneously. We establish variable selection consistency, as well as estimation and prediction error bounds for CONCERT. Our theory demonstrates the covariate-specific benefit of transfer learning. To ensure the scalability of the algorithm, we adopt the variational Bayes framework to facilitate implementation. Extensive experiments and two real data applications showcase the validity and advantages of CONCERT over existing cutting-edge transfer learning methods.", "date": "2024-03-30", "categories": []}
{"id": "2408.00681", "title": "Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification", "abstract": "We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) using R\\'enyi's $\\alpha$-divergence to learn complex operators while quantifying uncertainty. By incorporating Bayesian neural networks as the building blocks for the branch and trunk networks, our framework endows DeepONet with uncertainty quantification. The use of R\\'enyi's $\\alpha$-divergence, instead of the Kullback-Leibler divergence (KLD), commonly used in standard variational inference, mitigates issues related to prior misspecification that are prevalent in Variational Bayesian DeepONets. This approach offers enhanced flexibility and robustness. We demonstrate that modifying the variational objective function yields superior results in terms of minimising the mean squared error and improving the negative log-likelihood on the test set. Our framework's efficacy is validated across various mechanical systems, where it outperforms both deterministic and standard KLD-based VI DeepONets in predictive accuracy and uncertainty quantification. The hyperparameter $\\alpha$, which controls the degree of robustness, can be tuned to optimise performance for specific problems. We apply this approach to a range of mechanics problems, including gravity pendulum, advection-diffusion, and diffusion-reaction systems. Our findings underscore the potential of $\\alpha$-VI DeepONet to advance the field of data-driven operator learning and its applications in engineering and scientific domains.", "date": "2024-08-01", "categories": []}
{"id": "2512.02217", "title": "Uncertainty Reasoning with Photonic Bayesian Machines", "abstract": "Artificial intelligence (AI) systems increasingly influence safety-critical aspects of society, from medical diagnosis to autonomous mobility, making uncertainty awareness a central requirement for trustworthy AI. We present a photonic Bayesian machine that leverages the inherent randomness of chaotic light sources to enable uncertainty reasoning within the framework of Bayesian Neural Networks. The analog processor features a 1.28 Tbit/s digital interface compatible with PyTorch, enabling probabilistic convolutions processing within 37.5 ps per convolution. We use the system for simultaneous classification and out-of-domain detection of blood cell microscope images and demonstrate reasoning between aleatoric and epistemic uncertainties. The photonic Bayesian machine removes the bottleneck of pseudo random number generation in digital systems, minimizes the cost of sampling for probabilistic models, and thus enables high-speed trustworthy AI systems.", "date": "2025-12-01", "categories": []}
{"id": "2509.21572", "title": "General Pruning Criteria for Fast SBL", "abstract": "Sparse Bayesian learning (SBL) associates to each weight in the underlying linear model a hyperparameter by assuming that each weight is Gaussian distributed with zero mean and precision (inverse variance) equal to its associated hyperparameter. The method estimates the hyperparameters by marginalizing out the weights and performing (marginalized) maximum likelihood (ML) estimation. SBL returns many hyperparameter estimates to diverge to infinity, effectively setting the estimates of the corresponding weights to zero (i.e., pruning the corresponding weights from the model) and thereby yielding a sparse estimate of the weight vector.\n  In this letter, we analyze the marginal likelihood as function of a single hyperparameter while keeping the others fixed, when the Gaussian assumptions on the noise samples and the weight distribution that underlies the derivation of SBL are weakened. We derive sufficient conditions that lead, on the one hand, to finite hyperparameter estimates and, on the other, to infinite ones. Finally, we show that in the Gaussian case, the two conditions are complementary and coincide with the pruning condition of fast SBL (F-SBL), thereby providing additional insights into this algorithm.", "date": "2025-09-25", "categories": []}
{"id": "2512.01500", "title": "Walking on the Fiber: A Simple Geometric Approximation for Bayesian Neural Networks", "abstract": "Bayesian Neural Networks provide a principled framework for uncertainty quantification by modeling the posterior distribution of network parameters. However, exact posterior inference is computationally intractable, and widely used approximations like the Laplace method struggle with scalability and posterior accuracy in modern deep networks. In this work, we revisit sampling techniques for posterior exploration, proposing a simple variation tailored to efficiently sample from the posterior in over-parameterized networks by leveraging the low-dimensional structure of loss minima. Building on this, we introduce a model that learns a deformation of the parameter space, enabling rapid posterior sampling without requiring iterative methods. Empirical results demonstrate that our approach achieves competitive posterior approximations with improved scalability compared to recent refinement techniques. These contributions provide a practical alternative for Bayesian inference in deep learning.", "date": "2025-12-01", "categories": []}
{"id": "2511.22296", "title": "Data-driven informative priors for Bayesian inference with quasi-periodic data", "abstract": "Bayesian computational strategies for inference can be inefficient in approximating the posterior distribution in models that exhibit some form of periodicity. This is because the probability mass of the marginal posterior distribution of the parameter representing the period is usually highly concentrated in a very small region of the parameter space. Therefore, it is necessary to provide as much information as possible to the inference method through the parameter prior distribution. We intend to show that it is possible to construct a prior distribution from the data by fitting a Gaussian process (GP) with a periodic kernel. More specifically, we want to show that it is possible to approximate the marginal posterior distribution of the hyperparameter corresponding to the period in the kernel. Subsequently, this distribution can be used as a prior distribution for the inference method. We use an adaptive importance sampling method to approximate the posterior distribution of the hyperparameters of the GP. Then, we use the marginal posterior distribution of the hyperparameter related to the periodicity in order to construct a prior distribution for the period of the parametric model. This workflow is empirical Bayes, implemented as a modular (cut) transfer of a GP posterior for the period to the parametric model. We applied the proposed methodology to both synthetic and real data. We approximated the posterior distribution of the period of the GP kernel and then passed it forward as a posterior-as-prior with no feedback. Finally, we analyzed its impact on the marginal posterior distribution.", "date": "2025-11-27", "categories": []}
{"id": "2511.23440", "title": "Accelerated Execution of Bayesian Neural Networks using a Single Probabilistic Forward Pass and Code Generation", "abstract": "Machine learning models perform well across domains such as diagnostics, weather forecasting, NLP, and autonomous driving, but their limited uncertainty handling restricts use in safety-critical settings. Traditional neural networks often fail to detect out-of-domain (OOD) data and may output confident yet incorrect predictions. Bayesian neural networks (BNNs) address this by providing probabilistic estimates, but incur high computational cost because predictions require sampling weight distributions and multiple forward passes. The Probabilistic Forward Pass (PFP) offers a highly efficient approximation to Stochastic Variational Inference (SVI) by assuming Gaussian-distributed weights and activations, enabling fully analytic uncertainty propagation and replacing sampling with a single deterministic forward pass. We present an end-to-end pipeline for training, compiling, optimizing, and deploying PFP-based BNNs on embedded ARM CPUs. Using the TVM deep learning compiler, we implement a dedicated library of Gaussian-propagating operators for multilayer perceptrons and convolutional neural networks, combined with manual and automated tuning strategies. Ablation studies show that PFP consistently outperforms SVI in computational efficiency, achieving speedups of up to 4200x for small mini-batches. PFP-BNNs match SVI-BNNs on Dirty-MNIST in accuracy, uncertainty estimation, and OOD detection while greatly reducing compute cost. These results highlight the potential of combining Bayesian approximations with code generation to enable efficient BNN deployment on resource-constrained systems.", "date": "2025-11-28", "categories": []}
{"id": "2501.14265", "title": "Bayesian Neural Networks for One-to-Many Mapping in Image Enhancement", "abstract": "In image enhancement tasks, such as low-light and underwater image enhancement, a degraded image can correspond to multiple plausible target images due to dynamic photography conditions. This naturally results in a one-to-many mapping problem. To address this, we propose a Bayesian Enhancement Model (BEM) that incorporates Bayesian Neural Networks (BNNs) to capture data uncertainty and produce diverse outputs. To enable fast inference, we introduce a BNN-DNN framework: a BNN is first employed to model the one-to-many mapping in a low-dimensional space, followed by a Deterministic Neural Network (DNN) that refines fine-grained image details. Extensive experiments on multiple low-light and underwater image enhancement benchmarks demonstrate the effectiveness of our method.", "date": "2025-01-24", "categories": []}
{"id": "2505.11355", "title": "Sparse Techniques for Regression in Deep Gaussian Processes", "abstract": "Gaussian processes (GPs) have gained popularity as flexible machine learning models for regression and function approximation with an in-built method for uncertainty quantification. However, GPs suffer when the amount of training data is large or when the underlying function contains multi-scale features that are difficult to represent by a stationary kernel. To address the former, training of GPs with large-scale data is often performed through inducing point approximations, also known as sparse GP regression (GPR), where the size of the covariance matrices in GPR is reduced considerably through a greedy search on the data set. To aid the latter, deep GPs have gained traction as hierarchical models that resolve multi-scale features by combining multiple GPs. Posterior inference in deep GPs requires a sampling or, more usual, a variational approximation. Variational approximations lead to large-scale stochastic, non-convex optimisation problems and the resulting approximation tends to represent uncertainty incorrectly. In this work, we combine variational learning with MCMC to develop a particle-based expectation-maximisation method to simultaneously find inducing points within the large-scale data (variationally) and accurately train the deep GPs (sampling-based). The result is a highly efficient and accurate methodology for deep GP training on large-scale data. We test our method on standard benchmark problems.", "date": "2025-05-16", "categories": []}
{"id": "2504.20319", "title": "Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach", "abstract": "Bayesian experimental design (BED) offers a principled framework for optimizing data acquisition by leveraging probabilistic inference. However, practical implementations of BED are often compromised by model discrepancy, i.e., the mismatch between predictive models and true physical systems, which can potentially lead to biased parameter estimates. While data-driven approaches have been recently explored to characterize the model discrepancy, the resulting high-dimensional parameter space poses severe challenges for both Bayesian updating and design optimization. In this work, we propose a hybrid BED framework enabled by auto-differentiable ensemble Kalman inversion (AD-EKI) that addresses these challenges by providing a computationally efficient, gradient-free alternative to estimate the information gain for high-dimensional network parameters. The AD-EKI allows a differentiable evaluation of the utility function in BED and thus facilitates the use of standard gradient-based methods for design optimization. In the proposed hybrid framework, we iteratively optimize experimental designs, decoupling the inference of low-dimensional physical parameters handled by standard BED methods, from the high-dimensional model discrepancy handled by AD-EKI. The identified optimal designs for the model discrepancy enable us to systematically collect informative data for its calibration. The performance of the proposed method is studied by a classical convection-diffusion BED example, and the hybrid framework enabled by AD-EKI efficiently identifies informative data to calibrate the model discrepancy and robustly infers the unknown physical parameters in the modeled system. Besides addressing the challenges of BED with model discrepancy, AD-EKI also potentially fosters efficient and scalable frameworks in many other areas with bilevel optimization, such as meta-learning and structure optimization.", "date": "2025-04-29", "categories": []}
{"id": "2511.16340", "title": "Improving Iterative Gaussian Processes via Warm Starting Sequential Posteriors", "abstract": "Scalable Gaussian process (GP) inference is essential for sequential decision-making tasks, yet improving GP scalability remains a challenging problem with many open avenues of research. This paper focuses on iterative GPs, where iterative linear solvers, such as conjugate gradients, stochastic gradient descent or alternative projections, are used to approximate the GP posterior. We propose a new method which improves solver convergence of a large linear system by leveraging the known solution to a smaller system contained within. This is significant for tasks with incremental data additions, and we show that our technique achieves speed-ups when solving to tolerance, as well as improved Bayesian optimisation performance under a fixed compute budget.", "date": "2025-11-20", "categories": []}
{"id": "2511.14049", "title": "SmallML: Bayesian Transfer Learning for Small-Data Predictive Analytics", "abstract": "Small and medium-sized enterprises (SMEs) represent 99.9% of U.S. businesses yet remain systematically excluded from AI due to a mismatch between their operational scale and modern machine learning's data requirements. This paper introduces SmallML, a Bayesian transfer learning framework achieving enterprise-level prediction accuracy with datasets as small as 50-200 observations.\n  We develop a three-layer architecture integrating transfer learning, hierarchical Bayesian modeling, and conformal prediction. Layer 1 extracts informative priors from 22,673 public records using a SHAP-based procedure transferring knowledge from gradient boosting to logistic regression. Layer 2 implements hierarchical pooling across J=5-50 SMEs with adaptive shrinkage, balancing population patterns with entity-specific characteristics. Layer 3 provides conformal sets with finite-sample coverage guarantees P(y in C(x)) >= 1-alpha for distribution-free uncertainty quantification.\n  Validation on customer churn data demonstrates 96.7% +/- 4.2% AUC with 100 observations per business -- a +24.2 point improvement over independent logistic regression (72.5% +/- 8.1%), with p < 0.000001. Conformal prediction achieves 92% empirical coverage at 90% target. Training completes in 33 minutes on standard CPU hardware. By enabling enterprise-grade predictions for 33 million U.S. SMEs previously excluded from machine learning, SmallML addresses a critical gap in AI democratization.\n  Keywords: Bayesian transfer learning, hierarchical models, conformal prediction, small-data analytics, SME machine learning", "date": "2025-11-18", "categories": []}
{"id": "2506.13613", "title": "Variational Inference with Mixtures of Isotropic Gaussians", "abstract": "Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. In this paper, we focus on the following parametric family: mixtures of isotropic Gaussians (i.e., with diagonal covariance matrices proportional to the identity) and uniform weights. We develop a variational framework and provide efficient algorithms suited for this family. In contrast with mixtures of Gaussian with generic covariance matrices, this choice presents a balance between accurate approximations of multimodal Bayesian posteriors, while being memory and computationally efficient. Our algorithms implement gradient descent on the location of the mixture components (the modes of the Gaussians), and either (an entropic) Mirror or Bures descent on their variance parameters. We illustrate the performance of our algorithms on numerical experiments.", "date": "2025-06-16", "categories": []}
{"id": "2511.11983", "title": "Bayesian--AI Fusion for Epidemiological Decision Making: Calibrated Risk, Honest Uncertainty, and Hyperparameter Intelligence", "abstract": "Modern epidemiological analytics increasingly use machine learning models that offer strong prediction but often lack calibrated uncertainty. Bayesian methods provide principled uncertainty quantification, yet are viewed as difficult to integrate with contemporary AI workflows. This paper proposes a unified Bayesian and AI framework that combines Bayesian prediction with Bayesian hyperparameter optimization.\n  We use Bayesian logistic regression to obtain calibrated individual-level disease risk and credible intervals on the Pima Indians Diabetes dataset. In parallel, we use Gaussian-process Bayesian optimization to tune penalized Cox survival models on the GBSG2 breast cancer cohort. This yields a two-layer system: a Bayesian predictive layer that represents risk as a posterior distribution, and a Bayesian optimization layer that treats model selection as inference over a black-box objective.\n  Simulation studies in low- and high-dimensional regimes show that the Bayesian layer provides reliable coverage and improved calibration, while Bayesian shrinkage improves AUC, Brier score, and log-loss. Bayesian optimization consistently pushes survival models toward near-oracle concordance. Overall, Bayesian reasoning enhances both what we infer and how we search, enabling calibrated risk and principled hyperparameter intelligence for epidemiological decision making.", "date": "2025-11-15", "categories": []}
{"id": "2511.12257", "title": "Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems", "abstract": "This paper proposes a novel Bayesian framework for solving Poisson inverse problems by devising a Monte Carlo sampling algorithm which accounts for the underlying non-Euclidean geometry. To address the challenges posed by the Poisson likelihood -- such as non-Lipschitz gradients and positivity constraints -- we derive a Bayesian model which leverages exact and asymptotically exact data augmentations. In particular, the augmented model incorporates two sets of splitting variables both derived through a Bregman divergence based on the Burg entropy. Interestingly the resulting augmented posterior distribution is characterized by conditional distributions which benefit from natural conjugacy properties and preserve the intrinsic geometry of the latent and splitting variables. This allows for efficient sampling via Gibbs steps, which can be performed explicitly for all conditionals, except the one incorporating the regularization potential. For this latter, we resort to a Hessian Riemannian Langevin Monte Carlo (HRLMC) algorithm which is well suited to handle priors with explicit or easily computable score functions. By operating on a mirror manifold, this Langevin step ensures that the sampling satisfies the positivity constraints and more accurately reflects the underlying problem structure. Performance results obtained on denoising, deblurring, and positron emission tomography (PET) experiments demonstrate that the method achieves competitive performance in terms of reconstruction quality compared to optimization- and sampling-based approaches.", "date": "2025-11-15", "categories": []}
{"id": "2406.17058", "title": "Bayesian ICA with super-Gaussian Source Priors", "abstract": "Independent Component Analysis (ICA) plays a central role in modern machine learning as a flexible framework for feature extraction. We introduce a horseshoe-type prior with a latent Polya-Gamma scale mixture representation, yielding scalable algorithms for both point estimation via expectation-maximization (EM) and full posterior inference via Markov chain Monte Carlo (MCMC). This hierarchical formulation unifies several previously disparate estimation strategies within a single Bayesian framework. We also establish the first theoretical guarantees for hierarchical Bayesian ICA, including posterior contraction and local asymptotic normality results for the unmixing matrix. Comprehensive simulation studies demonstrate that our methods perform competitively with widely used ICA tools. We further discuss implementation of conditional posteriors, envelope-based optimization, and possible extensions to flow-based architectures for nonlinear feature extraction and deep learning. Finally, we outline several promising directions for future work.", "date": "2024-06-24", "categories": []}
{"id": "2406.00713", "title": "Logistic Variational Bayes Revisited", "abstract": "Variational logistic regression is a popular method for approximate Bayesian inference seeing wide-spread use in many areas of machine learning including: Bayesian optimization, reinforcement learning and multi-instance learning to name a few. However, due to the intractability of the Evidence Lower Bound, authors have turned to the use of Monte Carlo, quadrature or bounds to perform inference, methods which are costly or give poor approximations to the true posterior.\n  In this paper we introduce a new bound for the expectation of softplus function and subsequently show how this can be applied to variational logistic regression and Gaussian process classification. Unlike other bounds, our proposal does not rely on extending the variational family, or introducing additional parameters to ensure the bound is tight. In fact, we show that this bound is tighter than the state-of-the-art, and that the resulting variational posterior achieves state-of-the-art performance, whilst being significantly faster to compute than Monte-Carlo methods.", "date": "2024-06-02", "categories": []}
{"id": "2511.09897", "title": "Theory and computation for structured variational inference", "abstract": "Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural setting of star-structured variational inference, where a root variable impacts all the other ones. We prove the first results for existence, uniqueness, and self-consistency of the variational approximation. In turn, we derive quantitative approximation error bounds for the variational approximation to the posterior, extending prior work from the mean-field setting to the star-structured setting. We also develop a gradient-based algorithm with provable guarantees for computing the variational approximation using ideas from optimal transport theory. We explore the implications of our results for Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. As a by-product of our analysis, we develop new stability results for star-separable transport maps which might be of independent interest.", "date": "2025-11-13", "categories": []}
{"id": "2502.12684", "title": "Federated Variational Inference for Bayesian Mixture Models", "abstract": "We present a federated learning approach for Bayesian model-based clustering of large-scale binary and categorical datasets. We introduce a principled 'divide and conquer' inference procedure using variational inference with local merge and delete moves within batches of the data in parallel, followed by 'global' merge moves across batches to find global clustering structures. We show that these merge moves require only summaries of the data in each batch, enabling federated learning across local nodes without requiring the full dataset to be shared. Empirical results on simulated and benchmark datasets demonstrate that our method performs well in comparison to existing clustering algorithms. We validate the practical utility of the method by applying it to large scale electronic health record (EHR) data.", "date": "2025-02-18", "categories": []}
{"id": "2511.07671", "title": "Robust Experimental Design via Generalised Bayesian Inference", "abstract": "Bayesian optimal experimental design is a principled framework for conducting experiments that leverages Bayesian inference to quantify how much information one can expect to gain from selecting a certain design. However, accurate Bayesian inference relies on the assumption that one's statistical model of the data-generating process is correctly specified. If this assumption is violated, Bayesian methods can lead to poor inference and estimates of information gain. Generalised Bayesian (or Gibbs) inference is a more robust probabilistic inference framework that replaces the likelihood in the Bayesian update by a suitable loss function. In this work, we present Generalised Bayesian Optimal Experimental Design (GBOED), an extension of Gibbs inference to the experimental design setting which achieves robustness in both design and inference. Using an extended information-theoretic framework, we derive a new acquisition function, the Gibbs expected information gain (Gibbs EIG). Our empirical results demonstrate that GBOED enhances robustness to outliers and incorrect assumptions about the outcome noise distribution.", "date": "2025-11-10", "categories": []}
{"id": "2504.05004", "title": "Stacking Variational Bayesian Monte Carlo", "abstract": "Approximate Bayesian inference for models with computationally expensive, black-box likelihoods poses a significant challenge, especially when the posterior distribution is complex. Many inference methods struggle to explore the parameter space efficiently under a limited budget of likelihood evaluations. Variational Bayesian Monte Carlo (VBMC) is a sample-efficient method that addresses this by building a local surrogate model of the log-posterior. However, its conservative exploration strategy, while promoting stability, can cause it to miss important regions of the posterior, such as distinct modes or long tails. In this work, we introduce Stacking Variational Bayesian Monte Carlo (S-VBMC), a method that overcomes this limitation by constructing a robust, global posterior approximation from multiple independent VBMC runs. Our approach merges these local approximations through a principled and inexpensive post-processing step that leverages VBMC's mixture posterior representation and per-component evidence estimates. Crucially, S-VBMC requires no additional likelihood evaluations and is naturally parallelisable, fitting seamlessly into existing inference workflows. We demonstrate its effectiveness on two synthetic problems designed to challenge VBMC's exploration and two real-world applications from computational neuroscience, showing substantial improvements in posterior approximation quality across all cases. Our code is available as a Python package at https://github.com/acerbilab/svbmc.", "date": "2025-04-07", "categories": []}
{"id": "2511.06407", "title": "Fast Riemannian-manifold Hamiltonian Monte Carlo for hierarchical Gaussian-process models", "abstract": "Hierarchical Bayesian models based on Gaussian processes are considered useful for describing complex nonlinear statistical dependencies among variables in real-world data. However, effective Monte Carlo algorithms for inference with these models have not yet been established, except for several simple cases. In this study, we show that, compared with the slow inference achieved with existing program libraries, the performance of Riemannian-manifold Hamiltonian Monte Carlo (RMHMC) can be drastically improved by optimising the computation order according to the model structure and dynamically programming the eigendecomposition. This improvement cannot be achieved when using an existing library based on a naive automatic differentiator. We numerically demonstrate that RMHMC effectively samples from the posterior, allowing the calculation of model evidence, in a Bayesian logistic regression on simulated data and in the estimation of propensity functions for the American national medical expenditure data using several Bayesian multiple-kernel models. These results lay a foundation for implementing effective Monte Carlo algorithms for analysing real-world data with Gaussian processes, and highlight the need to develop a customisable library set that allows users to incorporate dynamically programmed objects and finely optimises the mode of automatic differentiation depending on the model structure.", "date": "2025-11-09", "categories": []}
{"id": "2511.06978", "title": "Fast Bayesian Updates via Harmonic Representations", "abstract": "Bayesian inference, while foundational to probabilistic reasoning, is often hampered by the computational intractability of posterior distributions, particularly through the challenging evidence integral. Conventional approaches like Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) face significant scalability and efficiency limitations. This paper introduces a novel, unifying framework for fast Bayesian updates by leveraging harmonic analysis. We demonstrate that representing the prior and likelihood in a suitable orthogonal basis transforms the Bayesian update rule into a spectral convolution. Specifically, the Fourier coefficients of the posterior are shown to be the normalized convolution of the prior and likelihood coefficients. To achieve computational feasibility, we introduce a spectral truncation scheme, which, for smooth functions, yields an exceptionally accurate finite-dimensional approximation and reduces the update to a circular convolution. This formulation allows us to exploit the Fast Fourier Transform (FFT), resulting in a deterministic algorithm with O(N log N) complexity -- a substantial improvement over the O(N^2) cost of naive methods. We establish rigorous mathematical criteria for the applicability of our method, linking its efficiency to the smoothness and spectral decay of the involved distributions. The presented work offers a paradigm shift, connecting Bayesian computation to signal processing and opening avenues for real-time, sequential inference in a wide class of problems.", "date": "2025-11-10", "categories": []}
{"id": "2505.24243", "title": "Model-Informed Flows for Bayesian Inference", "abstract": "Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow-based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model's prior. Guided by this theoretical insight, we introduce the Model-Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state-of-the-art performance across a suite of hierarchical and non-hierarchical benchmarks.", "date": "2025-05-30", "categories": []}
{"id": "2301.03655", "title": "Bayesian Additive Main Effects and Multiplicative Interaction Models using Tensor Regression for Multi-environmental Trials", "abstract": "We propose a Bayesian tensor regression model to accommodate the effect of multiple factors on phenotype prediction. We adopt a set of prior distributions that resolve identifiability issues that may arise between the parameters in the model. Further, we incorporate a spike-and-slab structure that identifies which interactions are relevant for inclusion in the linear predictor, even when they form a subset of the available variables. Simulation experiments show that our method outperforms previous related models and machine learning algorithms under different sample sizes and degrees of complexity. We further explore the applicability of our model by analysing real-world data related to wheat production across Ireland from 2010 to 2019. Our model performs competitively and overcomes key limitations found in other analogous approaches. Finally, we adapt a set of visualisations for the posterior distribution of the tensor effects that facilitate the identification of optimal interactions between the tensor variables, whilst accounting for the uncertainty in the posterior distribution.", "date": "2023-01-09", "categories": []}
{"id": "2404.11350", "title": "Calibrating Bayesian Learning via Regularization, Confidence Minimization, and Selective Inference", "abstract": "The application of artificial intelligence (AI) models in fields such as engineering is limited by the known difficulty of quantifying the reliability of an AI's decision. A well-calibrated AI model must correctly report its accuracy on in-distribution (ID) inputs, while also enabling the detection of out-of-distribution (OOD) inputs. A conventional approach to improve calibration is the application of Bayesian ensembling. However, owing to computational limitations and model misspecification, practical ensembling strategies do not necessarily enhance calibration. This paper proposes an extension of variational inference (VI)-based Bayesian learning that integrates calibration regularization for improved ID performance, confidence minimization for OOD detection, and selective calibration to ensure a synergistic use of calibration regularization and confidence minimization. The scheme is constructed successively by first introducing calibration-regularized Bayesian learning (CBNN), then incorporating out-of-distribution confidence minimization (OCM) to yield CBNN-OCM, and finally integrating also selective calibration to produce selective CBNN-OCM (SCBNN-OCM). Selective calibration rejects inputs for which the calibration performance is expected to be insufficient. Numerical results illustrate the trade-offs between ID accuracy, ID calibration, and OOD calibration attained by both frequentist and Bayesian learning methods. Among the main conclusions, SCBNN-OCM is seen to achieve best ID and OOD performance as compared to existing state-of-the-art approaches at the cost of rejecting a sufficiently large number of inputs.", "date": "2024-04-17", "categories": []}
{"id": "2410.11067", "title": "Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix", "abstract": "Given an intractable target density $p$, variational inference (VI) attempts to find the best approximation $q$ from a tractable family $Q$. This is typically done by minimizing the exclusive Kullback-Leibler divergence, $\\text{KL}(q||p)$. In practice, $Q$ is not rich enough to contain $p$, and the approximation is misspecified even when it is a unique global minimizer of $\\text{KL}(q||p)$. In this paper, we analyze the robustness of VI to these misspecifications when $p$ exhibits certain symmetries and $Q$ is a location-scale family that shares these symmetries. We prove strong guarantees for VI not only under mild regularity conditions but also in the face of severe misspecifications. Namely, we show that (i) VI recovers the mean of $p$ when $p$ exhibits an \\textit{even} symmetry, and (ii) it recovers the correlation matrix of $p$ when in addition~$p$ exhibits an \\textit{elliptical} symmetry. These guarantees hold for the mean even when $q$ is factorized and $p$ is not, and for the correlation matrix even when~$q$ and~$p$ behave differently in their tails. We analyze various regimes of Bayesian inference where these symmetries are useful idealizations, and we also investigate experimentally how VI behaves in their absence.", "date": "2024-10-14", "categories": []}
{"id": "2511.01628", "title": "Partial Trace-Class Bayesian Neural Networks", "abstract": "Bayesian neural networks (BNNs) allow rigorous uncertainty quantification in deep learning, but often come at a prohibitive computational cost. We propose three different innovative architectures of partial trace-class Bayesian neural networks (PaTraC BNNs) that enable uncertainty quantification comparable to standard BNNs but use significantly fewer Bayesian parameters. These PaTraC BNNs have computational and statistical advantages over standard Bayesian neural networks in terms of speed and memory requirements. Our proposed methodology therefore facilitates reliable, robust, and scalable uncertainty quantification in neural networks. The three architectures build on trace-class neural network priors which induce an ordering of the neural network parameters, and are thus a natural choice in our framework. In a numerical simulation study, we verify the claimed benefits, and further illustrate the performance of our proposed methodology on a real-world dataset.", "date": "2025-11-03", "categories": []}
{"id": "2510.25001", "title": "Bayesian Neural Networks vs. Mixture Density Networks: Theoretical and Empirical Insights for Uncertainty-Aware Nonlinear Modeling", "abstract": "This paper investigates two prominent probabilistic neural modeling paradigms: Bayesian Neural Networks (BNNs) and Mixture Density Networks (MDNs) for uncertainty-aware nonlinear regression. While BNNs incorporate epistemic uncertainty by placing prior distributions over network parameters, MDNs directly model the conditional output distribution, thereby capturing multimodal and heteroscedastic data-generating mechanisms. We present a unified theoretical and empirical framework comparing these approaches. On the theoretical side, we derive convergence rates and error bounds under H\\\"older smoothness conditions, showing that MDNs achieve faster Kullback-Leibler (KL) divergence convergence due to their likelihood-based nature, whereas BNNs exhibit additional approximation bias induced by variational inference. Empirically, we evaluate both architectures on synthetic nonlinear datasets and a radiographic benchmark (RSNA Pediatric Bone Age Challenge). Quantitative and qualitative results demonstrate that MDNs more effectively capture multimodal responses and adaptive uncertainty, whereas BNNs provide more interpretable epistemic uncertainty under limited data. Our findings clarify the complementary strengths of posterior-based and likelihood-based probabilistic learning, offering guidance for uncertainty-aware modeling in nonlinear systems.", "date": "2025-10-28", "categories": []}
{"id": "2510.25254", "title": "Scaling Up Bayesian DAG Sampling", "abstract": "Bayesian inference of Bayesian network structures is often performed by sampling directed acyclic graphs along an appropriately constructed Markov chain. We present two techniques to improve sampling. First, we give an efficient implementation of basic moves, which add, delete, or reverse a single arc. Second, we expedite summing over parent sets, an expensive task required for more sophisticated moves: we devise a preprocessing method to prune possible parent sets so as to approximately preserve the sums. Our empirical study shows that our techniques can yield substantial efficiency gains compared to previous methods.", "date": "2025-10-29", "categories": []}
{"id": "2510.23684", "title": "VIKING: Deep variational inference with stochastic projections", "abstract": "Variational mean field approximations tend to struggle with contemporary overparametrized deep neural networks. Where a Bayesian treatment is usually associated with high-quality predictions and uncertainties, the practical reality has been the opposite, with unstable training, poor predictive power, and subpar calibration. Building upon recent work on reparametrizations of neural networks, we propose a simple variational family that considers two independent linear subspaces of the parameter space. These represent functional changes inside and outside the support of training data. This allows us to build a fully-correlated approximate posterior reflecting the overparametrization that tunes easy-to-interpret hyperparameters. We develop scalable numerical routines that maximize the associated evidence lower bound (ELBO) and sample from the approximate posterior. Empirically, we observe state-of-the-art performance across tasks, models, and datasets compared to a wide array of baseline methods. Our results show that approximate Bayesian inference applied to deep neural networks is far from a lost cause when constructing inference mechanisms that reflect the geometry of reparametrizations.", "date": "2025-10-27", "categories": []}
{"id": "2510.23831", "title": "Testing-driven Variable Selection in Bayesian Modal Regression", "abstract": "We propose a Bayesian variable selection method in the framework of modal regression for heavy-tailed responses. An efficient expectation-maximization algorithm is employed to expedite parameter estimation. A test statistic is constructed to exploit the shape of the model error distribution to effectively separate informative covariates from unimportant ones. Through simulations, we demonstrate and evaluate the efficacy of the proposed method in identifying important covariates in the presence of non-Gaussian model errors. Finally, we apply the proposed method to analyze two datasets arising in genetic and epigenetic studies.", "date": "2025-10-27", "categories": []}
{"id": "2412.01993", "title": "Generalized EXTRA stochastic gradient Langevin dynamics", "abstract": "Langevin algorithms are popular Markov Chain Monte Carlo methods for Bayesian learning, particularly when the aim is to sample from the posterior distribution of a parametric model, given the input data and the prior distribution over the model parameters. Their stochastic versions such as stochastic gradient Langevin dynamics (SGLD) allow iterative learning based on randomly sampled mini-batches of large datasets and are scalable to large datasets. However, when data is decentralized across a network of agents subject to communication and privacy constraints, standard SGLD algorithms cannot be applied. Instead, we employ decentralized SGLD (DE-SGLD) algorithms, where Bayesian learning is performed collaboratively by a network of agents without sharing individual data. Nonetheless, existing DE-SGLD algorithms induce a bias at every agent that can negatively impact performance; this bias persists even when using full batches and is attributable to network effects. Motivated by the EXTRA algorithm and its generalizations for decentralized optimization, we propose the generalized EXTRA stochastic gradient Langevin dynamics, which eliminates this bias in the full-batch setting. Moreover, we show that, in the mini-batch setting, our algorithm provides performance bounds that significantly improve upon those of standard DE-SGLD algorithms in the literature. Our numerical results also demonstrate the efficiency of the proposed approach.", "date": "2024-12-02", "categories": []}
{"id": "2510.22054", "title": "Input Adaptive Bayesian Model Averaging", "abstract": "This paper studies prediction with multiple candidate models, where the goal is to combine their outputs. This task is especially challenging in heterogeneous settings, where different models may be better suited to different inputs. We propose input adaptive Bayesian Model Averaging (IA-BMA), a Bayesian method that assigns model weights conditional on the input. IA-BMA employs an input adaptive prior, and yields a posterior distribution that adapts to each prediction, which we estimate with amortized variational inference. We derive formal guarantees for its performance, relative to any single predictor selected per input. We evaluate IABMA across regression and classification tasks, studying data from personalized cancer treatment, credit-card fraud detection, and UCI datasets. IA-BMA consistently delivers more accurate and better-calibrated predictions than both non-adaptive baselines and existing adaptive methods.", "date": "2025-10-24", "categories": []}
{"id": "2510.22651", "title": "Variational Polya Tree", "abstract": "Density estimation is essential for generative modeling, particularly with the rise of modern neural networks. While existing methods capture complex data distributions, they often lack interpretability and uncertainty quantification. Bayesian nonparametric methods, especially the \\polya tree, offer a robust framework that addresses these issues by accurately capturing function behavior over small intervals. Traditional techniques like Markov chain Monte Carlo (MCMC) face high computational complexity and scalability limitations, hindering the use of Bayesian nonparametric methods in deep learning. To tackle this, we introduce the variational \\polya tree (VPT) model, which employs stochastic variational inference to compute posterior distributions. This model provides a flexible, nonparametric Bayesian prior that captures latent densities and works well with stochastic gradient optimization. We also leverage the joint distribution likelihood for a more precise variational posterior approximation than traditional mean-field methods. We evaluate the model performance on both real data and images, and demonstrate its competitiveness with other state-of-the-art deep density estimation methods. We also explore its ability in enhancing interpretability and uncertainty quantification. Code is available at https://github.com/howardchanth/var-polya-tree.", "date": "2025-10-26", "categories": []}
{"id": "2410.05757", "title": "Temperature Optimization for Bayesian Deep Learning", "abstract": "The Cold Posterior Effect (CPE) is a phenomenon in Bayesian Deep Learning (BDL), where tempering the posterior to a cold temperature often improves the predictive performance of the posterior predictive distribution (PPD). Although the term `CPE' suggests colder temperatures are inherently better, the BDL community increasingly recognizes that this is not always the case. Despite this, there remains no systematic method for finding the optimal temperature beyond grid search. In this work, we propose a data-driven approach to select the temperature that maximizes test log-predictive density, treating the temperature as a model parameter and estimating it directly from the data. We empirically demonstrate that our method performs comparably to grid search, at a fraction of the cost, across both regression and classification tasks. Finally, we highlight the differing perspectives on CPE between the BDL and Generalized Bayes communities: while the former primarily emphasizes the predictive performance of the PPD, the latter prioritizes the utility of the posterior under model misspecification; these distinct objectives lead to different temperature preferences.", "date": "2024-10-08", "categories": []}
{"id": "2506.19144", "title": "Posterior Contraction for Sparse Neural Networks in Besov Spaces with Intrinsic Dimensionality", "abstract": "This work establishes that sparse Bayesian neural networks achieve optimal posterior contraction rates over anisotropic Besov spaces and their hierarchical compositions. These structures reflect the intrinsic dimensionality of the underlying function, thereby mitigating the curse of dimensionality. Our analysis shows that Bayesian neural networks equipped with either sparse or continuous shrinkage priors attain the optimal rates which are dependent on the intrinsic dimension of the true structures. Moreover, we show that these priors enable rate adaptation, allowing the posterior to contract at the optimal rate even when the smoothness level of the true function is unknown. The proposed framework accommodates a broad class of functions, including additive and multiplicative Besov functions as special cases. These results advance the theoretical foundations of Bayesian neural networks and provide rigorous justification for their practical effectiveness in high-dimensional, structured estimation problems.", "date": "2025-06-23", "categories": []}
{"id": "2509.20753", "title": "RAPTOR-GEN: RApid PosTeriOR GENerator for Bayesian Learning in Biomanufacturing", "abstract": "Biopharmaceutical manufacturing is vital to public health but lacks the agility for rapid, on-demand production of biotherapeutics due to the complexity and variability of bioprocesses. To overcome this, we introduce RApid PosTeriOR GENerator (RAPTOR-GEN), a mechanism-informed Bayesian learning framework designed to accelerate intelligent digital twin development from sparse and heterogeneous experimental data. This framework is built on a multi-scale probabilistic knowledge graph (pKG), formulated as a stochastic differential equation (SDE)-based foundational model that captures the nonlinear dynamics of bioprocesses. RAPTOR-GEN consists of two ingredients: (i) an interpretable metamodel integrating linear noise approximation (LNA) that exploits the structural information of bioprocessing mechanisms and a sequential learning strategy to fuse heterogeneous and sparse data, enabling inference of latent state variables and explicit approximation of the intractable likelihood function; and (ii) an efficient Bayesian posterior sampling method that utilizes Langevin diffusion (LD) to accelerate posterior exploration by exploiting the gradients of the derived likelihood. It generalizes the LNA approach to circumvent the challenge of step size selection, facilitating robust learning of mechanistic parameters with provable finite-sample performance guarantees. We develop a fast and robust RAPTOR-GEN algorithm with controllable error. Numerical experiments demonstrate its effectiveness in uncovering the underlying regulatory mechanisms of biomanufacturing processes.", "date": "2025-09-25", "categories": []}
{"id": "2510.21598", "title": "Fisher meets Feynman: score-based variational inference with a product of experts", "abstract": "We introduce a highly expressive yet distinctly tractable family for black-box variational inference (BBVI). Each member of this family is a weighted product of experts (PoE), and each weighted expert in the product is proportional to a multivariate $t$-distribution. These products of experts can model distributions with skew, heavy tails, and multiple modes, but to use them for BBVI, we must be able to sample from their densities. We show how to do this by reformulating these products of experts as latent variable models with auxiliary Dirichlet random variables. These Dirichlet variables emerge from a Feynman identity, originally developed for loop integrals in quantum field theory, that expresses the product of multiple fractions (or in our case, $t$-distributions) as an integral over the simplex. We leverage this simplicial latent space to draw weighted samples from these products of experts -- samples which BBVI then uses to find the PoE that best approximates a target density. Given a collection of experts, we derive an iterative procedure to optimize the exponents that determine their geometric weighting in the PoE. At each iteration, this procedure minimizes a regularized Fisher divergence to match the scores of the variational and target densities at a batch of samples drawn from the current approximation. This minimization reduces to a convex quadratic program, and we prove under general conditions that these updates converge exponentially fast to a near-optimal weighting of experts. We conclude by evaluating this approach on a variety of synthetic and real-world target distributions.", "date": "2025-10-24", "categories": []}
{"id": "2510.19425", "title": "Neural Variational Dropout Processes", "abstract": "Learning to infer the conditional posterior model is a key step for robust meta-learning. This paper presents a new Bayesian meta-learning approach called Neural Variational Dropout Processes (NVDPs). NVDPs model the conditional posterior distribution based on a task-specific dropout; a low-rank product of Bernoulli experts meta-model is utilized for a memory-efficient mapping of dropout rates from a few observed contexts. It allows for a quick reconfiguration of a globally learned and shared neural network for new tasks in multi-task few-shot learning. In addition, NVDPs utilize a novel prior conditioned on the whole task data to optimize the conditional \\textit{dropout} posterior in the amortized variational inference. Surprisingly, this enables the robust approximation of task-specific dropout rates that can deal with a wide range of functional ambiguities and uncertainties. We compared the proposed method with other meta-learning approaches in the few-shot learning tasks such as 1D stochastic regression, image inpainting, and classification. The results show the excellent performance of NVDPs.", "date": "2025-10-22", "categories": []}
{"id": "2409.01163", "title": "PACSBO: Probably approximately correct safe Bayesian optimization", "abstract": "Safe Bayesian optimization (BO) algorithms promise to find optimal control policies without knowing the system dynamics while at the same time guaranteeing safety with high probability. In exchange for those guarantees, popular algorithms require a smoothness assumption: a known upper bound on a norm in a reproducing kernel Hilbert space (RKHS). The RKHS is a potentially infinite-dimensional space, and it is unclear how to, in practice, obtain an upper bound of an unknown function in its corresponding RKHS. In response, we propose an algorithm that estimates an upper bound on the RKHS norm of an unknown function from data and investigate its theoretical properties. Moreover, akin to Lipschitz-based methods, we treat the RKHS norm as a local rather than a global object, and thus reduce conservatism. Integrating the RKHS norm estimation and the local interpretation of the RKHS norm into a safe BO algorithm yields PACSBO, an algorithm for probably approximately correct safe Bayesian optimization, for which we provide numerical and hardware experiments that demonstrate its applicability and benefits over popular safe BO algorithms.", "date": "2024-09-02", "categories": []}
{"id": "2506.07259", "title": "ALINE: Joint Amortization for Bayesian Inference and Active Data Acquisition", "abstract": "Many critical applications, from autonomous scientific discovery to personalized medicine, demand systems that can both strategically acquire the most informative data and instantaneously perform inference based upon it. While amortized methods for Bayesian inference and experimental design offer part of the solution, neither approach is optimal in the most general and challenging task, where new data needs to be collected for instant inference. To tackle this issue, we introduce the Amortized Active Learning and Inference Engine (ALINE), a unified framework for amortized Bayesian inference and active data acquisition. ALINE leverages a transformer architecture trained via reinforcement learning with a reward based on self-estimated information gain provided by its own integrated inference component. This allows it to strategically query informative data points while simultaneously refining its predictions. Moreover, ALINE can selectively direct its querying strategy towards specific subsets of model parameters or designated predictive tasks, optimizing for posterior estimation, data prediction, or a mixture thereof. Empirical results on regression-based active learning, classical Bayesian experimental design benchmarks, and a psychometric model with selectively targeted parameters demonstrate that ALINE delivers both instant and accurate inference along with efficient selection of informative points.", "date": "2025-06-08", "categories": []}
{"id": "2403.13748", "title": "Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs", "abstract": "Given an intractable distribution $p$, the problem of variational inference (VI) is to find the best approximation from some more tractable family $Q$. Commonly, one chooses $Q$ to be a family of factorized distributions (i.e., the mean-field assumption), even though $p$ itself does not factorize. We show that this mismatch can lead to an impossibility theorem: if $p$ does not factorize and furthermore has a non-diagonal covariance matrix, then any factorized approximation $q\\!\\in\\!Q$ can correctly estimate at most one of the following three measures of uncertainty: (i) the marginal variances, (ii) the marginal precisions, or (iii) the generalized variance (which for elliptical distributions is closely related to the entropy). In practice, the best variational approximation in $Q$ is found by minimizing some divergence $D(q,p)$ between distributions, and so we ask: how does the choice of divergence determine which measure of uncertainty, if any, is correctly estimated by VI? We consider the classic Kullback-Leibler divergences, the more general $\\alpha$-divergences, and a score-based divergence which compares $\\nabla \\log p$ and $\\nabla \\log q$. We thoroughly analyze the case where $p$ is a Gaussian and $q$ is a (factorized) Gaussian. In this setting, we show that all the considered divergences can be ordered based on the estimates of uncertainty they yield as objective functions for VI. Finally, we empirically evaluate the validity of this ordering when the target distribution $p$ is not Gaussian.", "date": "2024-03-20", "categories": []}
{"id": "2502.18300", "title": "Bayesian Computation in Deep Learning", "abstract": "Bayesian methods have shown success in deep learning applications. For example, in predictive tasks, Bayesian neural networks leverage Bayesian reasoning of model uncertainty to improve the reliability and uncertainty awareness of deep neural networks. In generative modeling domain, many widely used deep generative models, such as deep latent variable models, require approximate Bayesian inference to infer their latent variables for the training. In this chapter, we provide an introduction to approximate inference techniques as Bayesian computation methods applied to deep learning models, with a focus on Bayesian neural networks and deep generative models. We review two arguably most popular approximate Bayesian computational methods, stochastic gradient Markov chain Monte Carlo (SG-MCMC) and variational inference (VI), and explain their unique challenges in posterior inference as well as the solutions when applied to deep learning models.", "date": "2025-02-25", "categories": []}
{"id": "2510.15548", "title": "Geometric Convergence Analysis of Variational Inference via Bregman Divergences", "abstract": "Variational Inference (VI) provides a scalable framework for Bayesian inference by optimizing the Evidence Lower Bound (ELBO), but convergence analysis remains challenging due to the objective's non-convexity and non-smoothness in Euclidean space. We establish a novel theoretical framework for analyzing VI convergence by exploiting the exponential family structure of distributions. We express negative ELBO as a Bregman divergence with respect to the log-partition function, enabling a geometric analysis of the optimization landscape. We show that this Bregman representation admits a weak monotonicity property that, while weaker than convexity, provides sufficient structure for rigorous convergence analysis. By deriving bounds on the objective function along rays in parameter space, we establish properties governed by the spectral characteristics of the Fisher information matrix. Under this geometric framework, we prove non-asymptotic convergence rates for gradient descent algorithms with both constant and diminishing step sizes.", "date": "2025-10-17", "categories": []}
{"id": "2507.07338", "title": "Bayesian Double Descent", "abstract": "Double descent is a phenomenon of over-parameterized statistical models such as deep neural networks which have a re-descending property in their risk function. As the complexity of the model increases, risk exhibits a U-shaped region due to the traditional bias-variance trade-off, then as the number of parameters equals the number of observations and the model becomes one of interpolation where the risk can be unbounded and finally, in the over-parameterized region, it re-descends -- the double descent effect. Our goal is to show that this has a natural Bayesian interpretation. We also show that this is not in conflict with the traditional Occam's razor -- simpler models are preferred to complex ones, all else being equal. Our theoretical foundations use Bayesian model selection, the Dickey-Savage density ratio, and connect generalized ridge regression and global-local shrinkage methods with double descent. We illustrate our approach for high dimensional neural networks and provide detailed treatments of infinite Gaussian means models and non-parametric regression. Finally, we conclude with directions for future research.", "date": "2025-07-09", "categories": []}
{"id": "2507.08922", "title": "The Bayesian Approach to Continual Learning: An Overview", "abstract": "Continual learning is an online paradigm where a learner continually accumulates knowledge from different tasks encountered over sequential time steps. Importantly, the learner is required to extend and update its knowledge without forgetting about the learning experience acquired from the past, and while avoiding the need to retrain from scratch. Given its sequential nature and its resemblance to the way humans think, continual learning offers an opportunity to address several challenges which currently stand in the way of widening the range of applicability of deep models to further real-world problems. The continual need to update the learner with data arriving sequentially strikes inherent congruence between continual learning and Bayesian inference which provides a principal platform to keep updating the prior beliefs of a model given new data, without completely forgetting the knowledge acquired from the old data. This survey inspects different settings of Bayesian continual learning, namely task-incremental learning and class-incremental learning. We begin by discussing definitions of continual learning along with its Bayesian setting, as well as the links with related fields, such as domain adaptation, transfer learning and meta-learning. Afterwards, we introduce a taxonomy offering a comprehensive categorization of algorithms belonging to the Bayesian continual learning paradigm. Meanwhile, we analyze the state-of-the-art while zooming in on some of the most prominent Bayesian continual learning algorithms to date. Furthermore, we shed some light on links between continual learning and developmental psychology, and correspondingly introduce analogies between both fields. We follow that with a discussion of current challenges, and finally conclude with potential areas for future research on Bayesian continual learning.", "date": "2025-07-11", "categories": []}
{"id": "2505.14161", "title": "Personalized Bayesian Federated Learning with Wasserstein Barycenter Aggregation", "abstract": "Personalized Bayesian federated learning (PBFL) handles non-i.i.d. client data and quantifies uncertainty by combining personalization with Bayesian inference. However, existing PBFL methods face two limitations: restrictive parametric assumptions in client posterior inference and naive parameter averaging for server aggregation. To overcome these issues, we propose FedWBA, a novel PBFL method that enhances both local inference and global aggregation. At the client level, we use particle-based variational inference for nonparametric posterior representation. At the server level, we introduce particle-based Wasserstein barycenter aggregation, offering a more geometrically meaningful approach. Theoretically, we provide local and global convergence guarantees for FedWBA. Locally, we prove a KL divergence decrease lower bound per iteration for variational inference convergence. Globally, we show that the Wasserstein barycenter converges to the true parameter as the client data size increases. Empirically, experiments show that FedWBA outperforms baselines in prediction accuracy, uncertainty calibration, and convergence rate, with ablation studies confirming its robustness.", "date": "2025-05-20", "categories": []}
{"id": "2510.11704", "title": "Bayesian Topological Convolutional Neural Nets", "abstract": "Convolutional neural networks (CNNs) have been established as the main workhorse in image data processing; nonetheless, they require large amounts of data to train, often produce overconfident predictions, and frequently lack the ability to quantify the uncertainty of their predictions. To address these concerns, we propose a new Bayesian topological CNN that promotes a novel interplay between topology-aware learning and Bayesian sampling. Specifically, it utilizes information from important manifolds to accelerate training while reducing calibration error by placing prior distributions on network parameters and properly learning appropriate posteriors. One important contribution of our work is the inclusion of a consistency condition in the learning cost, which can effectively modify the prior distributions to improve the performance of our novel network architecture. We evaluate the model on benchmark image classification datasets and demonstrate its superiority over conventional CNNs, Bayesian neural networks (BNNs), and topological CNNs. In particular, we supply evidence that our method provides an advantage in situations where training data is limited or corrupted. Furthermore, we show that the new model allows for better uncertainty quantification than standard BNNs since it can more readily identify examples of out-of-distribution data on which it has not been trained. Our results highlight the potential of our novel hybrid approach for more efficient and robust image classification.", "date": "2025-10-13", "categories": []}
{"id": "2502.06379", "title": "Solving Linear-Gaussian Bayesian Inverse Problems with Decoupled Diffusion Sequential Monte Carlo", "abstract": "A recent line of research has exploited pre-trained generative diffusion models as priors for solving Bayesian inverse problems. We contribute to this research direction by designing a sequential Monte Carlo method for linear-Gaussian inverse problems which builds on \"decoupled diffusion\", where the generative process is designed such that larger updates to the sample are possible. The method is asymptotically exact and we demonstrate the effectiveness of our Decoupled Diffusion Sequential Monte Carlo (DDSMC) algorithm on both synthetic as well as protein and image data. Further, we demonstrate how the approach can be extended to discrete data.", "date": "2025-02-10", "categories": []}
{"id": "2506.11898", "title": "Martingale Posterior Neural Networks for Fast Sequential Decision Making", "abstract": "We introduce scalable algorithms for online learning of neural network parameters and Bayesian sequential decision making. Unlike classical Bayesian neural networks, which induce predictive uncertainty through a posterior over model parameters, our methods adopt a predictive-first perspective based on martingale posteriors. In particular, we work directly with the one-step-ahead posterior predictive, which we parameterize with a neural network and update sequentially with incoming observations. This decouples Bayesian decision-making from parameter-space inference: we sample from the posterior predictive for decision making, and update the parameters of the posterior predictive via fast, frequentist Kalman-filter-like recursions. Our algorithms operate in a fully online, replay-free setting, providing principled uncertainty quantification without costly posterior sampling. Empirically, they achieve competitive performance-speed trade-offs in non-stationary contextual bandits and Bayesian optimization, offering 10-100 times faster inference than classical Thompson sampling while maintaining comparable or superior decision performance.", "date": "2025-06-13", "categories": []}
{"id": "2510.07965", "title": "Stick-Breaking Mixture Normalizing Flows with Component-Wise Tail Adaptation for Variational Inference", "abstract": "Normalizing flows with a Gaussian base provide a computationally efficient way to approximate posterior distributions in Bayesian inference, but they often struggle to capture complex posteriors with multimodality and heavy tails. We propose a stick-breaking mixture base with component-wise tail adaptation (StiCTAF) for posterior approximation. The method first learns a flexible mixture base to mitigate the mode-seeking bias of reverse KL divergence through a weighted average of component-wise ELBOs. It then estimates local tail indices of unnormalized densities and finally refines each mixture component using a shared backbone combined with component-specific tail transforms calibrated by the estimated indices. This design enables accurate mode coverage and anisotropic tail modeling while retaining exact density evaluation and stable optimization. Experiments on synthetic posteriors demonstrate improved tail recovery and better coverage of multiple modes compared to benchmark models. We also present a real-data analysis illustrating the practical benefits of our approach for posterior inference.", "date": "2025-10-09", "categories": []}
{"id": "2510.08113", "title": "Bayesian Decision Making around Experts", "abstract": "Complex learning agents are increasingly deployed alongside existing experts, such as human operators or previously trained agents. However, it remains unclear how should learners optimally incorporate certain forms of expert data, which may differ in structure from the learner's own action-outcome experiences. We study this problem in the context of Bayesian multi-armed bandits, considering: (i) offline settings, where the learner receives a dataset of outcomes from the expert's optimal policy before interaction, and (ii) simultaneous settings, where the learner must choose at each step whether to update its beliefs based on its own experience, or based on the outcome simultaneously achieved by an expert. We formalize how expert data influences the learner's posterior, and prove that pretraining on expert outcomes tightens information-theoretic regret bounds by the mutual information between the expert data and the optimal action. For the simultaneous setting, we propose an information-directed rule where the learner processes the data source that maximizes their one-step information gain about the optimal action. Finally, we propose strategies for how the learner can infer when to trust the expert and when not to, safeguarding the learner for the cases where the expert is ineffective or compromised. By quantifying the value of expert data, our framework provides practical, information-theoretic algorithms for agents to intelligently decide when to learn from others.", "date": "2025-10-09", "categories": []}
{"id": "2509.07030", "title": "A Minimalist Bayesian Framework for Stochastic Optimization", "abstract": "The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the component of interest, such as the location of the optimum. Nuisance parameters are eliminated via profile likelihood, which naturally handles constraints. As a direct instantiation, we develop a MINimalist Thompson Sampling (MINTS) algorithm. Our framework accommodates structured problems, including continuum-armed Lipschitz bandits and dynamic pricing. It also provides a probabilistic lens on classical convex optimization algorithms such as the center of gravity and ellipsoid methods. We further analyze MINTS for multi-armed bandits and establish near-optimal regret guarantees.", "date": "2025-09-07", "categories": []}
{"id": "2510.06919", "title": "Bayesian Nonparametric Dynamical Clustering of Time Series", "abstract": "We present a method that models the evolution of an unbounded number of time series clusters by switching among an unknown number of regimes with linear dynamics. We develop a Bayesian non-parametric approach using a hierarchical Dirichlet process as a prior on the parameters of a Switching Linear Dynamical System and a Gaussian process prior to model the statistical variations in amplitude and temporal alignment within each cluster. By modeling the evolution of time series patterns, the method avoids unnecessary proliferation of clusters in a principled manner. We perform inference by formulating a variational lower bound for off-line and on-line scenarios, enabling efficient learning through optimization. We illustrate the versatility and effectiveness of the approach through several case studies of electrocardiogram analysis using publicly available databases.", "date": "2025-10-08", "categories": []}
{"id": "2208.12830", "title": "Mixtures of Gaussian Process Experts with SMC$^2$", "abstract": "Gaussian processes are a key component of many flexible statistical and machine learning models. However, they exhibit cubic computational complexity and high memory constraints due to the need of inverting and storing a full covariance matrix. To circumvent this, mixtures of Gaussian process experts have been considered where data points are assigned to independent experts, reducing the complexity by allowing inference based on smaller, local covariance matrices. Moreover, mixtures of Gaussian process experts substantially enrich the model's flexibility, allowing for behaviors such as non-stationarity, heteroscedasticity, and discontinuities. In this work, we construct a novel inference approach based on nested sequential Monte Carlo samplers to simultaneously infer both the gating network and Gaussian process expert parameters. This greatly improves inference compared to importance sampling, particularly in settings when a stationary Gaussian process is inappropriate, while still being thoroughly parallelizable.", "date": "2022-08-26", "categories": []}
{"id": "2503.01072", "title": "Vector Copula Variational Inference and Dependent Block Posterior Approximations", "abstract": "The key to VI is the selection of a tractable density to approximate the Bayesian posterior. For large and complex models a common choice is to assume independence between multivariate blocks in a partition of the parameter space. While this simplifies the problem it can reduce accuracy. This paper proposes using vector copulas to capture dependence between the blocks parsimoniously. Tailored multivariate marginals are constructed using learnable transport maps. We call the resulting joint distribution a ``dependent block posterior'' approximation. Vector copula models are suggested that make tractable and flexible variational approximations. They allow for differing marginals, numbers of blocks, block sizes and forms of between block dependence. They also allow for solution of the variational optimization using efficient stochastic gradient methods. The approach is demonstrated using four different statistical models and 16 datasets which have posteriors that are challenging to approximate. This includes models that use global-local shrinkage priors for regularization, and hierarchical models for smoothing and heteroscedastic time series. In all cases, our method produces more accurate posterior approximations than benchmark VI methods that either assume block independence or factor-based dependence, at limited additional computational cost. A python package implementing the method is available on GitHub at https://github.com/YuFuOliver/VCVI_Rep_PyPackage.", "date": "2025-03-03", "categories": []}
{"id": "2509.17153", "title": "Flow-Induced Diagonal Gaussian Processes", "abstract": "We present Flow-Induced Diagonal Gaussian Processes (FiD-GP), a compression framework that incorporates a compact inducing weight matrix to project a neural network's weight uncertainty into a lower-dimensional subspace. Critically, FiD-GP relies on normalising-flow priors and spectral regularisations to augment its expressiveness and align the inducing subspace with feature-gradient geometry through a numerically stable projection mechanism objective. Furthermore, we demonstrate how the prediction framework in FiD-GP can help to design a single-pass projection for Out-of-Distribution (OoD) detection. Our analysis shows that FiD-GP improves uncertainty estimation ability on various tasks compared with SVGP-based baselines, satisfies tight spectral residual bounds with theoretically guaranteed OoD detection, and significantly compresses the neural network's storage requirements at the cost of increased inference computation dependent on the number of inducing weights employed. Specifically, in a comprehensive empirical study spanning regression, image classification, semantic segmentation, and out-of-distribution detection benchmarks, it cuts Bayesian training cost by several orders of magnitude, compresses parameters by roughly 51%, reduces model size by about 75%, and matches state-of-the-art accuracy and uncertainty estimation.", "date": "2025-09-21", "categories": []}
{"id": "2412.11646", "title": "Information-Geometric Barycenters for Bayesian Federated Learning", "abstract": "Federated learning (FL) is a widely used and impactful distributed optimization framework that achieves consensus through averaging locally trained models. While effective, this approach may not align well with Bayesian inference, where the model space has the structure of a distribution space. Taking an information-geometric perspective, we reinterpret FL aggregation as the problem of finding the barycenter of local posteriors using a prespecified divergence metric, minimizing the average discrepancy across clients. This perspective provides a unifying framework that generalizes many existing methods and offers crisp insights into their theoretical underpinnings. We then propose BA-BFL, an algorithm that retains the convergence properties of Federated Averaging in non-convex settings. In non-independent and identically distributed scenarios, we conduct extensive comparisons with statistical aggregation techniques, showing that BA-BFL achieves performance comparable to state-of-the-art methods while offering a geometric interpretation of the aggregation phase. Additionally, we extend our analysis to Hybrid Bayesian Deep Learning, exploring the impact of Bayesian layers on uncertainty quantification and model calibration.", "date": "2024-12-16", "categories": []}
{"id": "2507.09828", "title": "Regret Analysis of Posterior Sampling-Based Expected Improvement for Bayesian Optimization", "abstract": "Bayesian optimization is a powerful tool for optimizing an expensive-to-evaluate black-box function. In particular, the effectiveness of expected improvement (EI) has been demonstrated in a wide range of applications. However, theoretical analyses of EI are limited compared with other theoretically established algorithms. This paper analyzes a randomized variant of EI, which evaluates the EI from the maximum of the posterior sample path. We show that this posterior sampling-based random EI achieves the sublinear Bayesian cumulative regret bounds under the assumption that the black-box function follows a Gaussian process. Finally, we demonstrate the effectiveness of the proposed method through numerical experiments.", "date": "2025-07-13", "categories": []}
{"id": "2509.23830", "title": "Bayesian Mixture-of-Experts: Towards Making LLMs Know What They Don't Know", "abstract": "The Mixture-of-Experts (MoE) architecture has enabled the creation of massive yet efficient Large Language Models (LLMs). However, the standard deterministic routing mechanism presents a significant limitation: its inherent brittleness is a key contributor to model miscalibration and overconfidence, resulting in systems that often do not know what they don't know.\n  This thesis confronts this challenge by proposing a structured \\textbf{Bayesian MoE routing framework}. Instead of forcing a single, deterministic expert selection, our approach models a probability distribution over the routing decision itself. We systematically investigate three families of methods that introduce this principled uncertainty at different stages of the routing pipeline: in the \\textbf{weight-space}, the \\textbf{logit-space}, and the final \\textbf{selection-space}.\n  Through a series of controlled experiments on a 3-billion parameter MoE model, we demonstrate that this framework significantly improves routing stability, in-distribution calibration, and out-of-distribution (OoD) detection. The results show that by targeting this core architectural component, we can create a more reliable internal uncertainty signal. This work provides a practical and computationally tractable pathway towards building more robust and self-aware LLMs, taking a crucial step towards making them know what they don't know.", "date": "2025-09-28", "categories": []}
{"id": "2509.24919", "title": "Meta-Learning Theory-Informed Inductive Biases using Deep Kernel Gaussian Processes", "abstract": "Normative and task-driven theories offer powerful top-down explanations for biological systems, yet the goals of quantitatively arbitrating between competing theories, and utilizing them as inductive biases to improve data-driven fits of real biological datasets are prohibitively laborious, and often impossible. To this end, we introduce a Bayesian meta-learning framework designed to automatically convert raw functional predictions from normative theories into tractable probabilistic models. We employ adaptive deep kernel Gaussian processes, meta-learning a kernel on synthetic data generated from a normative theory. This Theory-Informed Kernel specifies a probabilistic model representing the theory predictions -- usable for both fitting data and rigorously validating the theory. As a demonstration, we apply our framework to the early visual system, using efficient coding as our normative theory. We show improved response prediction accuracy in ex vivo recordings of mouse retinal ganglion cells stimulated by natural scenes compared to conventional data-driven baselines, while providing well-calibrated uncertainty estimates and interpretable representations. Using exact Bayesian model selection, we also show that our informed kernel can accurately infer the degree of theory-match from data, confirming faithful encapsulation of theory structure. This work provides a more general, scalable, and automated approach for integrating theoretical knowledge into data-driven scientific inquiry in neuroscience and beyond.", "date": "2025-09-29", "categories": []}
{"id": "2412.05723", "title": "Training-Free Bayesianization for Low-Rank Adapters of Large Language Models", "abstract": "Estimating the uncertainty of responses from Large Language Models (LLMs) remains a critical challenge. While recent Bayesian methods have demonstrated effectiveness in quantifying uncertainty through low-rank weight updates, they typically require complex fine-tuning or post-training procedures. In this paper, we propose Training-Free Bayesianization (TFB), a simple yet theoretically grounded framework that efficiently transforms trained low-rank adapters into Bayesian ones without additional training. TFB systematically searches for the maximally acceptable level of variance in the weight posterior, constrained within a family of low-rank isotropic Gaussian distributions. Our theoretical analysis shows that under mild conditions, this search process is equivalent to KL-regularized variational optimization, a generalized form of variational inference. Through comprehensive experiments, we show that TFB achieves superior uncertainty estimation and generalization compared to existing methods while eliminating the need for complex Bayesianization training procedures. Code will be available at https://github.com/Wang-ML-Lab/bayesian-peft.", "date": "2024-12-07", "categories": []}
{"id": "2509.22111", "title": "Modeling Psychological Profiles in Volleyball via Mixed-Type Bayesian Networks", "abstract": "Psychological attributes rarely operate in isolation: coaches reason about networks of related traits. We analyze a new dataset of 164 female volleyball players from Italy's C and D leagues that combines standardized psychological profiling with background information. To learn directed relationships among mixed-type variables (ordinal questionnaire scores, categorical demographics, continuous indicators), we introduce latent MMHC, a hybrid structure learner that couples a latent Gaussian copula and a constraint-based skeleton with a constrained score-based refinement to return a single DAG. We also study a bootstrap-aggregated variant for stability. In simulations spanning sample size, sparsity, and dimension, latent Max-Min Hill-Climbing (MMHC) attains lower structural Hamming distance and higher edge recall than recent copula-based learners while maintaining high specificity. Applied to volleyball, the learned network organizes mental skills around goal setting and self-confidence, with emotional arousal linking motivation and anxiety, and locates Big-Five traits (notably neuroticism and extraversion) upstream of skill clusters. Scenario analyses quantify how improvements in specific skills propagate through the network to shift preparation, confidence, and self-esteem. The approach provides an interpretable, data-driven framework for profiling psychological traits in sport and for decision support in athlete development.", "date": "2025-09-26", "categories": []}
{"id": "2509.20574", "title": "The Sensitivity of Variational Bayesian Neural Network Performance to Hyperparameters", "abstract": "In scientific applications, predictive modeling is often of limited use without accurate uncertainty quantification (UQ) to indicate when a model may be extrapolating or when more data needs to be collected. Bayesian Neural Networks (BNNs) produce predictive uncertainty by propagating uncertainty in neural network (NN) weights and offer the promise of obtaining not only an accurate predictive model but also accurate UQ. However, in practice, obtaining accurate UQ with BNNs is difficult due in part to the approximations used for practical model training and in part to the need to choose a suitable set of hyperparameters; these hyperparameters outnumber those needed for traditional NNs and often have opaque effects on the results. We aim to shed light on the effects of hyperparameter choices for BNNs by performing a global sensitivity analysis of BNN performance under varying hyperparameter settings. Our results indicate that many of the hyperparameters interact with each other to affect both predictive accuracy and UQ. For improved usage of BNNs in real-world applications, we suggest that global sensitivity analysis, or related methods such as Bayesian optimization, should be used to aid in dimensionality reduction and selection of hyperparameters to ensure accurate UQ in BNNs.", "date": "2025-09-24", "categories": []}
{"id": "2502.13105", "title": "Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problems", "abstract": "Among other uses, neural networks are a powerful tool for solving deterministic and Bayesian inverse problems in real-time, where variational autoencoders, a specialized type of neural network, enable the Bayesian estimation of model parameters and their distribution from observational data allowing real-time inverse uncertainty quantification. In this work, we build upon existing research [Goh, H. et al., Proceedings of Machine Learning Research, 2022] by proposing a novel loss function to train variational autoencoders for Bayesian inverse problems. When the forward map is affine, we provide a theoretical proof of the convergence of the latent states of variational autoencoders to the posterior distribution of the model parameters. We validate this theoretical result through numerical tests and we compare the proposed variational autoencoder with the existing one in the literature both in terms of accuracy and generalization properties. Finally, we test the proposed variational autoencoder on a Laplace equation, with comparison to the original one and Markov Chains Monte Carlo.", "date": "2025-02-18", "categories": []}
{"id": "2405.13535", "title": "Addressing the Inconsistency in Bayesian Deep Learning via Generalized Laplace Approximation", "abstract": "In recent years, inconsistency in Bayesian deep learning has attracted significant attention. Tempered or generalized posterior distributions are frequently employed as direct and effective solutions. Nonetheless, the underlying mechanisms and the effectiveness of generalized posteriors remain active research topics. In this work, we interpret posterior tempering as a correction for model misspecification via adjustments to the joint probability, and as a recalibration of priors by reducing aleatoric uncertainty. We also introduce the generalized Laplace approximation, which requires only a simple modification to the Hessian calculation of the regularized loss and provides a flexible and scalable framework for high-quality posterior inference. We evaluate the proposed method on state-of-the-art neural networks and real-world datasets, demonstrating that the generalized Laplace approximation enhances predictive performance.", "date": "2024-05-22", "categories": []}
{"id": "2504.13320", "title": "Gradient-Free Sequential Bayesian Experimental Design via Interacting Particle Systems", "abstract": "We introduce a gradient-free framework for Bayesian Optimal Experimental Design (BOED) in sequential settings, aimed at complex systems where gradient information is unavailable. Our method combines Ensemble Kalman Inversion (EKI) for design optimization with the Affine-Invariant Langevin Dynamics (ALDI) sampler for efficient posterior sampling-both of which are derivative-free and ensemble-based. To address the computational challenges posed by nested expectations in BOED, we propose variational Gaussian and parametrized Laplace approximations that provide tractable upper and lower bounds on the Expected Information Gain (EIG). These approximations enable scalable utility estimation in high-dimensional spaces and PDE-constrained inverse problems. We demonstrate the performance of our framework through numerical experiments ranging from linear Gaussian models to PDE-based inference tasks, highlighting the method's robustness, accuracy, and efficiency in information-driven experimental design.", "date": "2025-04-17", "categories": []}
{"id": "2509.15641", "title": "Information Geometry of Variational Bayes", "abstract": "We highlight a fundamental connection between information geometry and variational Bayes (VB) and discuss its consequences for machine learning. Under certain conditions, a VB solution always requires estimation or computation of natural gradients. We show several consequences of this fact by using the natural-gradient descent algorithm of Khan and Rue (2023) called the Bayesian Learning Rule (BLR). These include (i) a simplification of Bayes' rule as addition of natural gradients, (ii) a generalization of quadratic surrogates used in gradient-based methods, and (iii) a large-scale implementation of VB algorithms for large language models. Neither the connection nor its consequences are new but we further emphasize the common origins of the two fields of information geometry and Bayes with a hope to facilitate more work at the intersection of the two fields.", "date": "2025-09-19", "categories": []}
{"id": "2509.08834", "title": "An Interval Type-2 Version of Bayes Theorem Derived from Interval Probability Range Estimates Provided by Subject Matter Experts", "abstract": "Bayesian inference is widely used in many different fields to test hypotheses against observations. In most such applications, an assumption is made of precise input values to produce a precise output value. However, this is unrealistic for real-world applications. Often the best available information from subject matter experts (SMEs) in a given field is interval range estimates of the input probabilities involved in Bayes Theorem. This paper provides two key contributions to extend Bayes Theorem to an interval type-2 (IT2) version. First, we develop an IT2 version of Bayes Theorem that uses a novel and conservative method to avoid potential inconsistencies in the input IT2 MFs that otherwise might produce invalid output results. We then describe a novel and flexible algorithm for encoding SME-provided intervals into IT2 fuzzy membership functions (MFs), which we can use to specify the input probabilities in Bayes Theorem. Our algorithm generalizes and extends previous work on this problem that primarily addressed the encoding of intervals into word MFs for Computing with Words applications.", "date": "2025-08-29", "categories": []}
{"id": "2507.14652", "title": "Accelerating Hamiltonian Monte Carlo for Bayesian Inference in Neural Networks and Neural Operators", "abstract": "Hamiltonian Monte Carlo (HMC) is a powerful and accurate method to sample from the posterior distribution in Bayesian inference. However, HMC techniques are computationally demanding for Bayesian neural networks due to the high dimensionality of the network's parameter space and the non-convexity of their posterior distributions. Therefore, various approximation techniques, such as variational inference (VI) or stochastic gradient MCMC, are often employed to infer the posterior distribution of the network parameters. Such approximations introduce inaccuracies in the inferred distributions, resulting in unreliable uncertainty estimates. In this work, we propose a hybrid approach that combines inexpensive VI and accurate HMC methods to efficiently and accurately quantify uncertainties in neural networks and neural operators. The proposed approach leverages an initial VI training on the full network. We examine the influence of individual parameters on the prediction uncertainty, which shows that a large proportion of the parameters do not contribute substantially to uncertainty in the network predictions. This information is then used to significantly reduce the dimension of the parameter space, and HMC is performed only for the subset of network parameters that strongly influence prediction uncertainties. This yields a framework for accelerating the full batch HMC for posterior inference in neural networks. We demonstrate the efficiency and accuracy of the proposed framework on deep neural networks and operator networks, showing that inference can be performed for large networks with tens to hundreds of thousands of parameters. We show that this method can effectively learn surrogates for complex physical systems by modeling the operator that maps from upstream conditions to wall-pressure data on a cone in hypersonic flow.", "date": "2025-07-19", "categories": []}
{"id": "2506.19726", "title": "Precise Bayesian Neural Networks", "abstract": "Despite its long history, Bayesian neural networks (BNNs) and variational training remain underused in practice: standard Gaussian posteriors misalign with network geometry, KL terms can be brittle in high dimensions, and implementations often add complexity without reliably improving uncertainty. We revisit the problem through the lens of normalization. Because normalization layers neutralize the influence of weight magnitude, we model uncertainty \\emph{only in weight directions} using a von Mises-Fisher posterior on the unit sphere. High-dimensional geometry then yields a single, interpretable scalar per layer--the effective post-normalization noise $\\sigma_{\\mathrm{eff}}$--that (i) corresponds to simple additive Gaussian noise in the forward pass and (ii) admits a compact, dimension-aware KL in closed form. We derive accurate, closed-form approximations linking concentration $\\kappa$ to activation variance and to $\\sigma_{\\mathrm{eff}}$ across regimes, producing a lightweight, implementation-ready variational unit that fits modern normalized architectures and improves calibration without sacrificing accuracy. This dimension awareness is critical for stable optimization in high dimensions. In short, by aligning the variational posterior with the network's intrinsic geometry, BNNs can be simultaneously principled, practical, and precise.", "date": "2025-06-24", "categories": []}
{"id": "2509.05732", "title": "Simulation Priors for Data-Efficient Deep Learning", "abstract": "How do we enable AI systems to efficiently learn in the real-world? First-principles models are widely used to simulate natural systems, but often fail to capture real-world complexity due to simplifying assumptions. In contrast, deep learning approaches can estimate complex dynamics with minimal assumptions but require large, representative datasets. We propose SimPEL, a method that efficiently combines first-principles models with data-driven learning by using low-fidelity simulators as priors in Bayesian deep learning. This enables SimPEL to benefit from simulator knowledge in low-data regimes and leverage deep learning's flexibility when more data is available, all the while carefully quantifying epistemic uncertainty. We evaluate SimPEL on diverse systems, including biological, agricultural, and robotic domains, showing superior performance in learning complex dynamics. For decision-making, we demonstrate that SimPEL bridges the sim-to-real gap in model-based reinforcement learning. On a high-speed RC car task, SimPEL learns a highly dynamic parking maneuver involving drifting with substantially less data than state-of-the-art baselines. These results highlight the potential of SimPEL for data-efficient learning and control in complex real-world environments.", "date": "2025-09-06", "categories": []}
{"id": "2502.12122", "title": "Minimal Ranks, Maximum Confidence: Parameter-efficient Uncertainty Quantification for LoRA", "abstract": "Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning of large language models by decomposing weight updates into low-rank matrices, significantly reducing storage and computational overhead. While effective, standard LoRA lacks mechanisms for uncertainty quantification, leading to overconfident and poorly calibrated models. Bayesian variants of LoRA address this limitation, but at the cost of a significantly increased number of trainable parameters, partially offsetting the original efficiency gains. Additionally, these models are harder to train and may suffer from unstable convergence. In this work, we propose a novel parameter-efficient Bayesian LoRA via subspace inference, demonstrating that effective uncertainty quantification can be achieved in very low-dimensional parameter spaces. The proposed method achieves strong performance with improved calibration and generalization while maintaining computational efficiency. Our empirical findings show that, with the appropriate projection of the weight space: (1) uncertainty can be effectively modeled in a low-dimensional space, and (2) weight covariances exhibit low ranks.", "date": "2025-02-17", "categories": []}
{"id": "2509.00263", "title": "Probit Monotone BART", "abstract": "Bayesian Additive Regression Trees (BART) of Chipman et al. (2010) has proven to be a powerful tool for nonparametric modeling and prediction. Monotone BART (Chipman et al., 2022) is a recent development that allows BART to be more precise in estimating monotonic functions. We further these developments by proposing probit monotone BART, which allows the monotone BART framework to estimate conditional mean functions when the outcome variable is binary.", "date": "2025-08-29", "categories": []}
{"id": "2509.00753", "title": "FBMS: An R Package for Flexible Bayesian Model Selection and Model Averaging", "abstract": "The FBMS R package facilitates Bayesian model selection and model averaging in complex regression settings by employing a variety of Monte Carlo model exploration methods. At its core, the package implements an efficient Mode Jumping Markov Chain Monte Carlo (MJMCMC) algorithm, designed to improve mixing in multi-modal posterior landscapes within Bayesian generalized linear models. In addition, it provides a genetically modified MJMCMC (GMJMCMC) algorithm that introduces nonlinear feature generation, thereby enabling the estimation of Bayesian generalized nonlinear models (BGNLMs). Within this framework, the algorithm maintains and updates populations of transformed features, computes their posterior probabilities, and evaluates the posteriors of models constructed from them. We demonstrate the effective use of FBMS for both inferential and predictive modeling in Gaussian regression, focusing on different instances of the BGNLM class of models. Furthermore, through a broad set of applications, we illustrate how the methodology can be extended to increasingly complex modeling scenarios, extending to other response distributions and mixed effect models.", "date": "2025-08-31", "categories": []}
{"id": "2509.02003", "title": "Bouncy particle sampler with infinite exchanging parallel tempering", "abstract": "Bayesian inference is useful to obtain a predictive distribution with a small generalization error. However, since posterior distributions are rarely evaluated analytically, we employ the variational Bayesian inference or sampling method to approximate posterior distributions. When we obtain samples from a posterior distribution, Hamiltonian Monte Carlo (HMC) has been widely used for the continuous variable part and Markov chain Monte Carlo (MCMC) for the discrete variable part. Another sampling method, the bouncy particle sampler (BPS), has been proposed, which combines uniform linear motion and stochastic reflection to perform sampling. BPS was reported to have the advantage of being easier to set simulation parameters than HMC. To accelerate the convergence to a posterior distribution, we introduced parallel tempering (PT) to BPS, and then proposed an algorithm when the inverse temperature exchange rate is set to infinity. We performed numerical simulations and demonstrated its effectiveness for multimodal distribution.", "date": "2025-09-02", "categories": []}
{"id": "2508.21488", "title": "Priors Matter: Addressing Misspecification in Bayesian Deep Q-Learning", "abstract": "Uncertainty quantification in reinforcement learning can greatly improve exploration and robustness. Approximate Bayesian approaches have recently been popularized to quantify uncertainty in model-free algorithms. However, so far the focus has been on improving the accuracy of the posterior approximation, instead of studying the accuracy of the prior and likelihood assumptions underlying the posterior. In this work, we demonstrate that there is a cold posterior effect in Bayesian deep Q-learning, where contrary to theory, performance increases when reducing the temperature of the posterior. To identify and overcome likely causes, we challenge common assumptions made on the likelihood and priors in Bayesian model-free algorithms. We empirically study prior distributions and show through statistical tests that the common Gaussian likelihood assumption is frequently violated. We argue that developing more suitable likelihoods and priors should be a key focus in future Bayesian reinforcement learning research and we offer simple, implementable solutions for better priors in deep Q-learning that lead to more performant Bayesian algorithms.", "date": "2025-08-29", "categories": []}
{"id": "2507.11535", "title": "Canonical Bayesian Linear System Identification", "abstract": "Standard Bayesian approaches for linear time-invariant (LTI) system identification are hindered by parameter non-identifiability; the resulting complex, multi-modal posteriors make inference inefficient and impractical. We solve this problem by embedding canonical forms of LTI systems within the Bayesian framework. We rigorously establish that inference in these minimal parameterizations fully captures all invariant system dynamics (e.g., transfer functions, eigenvalues, predictive distributions of system outputs) while resolving identifiability. This approach unlocks the use of meaningful, structure-aware priors (e.g., enforcing stability via eigenvalues) and ensures conditions for a Bernstein--von Mises theorem -- a link between Bayesian and frequentist large-sample asymptotics that is broken in standard forms. Extensive simulations with modern MCMC methods highlight advantages over standard parameterizations: canonical forms achieve higher computational efficiency, generate interpretable and well-behaved posteriors, and provide robust uncertainty estimates, particularly from limited data.", "date": "2025-07-15", "categories": []}
{"id": "2503.03007", "title": "Benchmarking Diffusion Annealing-Based Bayesian Inverse Problem Solvers", "abstract": "In recent years, the ascendance of diffusion modeling as a state-of-the-art generative modeling approach has spurred significant interest in their use as priors in Bayesian inverse problems. However, it is unclear how to optimally integrate a diffusion model trained on the prior distribution with a given likelihood function to obtain posterior samples. While algorithms developed for this purpose can produce high-quality, diverse point estimates of the unknown parameters of interest, they are often tested on problems where the prior distribution is analytically unknown, making it difficult to assess their performance in providing rigorous uncertainty quantification. Motivated by this challenge, this work introduces three benchmark problems for evaluating the performance of diffusion model based samplers. The benchmark problems, which are inspired by problems in image inpainting, x-ray tomography, and phase retrieval, have a posterior density that is analytically known. In this setting, approximate ground-truth posterior samples can be obtained, enabling principled evaluation of the performance of posterior sampling algorithms. This work also introduces a general framework for diffusion model based posterior sampling, Bayesian Inverse Problem Solvers through Diffusion Annealing (BIPSDA). This framework unifies several recently proposed diffusion-model-based posterior sampling algorithms and contains novel algorithms that can be realized through flexible combinations of design choices. We tested the performance of a set of BIPSDA algorithms, including previously proposed state-of-the-art approaches, on the proposed benchmark problems. The results provide insight into the strengths and limitations of existing diffusion-model based posterior samplers, while the benchmark problems provide a testing ground for future algorithmic developments.", "date": "2025-03-04", "categories": []}
{"id": "2411.03263", "title": "Proxy-informed Bayesian transfer learning with unknown sources", "abstract": "Generalization outside the scope of one's training data requires leveraging prior knowledge about the effects that transfer, and the effects that don't, between different data sources. Transfer learning is a framework for specifying and refining this knowledge about sets of source (training) and target (prediction) data. A challenging open problem is addressing the empirical phenomenon of negative transfer, whereby the transfer learner performs worse on the target data after taking the source data into account than before. We first introduce a Bayesian perspective on negative transfer, and then a method to address it. The key insight from our formulation is that negative transfer can stem from misspecified prior information about non-transferable causes of the source data. Our proposed method, proxy-informed robust method for probabilistic transfer learning (PROMPT), does not require prior knowledge of the source data (the data sources may be \"unknown\"). PROMPT is thus applicable when differences between tasks are unobserved, such as in the presence of latent confounders. Moreover, the learner need not have access to observations in the target task (may not have the ability to \"fine-tune\"), and instead makes use of proxy (indirect) information. Our theoretical results show that the threat of negative transfer does not depend on the informativeness of the proxy information, highlighting the usefulness of PROMPT in cases where only noisy indirect information, such as human feedback, is available.", "date": "2024-11-05", "categories": []}
{"id": "2505.13585", "title": "Scalable Bayesian Monte Carlo: fast uncertainty estimation beyond deep ensembles", "abstract": "This work introduces a new method designed for Bayesian deep learning called scalable Bayesian Monte Carlo (SBMC). The method is comprised of a model and an algorithm. The model interpolates between a point estimator and the posterior. The algorithm is a parallel implementation of sequential Monte Carlo sampler (SMC$_\\parallel$) or Markov chain Monte Carlo (MCMC$_\\parallel$). We collectively refer to these consistent (asymptotically unbiased) algorithms as Bayesian Monte Carlo (BMC), and any such algorithm can be used in our SBMC method. The utility of the method is demonstrated on practical examples: MNIST, CIFAR, IMDb. A systematic numerical study reveals that for the same wall-clock time as state-of-the-art (SOTA) methods like deep ensembles (DE), SBMC achieves comparable or better accuracy and substantially improved uncertainty quantification (UQ)--in particular, epistemic UQ. This is demonstrated on the downstream task of estimating the confidence in predictions, which can be used for reliability assessment or abstention decisions.", "date": "2025-05-19", "categories": []}
{"id": "2402.06173", "title": "Comparison of parallel SMC and MCMC for Bayesian deep learning", "abstract": "This work systematically compares parallel implementations of consistent (asymptotically unbiased) Bayesian deep learning algorithms: sequential Monte Carlo sampler (SMC$_\\parallel$) or Markov chain Monte Carlo (MCMC$_\\parallel$). We provide a proof of convergence for SMC$_\\parallel$ showing that it theoretically achieves the same level of convergence as a single monolithic SMC sampler, while the reduced communication lowers wall-clock time. It is well-known that the first samples from MCMC need to be discarded to eliminate initialization bias, and that the number of discarded samples must grow like the logarithm of the number of parallel chains to control that bias for MCMC$_\\parallel$. A systematic empirical numerical study on MNIST, CIFAR, and IMDb, reveals that parallel implementations of both methods perform comparably to non-parallel implementations in terms of performance and total cost, and also comparably to each other. However, both methods still require a large wall-clock time, and suffer from catastrophic non-convergence if they aren't run for long enough.", "date": "2024-02-09", "categories": []}
{"id": "2505.02277", "title": "Epistemic Wrapping for Uncertainty Quantification", "abstract": "Uncertainty estimation is pivotal in machine learning, especially for classification tasks, as it improves the robustness and reliability of models. We introduce a novel `Epistemic Wrapping' methodology aimed at improving uncertainty estimation in classification. Our approach uses Bayesian Neural Networks (BNNs) as a baseline and transforms their outputs into belief function posteriors, effectively capturing epistemic uncertainty and offering an efficient and general methodology for uncertainty quantification. Comprehensive experiments employing a Bayesian Neural Network (BNN) baseline and an Interval Neural Network for inference on the MNIST, Fashion-MNIST, CIFAR-10 and CIFAR-100 datasets demonstrate that our Epistemic Wrapper significantly enhances generalisation and uncertainty quantification.", "date": "2025-05-04", "categories": []}
{"id": "2508.13598", "title": "Approximate Bayesian Inference via Bitstring Representations", "abstract": "The machine learning community has recently put effort into quantized or low-precision arithmetics to scale large models. This paper proposes performing probabilistic inference in the quantized, discrete parameter space created by these representations, effectively enabling us to learn a continuous distribution using discrete parameters. We consider both 2D densities and quantized neural networks, where we introduce a tractable learning approach using probabilistic circuits. This method offers a scalable solution to manage complex distributions and provides clear insights into model behavior. We validate our approach with various models, demonstrating inference efficiency without sacrificing accuracy. This work advances scalable, interpretable machine learning by utilizing discrete approximations for probabilistic computations.", "date": "2025-08-19", "categories": []}
{"id": "2508.10055", "title": "Bayesian Models for Joint Selection of Features and Auto-Regressive Lags: Theory and Applications in Environmental and Financial Forecasting", "abstract": "We develop a Bayesian framework for variable selection in linear regression with autocorrelated errors, accommodating lagged covariates and autoregressive structures. This setting occurs in time series applications where responses depend on contemporaneous or past explanatory variables and persistent stochastic shocks, including financial modeling, hydrological forecasting, and meteorological applications requiring temporal dependency capture. Our methodology uses hierarchical Bayesian models with spike-and-slab priors to simultaneously select relevant covariates and lagged error terms. We propose an efficient two-stage MCMC algorithm separating sampling of variable inclusion indicators and model parameters to address high-dimensional computational challenges. Theoretical analysis establishes posterior selection consistency under mild conditions, even when candidate predictors grow exponentially with sample size, common in modern time series with many potential lagged variables. Through simulations and real applications (groundwater depth prediction, S&P 500 log returns modeling), we demonstrate substantial gains in variable selection accuracy and predictive performance. Compared to existing methods, our framework achieves lower MSPE, improved true model component identification, and greater robustness with autocorrelated noise, underscoring practical utility for model interpretation and forecasting in autoregressive settings.", "date": "2025-08-12", "categories": []}
{"id": "2508.11145", "title": "Towards the Next-generation Bayesian Network Classifiers", "abstract": "Bayesian network classifiers provide a feasible solution to tabular data classification, with a number of merits like high time and memory efficiency, and great explainability. However, due to the parameter explosion and data sparsity issues, Bayesian network classifiers are restricted to low-order feature dependency modeling, making them struggle in extrapolating the occurrence probabilities of complex real-world data. In this paper, we propose a novel paradigm to design high-order Bayesian network classifiers, by learning distributional representations for feature values, as what has been done in word embedding and graph representation learning. The learned distributional representations are encoded with the semantic relatedness between different features through their observed co-occurrence patterns in training data, which then serve as a hallmark to extrapolate the occurrence probabilities of new test samples. As a classifier design realization, we remake the K-dependence Bayesian classifier (KDB) by extending it into a neural version, i.e., NeuralKDB, where a novel neural network architecture is designed to learn distributional representations of feature values and parameterize the conditional probabilities between interdependent features. A stochastic gradient descent based algorithm is designed to train the NeuralKDB model efficiently. Extensive classification experiments on 60 UCI datasets demonstrate that the proposed NeuralKDB classifier excels in capturing high-order feature dependencies and significantly outperforms the conventional Bayesian network classifiers, as well as other competitive classifiers, including two neural network based classifiers without distributional representation learning.", "date": "2025-08-15", "categories": []}
{"id": "2508.09792", "title": "Bayesian autoregression to optimize temporal Mat\\'ern kernel Gaussian process hyperparameters", "abstract": "Gaussian processes are important models in the field of probabilistic numerics. We present a procedure for optimizing Mat\\'ern kernel temporal Gaussian processes with respect to the kernel covariance function's hyperparameters. It is based on casting the optimization problem as a recursive Bayesian estimation procedure for the parameters of an autoregressive model. We demonstrate that the proposed procedure outperforms maximizing the marginal likelihood as well as Hamiltonian Monte Carlo sampling, both in terms of runtime and ultimate root mean square error in Gaussian process regression.", "date": "2025-08-13", "categories": []}
{"id": "2309.12658", "title": "Neural Operator Variational Inference based on Regularized Stein Discrepancy for Deep Gaussian Processes", "abstract": "Deep Gaussian Process (DGP) models offer a powerful nonparametric approach for Bayesian inference, but exact inference is typically intractable, motivating the use of various approximations. However, existing approaches, such as mean-field Gaussian assumptions, limit the expressiveness and efficacy of DGP models, while stochastic approximation can be computationally expensive. To tackle these challenges, we introduce Neural Operator Variational Inference (NOVI) for Deep Gaussian Processes. NOVI uses a neural generator to obtain a sampler and minimizes the Regularized Stein Discrepancy in L2 space between the generated distribution and true posterior. We solve the minimax problem using Monte Carlo estimation and subsampling stochastic optimization techniques. We demonstrate that the bias introduced by our method can be controlled by multiplying the Fisher divergence with a constant, which leads to robust error control and ensures the stability and precision of the algorithm. Our experiments on datasets ranging from hundreds to tens of thousands demonstrate the effectiveness and the faster convergence rate of the proposed method. We achieve a classification accuracy of 93.56 on the CIFAR10 dataset, outperforming SOTA Gaussian process methods. Furthermore, our method guarantees theoretically controlled prediction error for DGP models and demonstrates remarkable performance on various datasets. We are optimistic that NOVI has the potential to enhance the performance of deep Bayesian nonparametric models and could have significant implications for various practical applications", "date": "2023-09-22", "categories": []}
{"id": "2504.04247", "title": "Randomised Postiterations for Calibrated BayesCG", "abstract": "The Bayesian conjugate gradient method offers probabilistic solutions to linear systems but suffers from poor calibration, limiting its utility in uncertainty quantification tasks. Recent approaches leveraging postiterations to construct priors have improved computational properties but failed to correct calibration issues. In this work, we propose a novel randomised postiteration strategy that enhances the calibration of the BayesCG posterior while preserving its favourable convergence characteristics. We present theoretical guarantees for the improved calibration, supported by results on the distribution of posterior errors. Numerical experiments demonstrate the efficacy of the method in both synthetic and inverse problem settings, showing enhanced uncertainty quantification and better propagation of uncertainties through computational pipelines.", "date": "2025-04-05", "categories": []}
{"id": "2508.08300", "title": "LLM-BI: Towards Fully Automated Bayesian Inference with Large Language Models", "abstract": "A significant barrier to the widespread adoption of Bayesian inference is the specification of prior distributions and likelihoods, which often requires specialized statistical expertise. This paper investigates the feasibility of using a Large Language Model (LLM) to automate this process. We introduce LLM-BI (Large Language Model-driven Bayesian Inference), a conceptual pipeline for automating Bayesian workflows. As a proof-of-concept, we present two experiments focused on Bayesian linear regression. In Experiment I, we demonstrate that an LLM can successfully elicit prior distributions from natural language. In Experiment II, we show that an LLM can specify the entire model structure, including both priors and the likelihood, from a single high-level problem description. Our results validate the potential of LLMs to automate key steps in Bayesian modeling, enabling the possibility of an automated inference pipeline for probabilistic programming.", "date": "2025-08-07", "categories": []}
{"id": "2508.08880", "title": "Hi-fi functional priors by learning activations", "abstract": "Function-space priors in Bayesian Neural Networks (BNNs) provide a more intuitive approach to embedding beliefs directly into the model's output, thereby enhancing regularization, uncertainty quantification, and risk-aware decision-making. However, imposing function-space priors on BNNs is challenging. We address this task through optimization techniques that explore how trainable activations can accommodate higher-complexity priors and match intricate target function distributions. We investigate flexible activation models, including Pade functions and piecewise linear functions, and discuss the learning challenges related to identifiability, loss construction, and symmetries. Our empirical findings indicate that even BNNs with a single wide hidden layer when equipped with flexible trainable activation, can effectively achieve desired function-space priors.", "date": "2025-08-12", "categories": []}
{"id": "2312.05568", "title": "Sparse Variational Student-t Processes", "abstract": "The theory of Bayesian learning incorporates the use of Student-t Processes to model heavy-tailed distributions and datasets with outliers. However, despite Student-t Processes having a similar computational complexity as Gaussian Processes, there has been limited emphasis on the sparse representation of this model. This is mainly due to the increased difficulty in modeling and computation compared to previous sparse Gaussian Processes. Our motivation is to address the need for a sparse representation framework that reduces computational complexity, allowing Student-t Processes to be more flexible for real-world datasets. To achieve this, we leverage the conditional distribution of Student-t Processes to introduce sparse inducing points. Bayesian methods and variational inference are then utilized to derive a well-defined lower bound, facilitating more efficient optimization of our model through stochastic gradient descent. We propose two methods for computing the variational lower bound, one utilizing Monte Carlo sampling and the other employing Jensen's inequality to compute the KL regularization term in the loss function. We propose adopting these approaches as viable alternatives to Gaussian processes when the data might contain outliers or exhibit heavy-tailed behavior, and we provide specific recommendations for their applicability. We evaluate the two proposed approaches on various synthetic and real-world datasets from UCI and Kaggle, demonstrating their effectiveness compared to baseline methods in terms of computational complexity and accuracy, as well as their robustness to outliers.", "date": "2023-12-09", "categories": []}
{"id": "2504.10076", "title": "Towards Scalable Bayesian Optimization via Gradient-Informed Bayesian Neural Networks", "abstract": "Bayesian optimization (BO) is a widely used method for data-driven optimization that generally relies on zeroth-order data of objective function to construct probabilistic surrogate models. These surrogates guide the exploration-exploitation process toward finding global optimum. While Gaussian processes (GPs) are commonly employed as surrogates of the unknown objective function, recent studies have highlighted the potential of Bayesian neural networks (BNNs) as scalable and flexible alternatives. Moreover, incorporating gradient observations into GPs, when available, has been shown to improve BO performance. However, the use of gradients within BNN surrogates remains unexplored. By leveraging automatic differentiation, gradient information can be seamlessly integrated into BNN training, resulting in more informative surrogates for BO. We propose a gradient-informed loss function for BNN training, effectively augmenting function observations with local gradient information. The effectiveness of this approach is demonstrated on well-known benchmarks in terms of improved BNN predictions and faster BO convergence as the number of decision variables increases.", "date": "2025-04-14", "categories": []}
{"id": "2508.03766", "title": "LLM-Prior: A Framework for Knowledge-Driven Prior Elicitation and Aggregation", "abstract": "The specification of prior distributions is fundamental in Bayesian inference, yet it remains a significant bottleneck. The prior elicitation process is often a manual, subjective, and unscalable task. We propose a novel framework which leverages Large Language Models (LLMs) to automate and scale this process. We introduce \\texttt{LLMPrior}, a principled operator that translates rich, unstructured contexts such as natural language descriptions, data or figures into valid, tractable probability distributions. We formalize this operator by architecturally coupling an LLM with an explicit, tractable generative model, such as a Gaussian Mixture Model (forming a LLM based Mixture Density Network), ensuring the resulting prior satisfies essential mathematical properties. We further extend this framework to multi-agent systems where Logarithmic Opinion Pooling is employed to aggregate prior distributions induced by decentralized knowledge. We present the federated prior aggregation algorithm, \\texttt{Fed-LLMPrior}, for aggregating distributed, context-dependent priors in a manner robust to agent heterogeneity. This work provides the foundation for a new class of tools that can potentially lower the barrier to entry for sophisticated Bayesian modeling.", "date": "2025-08-05", "categories": []}
{"id": "2508.02337", "title": "Posterior Sampling of Probabilistic Word Embeddings", "abstract": "Quantifying uncertainty in word embeddings is crucial for reliable inference from textual data. However, existing Bayesian methods such as Hamiltonian Monte Carlo (HMC) and mean-field variational inference (MFVI) are either computationally infeasible for large data or rely on restrictive assumptions.\n  We propose a scalable Gibbs sampler using Polya-Gamma augmentation as well as Laplace approximation and compare them with MFVI and HMC for word embeddings. In addition, we address non-identifiability in word embeddings. Our Gibbs sampler and HMC correctly estimate uncertainties, while MFVI does not, and Laplace approximation only does so on large sample sizes, as expected. Applying the Gibbs sampler to the US Congress and the Movielens datasets, we demonstrate the feasibility on larger real data. Finally, as a result of having draws from the full posterior, we show that the posterior mean of word embeddings improves over maximum a posteriori (MAP) estimates in terms of hold-out likelihood, especially for smaller sampling sizes, further strengthening the need for posterior sampling of word embeddings.", "date": "2025-08-04", "categories": []}
{"id": "2508.00617", "title": "Constructive Disintegration and Conditional Modes", "abstract": "Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a \"conditional mode\" does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.", "date": "2025-08-01", "categories": []}
{"id": "2507.23768", "title": "Formal Bayesian Transfer Learning via the Total Risk Prior", "abstract": "In analyses with severe data-limitations, augmenting the target dataset with information from ancillary datasets in the application domain, called source datasets, can lead to significantly improved statistical procedures. However, existing methods for this transfer learning struggle to deal with situations where the source datasets are also limited and not guaranteed to be well-aligned with the target dataset. A typical strategy is to use the empirical loss minimizer on the source data as a prior mean for the target parameters, which places the estimation of source parameters outside of the Bayesian formalism. Our key conceptual contribution is to use a risk minimizer conditional on source parameters instead. This allows us to construct a single joint prior distribution for all parameters from the source datasets as well as the target dataset. As a consequence, we benefit from full Bayesian uncertainty quantification and can perform model averaging via Gibbs sampling over indicator variables governing the inclusion of each source dataset. We show how a particular instantiation of our prior leads to a Bayesian Lasso in a transformed coordinate system and discuss computational techniques to scale our approach to moderately sized datasets. We also demonstrate that recently proposed minimax-frequentist transfer learning techniques may be viewed as an approximate Maximum a Posteriori approach to our model. Finally, we demonstrate superior predictive performance relative to the frequentist baseline on a genetics application, especially when the source data are limited.", "date": "2025-07-31", "categories": []}
{"id": "2507.21449", "title": "From Global to Local: A Scalable Benchmark for Local Posterior Sampling", "abstract": "Degeneracy is an inherent feature of the loss landscape of neural networks, but it is not well understood how stochastic gradient MCMC (SGMCMC) algorithms interact with this degeneracy. In particular, current global convergence guarantees for common SGMCMC algorithms rely on assumptions which are likely incompatible with degenerate loss landscapes. In this paper, we argue that this gap requires a shift in focus from global to local posterior sampling, and, as a first step, we introduce a novel scalable benchmark for evaluating the local sampling performance of SGMCMC algorithms. We evaluate a number of common algorithms, and find that RMSProp-preconditioned SGLD is most effective at faithfully representing the local geometry of the posterior distribution. Although we lack theoretical guarantees about global sampler convergence, our empirical results show that we are able to extract non-trivial local information in models with up to O(100M) parameters.", "date": "2025-07-29", "categories": []}
{"id": "2506.22607", "title": "Learning Individual Reproductive Behavior from Aggregate Fertility Rates via Neural Posterior Estimation", "abstract": "Age-specific fertility rates (ASFRs) provide the most extensive record of reproductive change, but their aggregate nature obscures the individual-level behavioral mechanisms that drive fertility trends. To bridge this micro-macro divide, we introduce a likelihood-free Bayesian framework that couples a demographically interpretable, individual-level simulation model of the reproductive process with Sequential Neural Posterior Estimation (SNPE). We show that this framework successfully recovers core behavioral parameters governing contemporary fertility, including preferences for family size, reproductive timing, and contraceptive failure, using only ASFRs. The framework's effectiveness is validated on cohorts from four countries with diverse fertility regimes. Most compellingly, the model, estimated solely on aggregate data, successfully predicts out-of-sample distributions of individual-level outcomes, including age at first sex, desired family size, and birth intervals. Because our framework yields complete synthetic life histories, it significantly reduces the data requirements for building microsimulation models and enables behaviorally explicit demographic forecasts.", "date": "2025-06-27", "categories": []}
{"id": "2303.05263", "title": "Fast post-process Bayesian inference with Variational Sparse Bayesian Quadrature", "abstract": "In applied Bayesian inference scenarios, users may have access to a large number of pre-existing model evaluations, for example from maximum-a-posteriori (MAP) optimization runs. However, traditional approximate inference techniques make little to no use of this available information. We propose the framework of post-process Bayesian inference as a means to obtain a quick posterior approximation from existing target density evaluations, with no further model calls. Within this framework, we introduce Variational Sparse Bayesian Quadrature (VSBQ), a method for post-process approximate inference for models with black-box and potentially noisy likelihoods. VSBQ reuses existing target density evaluations to build a sparse Gaussian process (GP) surrogate model of the log posterior density function. Subsequently, we leverage sparse-GP Bayesian quadrature combined with variational inference to achieve fast approximate posterior inference over the surrogate. We validate our method on challenging synthetic scenarios and real-world applications from computational neuroscience. The experiments show that VSBQ builds high-quality posterior approximations by post-processing existing optimization traces, with no further model evaluations.", "date": "2023-03-09", "categories": []}
{"id": "2507.17013", "title": "laplax -- Laplace Approximations with JAX", "abstract": "The Laplace approximation provides a scalable and efficient means of quantifying weight-space uncertainty in deep neural networks, enabling the application of Bayesian tools such as predictive uncertainty and model selection via Occam's razor. In this work, we introduce laplax, a new open-source Python package for performing Laplace approximations with jax. Designed with a modular and purely functional architecture and minimal external dependencies, laplax offers a flexible and researcher-friendly framework for rapid prototyping and experimentation. Its goal is to facilitate research on Bayesian neural networks, uncertainty quantification for deep learning, and the development of improved Laplace approximation techniques.", "date": "2025-07-22", "categories": []}
{"id": "2310.14968", "title": "Bayesian Active Learning in the Presence of Nuisance Parameters", "abstract": "In many settings, such as scientific inference, optimization, and transfer learning, the learner has a well-defined objective, which can be treated as estimation of a target parameter, and no intrinsic interest in characterizing the entire data-generating process. Usually, the learner must also contend with additional sources of uncertainty or variables -- with nuisance parameters. Bayesian active learning, or sequential optimal experimental design, can straightforwardly accommodate the presence of nuisance parameters, and so is a natural active learning framework for such problems. However, the introduction of nuisance parameters can lead to bias in the Bayesian learner's estimate of the target parameters, a phenomenon we refer to as negative interference. We characterize the threat of negative interference and how it fundamentally changes the nature of the Bayesian active learner's task. We show that the extent of negative interference can be extremely large, and that accurate estimation of the nuisance parameters is critical to reducing it. The Bayesian active learner is confronted with a dilemma: whether to spend a finite acquisition budget in pursuit of estimation of the target or of the nuisance parameters. Our setting encompasses Bayesian transfer learning as a special case, and our results shed light on the phenomenon of negative transfer between learning environments.", "date": "2023-10-23", "categories": []}
{"id": "2411.18425", "title": "Streamlining Prediction in Bayesian Deep Learning", "abstract": "The rising interest in Bayesian deep learning (BDL) has led to a plethora of methods for estimating the posterior distribution. However, efficient computation of inferences, such as predictions, has been largely overlooked with Monte Carlo integration remaining the standard. In this work we examine streamlining prediction in BDL through a single forward pass without sampling. For this we use local linearisation on activation functions and local Gaussian approximations at linear layers. Thus allowing us to analytically compute an approximation to the posterior predictive distribution. We showcase our approach for both MLP and transformers, such as ViT and GPT-2, and assess its performance on regression and classification tasks.\n  Open-source library: https://github.com/AaltoML/SUQ", "date": "2024-11-27", "categories": []}
{"id": "2501.18665", "title": "BARNN: A Bayesian Autoregressive and Recurrent Neural Network", "abstract": "Autoregressive and recurrent networks have achieved remarkable progress across various fields, from weather forecasting to molecular generation and Large Language Models. Despite their strong predictive capabilities, these models lack a rigorous framework for addressing uncertainty, which is key in scientific applications such as PDE solving, molecular generation and Machine Learning Force Fields. To address this shortcoming we present BARNN: a variational Bayesian Autoregressive and Recurrent Neural Network. BARNNs aim to provide a principled way to turn any autoregressive or recurrent model into its Bayesian version. BARNN is based on the variational dropout method, allowing to apply it to large recurrent neural networks as well. We also introduce a temporal version of the \"Variational Mixtures of Posteriors\" prior (tVAMP-prior) to make Bayesian inference efficient and well-calibrated. Extensive experiments on PDE modelling and molecular generation demonstrate that BARNN not only achieves comparable or superior accuracy compared to existing methods, but also excels in uncertainty quantification and modelling long-range dependencies.", "date": "2025-01-30", "categories": []}
{"id": "2501.17965", "title": "Variational Combinatorial Sequential Monte Carlo for Bayesian Phylogenetics in Hyperbolic Space", "abstract": "Hyperbolic space naturally encodes hierarchical structures such as phylogenies (binary trees), where inward-bending geodesics reflect paths through least common ancestors, and the exponential growth of neighborhoods mirrors the super-exponential scaling of topologies. This scaling challenge limits the efficiency of Euclidean-based approximate inference methods. Motivated by the geometric connections between trees and hyperbolic space, we develop novel hyperbolic extensions of two sequential search algorithms: Combinatorial and Nested Combinatorial Sequential Monte Carlo (\\textsc{Csmc} and \\textsc{Ncsmc}). Our approach introduces consistent and unbiased estimators, along with variational inference methods (\\textsc{H-Vcsmc} and \\textsc{H-Vncsmc}), which outperform their Euclidean counterparts. Empirical results demonstrate improved speed, scalability and performance in high-dimensional phylogenetic inference tasks.", "date": "2025-01-29", "categories": []}
{"id": "2507.11136", "title": "Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection", "abstract": "Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabilistic framework that uses sparsity-inducing hierarchical priors on TN factors to automatically infer model complexity. This enables automatic inference of tensor rank and feature dimensions, while also identifying the most relevant features for prediction, thereby enhancing model interpretability. All the model parameters and hyperparameters are treated as latent variables with corresponding priors. Given the Bayesian approach and latent variable dependencies, we apply a mean-field variational inference to approximate their posteriors. We show that applying a mean-field approximation to TN factors yields a Bayesian ALS algorithm with the same computational complexity as its deterministic counterpart, enabling uncertainty quantification at no extra computational cost. Experiments on synthetic and real-world datasets demonstrate the superior performance of our model in prediction accuracy, uncertainty quantification, interpretability, and scalability.", "date": "2025-07-15", "categories": []}
{"id": "2507.08905", "title": "Last Layer Hamiltonian Monte Carlo", "abstract": "We explore the use of Hamiltonian Monte Carlo (HMC) sampling as a probabilistic last layer approach for deep neural networks (DNNs). While HMC is widely regarded as a gold standard for uncertainty estimation, the computational demands limit its application to large-scale datasets and large DNN architectures. Although the predictions from the sampled DNN parameters can be parallelized, the computational cost still scales linearly with the number of samples (similar to an ensemble). Last layer HMC (LL--HMC) reduces the required computations by restricting the HMC sampling to the final layer of a DNN, making it applicable to more data-intensive scenarios with limited computational resources. In this paper, we compare LL-HMC against five last layer probabilistic deep learning (LL-PDL) methods across three real-world video datasets for driver action and intention. We evaluate the in-distribution classification performance, calibration, and out-of-distribution (OOD) detection. Due to the stochastic nature of the probabilistic evaluations, we performed five grid searches for different random seeds to avoid being reliant on a single initialization for the hyperparameter configurations. The results show that LL--HMC achieves competitive in-distribution classification and OOD detection performance. Additional sampled last layer parameters do not improve the classification performance, but can improve the OOD detection. Multiple chains or starting positions did not yield consistent improvements.", "date": "2025-07-11", "categories": []}
{"id": "2410.20289", "title": "On the Gaussian process limit of Bayesian Additive Regression Trees", "abstract": "Bayesian Additive Regression Trees (BART) is a nonparametric Bayesian regression technique of rising fame. It is a sum-of-decision-trees model, and is in some sense the Bayesian version of boosting. In the limit of infinite trees, it becomes equivalent to Gaussian process (GP) regression. This limit is known but has not yet led to any useful analysis or application. For the first time, I derive and compute the exact BART prior covariance function. With it I implement the infinite trees limit of BART as GP regression. Through empirical tests, I show that this limit is worse than standard BART in a fixed configuration, but also that tuning its hyperparameters in the natural GP way makes it competitive with BART. The advantage of using a GP surrogate of BART is the analytical likelihood, which simplifies model building and sidesteps the complex BART MCMC algorithm. More generally, this study opens new ways to understand and develop BART and GP regression. The implementation of BART as GP is available in the Python package lsqfitgp.", "date": "2024-10-26", "categories": []}
{"id": "2507.07461", "title": "Hess-MC2: Sequential Monte Carlo Squared using Hessian Information and Second Order Proposals", "abstract": "When performing Bayesian inference using Sequential Monte Carlo (SMC) methods, two considerations arise: the accuracy of the posterior approximation and computational efficiency. To address computational demands, Sequential Monte Carlo Squared (SMC$^2$) is well-suited for high-performance computing (HPC) environments. The design of the proposal distribution within SMC$^2$ can improve accuracy and exploration of the posterior as poor proposals may lead to high variance in importance weights and particle degeneracy. The Metropolis-Adjusted Langevin Algorithm (MALA) uses gradient information so that particles preferentially explore regions of higher probability. In this paper, we extend this idea by incorporating second-order information, specifically the Hessian of the log-target. While second-order proposals have been explored previously in particle Markov Chain Monte Carlo (p-MCMC) methods, we are the first to introduce them within the SMC$^2$ framework. Second-order proposals not only use the gradient (first-order derivative), but also the curvature (second-order derivative) of the target distribution. Experimental results on synthetic models highlight the benefits of our approach in terms of step-size selection and posterior approximation accuracy when compared to other proposals.", "date": "2025-07-10", "categories": []}
{"id": "2410.23244", "title": "Very fast Bayesian Additive Regression Trees on GPU", "abstract": "Bayesian Additive Regression Trees (BART) is a nonparametric Bayesian regression technique based on an ensemble of decision trees. It is part of the toolbox of many statisticians. The overall statistical quality of the regression is typically higher than other generic alternatives, and it requires less manual tuning, making it a good default choice. However, it is a niche method compared to its natural competitor XGBoost, due to the longer running time, making sample sizes above 10,000-100,000 a nuisance. I present a GPU-enabled implementation of BART, faster by up to 200x relative to a single CPU core, making BART competitive in running time with XGBoost. This implementation is available in the Python package bartz.", "date": "2024-10-30", "categories": []}
{"id": "2507.06839", "title": "Scalable Gaussian Processes: Advances in Iterative Methods and Pathwise Conditioning", "abstract": "Gaussian processes are a powerful framework for uncertainty-aware function approximation and sequential decision-making. Unfortunately, their classical formulation does not scale gracefully to large amounts of data and modern hardware for massively-parallel computation, prompting many researchers to develop techniques which improve their scalability. This dissertation focuses on the powerful combination of iterative methods and pathwise conditioning to develop methodological contributions which facilitate the use of Gaussian processes in modern large-scale settings. By combining these two techniques synergistically, expensive computations are expressed as solutions to systems of linear equations and obtained by leveraging iterative linear system solvers. This drastically reduces memory requirements, facilitating application to significantly larger amounts of data, and introduces matrix multiplication as the main computational operation, which is ideal for modern hardware.", "date": "2025-07-09", "categories": []}
{"id": "2506.21964", "title": "Using Large Language Models to Suggest Informative Prior Distributions in Bayesian Statistics", "abstract": "Selecting prior distributions in Bayesian statistics is challenging, resource-intensive, and subjective. We analyze using large-language models (LLMs) to suggest suitable, knowledge-based informative priors. We developed an extensive prompt asking LLMs not only to suggest priors but also to verify and reflect on their choices.\n  We evaluated Claude Opus, Gemini 2.5 Pro, and ChatGPT-4o-mini on two real datasets: heart disease risk and concrete strength. All LLMs correctly identified the direction for all associations (e.g., that heart disease risk is higher for males). The quality of suggested priors was measured by their Kullback-Leibler divergence from the maximum likelihood estimator's distribution.\n  The LLMs suggested both moderately and weakly informative priors. The moderate priors were often overconfident, resulting in distributions misaligned with the data. In our experiments, Claude and Gemini provided better priors than ChatGPT. For weakly informative priors, a key performance difference emerged: ChatGPT and Gemini defaulted to an \"unnecessarily vague\" mean of 0, while Claude did not, demonstrating a significant advantage.\n  The ability of LLMs to identify correct associations shows their great potential as an efficient, objective method for developing informative priors. However, the primary challenge remains in calibrating the width of these priors to avoid over- and under-confidence.", "date": "2025-06-27", "categories": []}
{"id": "2506.21408", "title": "Scalable Bayesian Low-Rank Adaptation of Large Language Models via Stochastic Variational Subspace Inference", "abstract": "Despite their widespread use, large language models (LLMs) are known to hallucinate incorrect information and be poorly calibrated. This makes the uncertainty quantification of these models of critical importance, especially in high-stakes domains, such as autonomy and healthcare. Prior work has made Bayesian deep learning-based approaches to this problem more tractable by performing inference over the low-rank adaptation (LoRA) parameters of a fine-tuned model. While effective, these approaches struggle to scale to larger LLMs due to requiring further additional parameters compared to LoRA. In this work we present $\\textbf{Scala}$ble $\\textbf{B}$ayesian $\\textbf{L}$ow-Rank Adaptation via Stochastic Variational Subspace Inference (ScalaBL). We perform Bayesian inference in an $r$-dimensional subspace, for LoRA rank $r$. By repurposing the LoRA parameters as projection matrices, we are able to map samples from this subspace into the full weight space of the LLM. This allows us to learn all the parameters of our approach using stochastic variational inference. Despite the low dimensionality of our subspace, we are able to achieve competitive performance with state-of-the-art approaches while only requiring ${\\sim}1000$ additional parameters. Furthermore, it allows us to scale up to the largest Bayesian LLM to date, with four times as a many base parameters as prior work.", "date": "2025-06-26", "categories": []}
{"id": "2406.03260", "title": "Feature learning in finite-width Bayesian deep linear networks with multiple outputs and convolutional layers", "abstract": "Deep linear networks have been extensively studied, as they provide simplified models of deep learning. However, little is known in the case of finite-width architectures with multiple outputs and convolutional layers. In this manuscript, we provide rigorous results for the statistics of functions implemented by the aforementioned class of networks, thus moving closer to a complete characterization of feature learning in the Bayesian setting. Our results include: (i) an exact and elementary non-asymptotic integral representation for the joint prior distribution over the outputs, given in terms of a mixture of Gaussians; (ii) an analytical formula for the posterior distribution in the case of squared error loss function (Gaussian likelihood); (iii) a quantitative description of the feature learning infinite-width regime, using large deviation theory. From a physical perspective, deep architectures with multiple outputs or convolutional layers represent different manifestations of kernel shape renormalization, and our work provides a dictionary that translates this physics intuition and terminology into rigorous Bayesian statistics.", "date": "2024-06-05", "categories": []}
{"id": "2408.08062", "title": "BINDy -- Bayesian identification of nonlinear dynamics with reversible-jump Markov-chain Monte-Carlo", "abstract": "Model parsimony is an important \\emph{cognitive bias} in data-driven modelling that aids interpretability and helps to prevent over-fitting. Sparse identification of nonlinear dynamics (SINDy) methods are able to learn sparse representations of complex dynamics directly from data, given a basis of library functions. In this work, a novel Bayesian treatment of dictionary learning system identification, as an alternative to SINDy, is envisaged. The proposed method -- Bayesian identification of nonlinear dynamics (BINDy) -- is distinct from previous approaches in that it targets the full joint posterior distribution over both the terms in the library and their parameterisation in the model. This formulation confers the advantage that an arbitrary prior may be placed over the model structure to produce models that are sparse in the model space rather than in parameter space. Because this posterior is defined over parameter vectors that can change in dimension, the inference cannot be performed by standard techniques. Instead, a Gibbs sampler based on reversible-jump Markov-chain Monte-Carlo is proposed. BINDy is shown to compare favourably to ensemble SINDy in three benchmark case-studies. In particular, it is seen that the proposed method is better able to assign high probability to correct model terms.", "date": "2024-08-15", "categories": []}
{"id": "2308.14048", "title": "A Bayesian Non-parametric Approach to Generative Models: Integrating Variational Autoencoder and Generative Adversarial Networks using Wasserstein and Maximum Mean Discrepancy", "abstract": "We propose a novel generative model within the Bayesian non-parametric learning (BNPL) framework to address some notable failure modes in generative adversarial networks (GANs) and variational autoencoders (VAEs)--these being overfitting in the GAN case and noisy samples in the VAE case. We will demonstrate that the BNPL framework enhances training stability and provides robustness and accuracy guarantees when incorporating the Wasserstein distance and maximum mean discrepancy measure (WMMD) into our model's loss function. Moreover, we introduce a so-called ``triple model'' that combines the GAN, the VAE, and further incorporates a code-GAN (CGAN) to explore the latent space of the VAE. This triple model design generates high-quality, diverse samples, while the BNPL framework, leveraging the WMMD loss function, enhances training stability. Together, these components enable our model to achieve superior performance across various generative tasks. These claims are supported by both theoretical analyses and empirical validation on a wide variety of datasets.", "date": "2023-08-27", "categories": []}
{"id": "2407.20722", "title": "Persistent Sampling: Enhancing the Efficiency of Sequential Monte Carlo", "abstract": "Sequential Monte Carlo (SMC) samplers are powerful tools for Bayesian inference but suffer from high computational costs due to their reliance on large particle ensembles for accurate estimates. We introduce persistent sampling (PS), an extension of SMC that systematically retains and reuses particles from all prior iterations to construct a growing, weighted ensemble. By leveraging multiple importance sampling and resampling from a mixture of historical distributions, PS mitigates the need for excessively large particle counts, directly addressing key limitations of SMC such as particle impoverishment and mode collapse. Crucially, PS achieves this without additional likelihood evaluations-weights for persistent particles are computed using cached likelihood values. This framework not only yields more accurate posterior approximations but also produces marginal likelihood estimates with significantly lower variance, enhancing reliability in model comparison. Furthermore, the persistent ensemble enables efficient adaptation of transition kernels by leveraging a larger, decorrelated particle pool. Experiments on high-dimensional Gaussian mixtures, hierarchical models, and non-convex targets demonstrate that PS consistently outperforms standard SMC and related variants, including recycled and waste-free SMC, achieving substantial reductions in mean squared error for posterior expectations and evidence estimates, all at reduced computational cost. PS thus establishes itself as a robust, scalable, and efficient alternative for complex Bayesian inference tasks.", "date": "2024-07-30", "categories": []}
{"id": "2506.18092", "title": "GRASP: Grouped Regression with Adaptive Shrinkage Priors", "abstract": "We introduce GRASP, a simple Bayesian framework for regression with grouped predictors, built on the normal beta prime (NBP) prior. The NBP prior is an adaptive generalization of the horseshoe prior with tunable hyperparameters that control tail behavior, enabling a flexible range of sparsity, from strong shrinkage to ridge-like regularization. Unlike prior work that introduced the group inverse-gamma gamma (GIGG) prior by decomposing the NBP prior into structured hierarchies, we show that directly controlling the tails is sufficient without requiring complex hierarchical constructions. Extending the non-tail adaptive grouped half-Cauchy hierarchy of Xu et al., GRASP assigns the NBP prior to both local and group shrinkage parameters allowing adaptive sparsity within and across groups. A key contribution of this work is a novel framework to explicitly quantify correlations among shrinkage parameters within a group, providing deeper insights into grouped shrinkage behavior. We also introduce an efficient Metropolis-Hastings sampler for hyperparameter estimation. Empirical results on simulated and real-world data demonstrate the robustness and versatility of GRASP across grouped regression problems with varying sparsity and signal-to-noise ratios.", "date": "2025-06-22", "categories": []}
{"id": "2410.18973", "title": "Tuning-Free Coreset Markov Chain Monte Carlo via Hot DoG", "abstract": "A Bayesian coreset is a small, weighted subset of a data set that replaces the full data during inference to reduce computational cost. The state-of-the-art coreset construction algorithm, Coreset Markov chain Monte Carlo (Coreset MCMC), uses draws from an adaptive Markov chain targeting the coreset posterior to train the coreset weights via stochastic gradient optimization. However, the quality of the constructed coreset, and thus the quality of its posterior approximation, is sensitive to the stochastic optimization learning rate. In this work, we propose a learning-rate-free stochastic gradient optimization procedure, Hot-start Distance over Gradient (Hot DoG), for training coreset weights in Coreset MCMC without user tuning effort. We provide a theoretical analysis of the convergence of the coreset weights produced by Hot DoG. We also provide empirical results demonstrate that Hot DoG provides higher quality posterior approximations than other learning-rate-free stochastic gradient methods, and performs competitively to optimally-tuned ADAM.", "date": "2024-10-24", "categories": []}
{"id": "2506.16844", "title": "Bandwidth Selectors on Semiparametric Bayesian Networks", "abstract": "Semiparametric Bayesian networks (SPBNs) integrate parametric and non-parametric probabilistic models, offering flexibility in learning complex data distributions from samples. In particular, kernel density estimators (KDEs) are employed for the non-parametric component. Under the assumption of data normality, the normal rule is used to learn the bandwidth matrix for the KDEs in SPBNs. This matrix is the key hyperparameter that controls the trade-off between bias and variance. However, real-world data often deviates from normality, potentially leading to suboptimal density estimation and reduced predictive performance. This paper first establishes the theoretical framework for the application of state-of-the-art bandwidth selectors and subsequently evaluates their impact on SPBN performance. We explore the approaches of cross-validation and plug-in selectors, assessing their effectiveness in enhancing the learning capability and applicability of SPBNs. To support this investigation, we have extended the open-source package PyBNesian for SPBNs with the additional bandwidth selection techniques and conducted extensive experimental analyses. Our results demonstrate that the proposed bandwidth selectors leverage increasing information more effectively than the normal rule, which, despite its robustness, stagnates with more data. In particular, unbiased cross-validation generally outperforms the normal rule, highlighting its advantage in high sample size scenarios.", "date": "2025-06-20", "categories": []}
{"id": "2406.15664", "title": "Flat Posterior Does Matter For Bayesian Model Averaging", "abstract": "Bayesian neural networks (BNNs) estimate the posterior distribution of model parameters and utilize posterior samples for Bayesian Model Averaging (BMA) in prediction. However, despite the crucial role of flatness in the loss landscape in improving the generalization of neural networks, its impact on BMA has been largely overlooked. In this work, we explore how posterior flatness influences BMA generalization and empirically demonstrate that (1) most approximate Bayesian inference methods fail to yield a flat posterior and (2) BMA predictions, without considering posterior flatness, are less effective at improving generalization. To address this, we propose Flat Posterior-aware Bayesian Model Averaging (FP-BMA), a novel training objective that explicitly encourages flat posteriors in a principled Bayesian manner. We also introduce a Flat Posterior-aware Bayesian Transfer Learning scheme that enhances generalization in downstream tasks. Empirically, we show that FP-BMA successfully captures flat posteriors, improving generalization performance.", "date": "2024-06-21", "categories": []}
{"id": "2411.11132", "title": "Variational Bayesian Bow tie Neural Networks with Shrinkage", "abstract": "Despite the dominant role of deep models in machine learning, limitations persist, including overconfident predictions, susceptibility to adversarial attacks, and underestimation of variability in predictions. The Bayesian paradigm provides a natural framework to overcome such issues and has become the gold standard for uncertainty estimation with deep models, also providing improved accuracy and a framework for tuning critical hyperparameters. However, exact Bayesian inference is challenging, typically involving variational algorithms that impose strong independence and distributional assumptions. Moreover, existing methods are sensitive to the architectural choice of the network. We address these issues by focusing on a stochastic relaxation of the standard feed-forward rectified neural network and using sparsity-promoting priors on the weights of the neural network for increased robustness to architectural design. Thanks to Polya-Gamma data augmentation tricks, which render a conditionally linear and Gaussian model, we derive a fast, approximate variational inference algorithm that avoids distributional assumptions and independence across layers. Suitable strategies to further improve scalability and account for multimodality are considered.", "date": "2024-11-17", "categories": []}
{"id": "2503.11808", "title": "Understanding the Trade-offs in Accuracy and Uncertainty Quantification: Architecture and Inference Choices in Bayesian Neural Networks", "abstract": "As modern neural networks get more complex, specifying a model with high predictive performance and sound uncertainty quantification becomes a more challenging task. Despite some promising theoretical results on the true posterior predictive distribution of Bayesian neural networks, the properties of even the most commonly used posterior approximations are often questioned. Computational burdens and intractable posteriors expose miscalibrated Bayesian neural networks to poor accuracy and unreliable uncertainty estimates. Approximate Bayesian inference aims to replace unknown and intractable posterior distributions with some simpler but feasible distributions. The dimensions of modern deep models, coupled with the lack of identifiability, make Markov chain Monte Carlo (MCMC) tremendously expensive and unable to fully explore the multimodal posterior. On the other hand, variational inference benefits from improved computational complexity but lacks the asymptotical guarantees of sampling-based inference and tends to concentrate around a single mode. The performance of both approaches heavily depends on architectural choices; this paper aims to shed some light on this by considering the computational costs, accuracy and uncertainty quantification in different scenarios including large width and out-of-sample data. To improve posterior exploration, different model averaging and ensembling techniques are studied, along with their benefits on predictive performance. In our experiments, variational inference overall provided better uncertainty quantification than MCMC; further, stacking and ensembles of variational approximations provided comparable accuracy to MCMC at a much-reduced cost.", "date": "2025-03-14", "categories": []}
{"id": "2408.13230", "title": "Amortized Bayesian Multilevel Models", "abstract": "Multilevel models (MLMs) are a central building block of the Bayesian workflow. They enable joint, interpretable modeling of data across hierarchical levels and provide a fully probabilistic quantification of uncertainty. Despite their well-recognized advantages, MLMs pose significant computational challenges, often rendering their estimation and evaluation intractable within reasonable time constraints. Recent advances in simulation-based inference offer promising solutions for addressing complex probabilistic models using deep generative networks. However, the utility and reliability of deep learning methods for estimating Bayesian MLMs remains largely unexplored, especially when compared with gold-standard samplers. To this end, we explore a family of neural network architectures that leverage the probabilistic factorization of multilevel models to facilitate efficient neural network training and subsequent near-instant posterior inference on unseen datasets. We test our method on several real-world case studies and provide comprehensive comparisons to Stan's gold standard sampler, where possible. Finally, we provide an open-source implementation of our methods to stimulate further research in the nascent field of amortized Bayesian inference.", "date": "2024-08-23", "categories": []}
{"id": "2506.09928", "title": "Course Project Report: Comparing MCMC and Variational Inference for Bayesian Probabilistic Matrix Factorization on the MovieLens Dataset", "abstract": "This is a course project report with complete methodology, experiments, references and mathematical derivations. Matrix factorization [1] is a widely used technique in recommendation systems. Probabilistic Matrix Factorization (PMF) [2] extends traditional matrix factorization by incorporating probability distributions over latent factors, allowing for uncertainty quantification. However, computing the posterior distribution is intractable due to the high-dimensional integral. To address this, we employ two Bayesian inference methods: Markov Chain Monte Carlo (MCMC) [3, 4] and Variational Inference (VI) [5, 6] to approximate the posterior. We evaluate their performance on MovieLens dataset [7] and compare their convergence speed, predictive accuracy, and computational efficiency. Experimental results demonstrate that VI offers faster convergence, while MCMC provides more accurate posterior estimates.", "date": "2025-06-11", "categories": []}
{"id": "2410.15777", "title": "Revisiting the Equivalence of Bayesian Neural Networks and Gaussian Processes: On the Importance of Learning Activations", "abstract": "Gaussian Processes (GPs) provide a convenient framework for specifying function-space priors, making them a natural choice for modeling uncertainty. In contrast, Bayesian Neural Networks (BNNs) offer greater scalability and extendability but lack the advantageous properties of GPs. This motivates the development of BNNs capable of replicating GP-like behavior. However, existing solutions are either limited to specific GP kernels or rely on heuristics.\n  We demonstrate that trainable activations are crucial for effective mapping of GP priors to wide BNNs. Specifically, we leverage the closed-form 2-Wasserstein distance for efficient gradient-based optimization of reparameterized priors and activations. Beyond learned activations, we also introduce trainable periodic activations that ensure global stationarity by design, and functional priors conditioned on GP hyperparameters to allow efficient model selection.\n  Empirically, our method consistently outperforms existing approaches or matches performance of the heuristic methods, while offering stronger theoretical foundations.", "date": "2024-10-21", "categories": []}
{"id": "2410.04196", "title": "Improving Generalization with Flat Hilbert Bayesian Inference", "abstract": "We introduce Flat Hilbert Bayesian Inference (FHBI), an algorithm designed to enhance generalization in Bayesian inference. Our approach involves an iterative two-step procedure with an adversarial functional perturbation step and a functional descent step within a reproducing kernel Hilbert space. This methodology is supported by a theoretical analysis that extends previous findings on generalization ability from finite-dimensional Euclidean spaces to infinite-dimensional functional spaces. To evaluate the effectiveness of FHBI, we conduct comprehensive comparisons against nine baseline methods on the \\texttt{VTAB-1K} benchmark, which encompasses 19 diverse datasets across various domains with diverse semantics. Empirical results demonstrate that FHBI consistently outperforms the baselines by notable margins, highlighting its practical efficacy.", "date": "2024-10-05", "categories": []}
{"id": "2501.04259", "title": "Stable Derivative Free Gaussian Mixture Variational Inference for Bayesian Inverse Problems", "abstract": "This paper is concerned with the approximation of probability distributions known up to normalization constants, with a focus on Bayesian inference for large-scale inverse problems in scientific computing. In this context, key challenges include costly repeated evaluations of forward models, multimodality, and inaccessible gradients for the forward model. To address them, we develop a variational inference framework that combines Fisher-Rao natural gradient with specialized quadrature rules to enable derivative free updates of Gaussian mixture variational families. The resulting method, termed Derivative Free Gaussian Mixture Variational Inference (DF-GMVI), guarantees covariance positivity and affine invariance, offering a stable and efficient framework for approximating complex posterior distributions. The effectiveness of DF-GMVI is demonstrated through numerical experiments on challenging scenarios, including distributions with multiple modes, infinitely many modes, and curved modes in spaces with up to 100 dimensions. The method's practicality is further demonstrated in a large-scale application, where it successfully recovers the initial conditions of the Navier-Stokes equations from solution data at positive times.", "date": "2025-01-08", "categories": []}
{"id": "2506.07247", "title": "Promoting Ensemble Diversity with Interactive Bayesian Distributional Robustness for Fine-tuning Foundation Models", "abstract": "We introduce Interactive Bayesian Distributional Robustness (IBDR), a novel Bayesian inference framework that allows modeling the interactions between particles, thereby enhancing ensemble quality through increased particle diversity. IBDR is grounded in a generalized theoretical framework that connects the distributional population loss with the approximate posterior, motivating a practical dual optimization procedure that enforces distributional robustness while fostering particle diversity. We evaluate IBDR's performance against various baseline methods using the VTAB-1K benchmark and the common reasoning language task. The results consistently show that IBDR outperforms these baselines, underscoring its effectiveness in real-world applications.", "date": "2025-06-08", "categories": []}
{"id": "2410.03085", "title": "Optimization Proxies using Limited Labeled Data and Training Time -- A Semi-Supervised Bayesian Neural Network Approach", "abstract": "Constrained optimization problems arise in various engineering systems such as inventory management and power grids. Standard deep neural network (DNN) based machine learning proxies are ineffective in practical settings where labeled data is scarce and training times are limited. We propose a semi-supervised Bayesian Neural Networks (BNNs) based optimization proxy for this complex regime, wherein training commences in a sandwiched fashion, alternating between a supervised learning step for minimizing cost, and an unsupervised learning step for enforcing constraint feasibility. We show that the proposed semi-supervised BNN outperforms DNN architectures on important non-convex constrained optimization problems from energy network operations, achieving up to a tenfold reduction in expected maximum equality gap and halving the inequality gaps. Further, the BNN's ability to provide posterior samples is leveraged to construct practically meaningful probabilistic confidence bounds on performance using a limited validation data, unlike prior methods. The implementation code for this study is available at: https://github.com/kaarthiksundar/BNN-OPF/.", "date": "2024-10-04", "categories": []}
{"id": "2503.05574", "title": "BARK: A Fully Bayesian Tree Kernel for Black-box Optimization", "abstract": "We perform Bayesian optimization using a Gaussian process perspective on Bayesian Additive Regression Trees (BART). Our BART Kernel (BARK) uses tree agreement to define a posterior over piecewise-constant functions, and we explore the space of tree kernels using a Markov chain Monte Carlo approach. Where BART only samples functions, the resulting BARK model obtains samples of Gaussian processes defining distributions over functions, which allow us to build acquisition functions for Bayesian optimization. Our tree-based approach enables global optimization over the surrogate, even for mixed-feature spaces. Moreover, where many previous tree-based kernels provide uncertainty quantification over function values, our sampling scheme captures uncertainty over the tree structure itself. Our experiments show the strong performance of BARK on both synthetic and applied benchmarks, due to the combination of our fully Bayesian surrogate and the optimization procedure.", "date": "2025-03-07", "categories": []}
{"id": "2406.04317", "title": "Regularized KL-Divergence for Well-Defined Function-Space Variational Inference in Bayesian neural networks", "abstract": "Bayesian neural networks (BNN) promise to combine the predictive performance of neural networks with principled uncertainty modeling important for safety-critical systems and decision making. However, posterior uncertainty estimates depend on the choice of prior, and finding informative priors in weight-space has proven difficult. This has motivated variational inference (VI) methods that pose priors directly on the function generated by the BNN rather than on weights. In this paper, we address a fundamental issue with such function-space VI approaches pointed out by Burt et al. (2020), who showed that the objective function (ELBO) is negative infinite for most priors of interest. Our solution builds on generalized VI (Knoblauch et al., 2019) with the regularized KL divergence (Quang, 2019) and is, to the best of our knowledge, the first well-defined variational objective for function-space inference in BNNs with Gaussian process (GP) priors. Experiments show that our method incorporates the properties specified by the GP prior on synthetic and small real-world data sets, and provides competitive uncertainty estimates for regression, classification and out-of-distribution detection compared to BNN baselines with both function and weight-space priors.", "date": "2024-06-06", "categories": []}
{"id": "2506.03670", "title": "Position: There Is No Free Bayesian Uncertainty Quantification", "abstract": "Due to their intuitive appeal, Bayesian methods of modeling and uncertainty quantification have become popular in modern machine and deep learning. When providing a prior distribution over the parameter space, it is straightforward to obtain a distribution over the parameters that is conventionally interpreted as uncertainty quantification of the model. We challenge the validity of such Bayesian uncertainty quantification by discussing the equivalent optimization-based representation of Bayesian updating, provide an alternative interpretation that is coherent with the optimization-based perspective, propose measures of the quality of the Bayesian inferential stage, and suggest directions for future work.", "date": "2025-06-04", "categories": []}
{"id": "2412.11743", "title": "Generalized Bayesian deep reinforcement learning", "abstract": "Bayesian reinforcement learning (BRL) is a method that merges principles from Bayesian statistics and reinforcement learning to make optimal decisions in uncertain environments. As a model-based RL method, it has two key components: (1) inferring the posterior distribution of the model for the data-generating process (DGP) and (2) policy learning using the learned posterior. We propose to model the dynamics of the unknown environment through deep generative models, assuming Markov dependence. In the absence of likelihood functions for these models, we train them by learning a generalized predictive-sequential (or prequential) scoring rule (SR) posterior. We used sequential Monte Carlo (SMC) samplers to draw samples from this generalized Bayesian posterior distribution. In conjunction, to achieve scalability in the high-dimensional parameter space of the neural networks, we use the gradient-based Markov kernels within SMC. To justify the use of the prequential scoring rule posterior, we prove a Bernstein-von Mises-type theorem. For policy learning, we propose expected Thompson sampling (ETS) to learn the optimal policy by maximising the expected value function with respect to the posterior distribution. This improves upon traditional Thompson sampling (TS) and its extensions, which utilize only one sample drawn from the posterior distribution. This improvement is studied both theoretically and using simulation studies, assuming a discrete action space. Finally, we successfully extended our setup for a challenging problem with a continuous action space without theoretical guarantees.", "date": "2024-12-16", "categories": []}
{"id": "2506.00270", "title": "Bayesian Data Sketching for Varying Coefficient Regression Models", "abstract": "Varying coefficient models are popular for estimating nonlinear regression functions in functional data models. Their Bayesian variants have received limited attention in large data applications, primarily due to prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We introduce Bayesian data sketching for varying coefficient models to obviate computational challenges presented by large sample sizes. To address the challenges of analyzing large data, we compress the functional response vector and predictor matrix by a random linear transformation to achieve dimension reduction and conduct inference on the compressed data. Our approach distinguishes itself from several existing methods for analyzing large functional data in that it requires neither the development of new models or algorithms, nor any specialized computational hardware while delivering fully model-based Bayesian inference. Well-established methods and algorithms for varying coefficient regression models can be applied to the compressed data.", "date": "2025-05-30", "categories": []}
{"id": "2506.00701", "title": "Bayesian Inference of Training Dataset Membership", "abstract": "Determining whether a dataset was part of a machine learning model's training data pool can reveal privacy vulnerabilities, a challenge often addressed through membership inference attacks (MIAs). Traditional MIAs typically require access to model internals or rely on computationally intensive shadow models. This paper proposes an efficient, interpretable and principled Bayesian inference method for membership inference. By analyzing post-hoc metrics such as prediction error, confidence (entropy), perturbation magnitude, and dataset statistics from a trained ML model, our approach computes posterior probabilities of membership without requiring extensive model training. Experimental results on synthetic datasets demonstrate the method's effectiveness in distinguishing member from non-member datasets. Beyond membership inference, this method can also detect distribution shifts, offering a practical and interpretable alternative to existing approaches.", "date": "2025-05-31", "categories": []}
{"id": "2505.23947", "title": "Position: The Future of Bayesian Prediction Is Prior-Fitted", "abstract": "Training neural networks on randomly generated artificial datasets yields Bayesian models that capture the prior defined by the dataset-generating distribution. Prior-data Fitted Networks (PFNs) are a class of methods designed to leverage this insight. In an era of rapidly increasing computational resources for pre-training and a near stagnation in the generation of new real-world data in many applications, PFNs are poised to play a more important role across a wide range of applications. They enable the efficient allocation of pre-training compute to low-data scenarios. Originally applied to small Bayesian modeling tasks, the field of PFNs has significantly expanded to address more complex domains and larger datasets. This position paper argues that PFNs and other amortized inference approaches represent the future of Bayesian inference, leveraging amortized learning to tackle data-scarce problems. We thus believe they are a fruitful area of research. In this position paper, we explore their potential and directions to address their current limitations.", "date": "2025-05-29", "categories": []}
{"id": "2505.24006", "title": "A2 Copula-Driven Spatial Bayesian Neural Network For Modeling Non-Gaussian Dependence: A Simulation Study", "abstract": "In this paper, we introduce the A2 Copula Spatial Bayesian Neural Network (A2-SBNN), a predictive spatial model designed to map coordinates to continuous fields while capturing both typical spatial patterns and extreme dependencies. By embedding the dual-tail novel Archimedean copula viz. A2 directly into the network's weight initialization, A2-SBNN naturally models complex spatial relationships, including rare co-movements in the data. The model is trained through a calibration-driven process combining Wasserstein loss, moment matching, and correlation penalties to refine predictions and manage uncertainty. Simulation results show that A2-SBNN consistently delivers high accuracy across a wide range of dependency strengths, offering a new, effective solution for spatial data modeling beyond traditional Gaussian-based approaches.", "date": "2025-05-29", "categories": []}
{"id": "2505.24556", "title": "Predictive posterior sampling from non-stationnary Gaussian process priors via Diffusion models with application to climate data", "abstract": "Bayesian models based on Gaussian processes (GPs) offer a flexible framework to predict spatially distributed variables with uncertainty. But the use of nonstationary priors, often necessary for capturing complex spatial patterns, makes sampling from the predictive posterior distribution (PPD) computationally intractable. In this paper, we propose a two-step approach based on diffusion generative models (DGMs) to mimic PPDs associated with non-stationary GP priors: we replace the GP prior by a DGM surrogate, and leverage recent advances on training-free guidance algorithms for DGMs to sample from the desired posterior distribution. We apply our approach to a rich non-stationary GP prior from which exact posterior sampling is untractable and validate that the issuing distributions are close to their GP counterpart using several statistical metrics. We also demonstrate how one can fine-tune the trained DGMs to target specific parts of the GP prior. Finally we apply the proposed approach to solve inverse problems arising in environmental sciences, thus yielding state-of-the-art predictions.", "date": "2025-05-23", "categories": []}
{"id": "2410.05612", "title": "A Bayesian Model Selection Criterion for Selecting Pretraining Checkpoints", "abstract": "Recent advances in artificial intelligence have been fueled by the development of foundation models such as BERT, GPT, T5, and Vision Transformers. These models are first pretrained on vast and diverse datasets and then adapted to specific downstream tasks, often with significantly less data. However, the mechanisms behind the success of this ubiquitous pretrain-then-adapt paradigm remain underexplored, particularly the characteristics of pretraining checkpoints that enhance downstream adaptation. We introduce a Bayesian model selection criterion, called the downstream free energy, which quantifies a checkpoint's adaptability by measuring the concentration of nearby favorable parameters for the downstream task. We demonstrate that this Bayesian model selection criterion can be effectively implemented without access to the downstream data or prior knowledge of the downstream task. Furthermore, we provide empirical evidence that the criterion reliably correlates with improved finetuning performance, offering a principled approach to predicting model adaptability.", "date": "2024-10-08", "categories": []}
{"id": "2503.11964", "title": "Entropy-regularized Gradient Estimators for Approximate Bayesian Inference", "abstract": "Effective uncertainty quantification is important for training modern predictive models with limited data, enhancing both accuracy and robustness. While Bayesian methods are effective for this purpose, they can be challenging to scale. When employing approximate Bayesian inference, ensuring the quality of samples from the posterior distribution in a computationally efficient manner is essential. This paper addresses the estimation of the Bayesian posterior to generate diverse samples by approximating the gradient flow of the Kullback-Leibler (KL) divergence and the cross entropy of the target approximation under the metric induced by the Stein Operator. It presents empirical evaluations on classification tasks to assess the method's performance and discuss its effectiveness for Model-Based Reinforcement Learning that uses uncertainty-aware network dynamics models.", "date": "2025-03-15", "categories": []}
{"id": "2505.23320", "title": "Efficient Parameter Estimation for Bayesian Network Classifiers using Hierarchical Linear Smoothing", "abstract": "Bayesian network classifiers (BNCs) possess a number of properties desirable for a modern classifier: They are easily interpretable, highly scalable, and offer adaptable complexity. However, traditional methods for learning BNCs have historically underperformed when compared to leading classification methods such as random forests. Recent parameter smoothing techniques using hierarchical Dirichlet processes (HDPs) have enabled BNCs to achieve performance competitive with random forests on categorical data, but these techniques are relatively inflexible, and require a complicated, specialized sampling process. In this paper, we introduce a novel method for parameter estimation that uses a log-linear regression to approximate the behaviour of HDPs. As a linear model, our method is remarkably flexible and simple to interpret, and can leverage the vast literature on learning linear models. Our experiments show that our method can outperform HDP smoothing while being orders of magnitude faster, remaining competitive with random forests on categorical data.", "date": "2025-05-29", "categories": []}
{"id": "2411.17284", "title": "AutoElicit: Using Large Language Models for Expert Prior Elicitation in Predictive Modelling", "abstract": "Large language models (LLMs) acquire a breadth of information across various domains. However, their computational complexity, cost, and lack of transparency often hinder their direct application for predictive tasks where privacy and interpretability are paramount. In fields such as healthcare, biology, and finance, specialised and interpretable linear models still hold considerable value. In such domains, labelled data may be scarce or expensive to obtain. Well-specified prior distributions over model parameters can reduce the sample complexity of learning through Bayesian inference; however, eliciting expert priors can be time-consuming. We therefore introduce AutoElicit to extract knowledge from LLMs and construct priors for predictive models. We show these priors are informative and can be refined using natural language. We perform a careful study contrasting AutoElicit with in-context learning and demonstrate how to perform model selection between the two methods. We find that AutoElicit yields priors that can substantially reduce error over uninformative priors, using fewer labels, and consistently outperform in-context learning. We show that AutoElicit saves over 6 months of labelling effort when building a new predictive model for urinary tract infections from sensor recordings of people living with dementia.", "date": "2024-11-26", "categories": []}
{"id": "2502.09844", "title": "Solving Empirical Bayes via Transformers", "abstract": "This work applies modern AI tools (transformers) to solving one of the oldest statistical problems: Poisson means under empirical Bayes (Poisson-EB) setting. In Poisson-EB a high-dimensional mean vector $\\theta$ (with iid coordinates sampled from an unknown prior $\\pi$) is estimated on the basis of $X=\\mathrm{Poisson}(\\theta)$. A transformer model is pre-trained on a set of synthetically generated pairs $(X,\\theta)$ and learns to do in-context learning (ICL) by adapting to unknown $\\pi$. Theoretically, we show that a sufficiently wide transformer can achieve vanishing regret with respect to an oracle estimator who knows $\\pi$ as dimension grows to infinity. Practically, we discover that already very small models (100k parameters) are able to outperform the best classical algorithm (non-parametric maximum likelihood, or NPMLE) both in runtime and validation loss, which we compute on out-of-distribution synthetic data as well as real-world datasets (NHL hockey, MLB baseball, BookCorpusOpen). Finally, by using linear probes, we confirm that the transformer's EB estimator appears to internally work differently from either NPMLE or Robbins' estimators.", "date": "2025-02-14", "categories": []}
{"id": "2505.21857", "title": "Revisiting Bayesian Model Averaging in the Era of Foundation Models", "abstract": "We revisit the classical, full-fledged Bayesian model averaging (BMA) paradigm to ensemble pre-trained and/or lightly-finetuned foundation models to enhance the classification performance on image and text data. To make BMA tractable under foundation models, we introduce trainable linear classifiers that take frozen features from the pre-trained foundation models as inputs. The model posteriors over the linear classifiers tell us which linear heads and frozen features are better suited for a given dataset, resulting in a principled model ensembling method. Furthermore, we propose a computationally cheaper, optimizable model averaging scheme (OMA). In OMA, we directly optimize the model ensemble weights, just like those weights based on model posterior distributions in BMA, by reducing the amount of surprise (expected entropy of the predictions) we get from predictions of ensembled models. With the rapid development of foundation models, these approaches will enable the incorporation of future, possibly significantly better foundation models to enhance the performance of challenging classification tasks.", "date": "2025-05-28", "categories": []}
{"id": "2505.22199", "title": "Enhancing Uncertainty Estimation and Interpretability via Bayesian Non-negative Decision Layer", "abstract": "Although deep neural networks have demonstrated significant success due to their powerful expressiveness, most models struggle to meet practical requirements for uncertainty estimation. Concurrently, the entangled nature of deep neural networks leads to a multifaceted problem, where various localized explanation techniques reveal that multiple unrelated features influence the decisions, thereby undermining interpretability. To address these challenges, we develop a Bayesian Non-negative Decision Layer (BNDL), which reformulates deep neural networks as a conditional Bayesian non-negative factor analysis. By leveraging stochastic latent variables, the BNDL can model complex dependencies and provide robust uncertainty estimation. Moreover, the sparsity and non-negativity of the latent variables encourage the model to learn disentangled representations and decision layers, thereby improving interpretability. We also offer theoretical guarantees that BNDL can achieve effective disentangled learning. In addition, we developed a corresponding variational inference method utilizing a Weibull variational inference network to approximate the posterior distribution of the latent variables. Our experimental results demonstrate that with enhanced disentanglement capabilities, BNDL not only improves the model's accuracy but also provides reliable uncertainty estimation and improved interpretability.", "date": "2025-05-28", "categories": []}
{"id": "2505.21468", "title": "Causal Posterior Estimation", "abstract": "We present Causal Posterior Estimation (CPE), a novel method for Bayesian inference in simulator models, i.e., models where the evaluation of the likelihood function is intractable or too computationally expensive, but where one can simulate model outputs given parameter values. CPE utilizes a normalizing flow-based (NF) approximation to the posterior distribution which carefully incorporates the conditional dependence structure induced by the graphical representation of the model into the neural network. Thereby it is possible to improve the accuracy of the approximation. We introduce both discrete and continuous NF architectures for CPE and propose a constant-time sampling procedure for the continuous case which reduces the computational complexity of drawing samples to O(1) as for discrete NFs. We show, through an extensive experimental evaluation, that by incorporating the conditional dependencies induced by the graphical model directly into the neural network, rather than learning them from data, CPE is able to conduct highly accurate posterior inference either outperforming or matching the state of the art in the field.", "date": "2025-05-27", "categories": []}
{"id": "2505.18280", "title": "Feature Preserving Shrinkage on Bayesian Neural Networks via the R2D2 Prior", "abstract": "Bayesian neural networks (BNNs) treat neural network weights as random variables, which aim to provide posterior uncertainty estimates and avoid overfitting by performing inference on the posterior weights. However, the selection of appropriate prior distributions remains a challenging task, and BNNs may suffer from catastrophic inflated variance or poor predictive performance when poor choices are made for the priors. Existing BNN designs apply different priors to weights, while the behaviours of these priors make it difficult to sufficiently shrink noisy signals or they are prone to overshrinking important signals in the weights. To alleviate this problem, we propose a novel R2D2-Net, which imposes the R^2-induced Dirichlet Decomposition (R2D2) prior to the BNN weights. The R2D2-Net can effectively shrink irrelevant coefficients towards zero, while preventing key features from over-shrinkage. To approximate the posterior distribution of weights more accurately, we further propose a variational Gibbs inference algorithm that combines the Gibbs updating procedure and gradient-based optimization. This strategy enhances stability and consistency in estimation when the variational objective involving the shrinkage parameters is non-convex. We also analyze the evidence lower bound (ELBO) and the posterior concentration rates from a theoretical perspective. Experiments on both natural and medical image classification and uncertainty estimation tasks demonstrate satisfactory performance of our method.", "date": "2025-05-23", "categories": []}
{"id": "2505.17856", "title": "Stochastic Weight Sharing for Bayesian Neural Networks", "abstract": "While offering a principled framework for uncertainty quantification in deep learning, the employment of Bayesian Neural Networks (BNNs) is still constrained by their increased computational requirements and the convergence difficulties when training very deep, state-of-the-art architectures. In this work, we reinterpret weight-sharing quantization techniques from a stochastic perspective in the context of training and inference with Bayesian Neural Networks (BNNs). Specifically, we leverage 2D adaptive Gaussian distributions, Wasserstein distance estimations, and alpha blending to encode the stochastic behaviour of a BNN in a lower dimensional, soft Gaussian representation. Through extensive empirical investigation, we demonstrate that our approach significantly reduces the computational overhead inherent in Bayesian learning by several orders of magnitude, enabling the efficient Bayesian training of large-scale models, such as ResNet-101 and Vision Transformer (VIT). On various computer vision benchmarks including CIFAR10, CIFAR100, and ImageNet1k. Our approach compresses model parameters by approximately 50x and reduces model size by 75, while achieving accuracy and uncertainty estimations comparable to the state-of-the-art.", "date": "2025-05-23", "categories": []}
{"id": "2407.12288", "title": "Information-Theoretic Foundations for Machine Learning", "abstract": "The progress of machine learning over the past decade is undeniable. In retrospect, it is both remarkable and unsettling that this progress was achievable with little to no rigorous theory to guide experimentation. Despite this fact, practitioners have been able to guide their future experimentation via observations from previous large-scale empirical investigations. In this work, we propose a theoretical framework which attempts to provide rigor to existing practices in machine learning. To the theorist, we provide a framework which is mathematically rigorous and leaves open many interesting ideas for future exploration. To the practitioner, we provide a framework whose results are simple, and provide intuition to guide future investigations across a wide range of learning paradigms. Concretely, we provide a theoretical framework rooted in Bayesian statistics and Shannon's information theory which is general enough to unify the analysis of many phenomena in machine learning. Our framework characterizes the performance of an optimal Bayesian learner as it learns from a stream of experience. Unlike existing analyses that weaken with increasing data complexity, our theoretical tools provide accurate insights across diverse machine learning settings. Throughout this work, we derive theoretical results and demonstrate their generality by apply them to derive insights specific to settings. These settings range from learning from data which is independently and identically distributed under an unknown distribution, to data which is sequential, to data which exhibits hierarchical structure amenable to meta-learning, and finally to data which is not fully explainable under the learner's beliefs (misspecification). These results are particularly relevant as we strive to understand and overcome increasingly difficult machine learning challenges in this endlessly complex world.", "date": "2024-07-17", "categories": []}
{"id": "2505.15888", "title": "Last Layer Empirical Bayes", "abstract": "The task of quantifying the inherent uncertainty associated with neural network predictions is a key challenge in artificial intelligence. Bayesian neural networks (BNNs) and deep ensembles are among the most prominent approaches to tackle this task. Both approaches produce predictions by computing an expectation of neural network outputs over some distribution on the corresponding weights; this distribution is given by the posterior in the case of BNNs, and by a mixture of point masses for ensembles. Inspired by recent work showing that the distribution used by ensembles can be understood as a posterior corresponding to a learned data-dependent prior, we propose last layer empirical Bayes (LLEB). LLEB instantiates a learnable prior as a normalizing flow, which is then trained to maximize the evidence lower bound; to retain tractability we use the flow only on the last layer. We show why LLEB is well motivated, and how it interpolates between standard BNNs and ensembles in terms of the strength of the prior that they use. LLEB performs on par with existing approaches, highlighting that empirical Bayes is a promising direction for future research in uncertainty quantification.", "date": "2025-05-21", "categories": []}
{"id": "2505.16145", "title": "Exponential Convergence of CAVI for Bayesian PCA", "abstract": "Probabilistic principal component analysis (PCA) and its Bayesian variant (BPCA) are widely used for dimension reduction in machine learning and statistics. The main advantage of probabilistic PCA over the traditional formulation is allowing uncertainty quantification. The parameters of BPCA are typically learned using mean-field variational inference, and in particular, the coordinate ascent variational inference (CAVI) algorithm. So far, the convergence speed of CAVI for BPCA has not been characterized. In our paper, we fill this gap in the literature. Firstly, we prove a precise exponential convergence result in the case where the model uses a single principal component (PC). Interestingly, this result is established through a connection with the classical $\\textit{power iteration algorithm}$ and it indicates that traditional PCA is retrieved as points estimates of the BPCA parameters. Secondly, we leverage recent tools to prove exponential convergence of CAVI for the model with any number of PCs, thus leading to a more general result, but one that is of a slightly different flavor. To prove the latter result, we additionally needed to introduce a novel lower bound for the symmetric Kullback--Leibler divergence between two multivariate normal distributions, which, we believe, is of independent interest in information theory.", "date": "2025-05-22", "categories": []}
{"id": "2505.16244", "title": "Generalized Power Priors for Improved Bayesian Inference with Historical Data", "abstract": "The power prior is a class of informative priors designed to incorporate historical data alongside current data in a Bayesian framework. It includes a power parameter that controls the influence of historical data, providing flexibility and adaptability. A key property of the power prior is that the resulting posterior minimizes a linear combination of KL divergences between two pseudo-posterior distributions: one ignoring historical data and the other fully incorporating it. We extend this framework by identifying the posterior distribution as the minimizer of a linear combination of Amari's $\\alpha$-divergence, a generalization of KL divergence. We show that this generalization can lead to improved performance by allowing for the data to adapt to appropriate choices of the $\\alpha$ parameter. Theoretical properties of this generalized power posterior are established, including behavior as a generalized geodesic on the Riemannian manifold of probability distributions, offering novel insights into its geometric interpretation.", "date": "2025-05-22", "categories": []}
{"id": "2502.19231", "title": "AI-Powered Bayesian Inference", "abstract": "The advent of Generative Artificial Intelligence (GAI) has heralded an inflection point that changed how society thinks about knowledge acquisition. While GAI cannot be fully trusted for decision-making, it may still provide valuable information that can be integrated into a decision pipeline. Rather than seeing the lack of certitude and inherent randomness of GAI as a problem, we view it as an opportunity. Indeed, variable answers to given prompts can be leveraged to construct a prior distribution which reflects assuredness of AI predictions. This prior distribution may be combined with tailored datasets for a fully Bayesian analysis with an AI-driven prior. In this paper, we explore such a possibility within a non-parametric Bayesian framework. The basic idea consists of assigning a Dirichlet process prior distribution on the data-generating distribution with AI generative model as its baseline. Hyper-parameters of the prior can be tuned out-of-sample to assess the informativeness of the AI prior. Posterior simulation is achieved by computing a suitably randomized functional on an augmented data that consists of observed (labeled) data as well as fake data whose labels have been imputed using AI. This strategy can be parallelized and rapidly produces iid samples from the posterior by optimization as opposed to sampling from conditionals. Our method enables (predictive) inference and uncertainty quantification leveraging AI predictions in a coherent probabilistic manner.", "date": "2025-02-26", "categories": []}
{"id": "2503.01375", "title": "Bayesian Inverse Problems Meet Flow Matching: Efficient and Flexible Inference via Transformers", "abstract": "The efficient resolution of Bayesian inverse problems remains challenging due to the high computational cost of traditional sampling methods. In this paper, we propose a novel framework that integrates Conditional Flow Matching (CFM) with a transformer-based architecture to enable fast and flexible sampling from complex posterior distributions. The proposed methodology involves the direct learning of conditional probability trajectories from the data, leveraging CFM's ability to bypass iterative simulation and transformers' capacity to process arbitrary numbers of observations. The efficacy of the proposed framework is demonstrated through its application to three problems: a simple nonlinear model, a disease dynamics framework, and a two-dimensional Darcy flow Partial Differential Equation. The primary outcomes demonstrate that the relative errors in parameters recovery are as low as 1.5%, and that the inference time is reduced by up to 2000 times on CPU in comparison with the Monte Carlo Markov Chain. This framework facilitates the expeditious resolution of Bayesian problems through the utilisation of sampling from the learned conditional distribution.", "date": "2025-03-03", "categories": []}
{"id": "2505.11671", "title": "Humble your Overconfident Networks: Unlearning Overfitting via Sequential Monte Carlo Tempered Deep Ensembles", "abstract": "Sequential Monte Carlo (SMC) methods offer a principled approach to Bayesian uncertainty quantification but are traditionally limited by the need for full-batch gradient evaluations. We introduce a scalable variant by incorporating Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) proposals into SMC, enabling efficient mini-batch based sampling. Our resulting SMCSGHMC algorithm outperforms standard stochastic gradient descent (SGD) and deep ensembles across image classification, out-of-distribution (OOD) detection, and transfer learning tasks. We further show that SMCSGHMC mitigates overfitting and improves calibration, providing a flexible, scalable pathway for converting pretrained neural networks into well-calibrated Bayesian models.", "date": "2025-05-16", "categories": []}
{"id": "2505.10448", "title": "Efficient MCMC Sampling with Expensive-to-Compute and Irregular Likelihoods", "abstract": "Bayesian inference with Markov Chain Monte Carlo (MCMC) is challenging when the likelihood function is irregular and expensive to compute. We explore several sampling algorithms that make use of subset evaluations to reduce computational overhead. We adapt the subset samplers for this setting where gradient information is not available or is unreliable. To achieve this, we introduce data-driven proxies in place of Taylor expansions and define a novel computation-cost aware adaptive controller. We undertake an extensive evaluation for a challenging disease modelling task and a configurable task with similar irregularity in the likelihood surface. We find our improved version of Hierarchical Importance with Nested Training Samples (HINTS), with adaptive proposals and a data-driven proxy, obtains the best sampling error in a fixed computational budget. We conclude that subset evaluations can provide cheap and naturally-tempered exploration, while a data-driven proxy can pre-screen proposals successfully in explored regions of the state space. These two elements combine through hierarchical delayed acceptance to achieve efficient, exact sampling.", "date": "2025-05-15", "categories": []}
{"id": "2405.17955", "title": "Efficient Prior Calibration From Indirect Data", "abstract": "Bayesian inversion is central to the quantification of uncertainty within problems arising from numerous applications in science and engineering. To formulate the approach, four ingredients are required: a forward model mapping the unknown parameter to an element of a solution space, often the solution space for a differential equation; an observation operator mapping an element of the solution space to the data space; a noise model describing how noise pollutes the observations; and a prior model describing knowledge about the unknown parameter before the data is acquired. This paper is concerned with learning the prior model from data; in particular, learning the prior from multiple realizations of indirect data obtained through the noisy observation process. The prior is represented, using a generative model, as the pushforward of a Gaussian in a latent space; the pushforward map is learned by minimizing an appropriate loss function. A metric that is well-defined under empirical approximation is used to define the loss function for the pushforward map to make an implementable methodology. Furthermore, an efficient residual-based neural operator approximation of the forward model is proposed and it is shown that this may be learned concurrently with the pushforward map, using a bilevel optimization formulation of the problem; this use of neural operator approximation has the potential to make prior learning from indirect data more computationally efficient, especially when the observation process is expensive, non-smooth or not known. The ideas are illustrated with the Darcy flow inverse problem of finding permeability from piezometric head measurements.", "date": "2024-05-28", "categories": []}
{"id": "2409.15511", "title": "Bayesian computation with generative diffusion models by Multilevel Monte Carlo", "abstract": "Generative diffusion models have recently emerged as a powerful strategy to perform stochastic sampling in Bayesian inverse problems, delivering remarkably accurate solutions for a wide range of challenging applications. However, diffusion models often require a large number of neural function evaluations per sample in order to deliver accurate posterior samples. As a result, using diffusion models as stochastic samplers for Monte Carlo integration in Bayesian computation can be highly computationally expensive, particularly in applications that require a substantial number of Monte Carlo samples for conducting uncertainty quantification analyses. This cost is especially high in large-scale inverse problems such as computational imaging, which rely on large neural networks that are expensive to evaluate. With quantitative imaging applications in mind, this paper presents a Multilevel Monte Carlo strategy that significantly reduces the cost of Bayesian computation with diffusion models. This is achieved by exploiting cost-accuracy trade-offs inherent to diffusion models to carefully couple models of different levels of accuracy in a manner that significantly reduces the overall cost of the calculation, without reducing the final accuracy. The proposed approach achieves a $4\\times$-to-$8\\times$ reduction in computational cost w.r.t. standard techniques across three benchmark imaging problems.", "date": "2024-09-23", "categories": []}
{"id": "2505.04603", "title": "Likelihood-Free Adaptive Bayesian Inference via Nonparametric\n  Distribution Matching", "abstract": "  When the likelihood is analytically unavailable and computationally\nintractable, approximate Bayesian computation (ABC) has emerged as a widely\nused methodology for approximate posterior inference; however, it suffers from\nsevere computational inefficiency in high-dimensional settings or under diffuse\npriors. To overcome these limitations, we propose Adaptive Bayesian Inference\n(ABI), a framework that bypasses traditional data-space discrepancies and\ninstead compares distributions directly in posterior space through\nnonparametric distribution matching. By leveraging a novel Marginally-augmented\nSliced Wasserstein (MSW) distance on posterior measures and exploiting its\nquantile representation, ABI transforms the challenging problem of measuring\ndivergence between posterior distributions into a tractable sequence of\none-dimensional conditional quantile regression tasks. Moreover, we introduce a\nnew adaptive rejection sampling scheme that iteratively refines the posterior\napproximation by updating the proposal distribution via generative density\nestimation. Theoretically, we establish parametric convergence rates for the\ntrimmed MSW distance and prove that the ABI posterior converges to the true\nposterior as the tolerance threshold vanishes. Through extensive empirical\nevaluation, we demonstrate that ABI significantly outperforms data-based\nWasserstein ABC, summary-based ABC, and state-of-the-art likelihood-free\nsimulators, especially in high-dimensional or dependent observation regimes.\n", "date": "2025-05-07", "categories": []}
{"id": "2505.01859", "title": "Bayesian learning of the optimal action-value function in a Markov\n  decision process", "abstract": "  The Markov Decision Process (MDP) is a popular framework for sequential\ndecision-making problems, and uncertainty quantification is an essential\ncomponent of it to learn optimal decision-making strategies. In particular, a\nBayesian framework is used to maintain beliefs about the optimal decisions and\nthe unknown ingredients of the model, which are also to be learned from the\ndata, such as the rewards and state dynamics. However, many existing Bayesian\napproaches for learning the optimal decision-making strategy are based on\nunrealistic modelling assumptions and utilise approximate inference techniques.\nThis raises doubts whether the benefits of Bayesian uncertainty quantification\nare fully realised or can be relied upon.\n  We focus on infinite-horizon and undiscounted MDPs, with finite state and\naction spaces, and a terminal state. We provide a full Bayesian framework, from\nmodelling to inference to decision-making. For modelling, we introduce a\nlikelihood function with minimal assumptions for learning the optimal\naction-value function based on Bellman's optimality equations, analyse its\nproperties, and clarify connections to existing works. For deterministic\nrewards, the likelihood is degenerate and we introduce artificial observation\nnoise to relax it, in a controlled manner, to facilitate more efficient Monte\nCarlo-based inference. For inference, we propose an adaptive sequential Monte\nCarlo algorithm to both sample from and adjust the sequence of relaxed\nposterior distributions. For decision-making, we choose actions using samples\nfrom the posterior distribution over the optimal strategies. While commonly\ndone, we provide new insight that clearly shows that it is a generalisation of\nThompson sampling from multi-arm bandit problems. Finally, we evaluate our\nframework on the Deep Sea benchmark problem and demonstrate the exploration\nbenefits of posterior sampling in MDPs.\n", "date": "2025-05-03", "categories": []}
{"id": "2410.01284", "title": "Deep Kernel Posterior Learning under Infinite Variance Prior Weights", "abstract": "  Neal (1996) proved that infinitely wide shallow Bayesian neural networks\n(BNN) converge to Gaussian processes (GP), when the network weights have\nbounded prior variance. Cho & Saul (2009) provided a useful recursive formula\nfor deep kernel processes for relating the covariance kernel of each layer to\nthe layer immediately below. Moreover, they worked out the form of the\nlayer-wise covariance kernel in an explicit manner for several common\nactivation functions. Recent works, including Aitchison et al. (2021), have\nhighlighted that the covariance kernels obtained in this manner are\ndeterministic and hence, precludes any possibility of representation learning,\nwhich amounts to learning a non-degenerate posterior of a random kernel given\nthe data. To address this, they propose adding artificial noise to the kernel\nto retain stochasticity, and develop deep kernel inverse Wishart processes.\nNonetheless, this artificial noise injection could be critiqued in that it\nwould not naturally emerge in a classic BNN architecture under an\ninfinite-width limit. To address this, we show that a Bayesian deep neural\nnetwork, where each layer width approaches infinity, and all network weights\nare elliptically distributed with infinite variance, converges to a process\nwith $\\alpha$-stable marginals in each layer that has a conditionally Gaussian\nrepresentation. These conditional random covariance kernels could be\nrecursively linked in the manner of Cho & Saul (2009), even though marginally\nthe process exhibits stable behavior, and hence covariances are not even\nnecessarily defined. We also provide useful generalizations of the recent\nresults of Lor\\'ia & Bhadra (2024) on shallow networks to multi-layer networks,\nand remedy the computational burden of their approach. The computational and\nstatistical benefits over competing approaches stand out in simulations and in\ndemonstrations on benchmark data sets.\n", "date": "2024-10-02", "categories": []}
{"id": "2402.01338", "title": "Inferring the Langevin Equation with Uncertainty via Bayesian Neural\n  Networks", "abstract": "  Pervasive across diverse domains, stochastic systems exhibit fluctuations in\nprocesses ranging from molecular dynamics to climate phenomena. The Langevin\nequation has served as a common mathematical model for studying such systems,\nenabling predictions of their temporal evolution and analyses of thermodynamic\nquantities, including absorbed heat, work done on the system, and entropy\nproduction. However, inferring the Langevin equation from observed trajectories\nis a challenging problem, and assessing the uncertainty associated with the\ninferred equation has yet to be accomplished. In this study, we present a\ncomprehensive framework that employs Bayesian neural networks for inferring\nLangevin equations in both overdamped and underdamped regimes. Our framework\nfirst provides the drift force and diffusion matrix separately and then\ncombines them to construct the Langevin equation. By providing a distribution\nof predictions instead of a single value, our approach allows us to assess\nprediction uncertainties, which can help prevent potential misunderstandings\nand erroneous decisions about the system. We demonstrate the effectiveness of\nour framework in inferring Langevin equations for various scenarios including a\nneuron model and microscopic engine, highlighting its versatility and potential\nimpact.\n", "date": "2024-02-02", "categories": []}
{"id": "2404.13056", "title": "Variational Bayesian Optimal Experimental Design with Normalizing Flows", "abstract": "  Bayesian optimal experimental design (OED) seeks experiments that maximize\nthe expected information gain (EIG) in model parameters. Directly estimating\nthe EIG using nested Monte Carlo is computationally expensive and requires an\nexplicit likelihood. Variational OED (vOED), in contrast, estimates a lower\nbound of the EIG without likelihood evaluations by approximating the posterior\ndistributions with variational forms, and then tightens the bound by optimizing\nits variational parameters. We introduce the use of normalizing flows (NFs) for\nrepresenting variational distributions in vOED; we call this approach vOED-NFs.\nSpecifically, we adopt NFs with a conditional invertible neural network\narchitecture built from compositions of coupling layers, and enhanced with a\nsummary network for data dimension reduction. We present Monte Carlo estimators\nto the lower bound along with gradient expressions to enable a gradient-based\nsimultaneous optimization of the variational parameters and the design\nvariables. The vOED-NFs algorithm is then validated in two benchmark problems,\nand demonstrated on a partial differential equation-governed application of\ncathodic electrophoretic deposition and an implicit likelihood case with\nstochastic modeling of aphid population. The findings suggest that a\ncomposition of 4--5 coupling layers is able to achieve lower EIG estimation\nbias, under a fixed budget of forward model runs, compared to previous\napproaches. The resulting NFs produce approximate posteriors that agree well\nwith the true posteriors, able to capture non-Gaussian and multi-modal features\neffectively.\n", "date": "2024-04-08", "categories": []}
{"id": "2402.14781", "title": "Effective Bayesian Causal Inference via Structural Marginalisation and\n  Autoregressive Orders", "abstract": "  The traditional two-stage approach to causal inference first identifies a\nsingle causal model (or equivalence class of models), which is then used to\nanswer causal queries. However, this neglects any epistemic model uncertainty.\nIn contrast, Bayesian causal inference does incorporate epistemic uncertainty\ninto query estimates via Bayesian marginalisation (posterior averaging) over\nall causal models. While principled, this marginalisation over entire causal\nmodels, i.e., both causal structures (graphs) and mechanisms, poses a\ntremendous computational challenge. In this work, we address this challenge by\ndecomposing structure marginalisation into the marginalisation over (i) causal\norders and (ii) directed acyclic graphs (DAGs) given an order. We can\nmarginalise the latter in closed form by limiting the number of parents per\nvariable and utilising Gaussian processes to model mechanisms. To marginalise\nover orders, we use a sampling-based approximation, for which we devise a novel\nauto-regressive distribution over causal orders (ARCO). Our method outperforms\nstate-of-the-art in structure learning on simulated non-linear additive noise\nbenchmarks, and yields competitive results on real-world data. Furthermore, we\ncan accurately infer interventional distributions and average causal effects.\n", "date": "2024-02-22", "categories": []}
{"id": "2504.17719", "title": "Evaluating Uncertainty in Deep Gaussian Processes", "abstract": "  Reliable uncertainty estimates are crucial in modern machine learning. Deep\nGaussian Processes (DGPs) and Deep Sigma Point Processes (DSPPs) extend GPs\nhierarchically, offering promising methods for uncertainty quantification\ngrounded in Bayesian principles. However, their empirical calibration and\nrobustness under distribution shift relative to baselines like Deep Ensembles\nremain understudied. This work evaluates these models on regression (CASP\ndataset) and classification (ESR dataset) tasks, assessing predictive\nperformance (MAE, Accu- racy), calibration using Negative Log-Likelihood (NLL)\nand Expected Calibration Error (ECE), alongside robustness under various\nsynthetic feature-level distribution shifts. Results indicate DSPPs provide\nstrong in-distribution calibration leveraging their sigma point approximations.\nHowever, compared to Deep Ensembles, which demonstrated superior robustness in\nboth per- formance and calibration under the tested shifts, the GP-based\nmethods showed vulnerabilities, exhibiting particular sensitivity in the\nobserved metrics. Our findings underscore ensembles as a robust baseline,\nsuggesting that while deep GP methods offer good in-distribution calibration,\ntheir practical robustness under distribution shift requires careful\nevaluation. To facilitate reproducibility, we make our code available at\nhttps://github.com/matthjs/xai-gp.\n", "date": "2025-04-24", "categories": []}
{"id": "2504.05490", "title": "Optimal Bayesian Affine Estimator and Active Learning for the Wiener\n  Model", "abstract": "  This paper presents a Bayesian estimation framework for Wiener models,\nfocusing on learning nonlinear output functions under known linear state\ndynamics. We derive a closed-form optimal affine estimator for the unknown\nparameters, characterized by the so-called \"dynamic basis statistics\" (DBS).\nSeveral features of the proposed estimator are studied, including Bayesian\nunbiasedness, closed-form posterior statistics, error monotonicity in\ntrajectory length, and consistency condition (also known as persistent\nexcitation). In the special case of Fourier basis functions, we demonstrate\nthat the closed-form description is computationally available, as the Fourier\nDBS enjoys explicit expressions. Furthermore, we identify an inherent\ninconsistency in the Fourier bases for single-trajectory measurements,\nregardless of the input excitation. Leveraging the closed-form estimation\nerror, we develop an active learning algorithm synthesizing input signals to\nminimize estimation error. Numerical experiments validate the efficacy of our\napproach, showing significant improvements over traditional regularized\nleast-squares methods.\n", "date": "2025-04-07", "categories": []}
{"id": "2504.12528", "title": "Robust and Scalable Variational Bayes", "abstract": "  We propose a robust and scalable framework for variational Bayes (VB) that\neffectively handles outliers and contamination of arbitrary nature in large\ndatasets. Our approach divides the dataset into disjoint subsets, computes the\nposterior for each subset, and applies VB approximation independently to these\nposteriors. The resulting variational posteriors with respect to the subsets\nare then aggregated using the geometric median of probability measures,\ncomputed with respect to the Wasserstein distance. This novel aggregation\nmethod yields the Variational Median Posterior (VM-Posterior) distribution. We\nrigorously demonstrate that the VM-Posterior preserves contraction properties\nakin to those of the true posterior, while accounting for approximation errors\nor the variational gap inherent in VB methods. We also provide provable\nrobustness guarantee of the VM-Posterior. Furthermore, we establish a\nvariational Bernstein-von Mises theorem for both multivariate Gaussian\ndistributions with general covariance structures and the mean-field variational\nfamily. To facilitate practical implementation, we adapt existing algorithms\nfor computing the VM-Posterior and evaluate its performance through extensive\nnumerical experiments. The results highlight its robustness and scalability,\nmaking it a reliable tool for Bayesian inference in the presence of complex,\ncontaminated datasets.\n", "date": "2025-04-16", "categories": []}
{"id": "2504.11554", "title": "Normalizing Flow Regression for Bayesian Inference with Offline\n  Likelihood Evaluations", "abstract": "  Bayesian inference with computationally expensive likelihood evaluations\nremains a significant challenge in many scientific domains. We propose\nnormalizing flow regression (NFR), a novel offline inference method for\napproximating posterior distributions. Unlike traditional surrogate approaches\nthat require additional sampling or inference steps, NFR directly yields a\ntractable posterior approximation through regression on existing log-density\nevaluations. We introduce training techniques specifically for flow regression,\nsuch as tailored priors and likelihood functions, to achieve robust posterior\nand model evidence estimation. We demonstrate NFR's effectiveness on synthetic\nbenchmarks and real-world applications from neuroscience and biology, showing\nsuperior or comparable performance to existing methods. NFR represents a\npromising approach for Bayesian inference when standard methods are\ncomputationally prohibitive or existing model evaluations can be recycled.\n", "date": "2025-04-15", "categories": []}
{"id": "2406.03369", "title": "Posterior and variational inference for deep neural networks with\n  heavy-tailed weights", "abstract": "  We consider deep neural networks in a Bayesian framework with a prior\ndistribution sampling the network weights at random. Following a recent idea of\nAgapiou and Castillo (2023), who show that heavy-tailed prior distributions\nachieve automatic adaptation to smoothness, we introduce a simple Bayesian deep\nlearning prior based on heavy-tailed weights and ReLU activation. We show that\nthe corresponding posterior distribution achieves near-optimal minimax\ncontraction rates, simultaneously adaptive to both intrinsic dimension and\nsmoothness of the underlying function, in a variety of contexts including\nnonparametric regression, geometric data and Besov spaces. While most works so\nfar need a form of model selection built-in within the prior distribution, a\nkey aspect of our approach is that it does not require to sample\nhyperparameters to learn the architecture of the network. We also provide\nvariational Bayes counterparts of the results, that show that mean-field\nvariational approximations still benefit from near-optimal theoretical support.\n", "date": "2024-06-05", "categories": []}
{"id": "2504.10770", "title": "Collaborative Bayesian Optimization via Wasserstein Barycenters", "abstract": "  Motivated by the growing need for black-box optimization and data privacy, we\nintroduce a collaborative Bayesian optimization (BO) framework that addresses\nboth of these challenges. In this framework agents work collaboratively to\noptimize a function they only have oracle access to. In order to mitigate\nagainst communication and privacy constraints, agents are not allowed to share\ntheir data but can share their Gaussian process (GP) surrogate models. To\nenable collaboration under these constraints, we construct a central model to\napproximate the objective function by leveraging the concept of Wasserstein\nbarycenters of GPs. This central model integrates the shared models without\naccessing the underlying data. A key aspect of our approach is a collaborative\nacquisition function that balances exploration and exploitation, allowing for\nthe optimization of decision variables collaboratively in each iteration. We\nprove that our proposed algorithm is asymptotically consistent and that its\nimplementation via Monte Carlo methods is numerically accurate. Through\nnumerical experiments, we demonstrate that our approach outperforms other\nbaseline collaborative frameworks and is competitive with centralized\napproaches that do not consider data privacy.\n", "date": "2025-04-15", "categories": []}
{"id": "2406.00104", "title": "Scalable Bayesian Learning with posteriors", "abstract": "  Although theoretically compelling, Bayesian learning with modern machine\nlearning models is computationally challenging since it requires approximating\na high dimensional posterior distribution. In this work, we (i) introduce\nposteriors, an easily extensible PyTorch library hosting general-purpose\nimplementations making Bayesian learning accessible and scalable to large data\nand parameter regimes; (ii) present a tempered framing of stochastic gradient\nMarkov chain Monte Carlo, as implemented in posteriors, that transitions\nseamlessly into optimization and unveils a minor modification to deep ensembles\nto ensure they are asymptotically unbiased for the Bayesian posterior, and\n(iii) demonstrate and compare the utility of Bayesian approximations through\nexperiments including an investigation into the cold posterior effect and\napplications with large language models.\n", "date": "2024-05-31", "categories": []}
{"id": "2404.10550", "title": "Analytical Approximation of the ELBO Gradient in the Context of the\n  Clutter Problem", "abstract": "  We propose an analytical solution for approximating the gradient of the\nEvidence Lower Bound (ELBO) in variational inference problems where the\nstatistical model is a Bayesian network consisting of observations drawn from a\nmixture of a Gaussian distribution embedded in unrelated clutter, known as the\nclutter problem. The method employs the reparameterization trick to move the\ngradient operator inside the expectation and relies on the assumption that,\nbecause the likelihood factorizes over the observed data, the variational\ndistribution is generally more compactly supported than the Gaussian\ndistribution in the likelihood factors. This allows efficient local\napproximation of the individual likelihood factors, which leads to an\nanalytical solution for the integral defining the gradient expectation. We\nintegrate the proposed gradient approximation as the expectation step in an EM\n(Expectation Maximization) algorithm for maximizing ELBO and test against\nclassical deterministic approaches in Bayesian inference, such as the Laplace\napproximation, Expectation Propagation and Mean-Field Variational Inference.\nThe proposed method demonstrates good accuracy and rate of convergence together\nwith linear computational complexity.\n", "date": "2024-04-16", "categories": []}
{"id": "2504.04013", "title": "Spatially-Heterogeneous Causal Bayesian Networks for Seismic\n  Multi-Hazard Estimation: A Variational Approach with Gaussian Processes and\n  Normalizing Flows", "abstract": "  Post-earthquake hazard and impact estimation are critical for effective\ndisaster response, yet current approaches face significant limitations.\nTraditional models employ fixed parameters regardless of geographical context,\nmisrepresenting how seismic effects vary across diverse landscapes, while\nremote sensing technologies struggle to distinguish between co-located hazards.\nWe address these challenges with a spatially-aware causal Bayesian network that\ndecouples co-located hazards by modeling their causal relationships with\nlocation-specific parameters. Our framework integrates sensing observations,\nlatent variables, and spatial heterogeneity through a novel combination of\nGaussian Processes with normalizing flows, enabling us to capture how same\nearthquake produces different effects across varied geological and\ntopographical features. Evaluations across three earthquakes demonstrate\nSpatial-VCBN achieves Area Under the Curve (AUC) improvements of up to 35.2%\nover existing methods. These results highlight the critical importance of\nmodeling spatial heterogeneity in causal mechanisms for accurate disaster\nassessment, with direct implications for improving emergency response resource\nallocation.\n", "date": "2025-04-05", "categories": []}
{"id": "2408.07379", "title": "Posterior Covariance Structures in Gaussian Processes", "abstract": "  In this paper, we present a comprehensive analysis of the posterior\ncovariance field in Gaussian processes, with applications to the posterior\ncovariance matrix. The analysis is based on the Gaussian prior covariance but\nthe approach also applies to other covariance kernels. Our geometric analysis\nreveals how the Gaussian kernel's bandwidth parameter and the spatial\ndistribution of the observations influence the posterior covariance as well as\nthe corresponding covariance matrix, enabling straightforward identification of\nareas with high or low covariance in magnitude. Drawing inspiration from the a\nposteriori error estimation techniques in adaptive finite element methods, we\nalso propose several estimators to efficiently measure the absolute posterior\ncovariance field, which can be used for efficient covariance matrix\napproximation and preconditioning. We conduct a wide range of experiments to\nillustrate our theoretical findings and their practical applications.\n", "date": "2024-08-14", "categories": []}
{"id": "2310.12781", "title": "Simulation-based Bayesian Inference from Privacy Protected Data", "abstract": "  Many modern statistical analysis and machine learning applications require\ntraining models on sensitive user data. Under a formal definition of privacy\nprotection, differentially private algorithms inject calibrated noise into the\nconfidential data or during the data analysis process to produce\nprivacy-protected datasets or queries. However, restricting access to only\nprivatized data during statistical analysis makes it computationally\nchallenging to make valid statistical inferences. In this work, we propose\nsimulation-based inference methods from privacy-protected datasets. In addition\nto sequential Monte Carlo approximate Bayesian computation, we adopt neural\nconditional density estimators as a flexible family of distributions to\napproximate the posterior distribution of model parameters given the observed\nprivate query results. We illustrate our methods on discrete time-series data\nunder an infectious disease model and with ordinary linear regression models.\nIllustrating the privacy-utility trade-off, our experiments and analysis\ndemonstrate the necessity and feasibility of designing valid statistical\ninference procedures to correct for biases introduced by the privacy-protection\nmechanisms.\n", "date": "2023-10-19", "categories": []}
{"id": "2311.16909", "title": "Multinomial belief networks for healthcare data", "abstract": "  Healthcare data from patient or population cohorts are often characterized by\nsparsity, high missingness and relatively small sample sizes. In addition,\nbeing able to quantify uncertainty is often important in a medical context. To\naddress these analytical requirements we propose a deep generative Bayesian\nmodel for multinomial count data. We develop a collapsed Gibbs sampling\nprocedure that takes advantage of a series of augmentation relations, inspired\nby the Zhou$\\unicode{x2013}$Cong$\\unicode{x2013}$Chen model. We visualise the\nmodel's ability to identify coherent substructures in the data using a dataset\nof handwritten digits. We then apply it to a large experimental dataset of DNA\nmutations in cancer and show that we can identify biologically meaningful\nclusters of mutational signatures in a fully data-driven way.\n", "date": "2023-11-28", "categories": []}
{"id": "2410.24079", "title": "Hamiltonian Monte Carlo Inference of Marginalized Linear Mixed-Effects\n  Models", "abstract": "  Bayesian reasoning in linear mixed-effects models (LMMs) is challenging and\noften requires advanced sampling techniques like Markov chain Monte Carlo\n(MCMC). A common approach is to write the model in a probabilistic programming\nlanguage and then sample via Hamiltonian Monte Carlo (HMC). However, there are\nmany ways a user can transform a model that make inference more or less\nefficient. In particular, marginalizing some variables can greatly improve\ninference but is difficult for users to do manually. We develop an algorithm to\neasily marginalize random effects in LMMs. A naive approach introduces cubic\ntime operations within an inference algorithm like HMC, but we reduce the\nrunning time to linear using fast linear algebra techniques. We show that\nmarginalization is always beneficial when applicable and highlight improvements\nin various models, especially ones from cognitive sciences.\n", "date": "2024-10-31", "categories": []}
{"id": "2402.09122", "title": "Weighted-Sum of Gaussian Process Latent Variable Models", "abstract": "  This work develops a Bayesian non-parametric approach to signal separation\nwhere the signals may vary according to latent variables. Our key contribution\nis to augment Gaussian Process Latent Variable Models (GPLVMs) for the case\nwhere each data point comprises the weighted sum of a known number of pure\ncomponent signals, observed across several input locations. Our framework\nallows arbitrary non-linear variations in the signals while being able to\nincorporate useful priors for the linear weights, such as summing-to-one. Our\ncontributions are particularly relevant to spectroscopy, where changing\nconditions may cause the underlying pure component signals to vary from sample\nto sample. To demonstrate the applicability to both spectroscopy and other\ndomains, we consider several applications: a near-infrared spectroscopy dataset\nwith varying temperatures, a simulated dataset for identifying flow\nconfiguration through a pipe, and a dataset for determining the type of rock\nfrom its reflectance.\n", "date": "2024-02-14", "categories": []}
{"id": "2406.13151", "title": "Bayesian Circular Regression with von Mises Quasi-Processes", "abstract": "  The need for regression models to predict circular values arises in many\nscientific fields. In this work we explore a family of expressive and\ninterpretable distributions over circle-valued random functions related to\nGaussian processes targeting two Euclidean dimensions conditioned on the unit\ncircle. The probability model has connections with continuous spin models in\nstatistical physics. Moreover, its density is very simple and has\nmaximum-entropy, unlike previous Gaussian process-based approaches, which use\nwrapping or radial marginalization. For posterior inference, we introduce a new\nStratonovich-like augmentation that lends itself to fast Gibbs sampling. We\nargue that transductive learning in these models favors a Bayesian approach to\nthe parameters and apply our sampling scheme to the Double Metropolis-Hastings\nalgorithm. We present experiments applying this model to the prediction of (i)\nwind directions and (ii) the percentage of the running gait cycle as a function\nof joint angles.\n", "date": "2024-06-19", "categories": []}
{"id": "2503.15150", "title": "Preference Construction: A Bayesian Interactive Preference Elicitation\n  Framework Based on Monte Carlo Tree Search", "abstract": "  We present a novel preference learning framework to capture participant\npreferences efficiently within limited interaction rounds. It involves three\nmain contributions. First, we develop a variational Bayesian approach to infer\nthe participant's preference model by estimating posterior distributions and\nmanaging uncertainty from limited information. Second, we propose an adaptive\nquestioning policy that maximizes cumulative uncertainty reduction, formulating\nquestioning as a finite Markov decision process and using Monte Carlo Tree\nSearch to prioritize promising question trajectories. By considering long-term\neffects and leveraging the efficiency of the Bayesian approach, the policy\navoids shortsightedness. Third, we apply the framework to Multiple Criteria\nDecision Aiding, with pairwise comparison as the preference information and an\nadditive value function as the preference model. We integrate the\nreparameterization trick to address high-variance issues, enhancing robustness\nand efficiency. Computational studies on real-world and synthetic datasets\ndemonstrate the framework's practical usability, outperforming baselines in\ncapturing preferences and achieving superior uncertainty reduction within\nlimited interactions.\n", "date": "2025-03-19", "categories": []}
{"id": "2503.13909", "title": "Quantification of Uncertainties in Probabilistic Deep Neural Network by\n  Implementing Boosting of Variational Inference", "abstract": "  Modern neural network architectures have achieved remarkable accuracies but\nremain highly dependent on their training data, often lacking interpretability\nin their learned mappings. While effective on large datasets, they tend to\noverfit on smaller ones. Probabilistic neural networks, such as those utilizing\nvariational inference, address this limitation by incorporating uncertainty\nestimation through weight distributions rather than point estimates. However,\nstandard variational inference often relies on a single-density approximation,\nwhich can lead to poor posterior estimates and hinder model performance. We\npropose Boosted Bayesian Neural Networks (BBNN), a novel approach that enhances\nneural network weight distribution approximations using Boosting Variational\nInference (BVI). By iteratively constructing a mixture of densities, BVI\nexpands the approximating family, enabling a more expressive posterior that\nleads to improved generalization and uncertainty estimation. While this\napproach increases computational complexity, it significantly enhances accuracy\nan essential tradeoff, particularly in high-stakes applications such as medical\ndiagnostics, where false negatives can have severe consequences. Our\nexperimental results demonstrate that BBNN achieves ~5% higher accuracy\ncompared to conventional neural networks while providing superior uncertainty\nquantification. This improvement highlights the effectiveness of leveraging a\nmixture-based variational family to better approximate the posterior\ndistribution, ultimately advancing probabilistic deep learning.\n", "date": "2025-03-18", "categories": []}
{"id": "2503.12315", "title": "Simulation-based Bayesian inference under model misspecification", "abstract": "  Simulation-based Bayesian inference (SBI) methods are widely used for\nparameter estimation in complex models where evaluating the likelihood is\nchallenging but generating simulations is relatively straightforward. However,\nthese methods commonly assume that the simulation model accurately reflects the\ntrue data-generating process, an assumption that is frequently violated in\nrealistic scenarios. In this paper, we focus on the challenges faced by SBI\nmethods under model misspecification. We consolidate recent research aimed at\nmitigating the effects of misspecification, highlighting three key strategies:\ni) robust summary statistics, ii) generalised Bayesian inference, and iii)\nerror modelling and adjustment parameters. To illustrate both the\nvulnerabilities of popular SBI methods and the effectiveness of\nmisspecification-robust alternatives, we present empirical results on an\nillustrative example.\n", "date": "2025-03-16", "categories": []}
{"id": "2503.13296", "title": "On Local Posterior Structure in Deep Ensembles", "abstract": "  Bayesian Neural Networks (BNNs) often improve model calibration and\npredictive uncertainty quantification compared to point estimators such as\nmaximum-a-posteriori (MAP). Similarly, deep ensembles (DEs) are also known to\nimprove calibration, and therefore, it is natural to hypothesize that deep\nensembles of BNNs (DE-BNNs) should provide even further improvements. In this\nwork, we systematically investigate this across a number of datasets, neural\nnetwork architectures, and BNN approximation methods and surprisingly find that\nwhen the ensembles grow large enough, DEs consistently outperform DE-BNNs on\nin-distribution data. To shine light on this observation, we conduct several\nsensitivity and ablation studies. Moreover, we show that even though DE-BNNs\noutperform DEs on out-of-distribution metrics, this comes at the cost of\ndecreased in-distribution performance. As a final contribution, we open-source\nthe large pool of trained models to facilitate further research on this topic.\n", "date": "2025-03-17", "categories": []}
{"id": "2410.11826", "title": "Bayesian Experimental Design via Contrastive Diffusions", "abstract": "  Bayesian Optimal Experimental Design (BOED) is a powerful tool to reduce the\ncost of running a sequence of experiments. When based on the Expected\nInformation Gain (EIG), design optimization corresponds to the maximization of\nsome intractable expected contrast between prior and posterior distributions.\nScaling this maximization to high dimensional and complex settings has been an\nissue due to BOED inherent computational complexity. In this work, we introduce\na pooled posterior distribution with cost-effective sampling properties and\nprovide a tractable access to the EIG contrast maximization via a new EIG\ngradient expression. Diffusion-based samplers are used to compute the dynamics\nof the pooled posterior and ideas from bi-level optimization are leveraged to\nderive an efficient joint sampling-optimization loop. The resulting efficiency\ngain allows to extend BOED to the well-tested generative capabilities of\ndiffusion models. By incorporating generative models into the BOED framework,\nwe expand its scope and its use in scenarios that were previously impractical.\nNumerical experiments and comparison with state-of-the-art methods show the\npotential of the approach.\n", "date": "2024-10-15", "categories": []}
{"id": "2503.10496", "title": "Explainable Bayesian deep learning through input-skip Latent Binary\n  Bayesian Neural Networks", "abstract": "  Modeling natural phenomena with artificial neural networks (ANNs) often\nprovides highly accurate predictions. However, ANNs often suffer from\nover-parameterization, complicating interpretation and raising uncertainty\nissues. Bayesian neural networks (BNNs) address the latter by representing\nweights as probability distributions, allowing for predictive uncertainty\nevaluation. Latent binary Bayesian neural networks (LBBNNs) further handle\nstructural uncertainty and sparsify models by removing redundant weights. This\narticle advances LBBNNs by enabling covariates to skip to any succeeding layer\nor be excluded, simplifying networks and clarifying input impacts on\npredictions. Ultimately, a linear model or even a constant can be found to be\noptimal for a specific problem at hand. Furthermore, the input-skip LBBNN\napproach reduces network density significantly compared to standard LBBNNs,\nachieving over 99% reduction for small networks and over 99.9% for larger ones,\nwhile still maintaining high predictive accuracy and uncertainty measurement.\nFor example, on MNIST, we reached 97% accuracy and great calibration with just\n935 weights, reaching state-of-the-art for compression of neural networks.\nFurthermore, the proposed method accurately identifies the true covariates and\nadjusts for system non-linearity. The main contribution is the introduction of\nactive paths, enhancing directly designed global and local explanations within\nthe LBBNN framework, that have theoretical guarantees and do not require post\nhoc external tools for explanations.\n", "date": "2025-03-13", "categories": []}
{"id": "2411.10153", "title": "A unifying framework for generalised Bayesian online learning in\n  non-stationary environments", "abstract": "  We propose a unifying framework for methods that perform probabilistic online\nlearning in non-stationary environments. We call the framework BONE, which\nstands for generalised (B)ayesian (O)nline learning in (N)on-stationary\n(E)nvironments. BONE provides a common structure to tackle a variety of\nproblems, including online continual learning, prequential forecasting, and\ncontextual bandits. The framework requires specifying three modelling choices:\n(i) a model for measurements (e.g., a neural network), (ii) an auxiliary\nprocess to model non-stationarity (e.g., the time since the last changepoint),\nand (iii) a conditional prior over model parameters (e.g., a multivariate\nGaussian). The framework also requires two algorithmic choices, which we use to\ncarry out approximate inference under this framework: (i) an algorithm to\nestimate beliefs (posterior distribution) about the model parameters given the\nauxiliary variable, and (ii) an algorithm to estimate beliefs about the\nauxiliary variable. We show how the modularity of our framework allows for many\nexisting methods to be reinterpreted as instances of BONE, and it allows us to\npropose new methods. We compare experimentally existing methods with our\nproposed new method on several datasets, providing insights into the situations\nthat make each method more suitable for a specific task. We provide a Jax open\nsource library to facilitate the adoption of this framework.\n", "date": "2024-11-15", "categories": []}
{"id": "2410.19780", "title": "Sampling from Bayesian Neural Network Posteriors with Symmetric\n  Minibatch Splitting Langevin Dynamics", "abstract": "  We propose a scalable kinetic Langevin dynamics algorithm for sampling\nparameter spaces of big data and AI applications. Our scheme combines a\nsymmetric forward/backward sweep over minibatches with a symmetric\ndiscretization of Langevin dynamics. For a particular Langevin splitting method\n(UBU), we show that the resulting Symmetric Minibatch Splitting-UBU (SMS-UBU)\nintegrator has bias $O(h^2 d^{1/2})$ in dimension $d>0$ with stepsize $h>0$,\ndespite only using one minibatch per iteration, thus providing excellent\ncontrol of the sampling bias as a function of the stepsize. We apply the\nalgorithm to explore local modes of the posterior distribution of Bayesian\nneural networks (BNNs) and evaluate the calibration performance of the\nposterior predictive probabilities for neural networks with convolutional\nneural network architectures for classification problems on three different\ndatasets (Fashion-MNIST, Celeb-A and chest X-ray). Our results indicate that\nBNNs sampled with SMS-UBU can offer significantly better calibration\nperformance compared to standard methods of training and stochastic weight\naveraging.\n", "date": "2024-10-14", "categories": []}
{"id": "2503.08264", "title": "Massively Parallel Expectation Maximization For Approximate Posteriors", "abstract": "  Bayesian inference for hierarchical models can be very challenging. MCMC\nmethods have difficulty scaling to large models with many observations and\nlatent variables. While variational inference (VI) and reweighted wake-sleep\n(RWS) can be more scalable, they are gradient-based methods and so often\nrequire many iterations to converge. Our key insight was that modern massively\nparallel importance weighting methods (Bowyer et al., 2024) give fast and\naccurate posterior moment estimates, and we can use these moment estimates to\nrapidly learn an approximate posterior. Specifically, we propose using\nexpectation maximization to fit the approximate posterior, which we call QEM.\nThe expectation step involves computing the posterior moments using\nhigh-quality massively parallel estimates from Bowyer et al. (2024). The\nmaximization step involves fitting the approximate posterior using these\nmoments, which can be done straightforwardly for simple approximate posteriors\nsuch as Gaussian, Gamma, Beta, Dirichlet, Binomial, Multinomial, Categorical,\netc. (or combinations thereof). We show that QEM is faster than\nstate-of-the-art, massively parallel variants of RWS and VI, and is invariant\nto reparameterizations of the model that dramatically slow down gradient based\nmethods.\n", "date": "2025-03-11", "categories": []}
{"id": "2408.14325", "title": "Function-Space MCMC for Bayesian Wide Neural Networks", "abstract": "  Bayesian Neural Networks represent a fascinating confluence of deep learning\nand probabilistic reasoning, offering a compelling framework for understanding\nuncertainty in complex predictive models. In this paper, we investigate the use\nof the preconditioned Crank-Nicolson algorithm and its Langevin version to\nsample from a reparametrised posterior distribution of the neural network's\nweights, as the widths grow larger. In addition to being robust in the\ninfinite-dimensional setting, we prove that the acceptance probabilities of the\nproposed algorithms approach 1 as the width of the network increases,\nindependently of any stepsize tuning. Moreover, we examine and compare how the\nmixing speeds of the underdamped Langevin Monte Carlo, the preconditioned\nCrank-Nicolson and the preconditioned Crank-Nicolson Langevin samplers are\ninfluenced by changes in the network width in some real-world cases. Our\nfindings suggest that, in wide Bayesian Neural Networks configurations, the\npreconditioned Crank-Nicolson algorithm allows for a scalable and more\nefficient sampling of the reparametrised posterior distribution, as also\nevidenced by a higher effective sample size and improved diagnostic results\ncompared with the other analysed algorithms.\n", "date": "2024-08-26", "categories": []}
{"id": "2502.07273", "title": "Variational Learning Induces Adaptive Label Smoothing", "abstract": "  We show that variational learning naturally induces an adaptive label\nsmoothing where label noise is specialized for each example. Such\nlabel-smoothing is useful to handle examples with labeling errors and\ndistribution shifts, but designing a good adaptivity strategy is not always\neasy. We propose to skip this step and simply use the natural adaptivity\ninduced during the optimization of a variational objective. We show empirical\nresults where a variational algorithm called IVON outperforms traditional label\nsmoothing and yields adaptivity strategies similar to those of an existing\napproach. By connecting Bayesian methods to label smoothing, our work provides\na new way to handle overconfident predictions.\n", "date": "2025-02-11", "categories": []}
{"id": "2503.02798", "title": "Spike-and-Slab Posterior Sampling in High Dimensions", "abstract": "  Posterior sampling with the spike-and-slab prior [MB88], a popular multimodal\ndistribution used to model uncertainty in variable selection, is considered the\ntheoretical gold standard method for Bayesian sparse linear regression [CPS09,\nRoc18]. However, designing provable algorithms for performing this sampling\ntask is notoriously challenging. Existing posterior samplers for Bayesian\nsparse variable selection tasks either require strong assumptions about the\nsignal-to-noise ratio (SNR) [YWJ16], only work when the measurement count grows\nat least linearly in the dimension [MW24], or rely on heuristic approximations\nto the posterior. We give the first provable algorithms for spike-and-slab\nposterior sampling that apply for any SNR, and use a measurement count\nsublinear in the problem dimension. Concretely, assume we are given a\nmeasurement matrix $\\mathbf{X} \\in \\mathbb{R}^{n\\times d}$ and noisy\nobservations $\\mathbf{y} = \\mathbf{X}\\mathbf{\\theta}^\\star + \\mathbf{\\xi}$ of a\nsignal $\\mathbf{\\theta}^\\star$ drawn from a spike-and-slab prior $\\pi$ with a\nGaussian diffuse density and expected sparsity k, where $\\mathbf{\\xi} \\sim\n\\mathcal{N}(\\mathbb{0}_n, \\sigma^2\\mathbf{I}_n)$. We give a polynomial-time\nhigh-accuracy sampler for the posterior $\\pi(\\cdot \\mid \\mathbf{X},\n\\mathbf{y})$, for any SNR $\\sigma^{-1}$ > 0, as long as $n \\geq k^3 \\cdot\n\\text{polylog}(d)$ and $X$ is drawn from a matrix ensemble satisfying the\nrestricted isometry property. We further give a sampler that runs in\nnear-linear time $\\approx nd$ in the same setting, as long as $n \\geq k^5 \\cdot\n\\text{polylog}(d)$. To demonstrate the flexibility of our framework, we extend\nour result to spike-and-slab posterior sampling with Laplace diffuse densities,\nachieving similar guarantees when $\\sigma = O(\\frac{1}{k})$ is bounded.\n", "date": "2025-03-04", "categories": []}
{"id": "2411.02728", "title": "Compositional simulation-based inference for time series", "abstract": "  Amortized simulation-based inference (SBI) methods train neural networks on\nsimulated data to perform Bayesian inference. While this strategy avoids the\nneed for tractable likelihoods, it often requires a large number of simulations\nand has been challenging to scale to time series data. Scientific simulators\nfrequently emulate real-world dynamics through thousands of single-state\ntransitions over time. We propose an SBI approach that can exploit such\nMarkovian simulators by locally identifying parameters consistent with\nindividual state transitions. We then compose these local results to obtain a\nposterior over parameters that align with the entire time series observation.\nWe focus on applying this approach to neural posterior score estimation but\nalso show how it can be applied, e.g., to neural likelihood (ratio) estimation.\nWe demonstrate that our approach is more simulation-efficient than directly\nestimating the global posterior on several synthetic benchmark tasks and\nsimulators used in ecology and epidemiology. Finally, we validate scalability\nand simulation efficiency of our approach by applying it to a high-dimensional\nKolmogorov flow simulator with around one million data dimensions.\n", "date": "2024-11-05", "categories": []}
{"id": "2503.00699", "title": "Parameter Expanded Stochastic Gradient Markov Chain Monte Carlo", "abstract": "  Bayesian Neural Networks (BNNs) provide a promising framework for modeling\npredictive uncertainty and enhancing out-of-distribution robustness (OOD) by\nestimating the posterior distribution of network parameters. Stochastic\nGradient Markov Chain Monte Carlo (SGMCMC) is one of the most powerful methods\nfor scalable posterior sampling in BNNs, achieving efficiency by combining\nstochastic gradient descent with second-order Langevin dynamics. However,\nSGMCMC often suffers from limited sample diversity in practice, which affects\nuncertainty estimation and model performance. We propose a simple yet effective\napproach to enhance sample diversity in SGMCMC without the need for tempering\nor running multiple chains. Our approach reparameterizes the neural network by\ndecomposing each of its weight matrices into a product of matrices, resulting\nin a sampling trajectory that better explores the target parameter space. This\napproach produces a more diverse set of samples, allowing faster mixing within\nthe same computational budget. Notably, our sampler achieves these improvements\nwithout increasing the inference cost compared to the standard SGMCMC.\nExtensive experiments on image classification tasks, including OOD robustness,\ndiversity, loss surface analyses, and a comparative study with Hamiltonian\nMonte Carlo, demonstrate the superiority of the proposed approach.\n", "date": "2025-03-02", "categories": []}
{"id": "2410.02490", "title": "Stochastic variance-reduced Gaussian variational inference on the\n  Bures-Wasserstein manifold", "abstract": "  Optimization in the Bures-Wasserstein space has been gaining popularity in\nthe machine learning community since it draws connections between variational\ninference and Wasserstein gradient flows. The variational inference objective\nfunction of Kullback-Leibler divergence can be written as the sum of the\nnegative entropy and the potential energy, making forward-backward Euler the\nmethod of choice. Notably, the backward step admits a closed-form solution in\nthis case, facilitating the practicality of the scheme. However, the forward\nstep is not exact since the Bures-Wasserstein gradient of the potential energy\ninvolves \"intractable\" expectations. Recent approaches propose using the Monte\nCarlo method -- in practice a single-sample estimator -- to approximate these\nterms, resulting in high variance and poor performance. We propose a novel\nvariance-reduced estimator based on the principle of control variates. We\ntheoretically show that this estimator has a smaller variance than the\nMonte-Carlo estimator in scenarios of interest. We also prove that variance\nreduction helps improve the optimization bounds of the current analysis. We\ndemonstrate that the proposed estimator gains order-of-magnitude improvements\nover the previous Bures-Wasserstein methods.\n", "date": "2024-10-03", "categories": []}
{"id": "2502.21143", "title": "Variational Bayesian Pseudo-Coreset", "abstract": "  The success of deep learning requires large datasets and extensive training,\nwhich can create significant computational challenges. To address these\nchallenges, pseudo-coresets, small learnable datasets that mimic the entire\ndata, have been proposed. Bayesian Neural Networks, which offer predictive\nuncertainty and probabilistic interpretation for deep neural networks, also\nface issues with large-scale datasets due to their high-dimensional parameter\nspace. Prior works on Bayesian Pseudo-Coresets (BPC) attempt to reduce the\ncomputational load for computing weight posterior distribution by a small\nnumber of pseudo-coresets but suffer from memory inefficiency during BPC\ntraining and sub-optimal results. To overcome these limitations, we propose\nVariational Bayesian Pseudo-Coreset (VBPC), a novel approach that utilizes\nvariational inference to efficiently approximate the posterior distribution,\nreducing memory usage and computational costs while improving performance\nacross benchmark datasets.\n", "date": "2025-02-28", "categories": []}
{"id": "2502.18279", "title": "Near-Optimal Approximations for Bayesian Inference in Function Space", "abstract": "  We propose a scalable inference algorithm for Bayes posteriors defined on a\nreproducing kernel Hilbert space (RKHS). Given a likelihood function and a\nGaussian random element representing the prior, the corresponding Bayes\nposterior measure $\\Pi_{\\text{B}}$ can be obtained as the stationary\ndistribution of an RKHS-valued Langevin diffusion. We approximate the\ninfinite-dimensional Langevin diffusion via a projection onto the first $M$\ncomponents of the Kosambi-Karhunen-Lo\\`eve expansion. Exploiting the thus\nobtained approximate posterior for these $M$ components, we perform inference\nfor $\\Pi_{\\text{B}}$ by relying on the law of total probability and a\nsufficiency assumption. The resulting method scales as $O(M^3+JM^2)$, where $J$\nis the number of samples produced from the posterior measure $\\Pi_{\\text{B}}$.\nInterestingly, the algorithm recovers the posterior arising from the sparse\nvariational Gaussian process (SVGP) (see Titsias, 2009) as a special case, owed\nto the fact that the sufficiency assumption underlies both methods. However,\nwhereas the SVGP is parametrically constrained to be a Gaussian process, our\nmethod is based on a non-parametric variational family\n$\\mathcal{P}(\\mathbb{R}^M)$ consisting of all probability measures on\n$\\mathbb{R}^M$. As a result, our method is provably close to the optimal\n$M$-dimensional variational approximation of the Bayes posterior\n$\\Pi_{\\text{B}}$ in $\\mathcal{P}(\\mathbb{R}^M)$ for convex and Lipschitz\ncontinuous negative log likelihoods, and coincides with SVGP for the special\ncase of a Gaussian error likelihood.\n", "date": "2025-02-25", "categories": []}
{"id": "2502.14298", "title": "Generalization Certificates for Adversarially Robust Bayesian Linear\n  Regression", "abstract": "  Adversarial robustness of machine learning models is critical to ensuring\nreliable performance under data perturbations. Recent progress has been on\npoint estimators, and this paper considers distributional predictors. First,\nusing the link between exponential families and Bregman divergences, we\nformulate an adversarial Bregman divergence loss as an adversarial negative\nlog-likelihood. Using the geometric properties of Bregman divergences, we\ncompute the adversarial perturbation for such models in closed-form. Second,\nunder such losses, we introduce \\emph{adversarially robust posteriors}, by\nexploiting the optimization-centric view of generalized Bayesian inference.\nThird, we derive the \\emph{first} rigorous generalization certificates in the\ncontext of an adversarial extension of Bayesian linear regression by leveraging\nthe PAC-Bayesian framework. Finally, experiments on real and synthetic datasets\ndemonstrate the superior robustness of the derived adversarially robust\nposterior over Bayes posterior, and also validate our theoretical guarantees.\n", "date": "2025-02-20", "categories": []}
{"id": "2502.13827", "title": "Bayesian Physics Informed Neural Networks for Linear Inverse problems", "abstract": "  Inverse problems arise almost everywhere in science and engineering where we\nneed to infer on a quantity from indirect observation. The cases of medical,\nbiomedical, and industrial imaging systems are the typical examples. A very\nhigh overview of classification of the inverse problems method can be: i)\nAnalytical, ii) Regularization, and iii) Bayesian inference methods. Even if\nthere are straight links between them, we can say that the Bayesian inference\nbased methods are the most powerful, as they give the possibility of accounting\nfor prior knowledge and can account for errors and uncertainties in general.\nOne of the main limitations stay in computational costs in particular for high\ndimensional imaging systems. Neural Networks (NN), and in particular Deep NNs\n(DNN), have been considered as a way to push farther this limit. Physics\nInformed Neural Networks (PINN) concept integrates physical laws with deep\nlearning techniques to enhance the speed, accuracy and efficiency of the above\nmentioned problems.\n  In this work, a new Bayesian framework for the concept of PINN (BPINN) is\npresented and discussed which includes the deterministic one if we use the\nMaximum A Posteriori (MAP) estimation framework. We consider two cases of\nsupervised and unsupervised for training step, obtain the expressions of the\nposterior probability of the unknown variables, and deduce the posterior laws\nof the NN's parameters. We also discuss about the challenges of implementation\nof these methods in real applications.\n", "date": "2025-02-18", "categories": []}
{"id": "2407.20891", "title": "Bayesian Low-Rank LeArning (Bella): A Practical Approach to Bayesian\n  Neural Networks", "abstract": "  Computational complexity of Bayesian learning is impeding its adoption in\npractical, large-scale tasks. Despite demonstrations of significant merits such\nas improved robustness and resilience to unseen or out-of-distribution inputs\nover their non- Bayesian counterparts, their practical use has faded to near\ninsignificance. In this study, we introduce an innovative framework to mitigate\nthe computational burden of Bayesian neural networks (BNNs). Our approach\nfollows the principle of Bayesian techniques based on deep ensembles, but\nsignificantly reduces their cost via multiple low-rank perturbations of\nparameters arising from a pre-trained neural network. Both vanilla version of\nensembles as well as more sophisticated schemes such as Bayesian learning with\nStein Variational Gradient Descent (SVGD), previously deemed impractical for\nlarge models, can be seamlessly implemented within the proposed framework,\ncalled Bayesian Low-Rank LeArning (Bella). In a nutshell, i) Bella achieves a\ndramatic reduction in the number of trainable parameters required to\napproximate a Bayesian posterior; and ii) it not only maintains, but in some\ninstances, surpasses the performance of conventional Bayesian learning methods\nand non-Bayesian baselines. Our results with large-scale tasks such as\nImageNet, CAMELYON17, DomainNet, VQA with CLIP, LLaVA demonstrate the\neffectiveness and versatility of Bella in building highly scalable and\npractical Bayesian deep models for real-world applications.\n", "date": "2024-07-30", "categories": []}
{"id": "2402.10727", "title": "From Risk to Uncertainty: Generating Predictive Uncertainty Measures via\n  Bayesian Estimation", "abstract": "  There are various measures of predictive uncertainty in the literature, but\ntheir relationships to each other remain unclear. This paper uses a\ndecomposition of statistical pointwise risk into components, associated with\ndifferent sources of predictive uncertainty, namely aleatoric uncertainty\n(inherent data variability) and epistemic uncertainty (model-related\nuncertainty). Together with Bayesian methods, applied as an approximation, we\nbuild a framework that allows one to generate different predictive uncertainty\nmeasures.\n  We validate our method on image datasets by evaluating its performance in\ndetecting out-of-distribution and misclassified instances using the AUROC\nmetric. The experimental results confirm that the measures derived from our\nframework are useful for the considered downstream tasks.\n", "date": "2024-02-16", "categories": []}
{"id": "2406.03334", "title": "Reparameterization invariance in approximate Bayesian inference", "abstract": "  Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a\ncrucial limitation: they fail to maintain invariance under reparameterization,\ni.e. BNNs assign different posterior densities to different parametrizations of\nidentical functions. This creates a fundamental flaw in the application of\nBayesian principles as it breaks the correspondence between uncertainty over\nthe parameters with uncertainty over the parametrized function. In this paper,\nwe investigate this issue in the context of the increasingly popular linearized\nLaplace approximation. Specifically, it has been observed that linearized\npredictives alleviate the common underfitting problems of the Laplace\napproximation. We develop a new geometric view of reparametrizations from which\nwe explain the success of linearization. Moreover, we demonstrate that these\nreparameterization invariance properties can be extended to the original neural\nnetwork predictive using a Riemannian diffusion process giving a\nstraightforward algorithm for approximate posterior sampling, which empirically\nimproves posterior fit.\n", "date": "2024-06-05", "categories": []}
{"id": "2407.17667", "title": "Tackling the Problem of Distributional Shifts: Correcting Misspecified,\n  High-Dimensional Data-Driven Priors for Inverse Problems", "abstract": "  Bayesian inference for inverse problems hinges critically on the choice of\npriors. In the absence of specific prior information, population-level\ndistributions can serve as effective priors for parameters of interest. With\nthe advent of machine learning, the use of data-driven population-level\ndistributions (encoded, e.g., in a trained deep neural network) as priors is\nemerging as an appealing alternative to simple parametric priors in a variety\nof inverse problems. However, in many astrophysical applications, it is often\ndifficult or even impossible to acquire independent and identically distributed\nsamples from the underlying data-generating process of interest to train these\nmodels. In these cases, corrupted data or a surrogate, e.g. a simulator, is\noften used to produce training samples, meaning that there is a risk of\nobtaining misspecified priors. This, in turn, can bias the inferred posteriors\nin ways that are difficult to quantify, which limits the potential\napplicability of these models in real-world scenarios. In this work, we propose\naddressing this issue by iteratively updating the population-level\ndistributions by retraining the model with posterior samples from different\nsets of observations, and we showcase the potential of this method on the\nproblem of background image reconstruction in strong gravitational lensing when\nscore-based models are used as data-driven priors. We show that, starting from\na misspecified prior distribution, the updated distribution becomes\nprogressively closer to the underlying population-level distribution, and the\nresulting posterior samples exhibit reduced bias after several updates.\n", "date": "2024-07-24", "categories": []}
{"id": "2502.06335", "title": "Microcanonical Langevin Ensembles: Advancing the Sampling of Bayesian\n  Neural Networks", "abstract": "  Despite recent advances, sampling-based inference for Bayesian Neural\nNetworks (BNNs) remains a significant challenge in probabilistic deep learning.\nWhile sampling-based approaches do not require a variational distribution\nassumption, current state-of-the-art samplers still struggle to navigate the\ncomplex and highly multimodal posteriors of BNNs. As a consequence, sampling\nstill requires considerably longer inference times than non-Bayesian methods\neven for small neural networks, despite recent advances in making software\nimplementations more efficient. Besides the difficulty of finding\nhigh-probability regions, the time until samplers provide sufficient\nexploration of these areas remains unpredictable. To tackle these challenges,\nwe introduce an ensembling approach that leverages strategies from optimization\nand a recently proposed sampler called Microcanonical Langevin Monte Carlo\n(MCLMC) for efficient, robust and predictable sampling performance. Compared to\napproaches based on the state-of-the-art No-U-Turn Sampler, our approach\ndelivers substantial speedups up to an order of magnitude, while maintaining or\nimproving predictive performance and uncertainty quantification across diverse\ntasks and data modalities. The suggested Microcanonical Langevin Ensembles and\nmodifications to MCLMC additionally enhance the method's predictability in\nresource requirements, facilitating easier parallelization. All in all, the\nproposed method offers a promising direction for practical, scalable inference\nfor BNNs.\n", "date": "2025-02-10", "categories": []}
{"id": "2502.06601", "title": "Amortized In-Context Bayesian Posterior Estimation", "abstract": "  Bayesian inference provides a natural way of incorporating prior beliefs and\nassigning a probability measure to the space of hypotheses. Current solutions\nrely on iterative routines like Markov Chain Monte Carlo (MCMC) sampling and\nVariational Inference (VI), which need to be re-run whenever new observations\nare available. Amortization, through conditional estimation, is a viable\nstrategy to alleviate such difficulties and has been the guiding principle\nbehind simulation-based inference, neural processes and in-context methods\nusing pre-trained models. In this work, we conduct a thorough comparative\nanalysis of amortized in-context Bayesian posterior estimation methods from the\nlens of different optimization objectives and architectural choices. Such\nmethods train an amortized estimator to perform posterior parameter inference\nby conditioning on a set of data examples passed as context to a sequence model\nsuch as a transformer. In contrast to language models, we leverage permutation\ninvariant architectures as the true posterior is invariant to the ordering of\ncontext examples. Our empirical study includes generalization to\nout-of-distribution tasks, cases where the assumed underlying model is\nmisspecified, and transfer from simulated to real problems. Subsequently, it\nhighlights the superiority of the reverse KL estimator for predictive problems,\nespecially when combined with the transformer architecture and normalizing\nflows.\n", "date": "2025-02-10", "categories": []}
{"id": "2408.03733", "title": "Bayes-optimal learning of an extensive-width neural network from\n  quadratically many samples", "abstract": "  We consider the problem of learning a target function corresponding to a\nsingle hidden layer neural network, with a quadratic activation function after\nthe first layer, and random weights. We consider the asymptotic limit where the\ninput dimension and the network width are proportionally large. Recent work\n[Cui & al '23] established that linear regression provides Bayes-optimal test\nerror to learn such a function when the number of available samples is only\nlinear in the dimension. That work stressed the open challenge of theoretically\nanalyzing the optimal test error in the more interesting regime where the\nnumber of samples is quadratic in the dimension. In this paper, we solve this\nchallenge for quadratic activations and derive a closed-form expression for the\nBayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE,\nwhich combines approximate message passing with rotationally invariant matrix\ndenoising, and that asymptotically achieves the optimal performance.\nTechnically, our result is enabled by establishing a link with recent works on\noptimal denoising of extensive-rank matrices and on the ellipsoid fitting\nproblem. We further show empirically that, in the absence of noise,\nrandomly-initialized gradient descent seems to sample the space of weights,\nleading to zero training loss, and averaging over initialization leads to a\ntest error equal to the Bayes-optimal one.\n", "date": "2024-08-07", "categories": []}
{"id": "2502.04247", "title": "Student-t processes as infinite-width limits of posterior Bayesian\n  neural networks", "abstract": "  The asymptotic properties of Bayesian Neural Networks (BNNs) have been\nextensively studied, particularly regarding their approximations by Gaussian\nprocesses in the infinite-width limit. We extend these results by showing that\nposterior BNNs can be approximated by Student-t processes, which offer greater\nflexibility in modeling uncertainty. Specifically, we show that, if the\nparameters of a BNN follow a Gaussian prior distribution, and the variance of\nboth the last hidden layer and the Gaussian likelihood function follows an\nInverse-Gamma prior distribution, then the resulting posterior BNN converges to\na Student-t process in the infinite-width limit. Our proof leverages the\nWasserstein metric to establish control over the convergence rate of the\nStudent-t process approximation.\n", "date": "2025-02-06", "categories": []}
{"id": "2502.02032", "title": "Heteroscedastic Double Bayesian Elastic Net", "abstract": "  In many practical applications, regression models are employed to uncover\nrelationships between predictors and a response variable, yet the common\nassumption of constant error variance is frequently violated. This issue is\nfurther compounded in high-dimensional settings where the number of predictors\nexceeds the sample size, necessitating regularization for effective estimation\nand variable selection. To address this problem, we propose the Heteroscedastic\nDouble Bayesian Elastic Net (HDBEN), a novel framework that jointly models the\nmean and log-variance using hierarchical Bayesian priors incorporating both\n$\\ell_1$ and $\\ell_2$ penalties. Our approach simultaneously induces sparsity\nand grouping in the regression coefficients and variance parameters, capturing\ncomplex variance structures in the data. Theoretical results demonstrate that\nproposed HDBEN achieves posterior concentration, variable selection\nconsistency, and asymptotic normality under mild conditions which justifying\nits behavior. Simulation studies further illustrate that HDBEN outperforms\nexisting methods, particularly in scenarios characterized by heteroscedasticity\nand high dimensionality.\n", "date": "2025-02-04", "categories": []}
{"id": "2502.02552", "title": "Hierarchical Sparse Bayesian Multitask Model with Scalable Inference for\n  Microbiome Analysis", "abstract": "  This paper proposes a hierarchical Bayesian multitask learning model that is\napplicable to the general multi-task binary classification learning problem\nwhere the model assumes a shared sparsity structure across different tasks. We\nderive a computationally efficient inference algorithm based on variational\ninference to approximate the posterior distribution. We demonstrate the\npotential of the new approach on various synthetic datasets and for predicting\nhuman health status based on microbiome profile. Our analysis incorporates data\npooled from multiple microbiome studies, along with a comprehensive comparison\nwith other benchmark methods. Results in synthetic datasets show that the\nproposed approach has superior support recovery property when the underlying\nregression coefficients share a common sparsity structure across different\ntasks. Our experiments on microbiome classification demonstrate the utility of\nthe method in extracting informative taxa while providing well-calibrated\npredictions with uncertainty quantification and achieving competitive\nperformance in terms of prediction metrics. Notably, despite the heterogeneity\nof the pooled datasets (e.g., different experimental objectives, laboratory\nsetups, sequencing equipment, patient demographics), our method delivers robust\nresults.\n", "date": "2025-02-04", "categories": []}
{"id": "2502.01527", "title": "Enhancing Bayesian Network Structural Learning with Monte Carlo Tree\n  Search", "abstract": "  This article presents MCTS-BN, an adaptation of the Monte Carlo Tree Search\n(MCTS) algorithm for the structural learning of Bayesian Networks (BNs).\nInitially designed for game tree exploration, MCTS has been repurposed to\naddress the challenge of learning BN structures by exploring the search space\nof potential ancestral orders in Bayesian Networks. Then, it employs Hill\nClimbing (HC) to derive a Bayesian Network structure from each order. In large\nBNs, where the search space for variable orders becomes vast, using completely\nrandom orders during the rollout phase is often unreliable and impractical. We\nadopt a semi-randomized approach to address this challenge by incorporating\nvariable orders obtained from other heuristic search algorithms such as Greedy\nEquivalent Search (GES), PC, or HC itself. This hybrid strategy mitigates the\ncomputational burden and enhances the reliability of the rollout process.\nExperimental evaluations demonstrate the effectiveness of MCTS-BN in improving\nBNs generated by traditional structural learning algorithms, exhibiting robust\nperformance even when base algorithm orders are suboptimal and surpassing the\ngold standard when provided with favorable orders.\n", "date": "2025-02-03", "categories": []}
{"id": "2501.17917", "title": "Deep Ensembles Secretly Perform Empirical Bayes", "abstract": "  Quantifying uncertainty in neural networks is a highly relevant problem which\nis essential to many applications. The two predominant paradigms to tackle this\ntask are Bayesian neural networks (BNNs) and deep ensembles. Despite some\nsimilarities between these two approaches, they are typically surmised to lack\na formal connection and are thus understood as fundamentally different. BNNs\nare often touted as more principled due to their reliance on the Bayesian\nparadigm, whereas ensembles are perceived as more ad-hoc; yet, deep ensembles\ntend to empirically outperform BNNs, with no satisfying explanation as to why\nthis is the case. In this work we bridge this gap by showing that deep\nensembles perform exact Bayesian averaging with a posterior obtained with an\nimplicitly learned data-dependent prior. In other words deep ensembles are\nBayesian, or more specifically, they implement an empirical Bayes procedure\nwherein the prior is learned from the data. This perspective offers two main\nbenefits: (i) it theoretically justifies deep ensembles and thus provides an\nexplanation for their strong empirical performance; and (ii) inspection of the\nlearned prior reveals it is given by a mixture of point masses -- the use of\nsuch a strong prior helps elucidate observed phenomena about ensembles.\nOverall, our work delivers a newfound understanding of deep ensembles which is\nnot only of interest in it of itself, but which is also likely to generate\nfuture insights that drive empirical improvements for these models.\n", "date": "2025-01-29", "categories": []}
{"id": "2406.11675", "title": "BLoB: Bayesian Low-Rank Adaptation by Backpropagation for Large Language\n  Models", "abstract": "  Large Language Models (LLMs) often suffer from overconfidence during\ninference, particularly when adapted to downstream domain-specific tasks with\nlimited data. Previous work addresses this issue by employing approximate\nBayesian estimation after the LLMs are trained, enabling them to quantify\nuncertainty. However, such post-training approaches' performance is severely\nlimited by the parameters learned during training. In this paper, we go beyond\npost-training Bayesianization and propose Bayesian Low-Rank Adaptation by\nBackpropagation (BLoB), an algorithm that continuously and jointly adjusts both\nthe mean and covariance of LLM parameters throughout the whole fine-tuning\nprocess. Our empirical results verify the effectiveness of BLoB in terms of\ngeneralization and uncertainty estimation, when evaluated on both\nin-distribution and out-of-distribution data.\n", "date": "2024-06-17", "categories": []}
{"id": "2501.15573", "title": "Approximate Message Passing for Bayesian Neural Networks", "abstract": "  Bayesian neural networks (BNNs) offer the potential for reliable uncertainty\nquantification and interpretability, which are critical for trustworthy AI in\nhigh-stakes domains. However, existing methods often struggle with issues such\nas overconfidence, hyperparameter sensitivity, and posterior collapse, leaving\nroom for alternative approaches. In this work, we advance message passing (MP)\nfor BNNs and present a novel framework that models the predictive posterior as\na factor graph. To the best of our knowledge, our framework is the first MP\nmethod that handles convolutional neural networks and avoids double-counting\ntraining data, a limitation of previous MP methods that causes overconfidence.\nWe evaluate our approach on CIFAR-10 with a convolutional neural network of\nroughly 890k parameters and find that it can compete with the SOTA baselines\nAdamW and IVON, even having an edge in terms of calibration. On synthetic data,\nwe validate the uncertainty estimates and observe a strong correlation (0.9)\nbetween posterior credible intervals and its probability of covering the true\ndata-generating function outside the training range. While our method scales to\nan MLP with 5.6 million parameters, further improvements are necessary to match\nthe scale and performance of state-of-the-art variational inference methods.\n", "date": "2025-01-26", "categories": []}
{"id": "2410.22948", "title": "ELBOing Stein: Variational Bayes with Stein Mixture Inference", "abstract": "  Stein variational gradient descent (SVGD) [Liu and Wang, 2016] performs\napproximate Bayesian inference by representing the posterior with a set of\nparticles. However, SVGD suffers from variance collapse, i.e. poor predictions\ndue to underestimating uncertainty [Ba et al., 2021], even for\nmoderately-dimensional models such as small Bayesian neural networks (BNNs). To\naddress this issue, we generalize SVGD by letting each particle parameterize a\ncomponent distribution in a mixture model. Our method, Stein Mixture Inference\n(SMI), optimizes a lower bound to the evidence (ELBO) and introduces\nuser-specified guides parameterized by particles. SMI extends the Nonlinear\nSVGD framework [Wang and Liu, 2019] to the case of variational Bayes. SMI\neffectively avoids variance collapse, judging by a previously described test\ndeveloped for this purpose, and performs well on standard data sets. In\naddition, SMI requires considerably fewer particles than SVGD to accurately\nestimate uncertainty for small BNNs. The synergistic combination of NSVGD, ELBO\noptimization and user-specified guides establishes a promising approach towards\nvariational Bayesian inference in the case of tall and wide data.\n", "date": "2024-10-30", "categories": []}
{"id": "2404.08969", "title": "Concentration properties of fractional posterior in 1-bit matrix\n  completion", "abstract": "  The problem of estimating a matrix based on a set of its observed entries is\ncommonly referred to as the matrix completion problem. In this work, we\nspecifically address the scenario of binary observations, often termed as 1-bit\nmatrix completion. While numerous studies have explored Bayesian and\nfrequentist methods for real-value matrix completion, there has been a lack of\ntheoretical exploration regarding Bayesian approaches in 1-bit matrix\ncompletion. We tackle this gap by considering a general, non-uniform sampling\nscheme and providing theoretical assurances on the efficacy of the fractional\nposterior. Our contributions include obtaining concentration results for the\nfractional posterior and demonstrating its effectiveness in recovering the\nunderlying parameter matrix. We accomplish this using two distinct types of\nprior distributions: low-rank factorization priors and a spectral scaled\nStudent prior, with the latter requiring fewer assumptions. Importantly, our\nresults exhibit an adaptive nature by not mandating prior knowledge of the rank\nof the parameter matrix. Our findings are comparable to those found in the\nfrequentist literature, yet demand fewer restrictive assumptions.\n", "date": "2024-04-13", "categories": []}
{"id": "2501.11773", "title": "Can Bayesian Neural Networks Make Confident Predictions?", "abstract": "  Bayesian inference promises a framework for principled uncertainty\nquantification of neural network predictions. Barriers to adoption include the\ndifficulty of fully characterizing posterior distributions on network\nparameters and the interpretability of posterior predictive distributions. We\ndemonstrate that under a discretized prior for the inner layer weights, we can\nexactly characterize the posterior predictive distribution as a Gaussian\nmixture. This setting allows us to define equivalence classes of network\nparameter values which produce the same likelihood (training error) and to\nrelate the elements of these classes to the network's scaling regime -- defined\nvia ratios of the training sample size, the size of each layer, and the number\nof final layer parameters. Of particular interest are distinct parameter\nrealizations that map to low training error and yet correspond to distinct\nmodes in the posterior predictive distribution. We identify settings that\nexhibit such predictive multimodality, and thus provide insight into the\naccuracy of unimodal posterior approximations. We also characterize the\ncapacity of a model to \"learn from data\" by evaluating contraction of the\nposterior predictive in different scaling regimes.\n", "date": "2025-01-20", "categories": []}
{"id": "2405.14440", "title": "Bayesian Adaptive Calibration and Optimal Design", "abstract": "  The process of calibrating computer models of natural phenomena is essential\nfor applications in the physical sciences, where plenty of domain knowledge can\nbe embedded into simulations and then calibrated against real observations.\nCurrent machine learning approaches, however, mostly rely on rerunning\nsimulations over a fixed set of designs available in the observed data,\npotentially neglecting informative correlations across the design space and\nrequiring a large amount of simulations. Instead, we consider the calibration\nprocess from the perspective of Bayesian adaptive experimental design and\npropose a data-efficient algorithm to run maximally informative simulations\nwithin a batch-sequential process. At each round, the algorithm jointly\nestimates the parameters of the posterior distribution and optimal designs by\nmaximising a variational lower bound of the expected information gain. The\nsimulator is modelled as a sample from a Gaussian process, which allows us to\ncorrelate simulations and observed data with the unknown calibration\nparameters. We show the benefits of our method when compared to related\napproaches across synthetic and real-data problems.\n", "date": "2024-05-23", "categories": []}
{"id": "2501.09900", "title": "SBAMDT: Bayesian Additive Decision Trees with Adaptive Soft\n  Semi-multivariate Split Rules", "abstract": "  Bayesian Additive Regression Trees [BART, Chipman et al., 2010] have gained\nsignificant popularity due to their remarkable predictive performance and\nability to quantify uncertainty. However, standard decision tree models rely on\nrecursive data splits at each decision node, using deterministic decision rules\nbased on a single univariate feature. This approach limits their ability to\neffectively capture complex decision boundaries, particularly in scenarios\ninvolving multiple features, such as spatial domains, or when transitions are\neither sharp or smoothly varying. In this paper, we introduce a novel\nprobabilistic additive decision tree model that employs a soft split rule. This\nmethod enables highly flexible splits that leverage both univariate and\nmultivariate features, while also respecting the geometric properties of the\nfeature domain. Notably, the probabilistic split rule adapts dynamically across\ndecision nodes, allowing the model to account for varying levels of smoothness\nin the regression function. We demonstrate the utility of the proposed model\nthrough comparisons with existing tree-based models on synthetic datasets and a\nNew York City education dataset.\n", "date": "2025-01-17", "categories": []}
{"id": "2501.08201", "title": "Globally Convergent Variational Inference", "abstract": "  In variational inference (VI), an approximation of the posterior distribution\nis selected from a family of distributions through numerical optimization. With\nthe most common variational objective function, known as the evidence lower\nbound (ELBO), only convergence to a local optimum can be guaranteed. In this\nwork, we instead establish the global convergence of a particular VI method.\nThis VI method, which may be considered an instance of neural posterior\nestimation (NPE), minimizes an expectation of the inclusive (forward) KL\ndivergence to fit a variational distribution that is parameterized by a neural\nnetwork. Our convergence result relies on the neural tangent kernel (NTK) to\ncharacterize the gradient dynamics that arise from considering the variational\nobjective in function space. In the asymptotic regime of a fixed,\npositive-definite neural tangent kernel, we establish conditions under which\nthe variational objective admits a unique solution in a reproducing kernel\nHilbert space (RKHS). Then, we show that the gradient descent dynamics in\nfunction space converge to this unique function. In ablation studies and\npractical problems, we demonstrate that our results explain the behavior of NPE\nin non-asymptotic finite-neuron settings, and show that NPE outperforms\nELBO-based optimization, which often converges to shallow local optima.\n", "date": "2025-01-14", "categories": []}
{"id": "2501.08223", "title": "Big Batch Bayesian Active Learning by Considering Predictive\n  Probabilities", "abstract": "  We observe that BatchBALD, a popular acquisition function for batch Bayesian\nactive learning for classification, can conflate epistemic and aleatoric\nuncertainty, leading to suboptimal performance. Motivated by this observation,\nwe propose to focus on the predictive probabilities, which only exhibit\nepistemic uncertainty. The result is an acquisition function that not only\nperforms better, but is also faster to evaluate, allowing for larger batches\nthan before.\n", "date": "2025-01-14", "categories": []}
{"id": "2501.08285", "title": "Can Bayesian Neural Networks Explicitly Model Input Uncertainty?", "abstract": "  Inputs to machine learning models can have associated noise or uncertainties,\nbut they are often ignored and not modelled. It is unknown if Bayesian Neural\nNetworks and their approximations are able to consider uncertainty in their\ninputs. In this paper we build a two input Bayesian Neural Network (mean and\nstandard deviation) and evaluate its capabilities for input uncertainty\nestimation across different methods like Ensembles, MC-Dropout, and Flipout.\nOur results indicate that only some uncertainty estimation methods for\napproximate Bayesian NNs can model input uncertainty, in particular Ensembles\nand Flipout.\n", "date": "2025-01-14", "categories": []}
{"id": "2501.06962", "title": "Compact Bayesian Neural Networks via pruned MCMC sampling", "abstract": "  Bayesian Neural Networks (BNNs) offer robust uncertainty quantification in\nmodel predictions, but training them presents a significant computational\nchallenge. This is mainly due to the problem of sampling multimodal posterior\ndistributions using Markov Chain Monte Carlo (MCMC) sampling and variational\ninference algorithms. Moreover, the number of model parameters scales\nexponentially with additional hidden layers, neurons, and features in the\ndataset. Typically, a significant portion of these densely connected parameters\nare redundant and pruning a neural network not only improves portability but\nalso has the potential for better generalisation capabilities. In this study,\nwe address some of the challenges by leveraging MCMC sampling with network\npruning to obtain compact probabilistic models having removed redundant\nparameters. We sample the posterior distribution of model parameters (weights\nand biases) and prune weights with low importance, resulting in a compact\nmodel. We ensure that the compact BNN retains its ability to estimate\nuncertainty via the posterior distribution while retaining the model training\nand generalisation performance accuracy by adapting post-pruning resampling. We\nevaluate the effectiveness of our MCMC pruning strategy on selected benchmark\ndatasets for regression and classification problems through empirical result\nanalysis. We also consider two coral reef drill-core lithology classification\ndatasets to test the robustness of the pruning model in complex real-world\ndatasets. We further investigate if refining compact BNN can retain any loss of\nperformance. Our results demonstrate the feasibility of training and pruning\nBNNs using MCMC whilst retaining generalisation performance with over 75%\nreduction in network size. This paves the way for developing compact BNN models\nthat provide uncertainty estimates for real-world applications.\n", "date": "2025-01-12", "categories": []}
{"id": "2501.05564", "title": "Analog Bayesian neural networks are insensitive to the shape of the\n  weight distribution", "abstract": "  Recent work has demonstrated that Bayesian neural networks (BNN's) trained\nwith mean field variational inference (MFVI) can be implemented in analog\nhardware, promising orders of magnitude energy savings compared to the standard\ndigital implementations. However, while Gaussians are typically used as the\nvariational distribution in MFVI, it is difficult to precisely control the\nshape of the noise distributions produced by sampling analog devices. This\npaper introduces a method for MFVI training using real device noise as the\nvariational distribution. Furthermore, we demonstrate empirically that the\npredictive distributions from BNN's with the same weight means and variances\nconverge to the same distribution, regardless of the shape of the variational\ndistribution. This result suggests that analog device designers do not need to\nconsider the shape of the device noise distribution when hardware-implementing\nBNNs performing MFVI.\n", "date": "2025-01-09", "categories": []}
{"id": "2501.02019", "title": "Benchmarking Constraint-Based Bayesian Structure Learning Algorithms:\n  Role of Network Topology", "abstract": "  Modeling the associations between real world entities from their multivariate\ncross-sectional profiles can provide cues into the concerted working of these\nentities as a system. Several techniques have been proposed for deciphering\nthese associations including constraint-based Bayesian structure learning (BSL)\nalgorithms that model them as directed acyclic graphs. Benchmarking these\nalgorithms have typically focused on assessing the variation in performance\nmeasures such as sensitivity as a function of the dimensionality represented by\nthe number of nodes in the DAG, and sample size. The present study elucidates\nthe importance of network topology in benchmarking exercises. More\nspecifically, it investigates variations in sensitivity across distinct network\ntopologies while constraining the nodes, edges, and sample-size to be\nidentical, eliminating these as potential confounders. Sensitivity of three\npopular constraint-based BSL algorithms (Peter-Clarke, Grow-Shrink, Incremental\nAssociation Markov Blanket) in learning the network structure from multivariate\ncross-sectional profiles sampled from network models with sub-linear, linear,\nand super-linear DAG topologies generated using preferential attachment is\ninvestigated. Results across linear and nonlinear models revealed statistically\nsignificant $(\\alpha=0.05)$ decrease in sensitivity estimates from sub-linear\nto super-linear topology constitutively across the three algorithms. These\nresults are demonstrated on networks with nodes $(N_{nods}=48,64)$, noise\nstrengths $(\\sigma =3,6)$ and sample size $(N = 2^{10})$. The findings\nelucidate the importance of accommodating the network topology in\nconstraint-based BSL benchmarking exercises.\n", "date": "2025-01-02", "categories": []}
{"id": "2407.08843", "title": "Inflationary Flows: Calibrated Bayesian Inference with Diffusion-Based\n  Models", "abstract": "  Beyond estimating parameters of interest from data, one of the key goals of\nstatistical inference is to properly quantify uncertainty in these estimates.\nIn Bayesian inference, this uncertainty is provided by the posterior\ndistribution, the computation of which typically involves an intractable\nhigh-dimensional integral. Among available approximation methods,\nsampling-based approaches come with strong theoretical guarantees but scale\npoorly to large problems, while variational approaches scale well but offer few\ntheoretical guarantees. In particular, variational methods are known to produce\noverconfident estimates of posterior uncertainty and are typically\nnon-identifiable, with many latent variable configurations generating\nequivalent predictions. Here, we address these challenges by showing how\ndiffusion-based models (DBMs), which have recently produced state-of-the-art\nperformance in generative modeling tasks, can be repurposed for performing\ncalibrated, identifiable Bayesian inference. By exploiting a previously\nestablished connection between the stochastic and probability flow ordinary\ndifferential equations (pfODEs) underlying DBMs, we derive a class of models,\ninflationary flows, that uniquely and deterministically map high-dimensional\ndata to a lower-dimensional Gaussian distribution via ODE integration. This map\nis both invertible and neighborhood-preserving, with controllable numerical\nerror, with the result that uncertainties in the data are correctly propagated\nto the latent space. We demonstrate how such maps can be learned via standard\nDBM training using a novel noise schedule and are effective at both preserving\nand reducing intrinsic data dimensionality. The result is a class of highly\nexpressive generative models, uniquely defined on a low-dimensional latent\nspace, that afford principled Bayesian inference.\n", "date": "2024-07-11", "categories": []}
{"id": "2409.05234", "title": "Empowering Bayesian Neural Networks with Functional Priors through\n  Anchored Ensembling for Mechanics Surrogate Modeling Applications", "abstract": "  In recent years, neural networks (NNs) have become increasingly popular for\nsurrogate modeling tasks in mechanics and materials modeling applications.\nWhile traditional NNs are deterministic functions that rely solely on data to\nlearn the input--output mapping, casting NN training within a Bayesian\nframework allows to quantify uncertainties, in particular epistemic\nuncertainties that arise from lack of training data, and to integrate a priori\nknowledge via the Bayesian prior. However, the high dimensionality and\nnon-physicality of the NN parameter space, and the complex relationship between\nparameters (NN weights) and predicted outputs, renders both prior design and\nposterior inference challenging. In this work we present a novel BNN training\nscheme based on anchored ensembling that can integrate a priori information\navailable in the function space, from e.g. low-fidelity models. The anchoring\nscheme makes use of low-rank correlations between NN parameters, learnt from\npre-training to realizations of the functional prior. We also perform a study\nto demonstrate how correlations between NN weights, which are often neglected\nin existing BNN implementations, is critical to appropriately transfer\nknowledge between the function-space and parameter-space priors. Performance of\nour novel BNN algorithm is first studied on a small 1D example to illustrate\nthe algorithm's behavior in both interpolation and extrapolation settings.\nThen, a thorough assessment is performed on a multi--input--output materials\nsurrogate modeling example, where we demonstrate the algorithm's capabilities\nboth in terms of accuracy and quality of the uncertainty estimation, for both\nin-distribution and out-of-distribution data.\n", "date": "2024-09-08", "categories": []}
{"id": "2310.12595", "title": "Bayesian Meta-Learning for Improving Generalizability of Health\n  Prediction Models With Similar Causal Mechanisms", "abstract": "  Machine learning strategies like multi-task learning, meta-learning, and\ntransfer learning enable efficient adaptation of machine learning models to\nspecific applications in healthcare, such as prediction of various diseases, by\nleveraging generalizable knowledge across large datasets and multiple domains.\nIn particular, Bayesian meta-learning methods pool data across related\nprediction tasks to learn prior distributions for model parameters, which are\nthen used to derive models for specific tasks. However, inter- and intra-task\nvariability due to disease heterogeneity and other patient-level differences\npose challenges of negative transfer during shared learning and poor\ngeneralizability to new patients. We introduce a novel Bayesian meta-learning\napproach that aims to address this in two key settings: (1) predictions for new\npatients (same population as the training set) and (2) adapting to new patient\npopulations. Our main contribution is in modeling similarity between causal\nmechanisms of the tasks, for (1) mitigating negative transfer during training\nand (2) fine-tuning that pools information from tasks that are expected to aid\ngeneralizability. We propose an algorithm for implementing this approach for\nBayesian deep learning, and apply it to a case study for stroke prediction\ntasks using electronic health record data. Experiments for the UK Biobank\ndataset as the training population demonstrated significant generalizability\nimprovements compared to standard meta-learning, non-causal task similarity\nmeasures, and local baselines (separate models for each task). This was\nassessed for a variety of tasks that considered both new patients from the\ntraining population (UK Biobank) and a new population (FinnGen).\n", "date": "2023-10-19", "categories": []}
{"id": "2412.10897", "title": "Task Diversity in Bayesian Federated Learning: Simultaneous Processing\n  of Classification and Regression", "abstract": "  This work addresses a key limitation in current federated learning\napproaches, which predominantly focus on homogeneous tasks, neglecting the task\ndiversity on local devices. We propose a principled integration of multi-task\nlearning using multi-output Gaussian processes (MOGP) at the local level and\nfederated learning at the global level. MOGP handles correlated classification\nand regression tasks, offering a Bayesian non-parametric approach that\nnaturally quantifies uncertainty. The central server aggregates the posteriors\nfrom local devices, updating a global MOGP prior redistributed for training\nlocal models until convergence. Challenges in performing posterior inference on\nlocal devices are addressed through the P\\'{o}lya-Gamma augmentation technique\nand mean-field variational inference, enhancing computational efficiency and\nconvergence rate. Experimental results on both synthetic and real data\ndemonstrate superior predictive performance, OOD detection, uncertainty\ncalibration and convergence rate, highlighting the method's potential in\ndiverse applications. Our code is publicly available at\nhttps://github.com/JunliangLv/task_diversity_BFL.\n", "date": "2024-12-14", "categories": []}
{"id": "2412.13286", "title": "Posterior Mean Matching: Generative Modeling through Online Bayesian\n  Inference", "abstract": "  This paper introduces posterior mean matching (PMM), a new method for\ngenerative modeling that is grounded in Bayesian inference. PMM uses conjugate\npairs of distributions to model complex data of various modalities like images\nand text, offering a flexible alternative to existing methods like diffusion\nmodels. PMM models iteratively refine noisy approximations of the target\ndistribution using updates from online Bayesian inference. PMM is flexible\nbecause its mechanics are based on general Bayesian models. We demonstrate this\nflexibility by developing specialized examples: a generative PMM model of\nreal-valued data using the Normal-Normal model, a generative PMM model of count\ndata using a Gamma-Poisson model, and a generative PMM model of discrete data\nusing a Dirichlet-Categorical model. For the Normal-Normal PMM model, we\nestablish a direct connection to diffusion models by showing that its\ncontinuous-time formulation converges to a stochastic differential equation\n(SDE). Additionally, for the Gamma-Poisson PMM, we derive a novel SDE driven by\na Cox process, which is a significant departure from traditional Brownian\nmotion-based generative models. PMMs achieve performance that is competitive\nwith generative models for language modeling and image generation.\n", "date": "2024-12-17", "categories": []}
{"id": "2412.15758", "title": "Function Space Diversity for Uncertainty Prediction via Repulsive\n  Last-Layer Ensembles", "abstract": "  Bayesian inference in function space has gained attention due to its\nrobustness against overparameterization in neural networks. However,\napproximating the infinite-dimensional function space introduces several\nchallenges. In this work, we discuss function space inference via particle\noptimization and present practical modifications that improve uncertainty\nestimation and, most importantly, make it applicable for large and pretrained\nnetworks. First, we demonstrate that the input samples, where particle\npredictions are enforced to be diverse, are detrimental to the model\nperformance. While diversity on training data itself can lead to underfitting,\nthe use of label-destroying data augmentation, or unlabeled out-of-distribution\ndata can improve prediction diversity and uncertainty estimates. Furthermore,\nwe take advantage of the function space formulation, which imposes no\nrestrictions on network parameterization other than sufficient flexibility.\nInstead of using full deep ensembles to represent particles, we propose a\nsingle multi-headed network that introduces a minimal increase in parameters\nand computation. This allows seamless integration to pretrained networks, where\nthis repulsive last-layer ensemble can be used for uncertainty aware\nfine-tuning at minimal additional cost. We achieve competitive results in\ndisentangling aleatoric and epistemic uncertainty for active learning,\ndetecting out-of-domain data, and providing calibrated uncertainty estimates\nunder distribution shifts with minimal compute and memory.\n", "date": "2024-12-20", "categories": []}
{"id": "2209.11366", "title": "Jensen-Shannon Divergence Based Novel Loss Functions for Bayesian Neural\n  Networks", "abstract": "  Bayesian neural networks (BNNs) are state-of-the-art machine learning methods\nthat can naturally regularize and systematically quantify uncertainties using\ntheir stochastic parameters. Kullback-Leibler (KL) divergence-based variational\ninference used in BNNs suffers from unstable optimization and challenges in\napproximating light-tailed posteriors due to the unbounded nature of the KL\ndivergence. To resolve these issues, we formulate a novel loss function for\nBNNs based on a new modification to the generalized Jensen-Shannon (JS)\ndivergence, which is bounded. In addition, we propose a Geometric JS\ndivergence-based loss, which is computationally efficient since it can be\nevaluated analytically. We found that the JS divergence-based variational\ninference is intractable, and hence employed a constrained optimization\nframework to formulate these losses. Our theoretical analysis and empirical\nexperiments on multiple regression and classification data sets suggest that\nthe proposed losses perform better than the KL divergence-based loss,\nespecially when the data sets are noisy or biased. Specifically, there are\napproximately 5% and 8% improvements in accuracy for a noise-added CIFAR-10\ndataset and a regression dataset, respectively. There is about a 13% reduction\nin false negative predictions of a biased histopathology dataset. In addition,\nwe quantify and compare the uncertainty metrics for the regression and\nclassification tasks.\n", "date": "2022-09-23", "categories": []}
{"id": "2312.14027", "title": "AdamMCMC: Combining Metropolis Adjusted Langevin with Momentum-based\n  Optimization", "abstract": "  Uncertainty estimation is a key issue when considering the application of\ndeep neural network methods in science and engineering. In this work, we\nintroduce a novel algorithm that quantifies epistemic uncertainty via Monte\nCarlo sampling from a tempered posterior distribution. It combines the well\nestablished Metropolis Adjusted Langevin Algorithm (MALA) with momentum-based\noptimization using Adam and leverages a prolate proposal distribution, to\nefficiently draw from the posterior. We prove that the constructed chain admits\nthe Gibbs posterior as invariant distribution and approximates this posterior\nin total variation distance. Furthermore, we demonstrate the efficiency of the\nresulting algorithm and the merit of the proposed changes on a state-of-the-art\nclassifier from high-energy particle physics.\n", "date": "2023-12-21", "categories": []}
{"id": "2404.12294", "title": "Bayesian evidence estimation from posterior samples with normalizing\n  flows", "abstract": "  We propose a novel method ($floZ$), based on normalizing flows, to estimate\nthe Bayesian evidence (and its numerical uncertainty) from a pre-existing set\nof samples drawn from the unnormalized posterior distribution. We validate it\non distributions whose evidence is known analytically, up to 15 parameter space\ndimensions, and compare with two state-of-the-art techniques for estimating the\nevidence: nested sampling (which computes the evidence as its main target) and\na $k$-nearest-neighbors technique that produces evidence estimates from\nposterior samples. Provided representative samples from the target posterior\nare available, our method is more robust to posterior distributions with sharp\nfeatures, especially in higher dimensions. For a simple multivariate Gaussian,\nwe demonstrate its accuracy for up to 200 dimensions with $10^5$ posterior\nsamples. $floZ$ has wide applicability, e.g., to estimate evidence from\nvariational inference, Markov Chain Monte Carlo samples, or any other method\nthat delivers samples and their likelihood from the unnormalized posterior\ndensity. As a physical application, we use $floZ$ to compute the Bayes factor\nfor the presence of the first overtone in the ringdown signal of the\ngravitational wave data of GW150914, finding good agreement with nested\nsampling.\n", "date": "2024-04-18", "categories": []}
{"id": "2412.03312", "title": "Path-Guided Particle-based Sampling", "abstract": "  Particle-based Bayesian inference methods by sampling from a partition-free\ntarget (posterior) distribution, e.g., Stein variational gradient descent\n(SVGD), have attracted significant attention. We propose a path-guided\nparticle-based sampling~(PGPS) method based on a novel Log-weighted Shrinkage\n(LwS) density path linking an initial distribution to the target distribution.\nWe propose to utilize a Neural network to learn a vector field motivated by the\nFokker-Planck equation of the designed density path. Particles, initiated from\nthe initial distribution, evolve according to the ordinary differential\nequation defined by the vector field. The distribution of these particles is\nguided along a density path from the initial distribution to the target\ndistribution. The proposed LwS density path allows for an efficient search of\nmodes of the target distribution while canonical methods fail. We theoretically\nanalyze the Wasserstein distance of the distribution of the PGPS-generated\nsamples and the target distribution due to approximation and discretization\nerrors. Practically, the proposed PGPS-LwS method demonstrates higher Bayesian\ninference accuracy and better calibration ability in experiments conducted on\nboth synthetic and real-world Bayesian learning tasks, compared to baselines,\nsuch as SVGD and Langevin dynamics, etc.\n", "date": "2024-12-04", "categories": []}
{"id": "2311.11126", "title": "Bayesian Neural Networks: A Min-Max Game Framework", "abstract": "  In deep learning, Bayesian neural networks (BNN) provide the role of\nrobustness analysis, and the minimax method is used to be a conservative choice\nin the traditional Bayesian field. In this paper, we study a conservative BNN\nwith the minimax method and formulate a two-player game between a deterministic\nneural network $f$ and a sampling stochastic neural network $f + r*\\xi$. From\nthis perspective, we understand the closed-loop neural networks with the\nminimax loss and reveal their connection to the BNN. We test the models on\nsimple data sets, study their robustness under noise perturbation, and report\nsome issues for searching $r$.\n", "date": "2023-11-18", "categories": []}
{"id": "2409.01464", "title": "Stein transport for Bayesian inference", "abstract": "  We introduce $\\textit{Stein transport}$, a novel methodology for Bayesian\ninference designed to efficiently push an ensemble of particles along a\npredefined curve of tempered probability distributions. The driving vector\nfield is chosen from a reproducing kernel Hilbert space and can be derived\neither through a suitable kernel ridge regression formulation or as an\ninfinitesimal optimal transport map in the Stein geometry. The update equations\nof Stein transport resemble those of Stein variational gradient descent (SVGD),\nbut introduce a time-varying score function as well as specific weights\nattached to the particles. While SVGD relies on convergence in the long-time\nlimit, Stein transport reaches its posterior approximation at finite time\n$t=1$. Studying the mean-field limit, we discuss the errors incurred by\nregularisation and finite-particle effects, and we connect Stein transport to\nbirth-death dynamics and Fisher-Rao gradient flows. In a series of experiments,\nwe show that in comparison to SVGD, Stein transport not only often reaches more\naccurate posterior approximations with a significantly reduced computational\nbudget, but that it also effectively mitigates the variance collapse phenomenon\ncommonly observed in SVGD.\n", "date": "2024-09-02", "categories": []}
{"id": "2409.05354", "title": "Recursive Nested Filtering for Efficient Amortized Bayesian Experimental\n  Design", "abstract": "  This paper introduces the Inside-Out Nested Particle Filter (IO-NPF), a\nnovel, fully recursive, algorithm for amortized sequential Bayesian\nexperimental design in the non-exchangeable setting. We frame policy\noptimization as maximum likelihood estimation in a non-Markovian state-space\nmodel, achieving (at most) $\\mathcal{O}(T^2)$ computational complexity in the\nnumber of experiments. We provide theoretical convergence guarantees and\nintroduce a backward sampling algorithm to reduce trajectory degeneracy. IO-NPF\noffers a practical, extensible, and provably consistent approach to sequential\nBayesian experimental design, demonstrating improved efficiency over existing\nmethods.\n", "date": "2024-09-09", "categories": []}
{"id": "2411.14202", "title": "Revised Regularization for Efficient Continual Learning through\n  Correlation-Based Parameter Update in Bayesian Neural Networks", "abstract": "  We propose a Bayesian neural network-based continual learning algorithm using\nVariational Inference, aiming to overcome several drawbacks of existing\nmethods. Specifically, in continual learning scenarios, storing network\nparameters at each step to retain knowledge poses challenges. This is\ncompounded by the crucial need to mitigate catastrophic forgetting,\nparticularly given the limited access to past datasets, which complicates\nmaintaining correspondence between network parameters and datasets across all\nsessions. Current methods using Variational Inference with KL divergence risk\ncatastrophic forgetting during uncertain node updates and coupled disruptions\nin certain nodes. To address these challenges, we propose the following\nstrategies. To reduce the storage of the dense layer parameters, we propose a\nparameter distribution learning method that significantly reduces the storage\nrequirements. In the continual learning framework employing variational\ninference, our study introduces a regularization term that specifically targets\nthe dynamics and population of the mean and variance of the parameters. This\nterm aims to retain the benefits of KL divergence while addressing related\nchallenges. To ensure proper correspondence between network parameters and the\ndata, our method introduces an importance-weighted Evidence Lower Bound term to\ncapture data and parameter correlations. This enables storage of common and\ndistinctive parameter hyperspace bases. The proposed method partitions the\nparameter space into common and distinctive subspaces, with conditions for\neffective backward and forward knowledge transfer, elucidating the\nnetwork-parameter dataset correspondence. The experimental results demonstrate\nthe effectiveness of our method across diverse datasets and various\ncombinations of sequential datasets, yielding superior performance compared to\nexisting approaches.\n", "date": "2024-11-21", "categories": []}
{"id": "2406.06946", "title": "Sparse Bayesian Networks: Efficient Uncertainty Quantification in\n  Medical Image Analysis", "abstract": "  Efficiently quantifying predictive uncertainty in medical images remains a\nchallenge. While Bayesian neural networks (BNN) offer predictive uncertainty,\nthey require substantial computational resources to train. Although Bayesian\napproximations such as ensembles have shown promise, they still suffer from\nhigh training and inference costs. Existing approaches mainly address the costs\nof BNN inference post-training, with little focus on improving training\nefficiency and reducing parameter complexity. This study introduces a training\nprocedure for a sparse (partial) Bayesian network. Our method selectively\nassigns a subset of parameters as Bayesian by assessing their deterministic\nsaliency through gradient sensitivity analysis. The resulting network combines\ndeterministic and Bayesian parameters, exploiting the advantages of both\nrepresentations to achieve high task-specific performance and minimize\npredictive uncertainty. Demonstrated on multi-label ChestMNIST for\nclassification and ISIC, LIDC-IDRI for segmentation, our approach achieves\ncompetitive performance and predictive uncertainty estimation by reducing\nBayesian parameters by over 95\\%, significantly reducing computational expenses\ncompared to fully Bayesian and ensemble methods.\n", "date": "2024-06-11", "categories": []}
{"id": "2408.00838", "title": "Calibrating Bayesian Generative Machine Learning for Bayesiamplification", "abstract": "  Recently, combinations of generative and Bayesian machine learning have been\nintroduced in particle physics for both fast detector simulation and inference\ntasks. These neural networks aim to quantify the uncertainty on the generated\ndistribution originating from limited training statistics. The interpretation\nof a distribution-wide uncertainty however remains ill-defined. We show a clear\nscheme for quantifying the calibration of Bayesian generative machine learning\nmodels. For a Continuous Normalizing Flow applied to a low-dimensional toy\nexample, we evaluate the calibration of Bayesian uncertainties from either a\nmean-field Gaussian weight posterior, or Monte Carlo sampling network weights,\nto gauge their behaviour on unsteady distribution edges. Well calibrated\nuncertainties can then be used to roughly estimate the number of uncorrelated\ntruth samples that are equivalent to the generated sample and clearly indicate\ndata amplification for smooth features of the distribution.\n", "date": "2024-08-01", "categories": []}
{"id": "2411.12102", "title": "BALI: Learning Neural Networks via Bayesian Layerwise Inference", "abstract": "  We introduce a new method for learning Bayesian neural networks, treating\nthem as a stack of multivariate Bayesian linear regression models. The main\nidea is to infer the layerwise posterior exactly if we know the target outputs\nof each layer. We define these pseudo-targets as the layer outputs from the\nforward pass, updated by the backpropagated gradients of the objective\nfunction. The resulting layerwise posterior is a matrix-normal distribution\nwith a Kronecker-factorized covariance matrix, which can be efficiently\ninverted. Our method extends to the stochastic mini-batch setting using an\nexponential moving average over natural-parameter terms, thus gradually\nforgetting older data. The method converges in few iterations and performs as\nwell as or better than leading Bayesian neural network methods on various\nregression, classification, and out-of-distribution detection benchmarks.\n", "date": "2024-11-18", "categories": []}
{"id": "2411.09483", "title": "Sparse Bayesian Generative Modeling for Compressive Sensing", "abstract": "  This work addresses the fundamental linear inverse problem in compressive\nsensing (CS) by introducing a new type of regularizing generative prior. Our\nproposed method utilizes ideas from classical dictionary-based CS and, in\nparticular, sparse Bayesian learning (SBL), to integrate a strong\nregularization towards sparse solutions. At the same time, by leveraging the\nnotion of conditional Gaussianity, it also incorporates the adaptability from\ngenerative models to training data. However, unlike most state-of-the-art\ngenerative models, it is able to learn from a few compressed and noisy data\nsamples and requires no optimization algorithm for solving the inverse problem.\nAdditionally, similar to Dirichlet prior networks, our model parameterizes a\nconjugate prior enabling its application for uncertainty quantification. We\nsupport our approach theoretically through the concept of variational inference\nand validate it empirically using different types of compressible signals.\n", "date": "2024-11-14", "categories": []}
{"id": "2411.08849", "title": "Oblique Bayesian additive regression trees", "abstract": "  Current implementations of Bayesian Additive Regression Trees (BART) are\nbased on axis-aligned decision rules that recursively partition the feature\nspace using a single feature at a time. Several authors have demonstrated that\noblique trees, whose decision rules are based on linear combinations of\nfeatures, can sometimes yield better predictions than axis-aligned trees and\nexhibit excellent theoretical properties. We develop an oblique version of BART\nthat leverages a data-adaptive decision rule prior that recursively partitions\nthe feature space along random hyperplanes. Using several synthetic and\nreal-world benchmark datasets, we systematically compared our oblique BART\nimplementation to axis-aligned BART and other tree ensemble methods, finding\nthat oblique BART was competitive with -- and sometimes much better than --\nthose methods.\n", "date": "2024-11-13", "categories": []}
{"id": "2306.02150", "title": "An information field theory approach to Bayesian state and parameter\n  estimation in dynamical systems", "abstract": "  Dynamical system state estimation and parameter calibration problems are\nubiquitous across science and engineering. Bayesian approaches to the problem\nare the gold standard as they allow for the quantification of uncertainties and\nenable the seamless fusion of different experimental modalities. When the\ndynamics are discrete and stochastic, one may employ powerful techniques such\nas Kalman, particle, or variational filters. Practitioners commonly apply these\nmethods to continuous-time, deterministic dynamical systems after discretizing\nthe dynamics and introducing fictitious transition probabilities. However,\napproaches based on time-discretization suffer from the curse of dimensionality\nsince the number of random variables grows linearly with the number of\ntime-steps. Furthermore, the introduction of fictitious transition\nprobabilities is an unsatisfactory solution because it increases the number of\nmodel parameters and may lead to inference bias. To address these drawbacks,\nthe objective of this paper is to develop a scalable Bayesian approach to state\nand parameter estimation suitable for continuous-time, deterministic dynamical\nsystems. Our methodology builds upon information field theory. Specifically, we\nconstruct a physics-informed prior probability measure on the function space of\nsystem responses so that functions that satisfy the physics are more likely.\nThis prior allows us to quantify model form errors. We connect the system's\nresponse to observations through a probabilistic model of the measurement\nprocess. The joint posterior over the system responses and all parameters is\ngiven by Bayes' rule. To approximate the intractable posterior, we develop a\nstochastic variational inference algorithm. In summary, the developed\nmethodology offers a powerful framework for Bayesian estimation in dynamical\nsystems.\n", "date": "2023-06-03", "categories": []}
{"id": "2411.05805", "title": "Variational Bayes Decomposition for Inverse Estimation with Superimposed\n  Multispectral Intensity", "abstract": "  A variational Bayesian inference for measured wave intensity, such as X-ray\nintensity, is proposed in this paper. The data is popular to obtain information\nabout unobservable features of an object, such as a material sample and the\ncomponents of it. The proposed method assumes particles represent the wave, and\ntheir behaviors are stochastically modeled. The inference is accurate even if\nthe data is noisy because of a smooth prior setting. Moreover, in this paper,\ntwo experimental results show feasibility of the proposed method.\n", "date": "2024-10-29", "categories": []}
{"id": "2411.05899", "title": "Streaming Bayes GFlowNets", "abstract": "  Bayes' rule naturally allows for inference refinement in a streaming fashion,\nwithout the need to recompute posteriors from scratch whenever new data\narrives. In principle, Bayesian streaming is straightforward: we update our\nprior with the available data and use the resulting posterior as a prior when\nprocessing the next data chunk. In practice, however, this recipe entails i)\napproximating an intractable posterior at each time step; and ii) encapsulating\nresults appropriately to allow for posterior propagation. For continuous state\nspaces, variational inference (VI) is particularly convenient due to its\nscalability and the tractability of variational posteriors. For discrete state\nspaces, however, state-of-the-art VI results in analytically intractable\napproximations that are ill-suited for streaming settings. To enable streaming\nBayesian inference over discrete parameter spaces, we propose streaming Bayes\nGFlowNets (abbreviated as SB-GFlowNets) by leveraging the recently proposed\nGFlowNets -- a powerful class of amortized samplers for discrete compositional\nobjects. Notably, SB-GFlowNet approximates the initial posterior using a\nstandard GFlowNet and subsequently updates it using a tailored procedure that\nrequires only the newly observed data. Our case studies in linear preference\nlearning and phylogenetic inference showcase the effectiveness of SB-GFlowNets\nin sampling from an unnormalized posterior in a streaming setting. As expected,\nwe also observe that SB-GFlowNets is significantly faster than repeatedly\ntraining a GFlowNet from scratch to sample from the full posterior.\n", "date": "2024-11-08", "categories": []}
{"id": "2411.07043", "title": "Unified Bayesian representation for high-dimensional multi-modal\n  biomedical data for small-sample classification", "abstract": "  We present BALDUR, a novel Bayesian algorithm designed to deal with\nmulti-modal datasets and small sample sizes in high-dimensional settings while\nproviding explainable solutions. To do so, the proposed model combines within a\ncommon latent space the different data views to extract the relevant\ninformation to solve the classification task and prune out the\nirrelevant/redundant features/data views. Furthermore, to provide generalizable\nsolutions in small sample size scenarios, BALDUR efficiently integrates dual\nkernels over the views with a small sample-to-feature ratio. Finally, its\nlinear nature ensures the explainability of the model outcomes, allowing its\nuse for biomarker identification. This model was tested over two different\nneurodegeneration datasets, outperforming the state-of-the-art models and\ndetecting features aligned with markers already described in the scientific\nliterature.\n", "date": "2024-11-11", "categories": []}
{"id": "2411.04278", "title": "The Recurrent Sticky Hierarchical Dirichlet Process Hidden Markov Model", "abstract": "  The Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) is a natural\nBayesian nonparametric extension of the classical Hidden Markov Model for\nlearning from (spatio-)temporal data. A sticky HDP-HMM has been proposed to\nstrengthen the self-persistence probability in the HDP-HMM. Then, disentangled\nsticky HDP-HMM has been proposed to disentangle the strength of the\nself-persistence prior and transition prior. However, the sticky HDP-HMM\nassumes that the self-persistence probability is stationary, limiting its\nexpressiveness. Here, we build on previous work on sticky HDP-HMM and\ndisentangled sticky HDP-HMM, developing a more general model: the recurrent\nsticky HDP-HMM (RS-HDP-HMM). We develop a novel Gibbs sampling strategy for\nefficient inference in this model. We show that RS-HDP-HMM outperforms\ndisentangled sticky HDP-HMM, sticky HDP-HMM, and HDP-HMM in both synthetic and\nreal data segmentation.\n", "date": "2024-11-06", "categories": []}
{"id": "2302.02420", "title": "Variational Inference on the Final-Layer Output of Neural Networks", "abstract": "  Traditional neural networks are simple to train but they typically produce\noverconfident predictions. In contrast, Bayesian neural networks provide good\nuncertainty quantification but optimizing them is time consuming due to the\nlarge parameter space. This paper proposes to combine the advantages of both\napproaches by performing Variational Inference in the Final layer Output space\n(VIFO), because the output space is much smaller than the parameter space. We\nuse neural networks to learn the mean and the variance of the probabilistic\noutput. Using the Bayesian formulation we incorporate collapsed variational\ninference with VIFO which significantly improves the performance in practice.\nOn the other hand, like standard, non-Bayesian models, VIFO enjoys simple\ntraining and one can use Rademacher complexity to provide risk bounds for the\nmodel. Experiments show that VIFO provides a good tradeoff in terms of run time\nand uncertainty quantification, especially for out of distribution data.\n", "date": "2023-02-05", "categories": []}
{"id": "2402.01810", "title": "Parameter uncertainties for imperfect surrogate models in the low-noise\n  regime", "abstract": "  Bayesian regression determines model parameters by minimizing the expected\nloss, an upper bound to the true generalization error. However, the loss\nignores misspecification, where models are imperfect. Parameter uncertainties\nfrom Bayesian regression are thus significantly underestimated and vanish in\nthe large data limit. This is particularly problematic when building models of\nlow-noise, or near-deterministic, calculations, as the main source of\nuncertainty is neglected. We analyze the generalization error of misspecified,\nnear-deterministic surrogate models, a regime of broad relevance in science and\nengineering. We show posterior distributions must cover every training point to\navoid a divergent generalization error and design an ansatz that respects this\nconstraint, which for linear models incurs minimal overhead. This is\ndemonstrated on model problems before application to thousand dimensional\ndatasets in atomistic machine learning. Our efficient misspecification-aware\nscheme gives accurate prediction and bounding of test errors where existing\nschemes fail, allowing this important source of uncertainty to be incorporated\nin computational workflows.\n", "date": "2024-02-02", "categories": []}
{"id": "2411.03768", "title": "A Bayesian Approach to Data Point Selection", "abstract": "  Data point selection (DPS) is becoming a critical topic in deep learning due\nto the ease of acquiring uncurated training data compared to the difficulty of\nobtaining curated or processed data. Existing approaches to DPS are\npredominantly based on a bi-level optimisation (BLO) formulation, which is\ndemanding in terms of memory and computation, and exhibits some theoretical\ndefects regarding minibatches. Thus, we propose a novel Bayesian approach to\nDPS. We view the DPS problem as posterior inference in a novel Bayesian model\nwhere the posterior distributions of the instance-wise weights and the main\nneural network parameters are inferred under a reasonable prior and likelihood\nmodel. We employ stochastic gradient Langevin MCMC sampling to learn the main\nnetwork and instance-wise weights jointly, ensuring convergence even with\nminibatches. Our update equation is comparable to the widely used SGD and much\nmore efficient than existing BLO-based methods. Through controlled experiments\nin both the vision and language domains, we present the proof-of-concept.\nAdditionally, we demonstrate that our method scales effectively to large\nlanguage models and facilitates automated per-task optimization for instruction\nfine-tuning datasets.\n", "date": "2024-11-06", "categories": []}
{"id": "2311.09997", "title": "Co-data Learning for Bayesian Additive Regression Trees", "abstract": "  Medical prediction applications often need to deal with small sample sizes\ncompared to the number of covariates. Such data pose problems for prediction\nand variable selection, especially when the covariate-response relationship is\ncomplicated. To address these challenges, we propose to incorporate co-data,\ni.e. external information on the covariates, into Bayesian additive regression\ntrees (BART), a sum-of-trees prediction model that utilizes priors on the tree\nparameters to prevent overfitting. To incorporate co-data, an empirical Bayes\n(EB) framework is developed that estimates, assisted by a co-data model, prior\ncovariate weights in the BART model. The proposed method can handle multiple\ntypes of co-data simultaneously. Furthermore, the proposed EB framework enables\nthe estimation of the other hyperparameters of BART as well, rendering an\nappealing alternative to cross-validation. We show that the method finds\nrelevant covariates and that it improves prediction compared to default BART in\nsimulations. If the covariate-response relationship is nonlinear, the method\nbenefits from the flexibility of BART to outperform regression-based co-data\nlearners. Finally, the use of co-data enhances prediction in an application to\ndiffuse large B-cell lymphoma prognosis based on clinical covariates, gene\nmutations, DNA translocations, and DNA copy number data.\n  Keywords: Bayesian additive regression trees; Empirical Bayes; Co-data;\nHigh-dimensional data; Omics; Prediction\n", "date": "2023-11-16", "categories": []}
{"id": "2411.01887", "title": "Stein Variational Newton Neural Network Ensembles", "abstract": "  Deep neural network ensembles are powerful tools for uncertainty\nquantification, which have recently been re-interpreted from a Bayesian\nperspective. However, current methods inadequately leverage second-order\ninformation of the loss landscape, despite the recent availability of efficient\nHessian approximations. We propose a novel approximate Bayesian inference\nmethod that modifies deep ensembles to incorporate Stein Variational Newton\nupdates. Our approach uniquely integrates scalable modern Hessian\napproximations, achieving faster convergence and more accurate posterior\ndistribution approximations. We validate the effectiveness of our method on\ndiverse regression and classification tasks, demonstrating superior performance\nwith a significantly reduced number of training epochs compared to existing\nensemble-based methods, while enhancing uncertainty quantification and\nrobustness against overfitting.\n", "date": "2024-11-04", "categories": []}
{"id": "2411.00259", "title": "Enhancing Diversity in Bayesian Deep Learning via Hyperspherical Energy\n  Minimization of CKA", "abstract": "  Particle-based Bayesian deep learning often requires a similarity metric to\ncompare two networks. However, naive similarity metrics lack permutation\ninvariance and are inappropriate for comparing networks. Centered Kernel\nAlignment (CKA) on feature kernels has been proposed to compare deep networks\nbut has not been used as an optimization objective in Bayesian deep learning.\nIn this paper, we explore the use of CKA in Bayesian deep learning to generate\ndiverse ensembles and hypernetworks that output a network posterior. Noting\nthat CKA projects kernels onto a unit hypersphere and that directly optimizing\nthe CKA objective leads to diminishing gradients when two networks are very\nsimilar. We propose adopting the approach of hyperspherical energy (HE) on top\nof CKA kernels to address this drawback and improve training stability.\nAdditionally, by leveraging CKA-based feature kernels, we derive feature\nrepulsive terms applied to synthetically generated outlier examples.\nExperiments on both diverse ensembles and hypernetworks show that our approach\nsignificantly outperforms baselines in terms of uncertainty quantification in\nboth synthetic and realistic outlier detection tasks.\n", "date": "2024-10-31", "categories": []}
{"id": "2411.00273", "title": "Efficient Model Compression for Bayesian Neural Networks", "abstract": "  Model Compression has drawn much attention within the deep learning community\nrecently. Compressing a dense neural network offers many advantages including\nlower computation cost, deployability to devices of limited storage and\nmemories, and resistance to adversarial attacks. This may be achieved via\nweight pruning or fully discarding certain input features. Here we demonstrate\na novel strategy to emulate principles of Bayesian model selection in a deep\nlearning setup. Given a fully connected Bayesian neural network with\nspike-and-slab priors trained via a variational algorithm, we obtain the\nposterior inclusion probability for every node that typically gets lost. We\nemploy these probabilities for pruning and feature selection on a host of\nsimulated and real-world benchmark data and find evidence of better\ngeneralizability of the pruned model in all our experiments.\n", "date": "2024-11-01", "categories": []}
{"id": "2407.13711", "title": "FSP-Laplace: Function-Space Priors for the Laplace Approximation in\n  Bayesian Deep Learning", "abstract": "  Laplace approximations are popular techniques for endowing deep networks with\nepistemic uncertainty estimates as they can be applied without altering the\npredictions of the trained network, and they scale to large models and\ndatasets. While the choice of prior strongly affects the resulting posterior\ndistribution, computational tractability and lack of interpretability of the\nweight space typically limit the Laplace approximation to isotropic Gaussian\npriors, which are known to cause pathological behavior as depth increases. As a\nremedy, we directly place a prior on function space. More precisely, since\nLebesgue densities do not exist on infinite-dimensional function spaces, we\nrecast training as finding the so-called weak mode of the posterior measure\nunder a Gaussian process (GP) prior restricted to the space of functions\nrepresentable by the neural network. Through the GP prior, one can express\nstructured and interpretable inductive biases, such as regularity or\nperiodicity, directly in function space, while still exploiting the implicit\ninductive biases that allow deep networks to generalize. After model\nlinearization, the training objective induces a negative log-posterior density\nto which we apply a Laplace approximation, leveraging highly scalable methods\nfrom matrix-free linear algebra. Our method provides improved results where\nprior knowledge is abundant (as is the case in many scientific inference\ntasks). At the same time, it stays competitive for black-box supervised\nlearning problems, where neural networks typically excel.\n", "date": "2024-07-18", "categories": []}
{"id": "2402.04646", "title": "Block Sparse Bayesian Learning: A Diversified Scheme", "abstract": "  This paper introduces a novel prior called Diversified Block Sparse Prior to\ncharacterize the widespread block sparsity phenomenon in real-world data. By\nallowing diversification on intra-block variance and inter-block correlation\nmatrices, we effectively address the sensitivity issue of existing block sparse\nlearning methods to pre-defined block information, which enables adaptive block\nestimation while mitigating the risk of overfitting. Based on this, a\ndiversified block sparse Bayesian learning method (DivSBL) is proposed,\nutilizing EM algorithm and dual ascent method for hyperparameter estimation.\nMoreover, we establish the global and local optimality theory of our model.\nExperiments validate the advantages of DivSBL over existing algorithms.\n", "date": "2024-02-07", "categories": []}
{"id": "2410.19131", "title": "Maximum a Posteriori Inference for Factor Graphs via Benders'\n  Decomposition", "abstract": "  Many Bayesian statistical inference problems come down to computing a maximum\na-posteriori (MAP) assignment of latent variables. Yet, standard methods for\nestimating the MAP assignment do not have a finite time guarantee that the\nalgorithm has converged to a fixed point. Previous research has found that MAP\ninference can be represented in dual form as a linear programming problem with\na non-polynomial number of constraints. A Lagrangian relaxation of the dual\nyields a statistical inference algorithm as a linear programming problem.\nHowever, the decision as to which constraints to remove in the relaxation is\noften heuristic. We present a method for maximum a-posteriori inference in\ngeneral Bayesian factor models that sequentially adds constraints to the fully\nrelaxed dual problem using Benders' decomposition. Our method enables the\nincorporation of expressive integer and logical constraints in clustering\nproblems such as must-link, cannot-link, and a minimum number of whole samples\nallocated to each cluster. Using this approach, we derive MAP estimation\nalgorithms for the Bayesian Gaussian mixture model and latent Dirichlet\nallocation. Empirical results show that our method produces a higher optimal\nposterior value compared to Gibbs sampling and variational Bayes methods for\nstandard data sets and provides certificate of convergence.\n", "date": "2024-10-24", "categories": []}
{"id": "2410.22065", "title": "Hamiltonian Monte Carlo on ReLU Neural Networks is Inefficient", "abstract": "  We analyze the error rates of the Hamiltonian Monte Carlo algorithm with\nleapfrog integrator for Bayesian neural network inference. We show that due to\nthe non-differentiability of activation functions in the ReLU family, leapfrog\nHMC for networks with these activation functions has a large local error rate\nof $\\Omega(\\epsilon)$ rather than the classical error rate of $O(\\epsilon^3)$.\nThis leads to a higher rejection rate of the proposals, making the method\ninefficient. We then verify our theoretical findings through empirical\nsimulations as well as experiments on a real-world dataset that highlight the\ninefficiency of HMC inference on ReLU-based neural networks compared to\nanalytical networks.\n", "date": "2024-10-29", "categories": []}
{"id": "2305.16905", "title": "Improving Neural Additive Models with Bayesian Principles", "abstract": "  Neural additive models (NAMs) enhance the transparency of deep neural\nnetworks by handling input features in separate additive sub-networks. However,\nthey lack inherent mechanisms that provide calibrated uncertainties and enable\nselection of relevant features and interactions. Approaching NAMs from a\nBayesian perspective, we augment them in three primary ways, namely by a)\nproviding credible intervals for the individual additive sub-networks; b)\nestimating the marginal likelihood to perform an implicit selection of features\nvia an empirical Bayes procedure; and c) facilitating the ranking of feature\npairs as candidates for second-order interaction in fine-tuned models. In\nparticular, we develop Laplace-approximated NAMs (LA-NAMs), which show improved\nempirical performance on tabular datasets and challenging real-world medical\ntasks.\n", "date": "2023-05-26", "categories": []}
{"id": "2410.20318", "title": "Low-rank Bayesian matrix completion via geodesic Hamiltonian Monte Carlo\n  on Stiefel manifolds", "abstract": "  We present a new sampling-based approach for enabling efficient computation\nof low-rank Bayesian matrix completion and quantifying the associated\nuncertainty. Firstly, we design a new prior model based on the\nsingular-value-decomposition (SVD) parametrization of low-rank matrices. Our\nprior is analogous to the seminal nuclear-norm regularization used in\nnon-Bayesian setting and enforces orthogonality in the factor matrices by\nconstraining them to Stiefel manifolds. Then, we design a geodesic Hamiltonian\nMonte Carlo (-within-Gibbs) algorithm for generating posterior samples of the\nSVD factor matrices. We demonstrate that our approach resolves the sampling\ndifficulties encountered by standard Gibbs samplers for the common two-matrix\nfactorization used in matrix completion. More importantly, the geodesic\nHamiltonian sampler allows for sampling in cases with more general likelihoods\nthan the typical Gaussian likelihood and Gaussian prior assumptions adopted in\nmost of the existing Bayesian matrix completion literature. We demonstrate an\napplications of our approach to fit the categorical data of a mice protein\ndataset and the MovieLens recommendation problem. Numerical examples\ndemonstrate superior sampling performance, including better mixing and faster\nconvergence to a stationary distribution. Moreover, they demonstrate improved\naccuracy on the two real-world benchmark problems we considered.\n", "date": "2024-10-27", "categories": []}
{"id": "2207.01524", "title": "Variational Neural Networks", "abstract": "  Bayesian Neural Networks (BNNs) provide a tool to estimate the uncertainty of\na neural network by considering a distribution over weights and sampling\ndifferent models for each input. In this paper, we propose a method for\nuncertainty estimation in neural networks which, instead of considering a\ndistribution over weights, samples outputs of each layer from a corresponding\nGaussian distribution, parametrized by the predictions of mean and variance\nsub-layers. In uncertainty quality estimation experiments, we show that the\nproposed method achieves better uncertainty quality than other single-bin\nBayesian Model Averaging methods, such as Monte Carlo Dropout or Bayes By\nBackpropagation methods.\n", "date": "2022-07-04", "categories": []}
{"id": "2410.17700", "title": "Scalable Random Feature Latent Variable Models", "abstract": "  Random feature latent variable models (RFLVMs) represent the state-of-the-art\nin latent variable models, capable of handling non-Gaussian likelihoods and\neffectively uncovering patterns in high-dimensional data. However, their heavy\nreliance on Monte Carlo sampling results in scalability issues which makes it\ndifficult to use these models for datasets with a massive number of\nobservations. To scale up RFLVMs, we turn to the optimization-based variational\nBayesian inference (VBI) algorithm which is known for its scalability compared\nto sampling-based methods. However, implementing VBI for RFLVMs poses\nchallenges, such as the lack of explicit probability distribution functions\n(PDFs) for the Dirichlet process (DP) in the kernel learning component, and the\nincompatibility of existing VBI algorithms with RFLVMs. To address these\nissues, we introduce a stick-breaking construction for DP to obtain an explicit\nPDF and a novel VBI algorithm called ``block coordinate descent variational\ninference\" (BCD-VI). This enables the development of a scalable version of\nRFLVMs, or in short, SRFLVM. Our proposed method shows scalability,\ncomputational efficiency, superior performance in generating informative latent\nrepresentations and the ability of imputing missing data across various\nreal-world datasets, outperforming state-of-the-art competitors.\n", "date": "2024-10-23", "categories": []}
{"id": "2302.09656", "title": "Credal Bayesian Deep Learning", "abstract": "  Uncertainty quantification and robustness to distribution shifts are\nimportant goals in machine learning and artificial intelligence. Although\nBayesian Neural Networks (BNNs) allow for uncertainty in the predictions to be\nassessed, different sources of predictive uncertainty cannot be distinguished\nproperly. We present Credal Bayesian Deep Learning (CBDL). Heuristically, CBDL\nallows to train an (uncountably) infinite ensemble of BNNs, using only finitely\nmany elements. This is possible thanks to prior and likelihood finitely\ngenerated credal sets (FGCSs), a concept from the imprecise probability\nliterature. Intuitively, convex combinations of a finite collection of\nprior-likelihood pairs are able to represent infinitely many such pairs. After\ntraining, CBDL outputs a set of posteriors on the parameters of the neural\nnetwork. At inference time, such posterior set is used to derive a set of\npredictive distributions that is in turn utilized to distinguish between\n(predictive) aleatoric and epistemic uncertainties, and to quantify them. The\npredictive set also produces either (i) a collection of outputs enjoying\ndesirable probabilistic guarantees, or (ii) the single output that is deemed\nthe best, that is, the one having the highest predictive lower probability --\nanother imprecise-probabilistic concept. CBDL is more robust than single BNNs\nto prior and likelihood misspecification, and to distribution shift. We show\nthat CBDL is better at quantifying and disentangling different types of\n(predictive) uncertainties than single BNNs and ensemble of BNNs. In addition,\nwe apply CBDL to two case studies to demonstrate its downstream tasks\ncapabilities: one, for motion prediction in autonomous driving scenarios, and\ntwo, to model blood glucose and insulin dynamics for artificial pancreas\ncontrol. We show that CBDL performs better when compared to an ensemble of BNNs\nbaseline.\n", "date": "2023-02-19", "categories": []}
{"id": "2410.16420", "title": "BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant\n  Neural Operator Framework", "abstract": "  Bayesian inference offers a robust framework for updating prior beliefs based\non new data using Bayes' theorem, but exact inference is often computationally\ninfeasible, necessitating approximate methods. Though widely used, these\nmethods struggle to estimate marginal likelihoods accurately, particularly due\nto the rigid functional structures of deterministic models like Gaussian\nprocesses and the limitations of small sample sizes in stochastic models like\nthe ensemble Kalman method. In this work, we introduce BI-EqNO, an equivariant\nneural operator framework for generalized approximate Bayesian inference,\ndesigned to enhance both deterministic and stochastic approaches. BI-EqNO\ntransforms priors into posteriors conditioned on observation data through\ndata-driven training. The framework is flexible, supporting diverse prior and\nposterior representations with arbitrary discretizations and varying numbers of\nobservations. Crucially, BI-EqNO's architecture ensures (1) permutation\nequivariance between prior and posterior representations, and (2) permutation\ninvariance with respect to observational data. We demonstrate BI-EqNO's utility\nthrough two examples: (1) as a generalized Gaussian process (gGP) for\nregression, and (2) as an ensemble neural filter (EnNF) for sequential data\nassimilation. Results show that gGP outperforms traditional Gaussian processes\nby offering a more flexible representation of covariance functions.\nAdditionally, EnNF not only outperforms the ensemble Kalman filter in\nsmall-ensemble settings but also has the potential to function as a \"super\"\nensemble filter, capable of representing and integrating multiple ensemble\nfilters for enhanced assimilation performance. This study highlights BI-EqNO's\nversatility and effectiveness, improving Bayesian inference through data-driven\ntraining while reducing computational costs across various applications.\n", "date": "2024-10-21", "categories": []}
{"id": "2410.16901", "title": "Bayes without Underfitting: Fully Correlated Deep Learning Posteriors\n  via Alternating Projections", "abstract": "  Bayesian deep learning all too often underfits so that the Bayesian\nprediction is less accurate than a simple point estimate. Uncertainty\nquantification then comes at the cost of accuracy. For linearized models, the\nnull space of the generalized Gauss-Newton matrix corresponds to parameters\nthat preserve the training predictions of the point estimate. We propose to\nbuild Bayesian approximations in this null space, thereby guaranteeing that the\nBayesian predictive does not underfit. We suggest a matrix-free algorithm for\nprojecting onto this null space, which scales linearly with the number of\nparameters and quadratically with the number of output dimensions. We further\npropose an approximation that only scales linearly with parameters to make the\nmethod applicable to generative models. An extensive empirical evaluation shows\nthat the approach scales to large models, including vision transformers with 28\nmillion parameters.\n", "date": "2024-10-22", "categories": []}
{"id": "2403.19381", "title": "On Uncertainty Quantification for Near-Bayes Optimal Algorithms", "abstract": "  Bayesian modelling allows for the quantification of predictive uncertainty\nwhich is crucial in safety-critical applications. Yet for many machine learning\n(ML) algorithms, it is difficult to construct or implement their Bayesian\ncounterpart. In this work we present a promising approach to address this\nchallenge, based on the hypothesis that commonly used ML algorithms are\nefficient across a wide variety of tasks and may thus be near Bayes-optimal\nw.r.t. an unknown task distribution. We prove that it is possible to recover\nthe Bayesian posterior defined by the task distribution, which is unknown but\noptimal in this setting, by building a martingale posterior using the\nalgorithm. We further propose a practical uncertainty quantification method\nthat apply to general ML algorithms. Experiments based on a variety of non-NN\nand NN algorithms demonstrate the efficacy of our method.\n", "date": "2024-03-28", "categories": []}
{"id": "2405.02063", "title": "Few-sample Variational Inference of Bayesian Neural Networks with\n  Arbitrary Nonlinearities", "abstract": "  Bayesian Neural Networks (BNNs) extend traditional neural networks to provide\nuncertainties associated with their outputs. On the forward pass through a BNN,\npredictions (and their uncertainties) are made either by Monte Carlo sampling\nnetwork weights from the learned posterior or by analytically propagating\nstatistical moments through the network. Though flexible, Monte Carlo sampling\nis computationally expensive and can be infeasible or impractical under\nresource constraints or for large networks. While moment propagation can\nameliorate the computational costs of BNN inference, it can be difficult or\nimpossible for networks with arbitrary nonlinearities, thereby restricting the\npossible set of network layers permitted with such a scheme. In this work, we\ndemonstrate a simple yet effective approach for propagating statistical moments\nthrough arbitrary nonlinearities with only 3 deterministic samples, enabling\nfew-sample variational inference of BNNs without restricting the set of network\nlayers used. Furthermore, we leverage this approach to demonstrate a novel\nnonlinear activation function that we use to inject physics-informed prior\ninformation into output nodes of a BNN.\n", "date": "2024-05-03", "categories": []}
{"id": "2410.15310", "title": "On Cold Posteriors of Probabilistic Neural Networks: Understanding the\n  Cold Posterior Effect and A New Way to Learn Cold Posteriors with Tight\n  Generalization Guarantees", "abstract": "  Bayesian inference provides a principled probabilistic framework for\nquantifying uncertainty by updating beliefs based on prior knowledge and\nobserved data through Bayes' theorem. In Bayesian deep learning, neural network\nweights are treated as random variables with prior distributions, allowing for\na probabilistic interpretation and quantification of predictive uncertainty.\nHowever, Bayesian methods lack theoretical generalization guarantees for unseen\ndata. PAC-Bayesian analysis addresses this limitation by offering a frequentist\nframework to derive generalization bounds for randomized predictors, thereby\ncertifying the reliability of Bayesian methods in machine learning.\n  Temperature $T$, or inverse-temperature $\\lambda = \\frac{1}{T}$, originally\nfrom statistical mechanics in physics, naturally arises in various areas of\nstatistical inference, including Bayesian inference and PAC-Bayesian analysis.\nIn Bayesian inference, when $T < 1$ (``cold'' posteriors), the likelihood is\nup-weighted, resulting in a sharper posterior distribution. Conversely, when $T\n> 1$ (``warm'' posteriors), the likelihood is down-weighted, leading to a more\ndiffuse posterior distribution. By balancing the influence of observed data and\nprior regularization, temperature adjustments can address issues of\nunderfitting or overfitting in Bayesian models, bringing improved predictive\nperformance.\n", "date": "2024-10-20", "categories": []}
{"id": "2212.14163", "title": "A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators", "abstract": "  Kernels are efficient in representing nonlocal dependence and they are widely\nused to design operators between function spaces. Thus, learning kernels in\noperators from data is an inverse problem of general interest. Due to the\nnonlocal dependence, the inverse problem can be severely ill-posed with a\ndata-dependent singular inversion operator. The Bayesian approach overcomes the\nill-posedness through a non-degenerate prior. However, a fixed non-degenerate\nprior leads to a divergent posterior mean when the observation noise becomes\nsmall, if the data induces a perturbation in the eigenspace of zero eigenvalues\nof the inversion operator. We introduce a data-adaptive prior to achieve a\nstable posterior whose mean always has a small noise limit. The data-adaptive\nprior's covariance is the inversion operator with a hyper-parameter selected\nadaptive to data by the L-curve method. Furthermore, we provide a detailed\nanalysis on the computational practice of the data-adaptive prior, and\ndemonstrate it on Toeplitz matrices and integral operators. Numerical tests\nshow that a fixed prior can lead to a divergent posterior mean in the presence\nof any of the four types of errors: discretization error, model error, partial\nobservation and wrong noise assumption. In contrast, the data-adaptive prior\nalways attains posterior means with small noise limits.\n", "date": "2022-12-29", "categories": []}
{"id": "2405.11780", "title": "General bounds on the quality of Bayesian coresets", "abstract": "  Bayesian coresets speed up posterior inference in the large-scale data regime\nby approximating the full-data log-likelihood function with a surrogate\nlog-likelihood based on a small, weighted subset of the data. But while\nBayesian coresets and methods for construction are applicable in a wide range\nof models, existing theoretical analysis of the posterior inferential error\nincurred by coreset approximations only apply in restrictive settings -- i.e.,\nexponential family models, or models with strong log-concavity and smoothness\nassumptions. This work presents general upper and lower bounds on the\nKullback-Leibler (KL) divergence of coreset approximations that reflect the\nfull range of applicability of Bayesian coresets. The lower bounds require only\nmild model assumptions typical of Bayesian asymptotic analyses, while the upper\nbounds require the log-likelihood functions to satisfy a generalized\nsubexponentiality criterion that is weaker than conditions used in earlier\nwork. The lower bounds are applied to obtain fundamental limitations on the\nquality of coreset approximations, and to provide a theoretical explanation for\nthe previously-observed poor empirical performance of importance sampling-based\nconstruction methods. The upper bounds are used to analyze the performance of\nrecent subsample-optimize methods. The flexibility of the theory is\ndemonstrated in validation experiments involving multimodal, unidentifiable,\nheavy-tailed Bayesian posterior distributions.\n", "date": "2024-05-20", "categories": []}
{"id": "2410.12984", "title": "Double-Bayesian Learning", "abstract": "  Contemporary machine learning methods will try to approach the Bayes error,\nas it is the lowest possible error any model can achieve. This paper postulates\nthat any decision is composed of not one but two Bayesian decisions and that\ndecision-making is, therefore, a double-Bayesian process. The paper shows how\nthis duality implies intrinsic uncertainty in decisions and how it incorporates\nexplainability. The proposed approach understands that Bayesian learning is\ntantamount to finding a base for a logarithmic function measuring uncertainty,\nwith solutions being fixed points. Furthermore, following this approach, the\ngolden ratio describes possible solutions satisfying Bayes' theorem. The\ndouble-Bayesian framework suggests using a learning rate and momentum weight\nwith values similar to those used in the literature to train neural networks\nwith stochastic gradient descent.\n", "date": "2024-10-16", "categories": []}
{"id": "2406.17699", "title": "Variance Reduction for the Independent Metropolis Sampler", "abstract": "  Assume that we would like to estimate the expected value of a function $F$\nwith respect to an intractable density $\\pi$, which is specified up to some\nunknown normalising constant. We prove that if $\\pi$ is close enough under KL\ndivergence to another density $q$, an independent Metropolis sampler estimator\nthat obtains samples from $\\pi$ with proposal density $q$, enriched with a\nvariance reduction computational strategy based on control variates, achieves\nsmaller asymptotic variance than i.i.d.\\ sampling from $\\pi$. The control\nvariates construction requires no extra computational effort but assumes that\nthe expected value of $F$ under $q$ is analytically available. We illustrate\nthis result by calculating the marginal likelihood in a linear regression model\nwith prior-likelihood conflict and a non-conjugate prior. Furthermore, we\npropose an adaptive independent Metropolis algorithm that adapts the proposal\ndensity such that its KL divergence with the target is being reduced. We\ndemonstrate its applicability in a Bayesian logistic and Gaussian process\nregression problems and we rigorously justify our asymptotic arguments under\neasily verifiable and essentially minimal conditions.\n", "date": "2024-06-25", "categories": []}
{"id": "2402.07160", "title": "PASOA- PArticle baSed Bayesian Optimal Adaptive design", "abstract": "  We propose a new procedure named PASOA, for Bayesian experimental design,\nthat performs sequential design optimization by simultaneously providing\naccurate estimates of successive posterior distributions for parameter\ninference. The sequential design process is carried out via a contrastive\nestimation principle, using stochastic optimization and Sequential Monte Carlo\n(SMC) samplers to maximise the Expected Information Gain (EIG). As larger\ninformation gains are obtained for larger distances between successive\nposterior distributions, this EIG objective may worsen classical SMC\nperformance. To handle this issue, tempering is proposed to have both a large\ninformation gain and an accurate SMC sampling, that we show is crucial for\nperformance. This novel combination of stochastic optimization and tempered SMC\nallows to jointly handle design optimization and parameter inference. We\nprovide a proof that the obtained optimal design estimators benefit from some\nconsistency property. Numerical experiments confirm the potential of the\napproach, which outperforms other recent existing procedures.\n", "date": "2024-02-11", "categories": []}
{"id": "2410.10395", "title": "Improved Depth Estimation of Bayesian Neural Networks", "abstract": "  This paper proposes improvements over earlier work by Nazareth and Blei\n(2022) for estimating the depth of Bayesian neural networks. Here, we propose a\ndiscrete truncated normal distribution over the network depth to independently\nlearn its mean and variance. Posterior distributions are inferred by minimizing\nthe variational free energy, which balances the model complexity and accuracy.\nOur method improves test accuracy on the spiral data set and reduces the\nvariance in posterior depth estimates.\n", "date": "2024-10-14", "categories": []}
{"id": "2210.09134", "title": "Principled Pruning of Bayesian Neural Networks through Variational Free\n  Energy Minimization", "abstract": "  Bayesian model reduction provides an efficient approach for comparing the\nperformance of all nested sub-models of a model, without re-evaluating any of\nthese sub-models. Until now, Bayesian model reduction has been applied mainly\nin the computational neuroscience community on simple models. In this paper, we\nformulate and apply Bayesian model reduction to perform principled pruning of\nBayesian neural networks, based on variational free energy minimization. Direct\napplication of Bayesian model reduction, however, gives rise to approximation\nerrors. Therefore, a novel iterative pruning algorithm is presented to\nalleviate the problems arising with naive Bayesian model reduction, as\nsupported experimentally on the publicly available UCI datasets for different\ninference algorithms. This novel parameter pruning scheme solves the\nshortcomings of current state-of-the-art pruning methods that are used by the\nsignal processing community. The proposed approach has a clear stopping\ncriterion and minimizes the same objective that is used during training. Next\nto these benefits, our experiments indicate better model performance in\ncomparison to state-of-the-art pruning schemes.\n", "date": "2022-10-17", "categories": []}
{"id": "2310.10379", "title": "Revisiting Logistic-softmax Likelihood in Bayesian Meta-Learning for\n  Few-Shot Classification", "abstract": "  Meta-learning has demonstrated promising results in few-shot classification\n(FSC) by learning to solve new problems using prior knowledge. Bayesian methods\nare effective at characterizing uncertainty in FSC, which is crucial in\nhigh-risk fields. In this context, the logistic-softmax likelihood is often\nemployed as an alternative to the softmax likelihood in multi-class Gaussian\nprocess classification due to its conditional conjugacy property. However, the\ntheoretical property of logistic-softmax is not clear and previous research\nindicated that the inherent uncertainty of logistic-softmax leads to suboptimal\nperformance. To mitigate these issues, we revisit and redesign the\nlogistic-softmax likelihood, which enables control of the \\textit{a priori}\nconfidence level through a temperature parameter. Furthermore, we theoretically\nand empirically show that softmax can be viewed as a special case of\nlogistic-softmax and logistic-softmax induces a larger family of data\ndistribution than softmax. Utilizing modified logistic-softmax, we integrate\nthe data augmentation technique into the deep kernel based Gaussian process\nmeta-learning framework, and derive an analytical mean-field approximation for\ntask-specific updates. Our approach yields well-calibrated uncertainty\nestimates and achieves comparable or superior results on standard benchmark\ndatasets. Code is publicly available at\n\\url{https://github.com/keanson/revisit-logistic-softmax}.\n", "date": "2023-10-16", "categories": []}
{"id": "2406.17263", "title": "Efficient, Multimodal, and Derivative-Free Bayesian Inference With\n  Fisher-Rao Gradient Flows", "abstract": "  In this paper, we study efficient approximate sampling for probability\ndistributions known up to normalization constants. We specifically focus on a\nproblem class arising in Bayesian inference for large-scale inverse problems in\nscience and engineering applications. The computational challenges we address\nwith the proposed methodology are: (i) the need for repeated evaluations of\nexpensive forward models; (ii) the potential existence of multiple modes; and\n(iii) the fact that gradient of, or adjoint solver for, the forward model might\nnot be feasible.\n  While existing Bayesian inference methods meet some of these challenges\nindividually, we propose a framework that tackles all three systematically. Our\napproach builds upon the Fisher-Rao gradient flow in probability space,\nyielding a dynamical system for probability densities that converges towards\nthe target distribution at a uniform exponential rate. This rapid convergence\nis advantageous for the computational burden outlined in (i). We apply Gaussian\nmixture approximations with operator splitting techniques to simulate the flow\nnumerically; the resulting approximation can capture multiple modes thus\naddressing (ii). Furthermore, we employ the Kalman methodology to facilitate a\nderivative-free update of these Gaussian components and their respective\nweights, addressing the issue in (iii).\n  The proposed methodology results in an efficient derivative-free sampler\nflexible enough to handle multi-modal distributions: Gaussian Mixture Kalman\nInversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically\nand numerically in several experiments with multimodal target distributions,\nincluding proof-of-concept and two-dimensional examples, as well as a\nlarge-scale application: recovering the Navier-Stokes initial condition from\nsolution data at positive times.\n", "date": "2024-06-25", "categories": []}
{"id": "2409.16632", "title": "Functional Stochastic Gradient MCMC for Bayesian Neural Networks", "abstract": "  Classical parameter-space Bayesian inference for Bayesian neural networks\n(BNNs) suffers from several unresolved prior issues, such as knowledge encoding\nintractability and pathological behaviours in deep networks, which can lead to\nimproper posterior inference. To address these issues, functional Bayesian\ninference has recently been proposed leveraging functional priors, such as the\nemerging functional variational inference. In addition to variational methods,\nstochastic gradient Markov Chain Monte Carlo (MCMC) is another scalable and\neffective inference method for BNNs to asymptotically generate samples from the\ntrue posterior by simulating continuous dynamics. However, existing MCMC\nmethods perform solely in parameter space and inherit the unresolved prior\nissues, while extending these dynamics to function space is a non-trivial\nundertaking. In this paper, we introduce novel functional MCMC schemes,\nincluding stochastic gradient versions, based on newly designed diffusion\ndynamics that can incorporate more informative functional priors. Moreover, we\nprove that the stationary measure of these functional dynamics is the target\nposterior over functions. Our functional MCMC schemes demonstrate improved\nperformance in both predictive accuracy and uncertainty quantification on\nseveral tasks compared to naive parameter-space MCMC and functional variational\ninference.\n", "date": "2024-09-25", "categories": []}
{"id": "2409.09614", "title": "HJ-sampler: A Bayesian sampler for inverse problems of a stochastic\n  process by leveraging Hamilton-Jacobi PDEs and score-based generative models", "abstract": "  The interplay between stochastic processes and optimal control has been\nextensively explored in the literature. With the recent surge in the use of\ndiffusion models, stochastic processes have increasingly been applied to sample\ngeneration. This paper builds on the log transform, known as the Cole-Hopf\ntransform in Brownian motion contexts, and extends it within a more abstract\nframework that includes a linear operator. Within this framework, we found that\nthe well-known relationship between the Cole-Hopf transform and optimal\ntransport is a particular instance where the linear operator acts as the\ninfinitesimal generator of a stochastic process. We also introduce a novel\nscenario where the linear operator is the adjoint of the generator, linking to\nBayesian inference under specific initial and terminal conditions. Leveraging\nthis theoretical foundation, we develop a new algorithm, named the HJ-sampler,\nfor Bayesian inference for the inverse problem of a stochastic differential\nequation with given terminal observations. The HJ-sampler involves two stages:\n(1) solving the viscous Hamilton-Jacobi partial differential equations, and (2)\nsampling from the associated stochastic optimal control problem. Our proposed\nalgorithm naturally allows for flexibility in selecting the numerical solver\nfor viscous HJ PDEs. We introduce two variants of the solver: the\nRiccati-HJ-sampler, based on the Riccati method, and the SGM-HJ-sampler, which\nutilizes diffusion models. We demonstrate the effectiveness and flexibility of\nthe proposed methods by applying them to solve Bayesian inverse problems\ninvolving various stochastic processes and prior distributions, including\napplications that address model misspecifications and quantifying model\nuncertainty.\n", "date": "2024-09-15", "categories": []}
{"id": "2410.05753", "title": "Pathwise Gradient Variance Reduction with Control Variates in\n  Variational Inference", "abstract": "  Variational inference in Bayesian deep learning often involves computing the\ngradient of an expectation that lacks a closed-form solution. In these cases,\npathwise and score-function gradient estimators are the most common approaches.\nThe pathwise estimator is often favoured for its substantially lower variance\ncompared to the score-function estimator, which typically requires variance\nreduction techniques. However, recent research suggests that even pathwise\ngradient estimators could benefit from variance reduction. In this work, we\nreview existing control-variates-based variance reduction methods for pathwise\ngradient estimators to assess their effectiveness. Notably, these methods often\nrely on integrand approximations and are applicable only to simple variational\nfamilies. To address this limitation, we propose applying zero-variance control\nvariates to pathwise gradient estimators. This approach offers the advantage of\nrequiring minimal assumptions about the variational distribution, other than\nbeing able to sample from it.\n", "date": "2024-10-08", "categories": []}
{"id": "2410.06349", "title": "Robust Domain Generalisation with Causal Invariant Bayesian Neural\n  Networks", "abstract": "  Deep neural networks can obtain impressive performance on various tasks under\nthe assumption that their training domain is identical to their target domain.\nPerformance can drop dramatically when this assumption does not hold. One\nexplanation for this discrepancy is the presence of spurious domain-specific\ncorrelations in the training data that the network exploits. Causal mechanisms,\nin the other hand, can be made invariant under distribution changes as they\nallow disentangling the factors of distribution underlying the data generation.\nYet, learning causal mechanisms to improve out-of-distribution generalisation\nremains an under-explored area. We propose a Bayesian neural architecture that\ndisentangles the learning of the the data distribution from the inference\nprocess mechanisms. We show theoretically and experimentally that our model\napproximates reasoning under causal interventions. We demonstrate the\nperformance of our method, outperforming point estimate-counterparts, on\nout-of-distribution image recognition tasks where the data distribution acts as\nstrong adversarial confounders.\n", "date": "2024-10-08", "categories": []}
{"id": "2410.01687", "title": "Uncertainty Quantification with Bayesian Higher Order ReLU KANs", "abstract": "  We introduce the first method of uncertainty quantification in the domain of\nKolmogorov-Arnold Networks, specifically focusing on (Higher Order) ReLUKANs to\nenhance computational efficiency given the computational demands of Bayesian\nmethods. The method we propose is general in nature, providing access to both\nepistemic and aleatoric uncertainties. It is also capable of generalization to\nother various basis functions. We validate our method through a series of\nclosure tests, including simple one-dimensional functions and application to\nthe domain of (Stochastic) Partial Differential Equations. Referring to the\nlatter, we demonstrate the method's ability to correctly identify functional\ndependencies introduced through the inclusion of a stochastic term. The code\nsupporting this work can be found at\nhttps://github.com/wmdataphys/Bayesian-HR-KAN\n", "date": "2024-10-02", "categories": []}
{"id": "2410.01793", "title": "Thermodynamic Bayesian Inference", "abstract": "  A fully Bayesian treatment of complicated predictive models (such as deep\nneural networks) would enable rigorous uncertainty quantification and the\nautomation of higher-level tasks including model selection. However, the\nintractability of sampling Bayesian posteriors over many parameters inhibits\nthe use of Bayesian methods where they are most needed. Thermodynamic computing\nhas emerged as a paradigm for accelerating operations used in machine learning,\nsuch as matrix inversion, and is based on the mapping of Langevin equations to\nthe dynamics of noisy physical systems. Hence, it is natural to consider the\nimplementation of Langevin sampling algorithms on thermodynamic devices. In\nthis work we propose electronic analog devices that sample from Bayesian\nposteriors by realizing Langevin dynamics physically. Circuit designs are given\nfor sampling the posterior of a Gaussian-Gaussian model and for Bayesian\nlogistic regression, and are validated by simulations. It is shown, under\nreasonable assumptions, that the Bayesian posteriors for these models can be\nsampled in time scaling with $\\ln(d)$, where $d$ is dimension. For the\nGaussian-Gaussian model, the energy cost is shown to scale with $ d \\ln(d)$.\nThese results highlight the potential for fast, energy-efficient Bayesian\ninference using thermodynamic computing.\n", "date": "2024-10-02", "categories": []}
{"id": "2409.17085", "title": "Parameter-efficient Bayesian Neural Networks for Uncertainty-aware Depth\n  Estimation", "abstract": "  State-of-the-art computer vision tasks, like monocular depth estimation\n(MDE), rely heavily on large, modern Transformer-based architectures. However,\ntheir application in safety-critical domains demands reliable predictive\nperformance and uncertainty quantification. While Bayesian neural networks\nprovide a conceptually simple approach to serve those requirements, they suffer\nfrom the high dimensionality of the parameter space. Parameter-efficient\nfine-tuning (PEFT) methods, in particular low-rank adaptations (LoRA), have\nemerged as a popular strategy for adapting large-scale models to down-stream\ntasks by performing parameter inference on lower-dimensional subspaces. In this\nwork, we investigate the suitability of PEFT methods for subspace Bayesian\ninference in large-scale Transformer-based vision models. We show that, indeed,\ncombining BitFit, DiffFit, LoRA, and CoLoRA, a novel LoRA-inspired PEFT method,\nwith Bayesian inference enables more robust and reliable predictive performance\nin MDE.\n", "date": "2024-09-25", "categories": []}
{"id": "2210.09141", "title": "Data Subsampling for Bayesian Neural Networks", "abstract": "  Markov Chain Monte Carlo (MCMC) algorithms do not scale well for large\ndatasets leading to difficulties in Neural Network posterior sampling. In this\npaper, we propose Penalty Bayesian Neural Networks - PBNNs, as a new algorithm\nthat allows the evaluation of the likelihood using subsampled batch data\n(mini-batches) in a Bayesian inference context towards addressing scalability.\nPBNN avoids the biases inherent in other naive subsampling techniques by\nincorporating a penalty term as part of a generalization of the Metropolis\nHastings algorithm. We show that it is straightforward to integrate PBNN with\nexisting MCMC frameworks, as the variance of the loss function merely reduces\nthe acceptance probability. By comparing with alternative sampling strategies\non both synthetic data and the MNIST dataset, we demonstrate that PBNN achieves\ngood predictive performance even for small mini-batch sizes of data. We show\nthat PBNN provides a novel approach for calibrating the predictive distribution\nby varying the mini-batch size, significantly reducing predictive\noverconfidence.\n", "date": "2022-10-17", "categories": []}
{"id": "2409.12301", "title": "Amortized Variational Inference for Deep Gaussian Processes", "abstract": "  Gaussian processes (GPs) are Bayesian nonparametric models for function\napproximation with principled predictive uncertainty estimates. Deep Gaussian\nprocesses (DGPs) are multilayer generalizations of GPs that can represent\ncomplex marginal densities as well as complex mappings. As exact inference is\neither computationally prohibitive or analytically intractable in GPs and\nextensions thereof, some existing methods resort to variational inference (VI)\ntechniques for tractable approximations. However, the expressivity of\nconventional approximate GP models critically relies on independent inducing\nvariables that might not be informative enough for some problems. In this work\nwe introduce amortized variational inference for DGPs, which learns an\ninference function that maps each observation to variational parameters. The\nresulting method enjoys a more expressive prior conditioned on fewer input\ndependent inducing variables and a flexible amortized marginal posterior that\nis able to model more complicated functions. We show with theoretical reasoning\nand experimental results that our method performs similarly or better than\nprevious approaches at less computational cost.\n", "date": "2024-09-18", "categories": []}
{"id": "2405.00385", "title": "Variational Bayesian Methods for a Tree-Structured Stick-Breaking\n  Process Mixture of Gaussians by Application of the Bayes Codes for Context\n  Tree Models", "abstract": "  The tree-structured stick-breaking process (TS-SBP) mixture model is a\nnon-parametric Bayesian model that can represent tree-like hierarchical\nstructures among the mixture components. For TS-SBP mixture models, only a\nMarkov chain Monte Carlo (MCMC) method has been proposed and any variational\nBayesian (VB) methods has not been proposed. In general, MCMC methods are\ncomputationally more expensive than VB methods. Therefore, we require a large\ncomputational cost to learn the TS-SBP mixture model. In this paper, we propose\na learning algorithm with less computational cost for the TS-SBP mixture of\nGaussians by using the VB method under an assumption of finite tree width and\ndepth. When constructing such VB method, the main challenge is efficient\ncalculation of a sum over all possible trees. To solve this challenge, we\nutilizes a subroutine in the Bayes coding algorithm for context tree models. We\nconfirm the computational efficiency of our VB method through an experiments on\na benchmark dataset.\n", "date": "2024-05-01", "categories": []}
{"id": "2406.03082", "title": "Learning Solutions of Stochastic Optimization Problems with Bayesian\n  Neural Networks", "abstract": "  Mathematical solvers use parametrized Optimization Problems (OPs) as inputs\nto yield optimal decisions. In many real-world settings, some of these\nparameters are unknown or uncertain. Recent research focuses on predicting the\nvalue of these unknown parameters using available contextual features, aiming\nto decrease decision regret by adopting end-to-end learning approaches.\nHowever, these approaches disregard prediction uncertainty and therefore make\nthe mathematical solver susceptible to provide erroneous decisions in case of\nlow-confidence predictions. We propose a novel framework that models prediction\nuncertainty with Bayesian Neural Networks (BNNs) and propagates this\nuncertainty into the mathematical solver with a Stochastic Programming\ntechnique. The differentiable nature of BNNs and differentiable mathematical\nsolvers allow for two different learning approaches: In the Decoupled learning\napproach, we update the BNN weights to increase the quality of the predictions'\ndistribution of the OP parameters, while in the Combined learning approach, we\nupdate the weights aiming to directly minimize the expected OP's cost function\nin a stochastic end-to-end fashion. We do an extensive evaluation using\nsynthetic data with various noise properties and a real dataset, showing that\ndecisions regret are generally lower (better) with both proposed methods.\n", "date": "2024-06-05", "categories": []}
{"id": "2408.06425", "title": "Bayesian Learning in a Nonlinear Multiscale State-Space Model", "abstract": "  The ubiquity of multiscale interactions in complex systems is\nwell-recognized, with development and heredity serving as a prime example of\nhow processes at different temporal scales influence one another. This work\nintroduces a novel multiscale state-space model to explore the dynamic\ninterplay between systems interacting across different time scales, with\nfeedback between each scale. We propose a Bayesian learning framework to\nestimate unknown states by learning the unknown process noise covariances\nwithin this multiscale model. We develop a Particle Gibbs with Ancestor\nSampling (PGAS) algorithm for inference and demonstrate through simulations the\nefficacy of our approach.\n", "date": "2024-08-12", "categories": []}
{"id": "2409.01293", "title": "Extracting Signal out of Chaos: Advancements on MAGI for Bayesian\n  Analysis of Dynamical Systems", "abstract": "  This work builds off the manifold-constrained Gaussian process inference\n(MAGI) method for Bayesian parameter inference and trajectory reconstruction of\nODE-based dynamical systems, focusing primarily on sparse and noisy data\nconditions. First, we introduce Pilot MAGI (pMAGI), a novel methodological\nupgrade on the base MAGI method that confers significantly-improved numerical\nstability, parameter inference, and trajectory reconstruction. Second, we\ndemonstrate, for the first time to our knowledge, how one can combine\nMAGI-based methods with dynamical systems theory to provide probabilistic\nclassifications of whether a system is stable or chaotic. Third, we demonstrate\nhow pMAGI performs favorably in many settings against much more\ncomputationally-expensive and overparameterized methods. Fourth, we introduce\nPilot MAGI Sequential Prediction (PMSP), a novel method building upon pMAGI\nthat allows one to predict the trajectory of ODE-based dynamical systems\nmultiple time steps into the future, given only sparse and noisy observations.\nWe show that PMSP can output accurate future predictions even on chaotic\ndynamical systems and significantly outperform PINN-based methods. Overall, we\ncontribute to the literature two novel methods, pMAGI and PMSP, that serve as\nBayesian, uncertainty-quantified competitors to the Physics-Informed Neural\nNetwork.\n", "date": "2024-08-20", "categories": []}
{"id": "2304.01762", "title": "Incorporating Unlabelled Data into Bayesian Neural Networks", "abstract": "  Conventional Bayesian Neural Networks (BNNs) are unable to leverage\nunlabelled data to improve their predictions. To overcome this limitation, we\nintroduce Self-Supervised Bayesian Neural Networks, which use unlabelled data\nto learn models with suitable prior predictive distributions. This is achieved\nby leveraging contrastive pretraining techniques and optimising a variational\nlower bound. We then show that the prior predictive distributions of\nself-supervised BNNs capture problem semantics better than conventional BNN\npriors. In turn, our approach offers improved predictive performance over\nconventional BNNs, especially in low-budget regimes.\n", "date": "2023-04-04", "categories": []}
{"id": "2309.01949", "title": "Variational Bayesian Imaging with an Efficient Surrogate Score-based\n  Prior", "abstract": "  We propose a surrogate function for efficient yet principled use of\nscore-based priors in Bayesian imaging. We consider ill-posed inverse imaging\nproblems in which one aims for a clean image posterior given incomplete or\nnoisy measurements. Since the measurements do not uniquely determine a true\nimage, a prior is needed to constrain the solution space. Recent work turned\nscore-based diffusion models into principled priors for solving ill-posed\nimaging problems by appealing to an ODE-based log-probability function.\nHowever, evaluating the ODE is computationally inefficient and inhibits\nposterior estimation of high-dimensional images. Our proposed surrogate prior\nis based on the evidence lower bound of a score-based diffusion model. We\ndemonstrate the surrogate prior on variational inference for efficient\napproximate posterior sampling of large images. Compared to the exact prior in\nprevious work, our surrogate accelerates optimization of the variational image\ndistribution by at least two orders of magnitude. We also find that our\nprincipled approach gives more accurate posterior estimation than\nnon-variational diffusion-based approaches that involve hyperparameter-tuning\nat inference. Our work establishes a practical path forward for using\nscore-based diffusion models as general-purpose image priors.\n", "date": "2023-09-05", "categories": []}
{"id": "2310.11122", "title": "Sensitivity-Aware Amortized Bayesian Inference", "abstract": "  Sensitivity analyses reveal the influence of various modeling choices on the\noutcomes of statistical analyses. While theoretically appealing, they are\noverwhelmingly inefficient for complex Bayesian models. In this work, we\npropose sensitivity-aware amortized Bayesian inference (SA-ABI), a multifaceted\napproach to efficiently integrate sensitivity analyses into simulation-based\ninference with neural networks. First, we utilize weight sharing to encode the\nstructural similarities between alternative likelihood and prior specifications\nin the training process with minimal computational overhead. Second, we\nleverage the rapid inference of neural networks to assess sensitivity to data\nperturbations and preprocessing steps. In contrast to most other Bayesian\napproaches, both steps circumvent the costly bottleneck of refitting the model\nfor each choice of likelihood, prior, or data set. Finally, we propose to use\ndeep ensembles to detect sensitivity arising from unreliable approximation\n(e.g., due to model misspecification). We demonstrate the effectiveness of our\nmethod in applied modeling problems, ranging from disease outbreak dynamics and\nglobal warming thresholds to human decision-making. Our results support\nsensitivity-aware inference as a default choice for amortized Bayesian\nworkflows, automatically providing modelers with insights into otherwise hidden\ndimensions.\n", "date": "2023-10-17", "categories": []}
{"id": "2408.14935", "title": "Quotient Normalized Maximum Likelihood Criterion for Learning Bayesian\n  Network Structures", "abstract": "  We introduce an information theoretic criterion for Bayesian network\nstructure learning which we call quotient normalized maximum likelihood (qNML).\nIn contrast to the closely related factorized normalized maximum likelihood\ncriterion, qNML satisfies the property of score equivalence. It is also\ndecomposable and completely free of adjustable hyperparameters. For practical\ncomputations, we identify a remarkably accurate approximation proposed earlier\nby Szpankowski and Weinberger. Experiments on both simulated and real data\ndemonstrate that the new criterion leads to parsimonious models with good\npredictive accuracy.\n", "date": "2024-08-27", "categories": []}
{"id": "2408.15136", "title": "Low-Budget Simulation-Based Inference with Bayesian Neural Networks", "abstract": "  Simulation-based inference methods have been shown to be inaccurate in the\ndata-poor regime, when training simulations are limited or expensive. Under\nthese circumstances, the inference network is particularly prone to\noverfitting, and using it without accounting for the computational uncertainty\narising from the lack of identifiability of the network weights can lead to\nunreliable results. To address this issue, we propose using Bayesian neural\nnetworks in low-budget simulation-based inference, thereby explicitly\naccounting for the computational uncertainty of the posterior approximation. We\ndesign a family of Bayesian neural network priors that are tailored for\ninference and show that they lead to well-calibrated posteriors on tested\nbenchmarks, even when as few as $O(10)$ simulations are available. This opens\nup the possibility of performing reliable simulation-based inference using very\nexpensive simulators, as we demonstrate on a problem from the field of\ncosmology where single simulations are computationally expensive. We show that\nBayesian neural networks produce informative and well-calibrated posterior\nestimates with only a few hundred simulations.\n", "date": "2024-08-27", "categories": []}
{"id": "2304.02595", "title": "Bayesian neural networks via MCMC: a Python-based tutorial", "abstract": "  Bayesian inference provides a methodology for parameter estimation and\nuncertainty quantification in machine learning and deep learning methods.\nVariational inference and Markov Chain Monte-Carlo (MCMC) sampling methods are\nused to implement Bayesian inference. In the past three decades, MCMC sampling\nmethods have faced some challenges in being adapted to larger models (such as\nin deep learning) and big data problems. Advanced proposal distributions that\nincorporate gradients, such as a Langevin proposal distribution, provide a\nmeans to address some of the limitations of MCMC sampling for Bayesian neural\nnetworks. Furthermore, MCMC methods have typically been constrained to\nstatisticians and currently not well-known among deep learning researchers. We\npresent a tutorial for MCMC methods that covers simple Bayesian linear and\nlogistic models, and Bayesian neural networks. The aim of this tutorial is to\nbridge the gap between theory and implementation via coding, given a general\nsparsity of libraries and tutorials to this end. This tutorial provides code in\nPython with data and instructions that enable their use and extension. We\nprovide results for some benchmark problems showing the strengths and\nweaknesses of implementing the respective Bayesian models via MCMC. We\nhighlight the challenges in sampling multi-modal posterior distributions for\nthe case of Bayesian neural networks and the need for further improvement of\nconvergence diagnosis methods.\n", "date": "2023-04-02", "categories": []}
{"id": "2308.09104", "title": "Spike-and-slab shrinkage priors for structurally sparse Bayesian neural\n  networks", "abstract": "  Network complexity and computational efficiency have become increasingly\nsignificant aspects of deep learning. Sparse deep learning addresses these\nchallenges by recovering a sparse representation of the underlying target\nfunction by reducing heavily over-parameterized deep neural networks.\nSpecifically, deep neural architectures compressed via structured sparsity\n(e.g. node sparsity) provide low latency inference, higher data throughput, and\nreduced energy consumption. In this paper, we explore two well-established\nshrinkage techniques, Lasso and Horseshoe, for model compression in Bayesian\nneural networks. To this end, we propose structurally sparse Bayesian neural\nnetworks which systematically prune excessive nodes with (i) Spike-and-Slab\nGroup Lasso (SS-GL), and (ii) Spike-and-Slab Group Horseshoe (SS-GHS) priors,\nand develop computationally tractable variational inference including\ncontinuous relaxation of Bernoulli variables. We establish the contraction\nrates of the variational posterior of our proposed models as a function of the\nnetwork topology, layer-wise node cardinalities, and bounds on the network\nweights. We empirically demonstrate the competitive performance of our models\ncompared to the baseline models in prediction accuracy, model compression, and\ninference latency.\n", "date": "2023-08-17", "categories": []}
{"id": "2206.00794", "title": "Sequential Bayesian Neural Subnetwork Ensembles", "abstract": "  Deep ensembles have emerged as a powerful technique for improving predictive\nperformance and enhancing model robustness across various applications by\nleveraging model diversity. However, traditional deep ensemble methods are\noften computationally expensive and rely on deterministic models, which may\nlimit their flexibility. Additionally, while sparse subnetworks of dense models\nhave shown promise in matching the performance of their dense counterparts and\neven enhancing robustness, existing methods for inducing sparsity typically\nincur training costs comparable to those of training a single dense model, as\nthey either gradually prune the network during training or apply thresholding\npost-training. In light of these challenges, we propose an approach for\nsequential ensembling of dynamic Bayesian neural subnetworks that consistently\nmaintains reduced model complexity throughout the training process while\ngenerating diverse ensembles in a single forward pass. Our approach involves an\ninitial exploration phase to identify high-performing regions within the\nparameter space, followed by multiple exploitation phases that take advantage\nof the compactness of the sparse model. These exploitation phases quickly\nconverge to different minima in the energy landscape, corresponding to\nhigh-performing subnetworks that together form a diverse and robust ensemble.\nWe empirically demonstrate that our proposed approach outperforms traditional\ndense and sparse deterministic and Bayesian ensemble models in terms of\nprediction accuracy, uncertainty estimation, out-of-distribution detection, and\nadversarial robustness.\n", "date": "2022-06-01", "categories": []}
{"id": "2408.09140", "title": "Learning to Explore for Stochastic Gradient MCMC", "abstract": "  Bayesian Neural Networks(BNNs) with high-dimensional parameters pose a\nchallenge for posterior inference due to the multi-modality of the posterior\ndistributions. Stochastic Gradient MCMC(SGMCMC) with cyclical learning rate\nscheduling is a promising solution, but it requires a large number of sampling\nsteps to explore high-dimensional multi-modal posteriors, making it\ncomputationally expensive. In this paper, we propose a meta-learning strategy\nto build \\gls{sgmcmc} which can efficiently explore the multi-modal target\ndistributions. Our algorithm allows the learned SGMCMC to quickly explore the\nhigh-density region of the posterior landscape. Also, we show that this\nexploration property is transferrable to various tasks, even for the ones\nunseen during a meta-training stage. Using popular image classification\nbenchmarks and a variety of downstream tasks, we demonstrate that our method\nsignificantly improves the sampling efficiency, achieving better performance\nthan vanilla \\gls{sgmcmc} without incurring significant computational overhead.\n", "date": "2024-08-17", "categories": []}
{"id": "2408.05496", "title": "Variational Inference Failures Under Model Symmetries: Permutation\n  Invariant Posteriors for Bayesian Neural Networks", "abstract": "  Weight space symmetries in neural network architectures, such as permutation\nsymmetries in MLPs, give rise to Bayesian neural network (BNN) posteriors with\nmany equivalent modes. This multimodality poses a challenge for variational\ninference (VI) techniques, which typically rely on approximating the posterior\nwith a unimodal distribution. In this work, we investigate the impact of weight\nspace permutation symmetries on VI. We demonstrate, both theoretically and\nempirically, that these symmetries lead to biases in the approximate posterior,\nwhich degrade predictive performance and posterior fit if not explicitly\naccounted for. To mitigate this behavior, we leverage the symmetric structure\nof the posterior and devise a symmetrization mechanism for constructing\npermutation invariant variational posteriors. We show that the symmetrized\ndistribution has a strictly better fit to the true posterior, and that it can\nbe trained using the original ELBO objective with a modified KL regularization\nterm. We demonstrate experimentally that our approach mitigates the\naforementioned biases and results in improved predictions and a higher ELBO.\n", "date": "2024-08-10", "categories": []}
{"id": "2408.03746", "title": "Flexible Bayesian Last Layer Models Using Implicit Priors and Diffusion\n  Posterior Sampling", "abstract": "  Bayesian Last Layer (BLL) models focus solely on uncertainty in the output\nlayer of neural networks, demonstrating comparable performance to more complex\nBayesian models. However, the use of Gaussian priors for last layer weights in\nBayesian Last Layer (BLL) models limits their expressive capacity when faced\nwith non-Gaussian, outlier-rich, or high-dimensional datasets. To address this\nshortfall, we introduce a novel approach that combines diffusion techniques and\nimplicit priors for variational learning of Bayesian last layer weights. This\nmethod leverages implicit distributions for modeling weight priors in BLL,\ncoupled with diffusion samplers for approximating true posterior predictions,\nthereby establishing a comprehensive Bayesian prior and posterior estimation\nstrategy. By delivering an explicit and computationally efficient variational\nlower bound, our method aims to augment the expressive abilities of BLL models,\nenhancing model accuracy, calibration, and out-of-distribution detection\nproficiency. Through detailed exploration and experimental validation, We\nshowcase the method's potential for improving predictive accuracy and\nuncertainty quantification while ensuring computational efficiency.\n", "date": "2024-08-07", "categories": []}
{"id": "2402.00809", "title": "Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI", "abstract": "  In the current landscape of deep learning research, there is a predominant\nemphasis on achieving high predictive accuracy in supervised tasks involving\nlarge image and language datasets. However, a broader perspective reveals a\nmultitude of overlooked metrics, tasks, and data types, such as uncertainty,\nactive and continual learning, and scientific data, that demand attention.\nBayesian deep learning (BDL) constitutes a promising avenue, offering\nadvantages across these diverse settings. This paper posits that BDL can\nelevate the capabilities of deep learning. It revisits the strengths of BDL,\nacknowledges existing challenges, and highlights some exciting research avenues\naimed at addressing these obstacles. Looking ahead, the discussion focuses on\npossible ways to combine large-scale foundation models with BDL to unlock their\nfull potential.\n", "date": "2024-02-01", "categories": []}
{"id": "2408.02706", "title": "Bayesian Kolmogorov Arnold Networks (Bayesian_KANs): A Probabilistic\n  Approach to Enhance Accuracy and Interpretability", "abstract": "  Because of its strong predictive skills, deep learning has emerged as an\nessential tool in many industries, including healthcare. Traditional deep\nlearning models, on the other hand, frequently lack interpretability and omit\nto take prediction uncertainty into account two crucial components of clinical\ndecision making. In order to produce explainable and uncertainty aware\npredictions, this study presents a novel framework called Bayesian Kolmogorov\nArnold Networks (BKANs), which combines the expressive capacity of Kolmogorov\nArnold Networks with Bayesian inference. We employ BKANs on two medical\ndatasets, which are widely used benchmarks for assessing machine learning\nmodels in medical diagnostics: the Pima Indians Diabetes dataset and the\nCleveland Heart Disease dataset. Our method provides useful insights into\nprediction confidence and decision boundaries and outperforms traditional deep\nlearning models in terms of prediction accuracy. Moreover, BKANs' capacity to\nrepresent aleatoric and epistemic uncertainty guarantees doctors receive more\nsolid and trustworthy decision support. Our Bayesian strategy improves the\ninterpretability of the model and considerably minimises overfitting, which is\nimportant for tiny and imbalanced medical datasets, according to experimental\nresults. We present possible expansions to further use BKANs in more\ncomplicated multimodal datasets and address the significance of these\ndiscoveries for future research in building reliable AI systems for healthcare.\nThis work paves the way for a new paradigm in deep learning model deployment in\nvital sectors where transparency and reliability are crucial.\n", "date": "2024-08-05", "categories": []}
{"id": "2407.20697", "title": "Weak neural variational inference for solving Bayesian inverse problems\n  without forward models: applications in elastography", "abstract": "  In this paper, we introduce a novel, data-driven approach for solving\nhigh-dimensional Bayesian inverse problems based on partial differential\nequations (PDEs), called Weak Neural Variational Inference (WNVI). The method\ncomplements real measurements with virtual observations derived from the\nphysical model. In particular, weighted residuals are employed as probes to the\ngoverning PDE in order to formulate and solve a Bayesian inverse problem\nwithout ever formulating nor solving a forward model. The formulation treats\nthe state variables of the physical model as latent variables, inferred using\nStochastic Variational Inference (SVI), along with the usual unknowns. The\napproximate posterior employed uses neural networks to approximate the inverse\nmapping from state variables to the unknowns. We illustrate the proposed method\nin a biomedical setting where we infer spatially varying material properties\nfrom noisy tissue deformation data. We demonstrate that WNVI is not only as\naccurate and more efficient than traditional methods that rely on repeatedly\nsolving the (non)linear forward problem as a black-box, but it can also handle\nill-posed forward problems (e.g., with insufficient boundary conditions).\n", "date": "2024-07-30", "categories": []}
{"id": "2309.12095", "title": "Bayesian sparsification for deep neural networks with Bayesian model\n  reduction", "abstract": "  Deep learning's immense capabilities are often constrained by the complexity\nof its models, leading to an increasing demand for effective sparsification\ntechniques. Bayesian sparsification for deep learning emerges as a crucial\napproach, facilitating the design of models that are both computationally\nefficient and competitive in terms of performance across various deep learning\napplications. The state-of-the-art -- in Bayesian sparsification of deep neural\nnetworks -- combines structural shrinkage priors on model weights with an\napproximate inference scheme based on stochastic variational inference.\nHowever, model inversion of the full generative model is exceptionally\ncomputationally demanding, especially when compared to standard deep learning\nof point estimates. In this context, we advocate for the use of Bayesian model\nreduction (BMR) as a more efficient alternative for pruning of model weights.\nAs a generalization of the Savage-Dickey ratio, BMR allows a post-hoc\nelimination of redundant model weights based on the posterior estimates under a\nstraightforward (non-hierarchical) generative model. Our comparative study\nhighlights the advantages of the BMR method relative to established approaches\nbased on hierarchical horseshoe priors over model weights. We illustrate the\npotential of BMR across various deep learning architectures, from classical\nnetworks like LeNet to modern frameworks such as Vision Transformers and\nMLP-Mixers.\n", "date": "2023-09-21", "categories": []}
{"id": "2407.19287", "title": "Bayesian meta learning for trustworthy uncertainty quantification", "abstract": "  We consider the problem of Bayesian regression with trustworthy uncertainty\nquantification. We define that the uncertainty quantification is trustworthy if\nthe ground truth can be captured by intervals dependent on the predictive\ndistributions with a pre-specified probability. Furthermore, we propose,\nTrust-Bayes, a novel optimization framework for Bayesian meta learning which is\ncognizant of trustworthy uncertainty quantification without explicit\nassumptions on the prior model/distribution of the functions. We characterize\nthe lower bounds of the probabilities of the ground truth being captured by the\nspecified intervals and analyze the sample complexity with respect to the\nfeasible probability for trustworthy uncertainty quantification. Monte Carlo\nsimulation of a case study using Gaussian process regression is conducted for\nverification and comparison with the Meta-prior algorithm.\n", "date": "2024-07-27", "categories": []}
{"id": "2407.18802", "title": "Log-Concave Coupling for Sampling Neural Net Posteriors", "abstract": "  In this work, we present a sampling algorithm for single hidden layer neural\nnetworks. This algorithm is built upon a recursive series of Bayesian\nposteriors using a method we call Greedy Bayes. Sampling of the Bayesian\nposterior for neuron weight vectors $w$ of dimension $d$ is challenging because\nof its multimodality. Our algorithm to tackle this problem is based on a\ncoupling of the posterior density for $w$ with an auxiliary random variable\n$\\xi$.\n  The resulting reverse conditional $w|\\xi$ of neuron weights given auxiliary\nrandom variable is shown to be log concave. In the construction of the\nposterior distributions we provide some freedom in the choice of the prior. In\nparticular, for Gaussian priors on $w$ with suitably small variance, the\nresulting marginal density of the auxiliary variable $\\xi$ is proven to be\nstrictly log concave for all dimensions $d$. For a uniform prior on the unit\n$\\ell_1$ ball, evidence is given that the density of $\\xi$ is again strictly\nlog concave for sufficiently large $d$.\n  The score of the marginal density of the auxiliary random variable $\\xi$ is\ndetermined by an expectation over $w|\\xi$ and thus can be computed by various\nrapidly mixing Markov Chain Monte Carlo methods. Moreover, the computation of\nthe score of $\\xi$ permits methods of sampling $\\xi$ by a stochastic diffusion\n(Langevin dynamics) with drift function built from this score. With such\ndynamics, information-theoretic methods pioneered by Bakry and Emery show that\naccurate sampling of $\\xi$ is obtained rapidly when its density is indeed\nstrictly log-concave. After which, one more draw from $w|\\xi$, produces neuron\nweights $w$ whose marginal distribution is from the desired posterior.\n", "date": "2024-07-26", "categories": []}
{"id": "2407.17033", "title": "Sparse Inducing Points in Deep Gaussian Processes: Enhancing Modeling\n  with Denoising Diffusion Variational Inference", "abstract": "  Deep Gaussian processes (DGPs) provide a robust paradigm for Bayesian deep\nlearning. In DGPs, a set of sparse integration locations called inducing points\nare selected to approximate the posterior distribution of the model. This is\ndone to reduce computational complexity and improve model efficiency. However,\ninferring the posterior distribution of inducing points is not straightforward.\nTraditional variational inference approaches to posterior approximation often\nlead to significant bias. To address this issue, we propose an alternative\nmethod called Denoising Diffusion Variational Inference (DDVI) that uses a\ndenoising diffusion stochastic differential equation (SDE) to generate\nposterior samples of inducing variables. We rely on score matching methods for\ndenoising diffusion model to approximate score functions with a neural network.\nFurthermore, by combining classical mathematical theory of SDEs with the\nminimization of KL divergence between the approximate and true processes, we\npropose a novel explicit variational lower bound for the marginal likelihood\nfunction of DGP. Through experiments on various datasets and comparisons with\nbaseline methods, we empirically demonstrate the effectiveness of DDVI for\nposterior inference of inducing points for DGP models.\n", "date": "2024-07-24", "categories": []}
{"id": "2407.17296", "title": "Enhanced SMC$^2$: Leveraging Gradient Information from Differentiable\n  Particle Filters Within Langevin Proposals", "abstract": "  Sequential Monte Carlo Squared (SMC$^2$) is a Bayesian method which can infer\nthe states and parameters of non-linear, non-Gaussian state-space models. The\nstandard random-walk proposal in SMC$^2$ faces challenges, particularly with\nhigh-dimensional parameter spaces. This study outlines a novel approach by\nharnessing first-order gradients derived from a Common Random Numbers -\nParticle Filter (CRN-PF) using PyTorch. The resulting gradients can be\nleveraged within a Langevin proposal without accept/reject. Including Langevin\ndynamics within the proposal can result in a higher effective sample size and\nmore accurate parameter estimates when compared with the random-walk. The\nresulting algorithm is parallelized on distributed memory using Message Passing\nInterface (MPI) and runs in $\\mathcal{O}(\\log_2N)$ time complexity. Utilizing\n64 computational cores we obtain a 51x speed-up when compared to a single core.\nA GitHub link is given which provides access to the code.\n", "date": "2024-07-24", "categories": []}
{"id": "2310.04395", "title": "Leveraging Self-Consistency for Data-Efficient Amortized Bayesian\n  Inference", "abstract": "  We propose a method to improve the efficiency and accuracy of amortized\nBayesian inference by leveraging universal symmetries in the joint\nprobabilistic model of parameters and data. In a nutshell, we invert Bayes'\ntheorem and estimate the marginal likelihood based on approximate\nrepresentations of the joint model. Upon perfect approximation, the marginal\nlikelihood is constant across all parameter values by definition. However,\nerrors in approximate inference lead to undesirable variance in the marginal\nlikelihood estimates across different parameter values. We penalize violations\nof this symmetry with a \\textit{self-consistency loss} which significantly\nimproves the quality of approximate inference in low data regimes and can be\nused to augment the training of popular neural density estimators. We apply our\nmethod to a number of synthetic problems and realistic scientific models,\ndiscovering notable advantages in the context of both neural posterior and\nlikelihood approximation.\n", "date": "2023-10-06", "categories": []}
{"id": "2405.03425", "title": "Gaussian Stochastic Weight Averaging for Bayesian Low-Rank Adaptation of\n  Large Language Models", "abstract": "  Fine-tuned Large Language Models (LLMs) often suffer from overconfidence and\npoor calibration, particularly when fine-tuned on small datasets. To address\nthese challenges, we propose a simple combination of Low-Rank Adaptation (LoRA)\nwith Gaussian Stochastic Weight Averaging (SWAG), facilitating approximate\nBayesian inference in LLMs. Through extensive testing across several Natural\nLanguage Processing (NLP) benchmarks, we demonstrate that our straightforward\nand computationally efficient approach improves model generalization and\ncalibration competitively with comparable, more sophisticated methods for\nBayesian inference in LLMs. We further show that our method exhibits greater\nrobustness against distribution shift, as reflected in its improved performance\non out-of-distribution tasks.\n", "date": "2024-05-06", "categories": []}
{"id": "2306.02729", "title": "Gibbs Sampling the Posterior of Neural Networks", "abstract": "  In this paper, we study sampling from a posterior derived from a neural\nnetwork. We propose a new probabilistic model consisting of adding noise at\nevery pre- and post-activation in the network, arguing that the resulting\nposterior can be sampled using an efficient Gibbs sampler. For small models,\nthe Gibbs sampler attains similar performances as the state-of-the-art Markov\nchain Monte Carlo (MCMC) methods, such as the Hamiltonian Monte Carlo (HMC) or\nthe Metropolis adjusted Langevin algorithm (MALA), both on real and synthetic\ndata. By framing our analysis in the teacher-student setting, we introduce a\nthermalization criterion that allows us to detect when an algorithm, when run\non data with synthetic labels, fails to sample from the posterior. The\ncriterion is based on the fact that in the teacher-student setting we can\ninitialize an algorithm directly at equilibrium.\n", "date": "2023-06-05", "categories": []}
{"id": "2407.12751", "title": "Scalable Monte Carlo for Bayesian Learning", "abstract": "  This book aims to provide a graduate-level introduction to advanced topics in\nMarkov chain Monte Carlo (MCMC) algorithms, as applied broadly in the Bayesian\ncomputational context. Most, if not all of these topics (stochastic gradient\nMCMC, non-reversible MCMC, continuous time MCMC, and new techniques for\nconvergence assessment) have emerged as recently as the last decade, and have\ndriven substantial recent practical and theoretical advances in the field. A\nparticular focus is on methods that are scalable with respect to either the\namount of data, or the data dimension, motivated by the emerging high-priority\napplication areas in machine learning and AI.\n", "date": "2024-07-17", "categories": []}
{"id": "2402.03495", "title": "Partially Stochastic Infinitely Deep Bayesian Neural Networks", "abstract": "  In this paper, we present Partially Stochastic Infinitely Deep Bayesian\nNeural Networks, a novel family of architectures that integrates partial\nstochasticity into the framework of infinitely deep neural networks. Our new\nclass of architectures is designed to improve the computational efficiency of\nexisting architectures at training and inference time. To do this, we leverage\nthe advantages of partial stochasticity in the infinite-depth limit which\ninclude the benefits of full stochasticity e.g. robustness, uncertainty\nquantification, and memory efficiency, whilst improving their limitations\naround computational complexity. We present a variety of architectural\nconfigurations, offering flexibility in network design including different\nmethods for weight partition. We also provide mathematical guarantees on the\nexpressivity of our models by establishing that our network family qualifies as\nUniversal Conditional Distribution Approximators. Lastly, empirical evaluations\nacross multiple tasks show that our proposed architectures achieve better\ndownstream task performance and uncertainty quantification than their\ncounterparts while being significantly more efficient. The code can be found at\n\\url{https://github.com/Sergio20f/part_stoch_inf_deep}\n", "date": "2024-02-05", "categories": []}
{"id": "2407.06935", "title": "Bayesian Federated Learning with Hamiltonian Monte Carlo: Algorithm and\n  Theory", "abstract": "  This work introduces a novel and efficient Bayesian federated learning\nalgorithm, namely, the Federated Averaging stochastic Hamiltonian Monte Carlo\n(FA-HMC), for parameter estimation and uncertainty quantification. We establish\nrigorous convergence guarantees of FA-HMC on non-iid distributed data sets,\nunder the strong convexity and Hessian smoothness assumptions. Our analysis\ninvestigates the effects of parameter space dimension, noise on gradients and\nmomentum, and the frequency of communication (between the central node and\nlocal nodes) on the convergence and communication costs of FA-HMC. Beyond that,\nwe establish the tightness of our analysis by showing that the convergence rate\ncannot be improved even for continuous FA-HMC process. Moreover, extensive\nempirical studies demonstrate that FA-HMC outperforms the existing Federated\nAveraging-Langevin Monte Carlo (FA-LD) algorithm.\n", "date": "2024-07-09", "categories": []}
{"id": "2407.05684", "title": "Multi-Fidelity Bayesian Neural Network for Uncertainty Quantification in\n  Transonic Aerodynamic Loads", "abstract": "  Multi-fidelity models are becoming more prevalent in engineering,\nparticularly in aerospace, as they combine both the computational efficiency of\nlow-fidelity models with the high accuracy of higher-fidelity simulations.\nVarious state-of-the-art techniques exist for fusing data from different\nfidelity sources, including Co-Kriging and transfer learning in neural\nnetworks. This paper aims to implement a multi-fidelity Bayesian neural network\nmodel that applies transfer learning to fuse data generated by models at\ndifferent fidelities. Bayesian neural networks use probability distributions\nover network weights, enabling them to provide predictions along with estimates\nof their confidence. This approach harnesses the predictive and data fusion\ncapabilities of neural networks while also quantifying uncertainty. The results\ndemonstrate that the multi-fidelity Bayesian model outperforms the\nstate-of-the-art Co-Kriging in terms of overall accuracy and robustness on\nunseen data.\n", "date": "2024-07-08", "categories": []}
{"id": "2402.02964", "title": "Mixed Noise and Posterior Estimation with Conditional DeepGEM", "abstract": "  Motivated by indirect measurements and applications from nanometrology with a\nmixed noise model, we develop a novel algorithm for jointly estimating the\nposterior and the noise parameters in Bayesian inverse problems. We propose to\nsolve the problem by an expectation maximization (EM) algorithm. Based on the\ncurrent noise parameters, we learn in the E-step a conditional normalizing flow\nthat approximates the posterior. In the M-step, we propose to find the noise\nparameter updates again by an EM algorithm, which has analytical formulas. We\ncompare the training of the conditional normalizing flow with the forward and\nreverse KL, and show that our model is able to incorporate information from\nmany measurements, unlike previous approaches.\n", "date": "2024-02-05", "categories": []}
{"id": "2312.12616", "title": "Online Variational Sequential Monte Carlo", "abstract": "  Being the most classical generative model for serial data, state-space models\n(SSM) are fundamental in AI and statistical machine learning. In SSM, any form\nof parameter learning or latent state inference typically involves the\ncomputation of complex latent-state posteriors. In this work, we build upon the\nvariational sequential Monte Carlo (VSMC) method, which provides\ncomputationally efficient and accurate model parameter estimation and Bayesian\nlatent-state inference by combining particle methods and variational inference.\nWhile standard VSMC operates in the offline mode, by re-processing repeatedly a\ngiven batch of data, we distribute the approximation of the gradient of the\nVSMC surrogate ELBO in time using stochastic approximation, allowing for online\nlearning in the presence of streams of data. This results in an algorithm,\nonline VSMC, that is capable of performing efficiently, entirely on-the-fly,\nboth parameter estimation and particle proposal adaptation. In addition, we\nprovide rigorous theoretical results describing the algorithm's convergence\nproperties as the number of data tends to infinity as well as numerical\nillustrations of its excellent convergence properties and usefulness also in\nbatch-processing settings.\n", "date": "2023-12-19", "categories": []}
{"id": "2407.02721", "title": "Model and Feature Diversity for Bayesian Neural Networks in Mutual\n  Learning", "abstract": "  Bayesian Neural Networks (BNNs) offer probability distributions for model\nparameters, enabling uncertainty quantification in predictions. However, they\noften underperform compared to deterministic neural networks. Utilizing mutual\nlearning can effectively enhance the performance of peer BNNs. In this paper,\nwe propose a novel approach to improve BNNs performance through deep mutual\nlearning. The proposed approaches aim to increase diversity in both network\nparameter distributions and feature distributions, promoting peer networks to\nacquire distinct features that capture different characteristics of the input,\nwhich enhances the effectiveness of mutual learning. Experimental results\ndemonstrate significant improvements in the classification accuracy, negative\nlog-likelihood, and expected calibration error when compared to traditional\nmutual learning for BNNs.\n", "date": "2024-07-03", "categories": []}
{"id": "2211.15771", "title": "Approximate Gibbs Sampler for Efficient Inference of Hierarchical\n  Bayesian Models for Grouped Count Data", "abstract": "  Hierarchical Bayesian Poisson regression models (HBPRMs) provide a flexible\nmodeling approach of the relationship between predictors and count response\nvariables. The applications of HBPRMs to large-scale datasets require efficient\ninference algorithms due to the high computational cost of inferring many model\nparameters based on random sampling. Although Markov Chain Monte Carlo (MCMC)\nalgorithms have been widely used for Bayesian inference, sampling using this\nclass of algorithms is time-consuming for applications with large-scale data\nand time-sensitive decision-making, partially due to the non-conjugacy of many\nmodels. To overcome this limitation, this research develops an approximate\nGibbs sampler (AGS) to efficiently learn the HBPRMs while maintaining the\ninference accuracy. In the proposed sampler, the data likelihood is\napproximated with Gaussian distribution such that the conditional posterior of\nthe coefficients has a closed-form solution. Numerical experiments using real\nand synthetic datasets with small and large counts demonstrate the superior\nperformance of AGS in comparison to the state-of-the-art sampling algorithm,\nespecially for large datasets.\n", "date": "2022-11-28", "categories": []}
{"id": "2405.17666", "title": "Structured Partial Stochasticity in Bayesian Neural Networks", "abstract": "  Bayesian neural network posterior distributions have a great number of modes\nthat correspond to the same network function. The abundance of such modes can\nmake it difficult for approximate inference methods to do their job. Recent\nwork has demonstrated the benefits of partial stochasticity for approximate\ninference in Bayesian neural networks; inference can be less costly and\nperformance can sometimes be improved. I propose a structured way to select the\ndeterministic subset of weights that removes neuron permutation symmetries, and\ntherefore the corresponding redundant posterior modes. With a drastically\nsimplified posterior distribution, the performance of existing approximate\ninference schemes is found to be greatly improved.\n", "date": "2024-05-27", "categories": []}
{"id": "2407.01985", "title": "The Epistemic Uncertainty Hole: an issue of Bayesian Neural Networks", "abstract": "  Bayesian Deep Learning (BDL) gives access not only to aleatoric uncertainty,\nas standard neural networks already do, but also to epistemic uncertainty, a\nmeasure of confidence a model has in its own predictions. In this article, we\nshow through experiments that the evolution of epistemic uncertainty metrics\nregarding the model size and the size of the training set, goes against\ntheoretical expectations. More precisely, we observe that the epistemic\nuncertainty collapses literally in the presence of large models and sometimes\nalso of little training data, while we expect the exact opposite behaviour.\nThis phenomenon, which we call \"epistemic uncertainty hole\", is all the more\nproblematic as it undermines the entire applicative potential of BDL, which is\nbased precisely on the use of epistemic uncertainty. As an example, we evaluate\nthe practical consequences of this uncertainty hole on one of the main\napplications of BDL, namely the detection of out-of-distribution samples\n", "date": "2024-07-02", "categories": []}
{"id": "2309.04742", "title": "Affine Invariant Ensemble Transform Methods to Improve Predictive\n  Uncertainty in Neural Networks", "abstract": "  We consider the problem of performing Bayesian inference for logistic\nregression using appropriate extensions of the ensemble Kalman filter. Two\ninteracting particle systems are proposed that sample from an approximate\nposterior and prove quantitative convergence rates of these interacting\nparticle systems to their mean-field limit as the number of particles tends to\ninfinity. Furthermore, we apply these techniques and examine their\neffectiveness as methods of Bayesian approximation for quantifying predictive\nuncertainty in neural networks.\n", "date": "2023-09-09", "categories": []}
{"id": "2406.17831", "title": "Empirical Bayes for Dynamic Bayesian Networks Using Generalized\n  Variational Inference", "abstract": "  In this work, we demonstrate the Empirical Bayes approach to learning a\nDynamic Bayesian Network. By starting with several point estimates of structure\nand weights, we can use a data-driven prior to subsequently obtain a model to\nquantify uncertainty. This approach uses a recent development of Generalized\nVariational Inference, and indicates the potential of sampling the uncertainty\nof a mixture of DAG structures as well as a parameter posterior.\n", "date": "2024-06-25", "categories": []}
{"id": "2407.00492", "title": "Fast Gibbs sampling for the local and global trend Bayesian exponential\n  smoothing model", "abstract": "  In Smyl et al. [Local and global trend Bayesian exponential smoothing models.\nInternational Journal of Forecasting, 2024.], a generalised exponential\nsmoothing model was proposed that is able to capture strong trends and\nvolatility in time series. This method achieved state-of-the-art performance in\nmany forecasting tasks, but its fitting procedure, which is based on the NUTS\nsampler, is very computationally expensive. In this work, we propose several\nmodifications to the original model, as well as a bespoke Gibbs sampler for\nposterior exploration; these changes improve sampling time by an order of\nmagnitude, thus rendering the model much more practically relevant. The new\nmodel, and sampler, are evaluated on the M3 dataset and are shown to be\ncompetitive, or superior, in terms of accuracy to the original method, while\nbeing substantially faster to run.\n", "date": "2024-06-29", "categories": []}
{"id": "2407.01015", "title": "Bayesian Entropy Neural Networks for Physics-Aware Prediction", "abstract": "  This paper addresses the need for deep learning models to integrate\nwell-defined constraints into their outputs, driven by their application in\nsurrogate models, learning with limited data and partial information, and\nscenarios requiring flexible model behavior to incorporate non-data sample\ninformation. We introduce Bayesian Entropy Neural Networks (BENN), a framework\ngrounded in Maximum Entropy (MaxEnt) principles, designed to impose constraints\non Bayesian Neural Network (BNN) predictions. BENN is capable of constraining\nnot only the predicted values but also their derivatives and variances,\nensuring a more robust and reliable model output. To achieve simultaneous\nuncertainty quantification and constraint satisfaction, we employ the method of\nmultipliers approach. This allows for the concurrent estimation of neural\nnetwork parameters and the Lagrangian multipliers associated with the\nconstraints. Our experiments, spanning diverse applications such as beam\ndeflection modeling and microstructure generation, demonstrate the\neffectiveness of BENN. The results highlight significant improvements over\ntraditional BNNs and showcase competitive performance relative to contemporary\nconstrained deep learning methods.\n", "date": "2024-07-01", "categories": []}
{"id": "2306.02568", "title": "Latent Optimal Paths by Gumbel Propagation for Variational Bayesian\n  Dynamic Programming", "abstract": "  We propose the stochastic optimal path which solves the classical optimal\npath problem by a probability-softening solution. This unified approach\ntransforms a wide range of DP problems into directed acyclic graphs in which\nall paths follow a Gibbs distribution. We show the equivalence of the Gibbs\ndistribution to a message-passing algorithm by the properties of the Gumbel\ndistribution and give all the ingredients required for variational Bayesian\ninference of a latent path, namely Bayesian dynamic programming (BDP). We\ndemonstrate the usage of BDP in the latent space of variational autoencoders\n(VAEs) and propose the BDP-VAE which captures structured sparse optimal paths\nas latent variables. This enables end-to-end training for generative tasks in\nwhich models rely on unobserved structural information. At last, we validate\nthe behavior of our approach and showcase its applicability in two real-world\napplications: text-to-speech and singing voice synthesis. Our implementation\ncode is available at\n\\url{https://github.com/XinleiNIU/LatentOptimalPathsBayesianDP}.\n", "date": "2023-06-05", "categories": []}
{"id": "2406.14808", "title": "On the estimation rate of Bayesian PINN for inverse problems", "abstract": "  Solving partial differential equations (PDEs) and their inverse problems\nusing Physics-informed neural networks (PINNs) is a rapidly growing approach in\nthe physics and machine learning community. Although several architectures\nexist for PINNs that work remarkably in practice, our theoretical understanding\nof their performances is somewhat limited. In this work, we study the behavior\nof a Bayesian PINN estimator of the solution of a PDE from $n$ independent\nnoisy measurement of the solution. We focus on a class of equations that are\nlinear in their parameters (with unknown coefficients $\\theta_\\star$). We show\nthat when the partial differential equation admits a classical solution (say\n$u_\\star$), differentiable to order $\\beta$, the mean square error of the\nBayesian posterior mean is at least of order $n^{-2\\beta/(2\\beta + d)}$.\nFurthermore, we establish a convergence rate of the linear coefficients of\n$\\theta_\\star$ depending on the order of the underlying differential operator.\nLast but not least, our theoretical results are validated through extensive\nsimulations.\n", "date": "2024-06-21", "categories": []}
{"id": "2406.14838", "title": "Bayesian neural networks for predicting uncertainty in full-field\n  material response", "abstract": "  Stress and material deformation field predictions are among the most\nimportant tasks in computational mechanics. These predictions are typically\nmade by solving the governing equations of continuum mechanics using finite\nelement analysis, which can become computationally prohibitive considering\ncomplex microstructures and material behaviors. Machine learning (ML) methods\noffer potentially cost effective surrogates for these applications. However,\nexisting ML surrogates are either limited to low-dimensional problems and/or do\nnot provide uncertainty estimates in the predictions. This work proposes an ML\nsurrogate framework for stress field prediction and uncertainty quantification\nfor diverse materials microstructures. A modified Bayesian U-net architecture\nis employed to provide a data-driven image-to-image mapping from initial\nmicrostructure to stress field with prediction (epistemic) uncertainty\nestimates. The Bayesian posterior distributions for the U-net parameters are\nestimated using three state-of-the-art inference algorithms: the posterior\nsampling-based Hamiltonian Monte Carlo method and two variational approaches,\nthe Monte-Carlo Dropout method and the Bayes by Backprop algorithm. A\nsystematic comparison of the predictive accuracy and uncertainty estimates for\nthese methods is performed for a fiber reinforced composite material and\npolycrystalline microstructure application. It is shown that the proposed\nmethods yield predictions of high accuracy compared to the FEA solution, while\nuncertainty estimates depend on the inference approach. Generally, the\nHamiltonian Monte Carlo and Bayes by Backprop methods provide consistent\nuncertainty estimates. Uncertainty estimates from Monte Carlo Dropout, on the\nother hand, are more difficult to interpret and depend strongly on the method's\ndesign.\n", "date": "2024-06-21", "categories": []}
{"id": "2302.09159", "title": "Bayesian Quantification with Black-Box Estimators", "abstract": "  Understanding how different classes are distributed in an unlabeled data set\nis an important challenge for the calibration of probabilistic classifiers and\nuncertainty quantification. Approaches like adjusted classify and count,\nblack-box shift estimators, and invariant ratio estimators use an auxiliary\n(and potentially biased) black-box classifier trained on a different (shifted)\ndata set to estimate the class distribution and yield asymptotic guarantees\nunder weak assumptions. We demonstrate that all these algorithms are closely\nrelated to the inference in a particular Bayesian model, approximating the\nassumed ground-truth generative process. Then, we discuss an efficient Markov\nChain Monte Carlo sampling scheme for the introduced model and show an\nasymptotic consistency guarantee in the large-data limit. We compare the\nintroduced model against the established point estimators in a variety of\nscenarios, and show it is competitive, and in some cases superior, with the\nstate of the art.\n", "date": "2023-02-17", "categories": []}
{"id": "2406.12353", "title": "Top-Down Bayesian Posterior Sampling for Sum-Product Networks", "abstract": "  Sum-product networks (SPNs) are probabilistic models characterized by exact\nand fast evaluation of fundamental probabilistic operations. Its superior\ncomputational tractability has led to applications in many fields, such as\nmachine learning with time constraints or accuracy requirements and real-time\nsystems. The structural constraints of SPNs supporting fast inference, however,\nlead to increased learning-time complexity and can be an obstacle to building\nhighly expressive SPNs. This study aimed to develop a Bayesian learning\napproach that can be efficiently implemented on large-scale SPNs. We derived a\nnew full conditional probability of Gibbs sampling by marginalizing multiple\nrandom variables to expeditiously obtain the posterior distribution. The\ncomplexity analysis revealed that our sampling algorithm works efficiently even\nfor the largest possible SPN. Furthermore, we proposed a hyperparameter tuning\nmethod that balances the diversity of the prior distribution and optimization\nefficiency in large-scale SPNs. Our method has improved learning-time\ncomplexity and demonstrated computational speed tens to more than one hundred\ntimes faster and superior predictive performance in numerical experiments on\nmore than 20 datasets.\n", "date": "2024-06-18", "categories": []}
{"id": "2406.12764", "title": "Quasi-Bayes meets Vines", "abstract": "  Recently proposed quasi-Bayesian (QB) methods initiated a new era in Bayesian\ncomputation by directly constructing the Bayesian predictive distribution\nthrough recursion, removing the need for expensive computations involved in\nsampling the Bayesian posterior distribution. This has proved to be\ndata-efficient for univariate predictions, but extensions to multiple\ndimensions rely on a conditional decomposition resulting from predefined\nassumptions on the kernel of the Dirichlet Process Mixture Model, which is the\nimplicit nonparametric model used. Here, we propose a different way to extend\nQuasi-Bayesian prediction to high dimensions through the use of Sklar's theorem\nby decomposing the predictive distribution into one-dimensional predictive\nmarginals and a high-dimensional copula. Thus, we use the efficient recursive\nQB construction for the one-dimensional marginals and model the dependence\nusing highly expressive vine copulas. Further, we tune hyperparameters using\nrobust divergences (eg. energy score) and show that our proposed Quasi-Bayesian\nVine (QB-Vine) is a fully non-parametric density estimator with \\emph{an\nanalytical form} and convergence rate independent of the dimension of data in\nsome situations. Our experiments illustrate that the QB-Vine is appropriate for\nhigh dimensional distributions ($\\sim$64), needs very few samples to train\n($\\sim$200) and outperforms state-of-the-art methods with analytical forms for\ndensity estimation and supervised tasks by a considerable margin.\n", "date": "2024-06-18", "categories": []}
{"id": "2404.04498", "title": "Bayesian Inference for Consistent Predictions in Overparameterized\n  Nonlinear Regression", "abstract": "  The remarkable generalization performance of large-scale models has been\nchallenging the conventional wisdom of the statistical learning theory.\nAlthough recent theoretical studies have shed light on this behavior in linear\nmodels and nonlinear classifiers, a comprehensive understanding of\noverparameterization in nonlinear regression models is still lacking. This\nstudy explores the predictive properties of overparameterized nonlinear\nregression within the Bayesian framework, extending the methodology of the\nadaptive prior considering the intrinsic spectral structure of the data.\nPosterior contraction is established for generalized linear and single-neuron\nmodels with Lipschitz continuous activation functions, demonstrating the\nconsistency in the predictions of the proposed approach. Moreover, the Bayesian\nframework enables uncertainty estimation of the predictions. The proposed\nmethod was validated via numerical simulations and a real data application,\nshowing its ability to achieve accurate predictions and reliable uncertainty\nestimates. This work provides a theoretical understanding of the advantages of\noverparameterization and a principled Bayesian approach to large nonlinear\nmodels.\n", "date": "2024-04-06", "categories": []}
{"id": "2406.09048", "title": "Central Limit Theorem for Bayesian Neural Network trained with\n  Variational Inference", "abstract": "  In this paper, we rigorously derive Central Limit Theorems (CLT) for Bayesian\ntwo-layerneural networks in the infinite-width limit and trained by variational\ninference on a regression task. The different networks are trained via\ndifferent maximization schemes of the regularized evidence lower bound: (i) the\nidealized case with exact estimation of a multiple Gaussian integral from the\nreparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling,\ncommonly known as Bayes-by-Backprop, and (iii) a computationally cheaper\nalgorithm named Minimal VI. The latter was recently introduced by leveraging\nthe information obtained at the level of the mean-field limit. Laws of large\nnumbers are already rigorously proven for the three schemes that admits the\nsame asymptotic limit. By deriving CLT, this work shows that the idealized and\nBayes-by-Backprop schemes have similar fluctuation behavior, that is different\nfrom the Minimal VI one. Numerical experiments then illustrate that the Minimal\nVI scheme is still more efficient, in spite of bigger variances, thanks to its\nimportant gain in computational complexity.\n", "date": "2024-06-10", "categories": []}
{"id": "2402.14758", "title": "Batch and match: black-box variational inference with a score-based\n  divergence", "abstract": "  Most leading implementations of black-box variational inference (BBVI) are\nbased on optimizing a stochastic evidence lower bound (ELBO). But such\napproaches to BBVI often converge slowly due to the high variance of their\ngradient estimates and their sensitivity to hyperparameters. In this work, we\npropose batch and match (BaM), an alternative approach to BBVI based on a\nscore-based divergence. Notably, this score-based divergence can be optimized\nby a closed-form proximal update for Gaussian variational families with full\ncovariance matrices. We analyze the convergence of BaM when the target\ndistribution is Gaussian, and we prove that in the limit of infinite batch size\nthe variational parameter updates converge exponentially quickly to the target\nmean and covariance. We also evaluate the performance of BaM on Gaussian and\nnon-Gaussian target distributions that arise from posterior inference in\nhierarchical and deep generative models. In these experiments, we find that BaM\ntypically converges in fewer (and sometimes significantly fewer) gradient\nevaluations than leading implementations of BBVI based on ELBO maximization.\n", "date": "2024-02-22", "categories": []}
{"id": "2402.07366", "title": "Bayesian Deep Learning Via Expectation Maximization and Turbo Deep\n  Approximate Message Passing", "abstract": "  Efficient learning and model compression algorithm for deep neural network\n(DNN) is a key workhorse behind the rise of deep learning (DL). In this work,\nwe propose a message passing based Bayesian deep learning algorithm called\nEM-TDAMP to avoid the drawbacks of traditional stochastic gradient descent\n(SGD) based learning algorithms and regularization-based model compression\nmethods. Specifically, we formulate the problem of DNN learning and compression\nas a sparse Bayesian inference problem, in which group sparse prior is employed\nto achieve structured model compression. Then, we propose an expectation\nmaximization (EM) framework to estimate posterior distributions for parameters\n(E-step) and update hyperparameters (M-step), where the E-step is realized by a\nnewly proposed turbo deep approximate message passing (TDAMP) algorithm. We\nfurther extend the EM-TDAMP and propose a novel Bayesian federated learning\nframework, in which and the clients perform TDAMP to efficiently calculate the\nlocal posterior distributions based on the local data, and the central server\nfirst aggregates the local posterior distributions to update the global\nposterior distributions and then update hyperparameters based on EM to\naccelerate convergence. We detail the application of EM-TDAMP to Boston housing\nprice prediction and handwriting recognition, and present extensive numerical\nresults to demonstrate the advantages of EM-TDAMP.\n", "date": "2024-02-12", "categories": []}
{"id": "2406.04012", "title": "Theoretical Guarantees for Variational Inference with Fixed-Variance\n  Mixture of Gaussians", "abstract": "  Variational inference (VI) is a popular approach in Bayesian inference, that\nlooks for the best approximation of the posterior distribution within a\nparametric family, minimizing a loss that is typically the (reverse)\nKullback-Leibler (KL) divergence. Despite its empirical success, the\ntheoretical properties of VI have only received attention recently, and mostly\nwhen the parametric family is the one of Gaussians. This work aims to\ncontribute to the theoretical study of VI in the non-Gaussian case by\ninvestigating the setting of Mixture of Gaussians with fixed covariance and\nconstant weights. In this view, VI over this specific family can be casted as\nthe minimization of a Mollified relative entropy, i.e. the KL between the\nconvolution (with respect to a Gaussian kernel) of an atomic measure supported\non Diracs, and the target distribution. The support of the atomic measure\ncorresponds to the localization of the Gaussian components. Hence, solving\nvariational inference becomes equivalent to optimizing the positions of the\nDiracs (the particles), which can be done through gradient descent and takes\nthe form of an interacting particle system. We study two sources of error of\nvariational inference in this context when optimizing the mollified relative\nentropy. The first one is an optimization result, that is a descent lemma\nestablishing that the algorithm decreases the objective at each iteration. The\nsecond one is an approximation error, that upper bounds the objective between\nan optimal finite mixture and the target distribution.\n", "date": "2024-06-06", "categories": []}
{"id": "2406.06307", "title": "Building Continuous Quantum-Classical Bayesian Neural Networks for a\n  Classical Clinical Dataset", "abstract": "  In this work, we are introducing a Quantum-Classical Bayesian Neural Network\n(QCBNN) that is capable to perform uncertainty-aware classification of\nclassical medical dataset. This model is a symbiosis of a classical\nConvolutional NN that performs ultra-sound image processing and a quantum\ncircuit that generates its stochastic weights, within a Bayesian learning\nframework. To test the utility of this idea for the possible future deployment\nin the medical sector we track multiple behavioral metrics that capture both\npredictive performance as well as model's uncertainty. It is our ambition to\ncreate a hybrid model that is capable to classify samples in a more uncertainty\naware fashion, which will advance the trustworthiness of these models and thus\nbring us step closer to utilizing them in the industry. We test multiple setups\nfor quantum circuit for this task, and our best architectures display bigger\nuncertainty gap between correctly and incorrectly identified samples than its\nclassical benchmark at an expense of a slight drop in predictive performance.\nThe innovation of this paper is two-fold: (1) combining of different approaches\nthat allow the stochastic weights from the quantum circuit to be continues thus\nallowing the model to classify application-driven dataset; (2) studying\narchitectural features of quantum circuit that make-or-break these models,\nwhich pave the way into further investigation of more informed architectural\ndesigns.\n", "date": "2024-06-10", "categories": []}
{"id": "2305.10664", "title": "Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks\n  under Weights with Unbounded Variance", "abstract": "  From the classical and influential works of Neal (1996), it is known that the\ninfinite width scaling limit of a Bayesian neural network with one hidden layer\nis a Gaussian process, when the network weights have bounded prior variance.\nNeal's result has been extended to networks with multiple hidden layers and to\nconvolutional neural networks, also with Gaussian process scaling limits. The\ntractable properties of Gaussian processes then allow straightforward posterior\ninference and uncertainty quantification, considerably simplifying the study of\nthe limit process compared to a network of finite width. Neural network weights\nwith unbounded variance, however, pose unique challenges. In this case, the\nclassical central limit theorem breaks down and it is well known that the\nscaling limit is an $\\alpha$-stable process under suitable conditions. However,\ncurrent literature is primarily limited to forward simulations under these\nprocesses and the problem of posterior inference under such a scaling limit\nremains largely unaddressed, unlike in the Gaussian process case. To this end,\nour contribution is an interpretable and computationally efficient procedure\nfor posterior inference, using a conditionally Gaussian representation, that\nthen allows full use of the Gaussian process machinery for tractable posterior\ninference and uncertainty quantification in the non-Gaussian regime.\n", "date": "2023-05-18", "categories": []}
{"id": "2406.02838", "title": "You Only Accept Samples Once: Fast, Self-Correcting Stochastic\n  Variational Inference", "abstract": "  We introduce YOASOVI, an algorithm for performing fast, self-correcting\nstochastic optimization for Variational Inference (VI) on large Bayesian\nheirarchical models. To accomplish this, we take advantage of available\ninformation on the objective function used for stochastic VI at each iteration\nand replace regular Monte Carlo sampling with acceptance sampling. Rather than\nspend computational resources drawing and evaluating over a large sample for\nthe gradient, we draw only one sample and accept it with probability\nproportional to the expected improvement in the objective. The following paper\ndevelops two versions of the algorithm: the first one based on a naive\nintuition, and another building up the algorithm as a Metropolis-type scheme.\nEmpirical results based on simulations and benchmark datasets for multivariate\nGaussian mixture models show that YOASOVI consistently converges faster (in\nclock time) and within better optimal neighborhoods than both regularized Monte\nCarlo and Quasi-Monte Carlo VI algorithms.\n", "date": "2024-06-05", "categories": []}
{"id": "2406.02177", "title": "One-Shot Federated Learning with Bayesian Pseudocoresets", "abstract": "  Optimization-based techniques for federated learning (FL) often come with\nprohibitive communication cost, as high dimensional model parameters need to be\ncommunicated repeatedly between server and clients. In this paper, we follow a\nBayesian approach allowing to perform FL with one-shot communication, by\nsolving the global inference problem as a product of local client posteriors.\nFor models with multi-modal likelihoods, such as neural networks, a naive\napplication of this scheme is hampered, since clients will capture different\nposterior modes, causing a destructive collapse of the posterior on the server\nside. Consequently, we explore approximate inference in the function-space\nrepresentation of client posteriors, hence suffering less or not at all from\nmulti-modality. We show that distributed function-space inference is tightly\nrelated to learning Bayesian pseudocoresets and develop a tractable Bayesian FL\nalgorithm on this insight. We show that this approach achieves prediction\nperformance competitive to state-of-the-art while showing a striking reduction\nin communication cost of up to two orders of magnitude. Moreover, due to its\nBayesian nature, our method also delivers well-calibrated uncertainty\nestimates.\n", "date": "2024-06-04", "categories": []}
{"id": "2210.04872", "title": "Sequential Neural Score Estimation: Likelihood-Free Inference with\n  Conditional Score Based Diffusion Models", "abstract": "  We introduce Sequential Neural Posterior Score Estimation (SNPSE), a\nscore-based method for Bayesian inference in simulator-based models. Our\nmethod, inspired by the remarkable success of score-based methods in generative\nmodelling, leverages conditional score-based diffusion models to generate\nsamples from the posterior distribution of interest. The model is trained using\nan objective function which directly estimates the score of the posterior. We\nembed the model into a sequential training procedure, which guides simulations\nusing the current approximation of the posterior at the observation of\ninterest, thereby reducing the simulation cost. We also introduce several\nalternative sequential approaches, and discuss their relative merits. We then\nvalidate our method, as well as its amortised, non-sequential, variant on\nseveral numerical examples, demonstrating comparable or superior performance to\nexisting state-of-the-art methods such as Sequential Neural Posterior\nEstimation (SNPE).\n", "date": "2022-10-10", "categories": []}
{"id": "2211.02763", "title": "Bayesian learning of Causal Structure and Mechanisms with GFlowNets and\n  Variational Bayes", "abstract": "  Bayesian causal structure learning aims to learn a posterior distribution\nover directed acyclic graphs (DAGs), and the mechanisms that define the\nrelationship between parent and child variables. By taking a Bayesian approach,\nit is possible to reason about the uncertainty of the causal model. The notion\nof modelling the uncertainty over models is particularly crucial for causal\nstructure learning since the model could be unidentifiable when given only a\nfinite amount of observational data. In this paper, we introduce a novel method\nto jointly learn the structure and mechanisms of the causal model using\nVariational Bayes, which we call Variational Bayes-DAG-GFlowNet (VBG). We\nextend the method of Bayesian causal structure learning using GFlowNets to\nlearn not only the posterior distribution over the structure, but also the\nparameters of a linear-Gaussian model. Our results on simulated data suggest\nthat VBG is competitive against several baselines in modelling the posterior\nover DAGs and mechanisms, while offering several advantages over existing\nmethods, including the guarantee to sample acyclic graphs, and the flexibility\nto generalize to non-linear causal mechanisms.\n", "date": "2022-11-04", "categories": []}
{"id": "2311.00463", "title": "Robust and Conjugate Gaussian Process Regression", "abstract": "  To enable closed form conditioning, a common assumption in Gaussian process\n(GP) regression is independent and identically distributed Gaussian observation\nnoise. This strong and simplistic assumption is often violated in practice,\nwhich leads to unreliable inferences and uncertainty quantification.\nUnfortunately, existing methods for robustifying GPs break closed-form\nconditioning, which makes them less attractive to practitioners and\nsignificantly more computationally expensive. In this paper, we demonstrate how\nto perform provably robust and conjugate Gaussian process (RCGP) regression at\nvirtually no additional cost using generalised Bayesian inference. RCGP is\nparticularly versatile as it enables exact conjugate closed form updates in all\nsettings where standard GPs admit them. To demonstrate its strong empirical\nperformance, we deploy RCGP for problems ranging from Bayesian optimisation to\nsparse variational Gaussian processes.\n", "date": "2023-11-01", "categories": []}
{"id": "2405.19747", "title": "Understanding and mitigating difficulties in posterior predictive\n  evaluation", "abstract": "  Predictive posterior densities (PPDs) are of interest in approximate Bayesian\ninference. Typically, these are estimated by simple Monte Carlo (MC) averages\nusing samples from the approximate posterior. We observe that the\nsignal-to-noise ratio (SNR) of such estimators can be extremely low. An\nanalysis for exact inference reveals SNR decays exponentially as there is an\nincrease in (a) the mismatch between training and test data, (b) the\ndimensionality of the latent space, or (c) the size of the test data relative\nto the training data. Further analysis extends these results to approximate\ninference. To remedy the low SNR problem, we propose replacing simple MC\nsampling with importance sampling using a proposal distribution optimized at\ntest time on a variational proxy for the SNR and demonstrate that this yields\ngreatly improved estimates.\n", "date": "2024-05-30", "categories": []}
{"id": "2402.01484", "title": "Connecting the Dots: Is Mode-Connectedness the Key to Feasible\n  Sample-Based Inference in Bayesian Neural Networks?", "abstract": "  A major challenge in sample-based inference (SBI) for Bayesian neural\nnetworks is the size and structure of the networks' parameter space. Our work\nshows that successful SBI is possible by embracing the characteristic\nrelationship between weight and function space, uncovering a systematic link\nbetween overparameterization and the difficulty of the sampling problem.\nThrough extensive experiments, we establish practical guidelines for sampling\nand convergence diagnosis. As a result, we present a deep ensemble initialized\napproach as an effective solution with competitive performance and uncertainty\nquantification.\n", "date": "2024-02-02", "categories": []}
{"id": "2405.18351", "title": "Evaluating Bayesian deep learning for radio galaxy classification", "abstract": "  The radio astronomy community is rapidly adopting deep learning techniques to\ndeal with the huge data volumes expected from the next generation of radio\nobservatories. Bayesian neural networks (BNNs) provide a principled way to\nmodel uncertainty in the predictions made by such deep learning models and will\nplay an important role in extracting well-calibrated uncertainty estimates on\ntheir outputs. In this work, we evaluate the performance of different BNNs\nagainst the following criteria: predictive performance, uncertainty calibration\nand distribution-shift detection for the radio galaxy classification problem.\n", "date": "2024-05-28", "categories": []}
{"id": "2303.16954", "title": "Leveraging joint sparsity in hierarchical Bayesian learning", "abstract": "  We present a hierarchical Bayesian learning approach to infer jointly sparse\nparameter vectors from multiple measurement vectors. Our model uses separate\nconditionally Gaussian priors for each parameter vector and common\ngamma-distributed hyper-parameters to enforce joint sparsity. The resulting\njoint-sparsity-promoting priors are combined with existing Bayesian inference\nmethods to generate a new family of algorithms. Our numerical experiments,\nwhich include a multi-coil magnetic resonance imaging application, demonstrate\nthat our new approach consistently outperforms commonly used hierarchical\nBayesian methods.\n", "date": "2023-03-29", "categories": []}
{"id": "2307.11018", "title": "Amortized Variational Inference: When and Why?", "abstract": "  In a probabilistic latent variable model, factorized (or mean-field)\nvariational inference (F-VI) fits a separate parametric distribution for each\nlatent variable. Amortized variational inference (A-VI) instead learns a common\ninference function, which maps each observation to its corresponding latent\nvariable's approximate posterior. Typically, A-VI is used as a step in the\ntraining of variational autoencoders, however it stands to reason that A-VI\ncould also be used as a general alternative to F-VI. In this paper we study\nwhen and why A-VI can be used for approximate Bayesian inference. We derive\nconditions on a latent variable model which are necessary, sufficient, and\nverifiable under which A-VI can attain F-VI's optimal solution, thereby closing\nthe amortization gap. We prove these conditions are uniquely verified by simple\nhierarchical models, a broad class that encompasses many models in machine\nlearning. We then show, on a broader class of models, how to expand the domain\nof AVI's inference function to improve its solution, and we provide examples,\ne.g. hidden Markov models, where the amortization gap cannot be closed.\n", "date": "2023-07-20", "categories": []}
{"id": "2302.12565", "title": "Variational Linearized Laplace Approximation for Bayesian Deep Learning", "abstract": "  The Linearized Laplace Approximation (LLA) has been recently used to perform\nuncertainty estimation on the predictions of pre-trained deep neural networks\n(DNNs). However, its widespread application is hindered by significant\ncomputational costs, particularly in scenarios with a large number of training\npoints or DNN parameters. Consequently, additional approximations of LLA, such\nas Kronecker-factored or diagonal approximate GGN matrices, are utilized,\npotentially compromising the model's performance. To address these challenges,\nwe propose a new method for approximating LLA using a variational sparse\nGaussian Process (GP). Our method is based on the dual RKHS formulation of GPs\nand retains, as the predictive mean, the output of the original DNN.\nFurthermore, it allows for efficient stochastic optimization, which results in\nsub-linear training time in the size of the training dataset. Specifically, its\ntraining cost is independent of the number of training points. We compare our\nproposed method against accelerated LLA (ELLA), which relies on the Nystr\\\"om\napproximation, as well as other LLA variants employing the sample-then-optimize\nprinciple. Experimental results, both on regression and classification\ndatasets, show that our method outperforms these already existing efficient\nvariants of LLA, both in terms of the quality of the predictive distribution\nand in terms of total computational time.\n", "date": "2023-02-24", "categories": []}
{"id": "2405.09817", "title": "Active Learning with Fully Bayesian Neural Networks for Discontinuous\n  and Nonstationary Data", "abstract": "  Active learning optimizes the exploration of large parameter spaces by\nstrategically selecting which experiments or simulations to conduct, thus\nreducing resource consumption and potentially accelerating scientific\ndiscovery. A key component of this approach is a probabilistic surrogate model,\ntypically a Gaussian Process (GP), which approximates an unknown functional\nrelationship between control parameters and a target property. However,\nconventional GPs often struggle when applied to systems with discontinuities\nand non-stationarities, prompting the exploration of alternative models. This\nlimitation becomes particularly relevant in physical science problems, which\nare often characterized by abrupt transitions between different system states\nand rapid changes in physical property behavior. Fully Bayesian Neural Networks\n(FBNNs) serve as a promising substitute, treating all neural network weights\nprobabilistically and leveraging advanced Markov Chain Monte Carlo techniques\nfor direct sampling from the posterior distribution. This approach enables\nFBNNs to provide reliable predictive distributions, crucial for making informed\ndecisions under uncertainty in the active learning setting. Although\ntraditionally considered too computationally expensive for 'big data'\napplications, many physical sciences problems involve small amounts of data in\nrelatively low-dimensional parameter spaces. Here, we assess the suitability\nand performance of FBNNs with the No-U-Turn Sampler for active learning tasks\nin the 'small data' regime, highlighting their potential to enhance predictive\naccuracy and reliability on test functions relevant to problems in physical\nsciences.\n", "date": "2024-05-16", "categories": []}
{"id": "2405.08999", "title": "Robust Approximate Sampling via Stochastic Gradient Barker Dynamics", "abstract": "  Stochastic Gradient (SG) Markov Chain Monte Carlo algorithms (MCMC) are\npopular algorithms for Bayesian sampling in the presence of large datasets.\nHowever, they come with little theoretical guarantees and assessing their\nempirical performances is non-trivial. In such context, it is crucial to\ndevelop algorithms that are robust to the choice of hyperparameters and to\ngradients heterogeneity since, in practice, both the choice of step-size and\nbehaviour of target gradients induce hard-to-control biases in the invariant\ndistribution. In this work we introduce the stochastic gradient Barker dynamics\n(SGBD) algorithm, extending the recently developed Barker MCMC scheme, a robust\nalternative to Langevin-based sampling algorithms, to the stochastic gradient\nframework. We characterize the impact of stochastic gradients on the Barker\ntransition mechanism and develop a bias-corrected version that, under suitable\nassumptions, eliminates the error due to the gradient noise in the proposal. We\nillustrate the performance on a number of high-dimensional examples, showing\nthat SGBD is more robust to hyperparameter tuning and to irregular behavior of\nthe target gradients compared to the popular stochastic gradient Langevin\ndynamics algorithm.\n", "date": "2024-05-14", "categories": []}
{"id": "2311.02495", "title": "Uncertainty Quantification in Multivariable Regression for Material\n  Property Prediction with Bayesian Neural Networks", "abstract": "  With the increased use of data-driven approaches and machine learning-based\nmethods in material science, the importance of reliable uncertainty\nquantification (UQ) of the predicted variables for informed decision-making\ncannot be overstated. UQ in material property prediction poses unique\nchallenges, including the multi-scale and multi-physics nature of advanced\nmaterials, intricate interactions between numerous factors, limited\navailability of large curated datasets for model training, etc. Recently,\nBayesian Neural Networks (BNNs) have emerged as a promising approach for UQ,\noffering a probabilistic framework for capturing uncertainties within neural\nnetworks. In this work, we introduce an approach for UQ within physics-informed\nBNNs, which integrates knowledge from governing laws in material modeling to\nguide the models toward physically consistent predictions. To evaluate the\neffectiveness of this approach, we present case studies for predicting the\ncreep rupture life of steel alloys. Experimental validation with three datasets\nof collected measurements from creep tests demonstrates the ability of BNNs to\nproduce accurate point and uncertainty estimates that are competitive or exceed\nthe performance of the conventional method of Gaussian Process Regression.\nSimilarly, we evaluated the suitability of BNNs for UQ in an active learning\napplication and reported competitive performance. The most promising framework\nfor creep life prediction is BNNs based on Markov Chain Monte Carlo\napproximation of the posterior distribution of network parameters, as it\nprovided more reliable results in comparison to BNNs based on variational\ninference approximation or related NNs with probabilistic outputs. The codes\nare available at:\nhttps://github.com/avakanski/Creep-uncertainty-quantification.\n", "date": "2023-11-04", "categories": []}
{"id": "2405.08179", "title": "Do Bayesian imaging methods report trustworthy probabilities?", "abstract": "  Bayesian statistics is a cornerstone of imaging sciences, underpinning many\nand varied approaches from Markov random fields to score-based denoising\ndiffusion models. In addition to powerful image estimation methods, the\nBayesian paradigm also provides a framework for uncertainty quantification and\nfor using image data as quantitative evidence. These probabilistic capabilities\nare important for the rigorous interpretation of experimental results and for\nrobust interfacing of quantitative imaging pipelines with scientific and\ndecision-making processes. However, are the probabilities delivered by existing\nBayesian imaging methods meaningful under replication of an experiment, or are\nthey only meaningful as subjective measures of belief? This paper presents a\nMonte Carlo method to explore this question. We then leverage the proposed\nMonte Carlo method and run a large experiment requiring 1,000 GPU-hours to\nprobe the accuracy of five canonical Bayesian imaging methods that are\nrepresentative of some of the main Bayesian imaging strategies from the past\ndecades (a score-based denoising diffusion technique, a plug-and-play Langevin\nalgorithm utilising a Lipschitz-regularised DnCNN denoiser, a Bayesian method\nwith a dictionary-based prior trained subject to a log-concavity constraint, an\nempirical Bayesian method with a total-variation prior, and a hierarchical\nBayesian Gibbs sampler based on a Gaussian Markov random field model). We find\nthat, a few cases, the probabilities reported by modern Bayesian imaging\ntechniques are in broad agreement with long-term averages as observed over a\nlarge number of replication of an experiment, but existing Bayesian imaging\nmethods are generally not able to deliver reliable uncertainty quantification\nresults.\n", "date": "2024-05-13", "categories": []}
{"id": "2405.05855", "title": "Compressed Bayesian Federated Learning for Reliable Passive Radio\n  Sensing in Industrial IoT", "abstract": "  Bayesian Federated Learning (FL) has been recently introduced to provide\nwell-calibrated Machine Learning (ML) models quantifying the uncertainty of\ntheir predictions. Despite their advantages compared to frequentist FL setups,\nBayesian FL tools implemented over decentralized networks are subject to high\ncommunication costs due to the iterated exchange of local posterior\ndistributions among cooperating devices. Therefore, this paper proposes a\ncommunication-efficient decentralized Bayesian FL policy to reduce the\ncommunication overhead without sacrificing final learning accuracy and\ncalibration. The proposed method integrates compression policies and allows\ndevices to perform multiple optimization steps before sending the local\nposterior distributions. We integrate the developed tool in an Industrial\nInternet of Things (IIoT) use case where collaborating nodes equipped with\nautonomous radar sensors are tasked to reliably localize human operators in a\nworkplace shared with robots. Numerical results show that the developed\napproach obtains highly accurate yet well-calibrated ML models compatible with\nthe ones provided by conventional (uncompressed) Bayesian FL tools while\nsubstantially decreasing the communication overhead (i.e., up to 99%).\nFurthermore, the proposed approach is advantageous when compared with\nstate-of-the-art compressed frequentist FL setups in terms of calibration,\nespecially when the statistical distribution of the testing dataset changes.\n", "date": "2024-05-09", "categories": []}
{"id": "2303.11278", "title": "Bayesian Pseudo-Coresets via Contrastive Divergence", "abstract": "  Bayesian methods provide an elegant framework for estimating parameter\nposteriors and quantification of uncertainty associated with probabilistic\nmodels. However, they often suffer from slow inference times. To address this\nchallenge, Bayesian Pseudo-Coresets (BPC) have emerged as a promising solution.\nBPC methods aim to create a small synthetic dataset, known as pseudo-coresets,\nthat approximates the posterior inference achieved with the original dataset.\nThis approximation is achieved by optimizing a divergence measure between the\ntrue posterior and the pseudo-coreset posterior. Various divergence measures\nhave been proposed for constructing pseudo-coresets, with forward\nKullback-Leibler (KL) divergence being the most successful. However, using\nforward KL divergence necessitates sampling from the pseudo-coreset posterior,\noften accomplished through approximate Gaussian variational distributions.\nAlternatively, one could employ Markov Chain Monte Carlo (MCMC) methods for\nsampling, but this becomes challenging in high-dimensional parameter spaces due\nto slow mixing. In this study, we introduce a novel approach for constructing\npseudo-coresets by utilizing contrastive divergence. Importantly, optimizing\ncontrastive divergence eliminates the need for approximations in the\npseudo-coreset construction process. Furthermore, it enables the use of\nfinite-step MCMC methods, alleviating the requirement for extensive mixing to\nreach a stationary distribution. To validate our method's effectiveness, we\nconduct extensive experiments on multiple datasets, demonstrating its\nsuperiority over existing BPC techniques.\n", "date": "2023-03-20", "categories": []}
{"id": "2405.05398", "title": "ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse\n  Problems", "abstract": "  Due to their uncertainty quantification, Bayesian solutions to inverse\nproblems are the framework of choice in applications that are risk averse.\nThese benefits come at the cost of computations that are in general,\nintractable. New advances in machine learning and variational inference (VI)\nhave lowered the computational barrier by learning from examples. Two VI\nparadigms have emerged that represent different tradeoffs: amortized and\nnon-amortized. Amortized VI can produce fast results but due to generalizing to\nmany observed datasets it produces suboptimal inference results. Non-amortized\nVI is slower at inference but finds better posterior approximations since it is\nspecialized towards a single observed dataset. Current amortized VI techniques\nrun into a sub-optimality wall that can not be improved without more expressive\nneural networks or extra training data. We present a solution that enables\niterative improvement of amortized posteriors that uses the same networks\narchitectures and training data. The benefits of our method requires extra\ncomputations but these remain frugal since they are based on physics-hybrid\nmethods and summary statistics. Importantly, these computations remain mostly\noffline thus our method maintains cheap and reusable online evaluation while\nbridging the approximation gap these two paradigms. We denote our proposed\nmethod ASPIRE - Amortized posteriors with Summaries that are Physics-based and\nIteratively REfined. We first validate our method on a stylized problem with a\nknown posterior then demonstrate its practical use on a high-dimensional and\nnonlinear transcranial medical imaging problem with ultrasound. Compared with\nthe baseline and previous methods from the literature our method stands out as\nan computationally efficient and high-fidelity method for posterior inference.\n", "date": "2024-05-08", "categories": []}
{"id": "2305.20028", "title": "A Study of Bayesian Neural Network Surrogates for Bayesian Optimization", "abstract": "  Bayesian optimization is a highly efficient approach to optimizing objective\nfunctions which are expensive to query. These objectives are typically\nrepresented by Gaussian process (GP) surrogate models which are easy to\noptimize and support exact inference. While standard GP surrogates have been\nwell-established in Bayesian optimization, Bayesian neural networks (BNNs) have\nrecently become practical function approximators, with many benefits over\nstandard GPs such as the ability to naturally handle non-stationarity and learn\nrepresentations for high-dimensional data. In this paper, we study BNNs as\nalternatives to standard GP surrogates for optimization. We consider a variety\nof approximate inference procedures for finite-width BNNs, including\nhigh-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics\nsuch as deep ensembles. We also consider infinite-width BNNs, linearized\nLaplace approximations, and partially stochastic models such as deep kernel\nlearning. We evaluate this collection of surrogate models on diverse problems\nwith varying dimensionality, number of objectives, non-stationarity, and\ndiscrete and continuous inputs. We find: (i) the ranking of methods is highly\nproblem dependent, suggesting the need for tailored inductive biases; (ii) HMC\nis the most successful approximate inference procedure for fully stochastic\nBNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is\nrelatively competitive; (iv) deep ensembles perform relatively poorly; (v)\ninfinite-width BNNs are particularly promising, especially in high dimensions.\n", "date": "2023-05-31", "categories": []}
{"id": "2305.19076", "title": "Approximate Bayesian Class-Conditional Models under Continuous\n  Representation Shift", "abstract": "  For models consisting of a classifier in some representation space, learning\nonline from a non-stationary data stream often necessitates changes in the\nrepresentation. So, the question arises of what is the best way to adapt the\nclassifier to shifts in representation. Current methods only slowly change the\nclassifier to representation shift, introducing noise into learning as the\nclassifier is misaligned to the representation. We propose DeepCCG, an\nempirical Bayesian approach to solve this problem. DeepCCG works by updating\nthe posterior of a class conditional Gaussian classifier such that the\nclassifier adapts in one step to representation shift. The use of a class\nconditional Gaussian classifier also enables DeepCCG to use a log conditional\nmarginal likelihood loss to update the representation. To perform the update to\nthe classifier and representation, DeepCCG maintains a fixed number of examples\nin memory and so a key part of DeepCCG is selecting what examples to store,\nchoosing the subset that minimises the KL divergence between the true posterior\nand the posterior induced by the subset. We explore the behaviour of DeepCCG in\nonline continual learning (CL), demonstrating that it performs well against a\nspectrum of online CL methods and that it reduces the change in performance due\nto representation shift.\n", "date": "2023-05-30", "categories": []}
{"id": "2405.01507", "title": "Accelerating Convergence in Bayesian Few-Shot Classification", "abstract": "  Bayesian few-shot classification has been a focal point in the field of\nfew-shot learning. This paper seamlessly integrates mirror descent-based\nvariational inference into Gaussian process-based few-shot classification,\naddressing the challenge of non-conjugate inference. By leveraging\nnon-Euclidean geometry, mirror descent achieves accelerated convergence by\nproviding the steepest descent direction along the corresponding manifold. It\nalso exhibits the parameterization invariance property concerning the\nvariational distribution. Experimental results demonstrate competitive\nclassification accuracy, improved uncertainty quantification, and faster\nconvergence compared to baseline models. Additionally, we investigate the\nimpact of hyperparameters and components. Code is publicly available at\nhttps://github.com/keanson/MD-BSFC.\n", "date": "2024-05-02", "categories": []}
{"id": "2405.01365", "title": "Dynamic Online Ensembles of Basis Expansions", "abstract": "  Practical Bayesian learning often requires (1) online inference, (2) dynamic\nmodels, and (3) ensembling over multiple different models. Recent advances have\nshown how to use random feature approximations to achieve scalable, online\nensembling of Gaussian processes with desirable theoretical properties and\nfruitful applications. One key to these methods' success is the inclusion of a\nrandom walk on the model parameters, which makes models dynamic. We show that\nthese methods can be generalized easily to any basis expansion model and that\nusing alternative basis expansions, such as Hilbert space Gaussian processes,\noften results in better performance. To simplify the process of choosing a\nspecific basis expansion, our method's generality also allows the ensembling of\nseveral entirely different models, for example, a Gaussian process and\npolynomial regression. Finally, we propose a novel method to ensemble static\nand dynamic models together.\n", "date": "2024-05-02", "categories": []}
{"id": "2404.19157", "title": "Scalable Bayesian Inference in the Era of Deep Learning: From Gaussian\n  Processes to Deep Neural Networks", "abstract": "  Large neural networks trained on large datasets have become the dominant\nparadigm in machine learning. These systems rely on maximum likelihood point\nestimates of their parameters, precluding them from expressing model\nuncertainty. This may result in overconfident predictions and it prevents the\nuse of deep learning models for sequential decision making. This thesis\ndevelops scalable methods to equip neural networks with model uncertainty. In\nparticular, we leverage the linearised Laplace approximation to equip\npre-trained neural networks with the uncertainty estimates provided by their\ntangent linear models. This turns the problem of Bayesian inference in neural\nnetworks into one of Bayesian inference in conjugate Gaussian-linear models.\nAlas, the cost of this remains cubic in either the number of network parameters\nor in the number of observations times output dimensions. By assumption,\nneither are tractable. We address this intractability by using stochastic\ngradient descent (SGD) -- the workhorse algorithm of deep learning -- to\nperform posterior sampling in linear models and their convex duals: Gaussian\nprocesses. With this, we turn back to linearised neural networks, finding the\nlinearised Laplace approximation to present a number of incompatibilities with\nmodern deep learning practices -- namely, stochastic optimisation, early\nstopping and normalisation layers -- when used for hyperparameter learning. We\nresolve these and construct a sample-based EM algorithm for scalable\nhyperparameter learning with linearised neural networks. We apply the above\nmethods to perform linearised neural network inference with ResNet-50 (25M\nparameters) trained on Imagenet (1.2M observations and 1000 output dimensions).\nAdditionally, we apply our methods to estimate uncertainty for 3d tomographic\nreconstructions obtained with the deep image prior network.\n", "date": "2024-04-29", "categories": []}
{"id": "2404.19640", "title": "Attacking Bayes: On the Adversarial Robustness of Bayesian Neural\n  Networks", "abstract": "  Adversarial examples have been shown to cause neural networks to fail on a\nwide range of vision and language tasks, but recent work has claimed that\nBayesian neural networks (BNNs) are inherently robust to adversarial\nperturbations. In this work, we examine this claim. To study the adversarial\nrobustness of BNNs, we investigate whether it is possible to successfully break\nstate-of-the-art BNN inference methods and prediction pipelines using even\nrelatively unsophisticated attacks for three tasks: (1) label prediction under\nthe posterior predictive mean, (2) adversarial example detection with Bayesian\npredictive uncertainty, and (3) semantic shift detection. We find that BNNs\ntrained with state-of-the-art approximate inference methods, and even BNNs\ntrained with Hamiltonian Monte Carlo, are highly susceptible to adversarial\nattacks. We also identify various conceptual and experimental errors in\nprevious works that claimed inherent adversarial robustness of BNNs and\nconclusively demonstrate that BNNs and uncertainty-aware Bayesian prediction\npipelines are not inherently robust against adversarial attacks.\n", "date": "2024-04-27", "categories": []}
{"id": "2305.13882", "title": "Subsampling Error in Stochastic Gradient Langevin Diffusions", "abstract": "  The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to\napproximate Bayesian posterior distributions in statistical learning procedures\nwith large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC)\nalgorithms, SGLD is not stationary with respect to the posterior distribution;\ntwo sources of error appear: The first error is introduced by an\nEuler--Maruyama discretisation of a Langevin diffusion process, the second\nerror comes from the data subsampling that enables its use in large-scale data\nsettings. In this work, we consider an idealised version of SGLD to analyse the\nmethod's pure subsampling error that we then see as a best-case error for\ndiffusion-based subsampling MCMC methods. Indeed, we introduce and study the\nStochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov\nprocess that follows the Langevin diffusion corresponding to a data subset and\nswitches this data subset after exponential waiting times. There, we show the\nexponential ergodicity of SLGDiff and that the Wasserstein distance between the\nposterior and the limiting distribution of SGLDiff is bounded above by a\nfractional power of the mean waiting time. We bring our results into context\nwith other analyses of SGLD.\n", "date": "2023-05-23", "categories": []}
{"id": "2404.18008", "title": "Implicit Generative Prior for Bayesian Neural Networks", "abstract": "  Predictive uncertainty quantification is crucial for reliable decision-making\nin various applied domains. Bayesian neural networks offer a powerful framework\nfor this task. However, defining meaningful priors and ensuring computational\nefficiency remain significant challenges, especially for complex real-world\napplications. This paper addresses these challenges by proposing a novel neural\nadaptive empirical Bayes (NA-EB) framework. NA-EB leverages a class of implicit\ngenerative priors derived from low-dimensional distributions. This allows for\nefficient handling of complex data structures and effective capture of\nunderlying relationships in real-world datasets. The proposed NA-EB framework\ncombines variational inference with a gradient ascent algorithm. This enables\nsimultaneous hyperparameter selection and approximation of the posterior\ndistribution, leading to improved computational efficiency. We establish the\ntheoretical foundation of the framework through posterior and classification\nconsistency. We demonstrate the practical applications of our framework through\nextensive evaluations on a variety of tasks, including the two-spiral problem,\nregression, 10 UCI datasets, and image classification tasks on both MNIST and\nCIFAR-10 datasets. The results of our experiments highlight the superiority of\nour proposed framework over existing methods, such as sparse variational\nBayesian and generative models, in terms of prediction accuracy and uncertainty\nquantification.\n", "date": "2024-04-27", "categories": []}
{"id": "2404.17249", "title": "Making Better Use of Unlabelled Data in Bayesian Active Learning", "abstract": "  Fully supervised models are predominant in Bayesian active learning. We argue\nthat their neglect of the information present in unlabelled data harms not just\npredictive performance but also decisions about what data to acquire. Our\nproposed solution is a simple framework for semi-supervised Bayesian active\nlearning. We find it produces better-performing models than either conventional\nBayesian active learning or semi-supervised learning with randomly acquired\ndata. It is also easier to scale up than the conventional approach. As well as\nsupporting a shift towards semi-supervised models, our findings highlight the\nimportance of studying models and acquisition methods in conjunction.\n", "date": "2024-04-26", "categories": []}
{"id": "2404.12478", "title": "A New Reliable & Parsimonious Learning Strategy Comprising Two Layers of\n  Gaussian Processes, to Address Inhomogeneous Empirical Correlation Structures", "abstract": "  We present a new strategy for learning the functional relation between a pair\nof variables, while addressing inhomogeneities in the correlation structure of\nthe available data, by modelling the sought function as a sample function of a\nnon-stationary Gaussian Process (GP), that nests within itself multiple other\nGPs, each of which we prove can be stationary, thereby establishing sufficiency\nof two GP layers. In fact, a non-stationary kernel is envisaged, with each\nhyperparameter set as dependent on the sample function drawn from the outer\nnon-stationary GP, such that a new sample function is drawn at every pair of\ninput values at which the kernel is computed. However, such a model cannot be\nimplemented, and we substitute this by recalling that the average effect of\ndrawing different sample functions from a given GP is equivalent to that of\ndrawing a sample function from each of a set of GPs that are rendered\ndifferent, as updated during the equilibrium stage of the undertaken inference\n(via MCMC). The kernel is fully non-parametric, and it suffices to learn one\nhyperparameter per layer of GP, for each dimension of the input variable. We\nillustrate this new learning strategy on a real dataset.\n", "date": "2024-04-18", "categories": []}
{"id": "2210.14598", "title": "Manifold Gaussian Variational Bayes on the Precision Matrix", "abstract": "  We propose an optimization algorithm for Variational Inference (VI) in\ncomplex models. Our approach relies on natural gradient updates where the\nvariational space is a Riemann manifold. We develop an efficient algorithm for\nGaussian Variational Inference whose updates satisfy the positive definite\nconstraint on the variational covariance matrix. Our Manifold Gaussian\nVariational Bayes on the Precision matrix (MGVBP) solution provides simple\nupdate rules, is straightforward to implement, and the use of the precision\nmatrix parametrization has a significant computational advantage. Due to its\nblack-box nature, MGVBP stands as a ready-to-use solution for VI in complex\nmodels. Over five datasets, we empirically validate our feasible approach on\ndifferent statistical and econometric models, discussing its performance with\nrespect to baseline methods.\n", "date": "2022-10-26", "categories": []}
{"id": "2404.11599", "title": "Variational Bayesian Last Layers", "abstract": "  We introduce a deterministic variational formulation for training Bayesian\nlast layer neural networks. This yields a sampling-free, single-pass model and\nloss that effectively improves uncertainty estimation. Our variational Bayesian\nlast layer (VBLL) can be trained and evaluated with only quadratic complexity\nin last layer width, and is thus (nearly) computationally free to add to\nstandard architectures. We experimentally investigate VBLLs, and show that they\nimprove predictive accuracy, calibration, and out of distribution detection\nover baselines across both regression and classification. Finally, we\ninvestigate combining VBLL layers with variational Bayesian feature learning,\nyielding a lower variance collapsed variational inference method for Bayesian\nneural networks.\n", "date": "2024-04-17", "categories": []}
{"id": "2404.10436", "title": "Tree Bandits for Generative Bayes", "abstract": "  In generative models with obscured likelihood, Approximate Bayesian\nComputation (ABC) is often the tool of last resort for inference. However, ABC\ndemands many prior parameter trials to keep only a small fraction that passes\nan acceptance test. To accelerate ABC rejection sampling, this paper develops a\nself-aware framework that learns from past trials and errors. We apply\nrecursive partitioning classifiers on the ABC lookup table to sequentially\nrefine high-likelihood regions into boxes. Each box is regarded as an arm in a\nbinary bandit problem treating ABC acceptance as a reward. Each arm has a\nproclivity for being chosen for the next ABC evaluation, depending on the prior\ndistribution and past rejections. The method places more splits in those areas\nwhere the likelihood resides, shying away from low-probability regions destined\nfor ABC rejections. We provide two versions: (1) ABC-Tree for posterior\nsampling, and (2) ABC-MAP for maximum a posteriori estimation. We demonstrate\naccurate ABC approximability at much lower simulation cost. We justify the use\nof our tree-based bandit algorithms with nearly optimal regret bounds. Finally,\nwe successfully apply our approach to the problem of masked image\nclassification using deep generative models.\n", "date": "2024-04-16", "categories": []}
{"id": "2206.08952", "title": "The Impact of Variable Ordering on Bayesian Network Structure Learning", "abstract": "  Causal Bayesian Networks provide an important tool for reasoning under\nuncertainty with potential application to many complex causal systems.\nStructure learning algorithms that can tell us something about the causal\nstructure of these systems are becoming increasingly important. In the\nliterature, the validity of these algorithms is often tested for sensitivity\nover varying sample sizes, hyper-parameters, and occasionally objective\nfunctions. In this paper, we show that the order in which the variables are\nread from data can have much greater impact on the accuracy of the algorithm\nthan these factors. Because the variable ordering is arbitrary, any significant\neffect it has on learnt graph accuracy is concerning, and this raises questions\nabout the validity of the results produced by algorithms that are sensitive to,\nbut have not been assessed against, different variable orderings.\n", "date": "2022-06-17", "categories": []}
{"id": "2305.07504", "title": "Calibration-Aware Bayesian Learning", "abstract": "  Deep learning models, including modern systems like large language models,\nare well known to offer unreliable estimates of the uncertainty of their\ndecisions. In order to improve the quality of the confidence levels, also known\nas calibration, of a model, common approaches entail the addition of either\ndata-dependent or data-independent regularization terms to the training loss.\nData-dependent regularizers have been recently introduced in the context of\nconventional frequentist learning to penalize deviations between confidence and\naccuracy. In contrast, data-independent regularizers are at the core of\nBayesian learning, enforcing adherence of the variational distribution in the\nmodel parameter space to a prior density. The former approach is unable to\nquantify epistemic uncertainty, while the latter is severely affected by model\nmisspecification. In light of the limitations of both methods, this paper\nproposes an integrated framework, referred to as calibration-aware Bayesian\nneural networks (CA-BNNs), that applies both regularizers while optimizing over\na variational distribution as in Bayesian learning. Numerical results validate\nthe advantages of the proposed approach in terms of expected calibration error\n(ECE) and reliability diagrams.\n", "date": "2023-05-12", "categories": []}
{"id": "2310.13027", "title": "Be Bayesian by Attachments to Catch More Uncertainty", "abstract": "  Bayesian Neural Networks (BNNs) have become one of the promising approaches\nfor uncertainty estimation due to the solid theorical foundations. However, the\nperformance of BNNs is affected by the ability of catching uncertainty. Instead\nof only seeking the distribution of neural network weights by in-distribution\n(ID) data, in this paper, we propose a new Bayesian Neural Network with an\nAttached structure (ABNN) to catch more uncertainty from out-of-distribution\n(OOD) data. We first construct a mathematical description for the uncertainty\nof OOD data according to the prior distribution, and then develop an attached\nBayesian structure to integrate the uncertainty of OOD data into the backbone\nnetwork. ABNN is composed of an expectation module and several distribution\nmodules. The expectation module is a backbone deep network which focuses on the\noriginal task, and the distribution modules are mini Bayesian structures which\nserve as attachments of the backbone. In particular, the distribution modules\naim at extracting the uncertainty from both ID and OOD data. We further provide\ntheoretical analysis for the convergence of ABNN, and experimentally validate\nits superiority by comparing with some state-of-the-art uncertainty estimation\nmethods Code will be made available.\n", "date": "2023-10-19", "categories": []}
{"id": "2403.04810", "title": "Restricted Bayesian Neural Network", "abstract": "  Modern deep learning tools are remarkably effective in addressing intricate\nproblems. However, their operation as black-box models introduces increased\nuncertainty in predictions. Additionally, they contend with various challenges,\nincluding the need for substantial storage space in large networks, issues of\noverfitting, underfitting, vanishing gradients, and more. This study explores\nthe concept of Bayesian Neural Networks, presenting a novel architecture\ndesigned to significantly alleviate the storage space complexity of a network.\nFurthermore, we introduce an algorithm adept at efficiently handling\nuncertainties, ensuring robust convergence values without becoming trapped in\nlocal optima, particularly when the objective function lacks perfect convexity.\n", "date": "2024-03-06", "categories": []}
{"id": "2404.04425", "title": "Bayesian Additive Regression Networks", "abstract": "  We apply Bayesian Additive Regression Tree (BART) principles to training an\nensemble of small neural networks for regression tasks. Using Markov Chain\nMonte Carlo, we sample from the posterior distribution of neural networks that\nhave a single hidden layer. To create an ensemble of these, we apply Gibbs\nsampling to update each network against the residual target value (i.e.\nsubtracting the effect of the other networks). We demonstrate the effectiveness\nof this technique on several benchmark regression problems, comparing it to\nequivalent shallow neural networks, BART, and ordinary least squares. Our\nBayesian Additive Regression Networks (BARN) provide more consistent and often\nmore accurate results. On test data benchmarks, BARN averaged between 5 to 20\npercent lower root mean square error. This error performance does come at the\ncost, however, of greater computation time. BARN sometimes takes on the order\nof a minute where competing methods take a second or less. But, BARN without\ncross-validated hyperparameter tuning takes about the same amount of\ncomputation time as tuned other methods. Yet BARN is still typically more\naccurate.\n", "date": "2024-04-05", "categories": []}
{"id": "2311.09491", "title": "Spatial Bayesian Neural Networks", "abstract": "  interpretable, and well understood models that are routinely employed even\nthough, as is revealed through prior and posterior predictive checks, these can\npoorly characterise the spatial heterogeneity in the underlying process of\ninterest. Here, we propose a new, flexible class of spatial-process models,\nwhich we refer to as spatial Bayesian neural networks (SBNNs). An SBNN\nleverages the representational capacity of a Bayesian neural network; it is\ntailored to a spatial setting by incorporating a spatial ``embedding layer''\ninto the network and, possibly, spatially-varying network parameters. An SBNN\nis calibrated by matching its finite-dimensional distribution at locations on a\nfine gridding of space to that of a target process of interest. That process\ncould be easy to simulate from or we may have many realisations from it. We\npropose several variants of SBNNs, most of which are able to match the\nfinite-dimensional distribution of the target process at the selected grid\nbetter than conventional BNNs of similar complexity. We also show that an SBNN\ncan be used to represent a variety of spatial processes often used in practice,\nsuch as Gaussian processes, lognormal processes, and max-stable processes. We\nbriefly discuss the tools that could be used to make inference with SBNNs, and\nwe conclude with a discussion of their advantages and limitations.\n", "date": "2023-11-16", "categories": []}
{"id": "2310.17463", "title": "Bayesian Neural Controlled Differential Equations for Treatment Effect\n  Estimation", "abstract": "  Treatment effect estimation in continuous time is crucial for personalized\nmedicine. However, existing methods for this task are limited to point\nestimates of the potential outcomes, whereas uncertainty estimates have been\nignored. Needless to say, uncertainty quantification is crucial for reliable\ndecision-making in medical applications. To fill this gap, we propose a novel\nBayesian neural controlled differential equation (BNCDE) for treatment effect\nestimation in continuous time. In our BNCDE, the time dimension is modeled\nthrough a coupled system of neural controlled differential equations and neural\nstochastic differential equations, where the neural stochastic differential\nequations allow for tractable variational Bayesian inference. Thereby, for an\nassigned sequence of treatments, our BNCDE provides meaningful posterior\npredictive distributions of the potential outcomes. To the best of our\nknowledge, ours is the first tailored neural method to provide uncertainty\nestimates of treatment effects in continuous time. As such, our method is of\ndirect practical value for promoting reliable decision-making in medicine.\n", "date": "2023-10-26", "categories": []}
{"id": "2404.02873", "title": "Gaussian Process Regression with Soft Inequality and Monotonicity\n  Constraints", "abstract": "  Gaussian process (GP) regression is a non-parametric, Bayesian framework to\napproximate complex models. Standard GP regression can lead to an unbounded\nmodel in which some points can take infeasible values. We introduce a new GP\nmethod that enforces the physical constraints in a probabilistic manner. This\nGP model is trained by the quantum-inspired Hamiltonian Monte Carlo (QHMC).\nQHMC is an efficient way to sample from a broad class of distributions. Unlike\nthe standard Hamiltonian Monte Carlo algorithm in which a particle has a fixed\nmass, QHMC allows a particle to have a random mass matrix with a probability\ndistribution. Introducing the QHMC method to the inequality and monotonicity\nconstrained GP regression in the probabilistic sense, our approach improves the\naccuracy and reduces the variance in the resulting GP model. According to our\nexperiments on several datasets, the proposed approach serves as an efficient\nmethod as it accelerates the sampling process while maintaining the accuracy,\nand it is applicable to high dimensional problems.\n", "date": "2024-04-03", "categories": []}
{"id": "2403.01389", "title": "Fusion of Gaussian Processes Predictions with Monte Carlo Sampling", "abstract": "  In science and engineering, we often work with models designed for accurate\nprediction of variables of interest. Recognizing that these models are\napproximations of reality, it becomes desirable to apply multiple models to the\nsame data and integrate their outcomes. In this paper, we operate within the\nBayesian paradigm, relying on Gaussian processes as our models. These models\ngenerate predictive probability density functions (pdfs), and the objective is\nto integrate them systematically, employing both linear and log-linear pooling.\nWe introduce novel approaches for log-linear pooling, determining\ninput-dependent weights for the predictive pdfs of the Gaussian processes. The\naggregation of the pdfs is realized through Monte Carlo sampling, drawing\nsamples of weights from their posterior. The performance of these methods, as\nwell as those based on linear pooling, is demonstrated using a synthetic\ndataset.\n", "date": "2024-03-03", "categories": []}
{"id": "2303.05101", "title": "Scalable Stochastic Gradient Riemannian Langevin Dynamics in\n  Non-Diagonal Metrics", "abstract": "  Stochastic-gradient sampling methods are often used to perform Bayesian\ninference on neural networks. It has been observed that the methods in which\nnotions of differential geometry are included tend to have better performances,\nwith the Riemannian metric improving posterior exploration by accounting for\nthe local curvature. However, the existing methods often resort to simple\ndiagonal metrics to remain computationally efficient. This loses some of the\ngains. We propose two non-diagonal metrics that can be used in\nstochastic-gradient samplers to improve convergence and exploration but have\nonly a minor computational overhead over diagonal metrics. We show that for\nfully connected neural networks (NNs) with sparsity-inducing priors and\nconvolutional NNs with correlated priors, using these metrics can provide\nimprovements. For some other choices the posterior is sufficiently easy also\nfor the simpler metrics.\n", "date": "2023-03-09", "categories": []}
{"id": "2404.00085", "title": "Bayesian Nonparametrics: An Alternative to Deep Learning", "abstract": "  Bayesian nonparametric models offer a flexible and powerful framework for\nstatistical model selection, enabling the adaptation of model complexity to the\nintricacies of diverse datasets. This survey intends to delve into the\nsignificance of Bayesian nonparametrics, particularly in addressing complex\nchallenges across various domains such as statistics, computer science, and\nelectrical engineering. By elucidating the basic properties and theoretical\nfoundations of these nonparametric models, this survey aims to provide a\ncomprehensive understanding of Bayesian nonparametrics and their relevance in\naddressing complex problems, particularly in the domain of multi-object\ntracking. Through this exploration, we uncover the versatility and efficacy of\nBayesian nonparametric methodologies, paving the way for innovative solutions\nto intricate challenges across diverse disciplines.\n", "date": "2024-03-29", "categories": []}
{"id": "2403.18147", "title": "Divide, Conquer, Combine Bayesian Decision Tree Sampling", "abstract": "  Decision trees are commonly used predictive models due to their flexibility\nand interpretability. This paper is directed at quantifying the uncertainty of\ndecision tree predictions by employing a Bayesian inference approach. This is\nchallenging because these approaches need to explore both the tree structure\nspace and the space of decision parameters associated with each tree structure.\nThis has been handled by using Markov Chain Monte Carlo (MCMC) methods, where a\nMarkov Chain is constructed to provide samples from the desired Bayesian\nestimate. Importantly, the structure and the decision parameters are tightly\ncoupled; small changes in the tree structure can demand vastly different\ndecision parameters to provide accurate predictions. A challenge for existing\nMCMC approaches is proposing joint changes in both the tree structure and the\ndecision parameters that result in efficient sampling. This paper takes a\ndifferent approach, where each distinct tree structure is associated with a\nunique set of decision parameters. The proposed approach, entitled DCC-Tree, is\ninspired by the work in Zhou et al. [23] for probabilistic programs and\nCochrane et al. [4] for Hamiltonian Monte Carlo (HMC) based sampling for\ndecision trees. Results show that DCC-Tree performs comparably to other\nHMC-based methods and better than existing Bayesian tree methods while\nimproving on consistency and reducing the per-proposal complexity.\n", "date": "2024-03-26", "categories": []}
{"id": "2310.05401", "title": "Entropy-MCMC: Sampling from Flat Basins with Ease", "abstract": "  Bayesian deep learning counts on the quality of posterior distribution\nestimation. However, the posterior of deep neural networks is highly\nmulti-modal in nature, with local modes exhibiting varying generalization\nperformance. Given a practical budget, targeting at the original posterior can\nlead to suboptimal performance, as some samples may become trapped in \"bad\"\nmodes and suffer from overfitting. Leveraging the observation that \"good\" modes\nwith low generalization error often reside in flat basins of the energy\nlandscape, we propose to bias sampling on the posterior toward these flat\nregions. Specifically, we introduce an auxiliary guiding variable, the\nstationary distribution of which resembles a smoothed posterior free from sharp\nmodes, to lead the MCMC sampler to flat basins. By integrating this guiding\nvariable with the model parameter, we create a simple joint distribution that\nenables efficient sampling with minimal computational overhead. We prove the\nconvergence of our method and further show that it converges faster than\nseveral existing flatness-aware methods in the strongly convex setting.\nEmpirical results demonstrate that our method can successfully sample from flat\nbasins of the posterior, and outperforms all compared baselines on multiple\nbenchmarks including classification, calibration, and out-of-distribution\ndetection.\n", "date": "2023-10-09", "categories": []}
{"id": "2403.16681", "title": "A note on generalization bounds for losses with finite moments", "abstract": "  This paper studies the truncation method from Alquier [1] to derive\nhigh-probability PAC-Bayes bounds for unbounded losses with heavy tails.\nAssuming that the $p$-th moment is bounded, the resulting bounds interpolate\nbetween a slow rate $1 / \\sqrt{n}$ when $p=2$, and a fast rate $1 / n$ when $p\n\\to \\infty$ and the loss is essentially bounded. Moreover, the paper derives a\nhigh-probability PAC-Bayes bound for losses with a bounded variance. This bound\nhas an exponentially better dependence on the confidence parameter and the\ndependency measure than previous bounds in the literature. Finally, the paper\nextends all results to guarantees in expectation and single-draw PAC-Bayes. In\norder to so, it obtains analogues of the PAC-Bayes fast rate bound for bounded\nlosses from [2] in these settings.\n", "date": "2024-03-25", "categories": []}
{"id": "2403.14225", "title": "Posterior concentrations of fully-connected Bayesian neural networks\n  with general priors on the weights", "abstract": "  Bayesian approaches for training deep neural networks (BNNs) have received\nsignificant interest and have been effectively utilized in a wide range of\napplications. There have been several studies on the properties of posterior\nconcentrations of BNNs. However, most of these studies only demonstrate results\nin BNN models with sparse or heavy-tailed priors. Surprisingly, no theoretical\nresults currently exist for BNNs using Gaussian priors, which are the most\ncommonly used one. The lack of theory arises from the absence of approximation\nresults of Deep Neural Networks (DNNs) that are non-sparse and have bounded\nparameters. In this paper, we present a new approximation theory for non-sparse\nDNNs with bounded parameters. Additionally, based on the approximation theory,\nwe show that BNNs with non-sparse general priors can achieve near-minimax\noptimal posterior concentration rates to the true model.\n", "date": "2024-03-21", "categories": []}
{"id": "2403.12729", "title": "Posterior Uncertainty Quantification in Neural Networks using Data\n  Augmentation", "abstract": "  In this paper, we approach the problem of uncertainty quantification in deep\nlearning through a predictive framework, which captures uncertainty in model\nparameters by specifying our assumptions about the predictive distribution of\nunseen future data. Under this view, we show that deep ensembling\n(Lakshminarayanan et al., 2017) is a fundamentally mis-specified model class,\nsince it assumes that future data are supported on existing observations only\n-- a situation rarely encountered in practice. To address this limitation, we\npropose MixupMP, a method that constructs a more realistic predictive\ndistribution using popular data augmentation techniques. MixupMP operates as a\ndrop-in replacement for deep ensembles, where each ensemble member is trained\non a random simulation from this predictive distribution. Grounded in the\nrecently-proposed framework of Martingale posteriors (Fong et al., 2023),\nMixupMP returns samples from an implicitly defined Bayesian posterior. Our\nempirical analysis showcases that MixupMP achieves superior predictive\nperformance and uncertainty quantification on various image classification\ndatasets, when compared with existing Bayesian and non-Bayesian approaches.\n", "date": "2024-03-18", "categories": []}
{"id": "2403.10610", "title": "Sequential Monte Carlo for Inclusive KL Minimization in Amortized\n  Variational Inference", "abstract": "  For training an encoder network to perform amortized variational inference,\nthe Kullback-Leibler (KL) divergence from the exact posterior to its\napproximation, known as the inclusive or forward KL, is an increasingly popular\nchoice of variational objective due to the mass-covering property of its\nminimizer. However, minimizing this objective is challenging. A popular\nexisting approach, Reweighted Wake-Sleep (RWS), suffers from heavily biased\ngradients and a circular pathology that results in highly concentrated\nvariational distributions. As an alternative, we propose SMC-Wake, a procedure\nfor fitting an amortized variational approximation that uses\nlikelihood-tempered sequential Monte Carlo samplers to estimate the gradient of\nthe inclusive KL divergence. We propose three gradient estimators, all of which\nare asymptotically unbiased in the number of iterations and two of which are\nstrongly consistent. Our method interleaves stochastic gradient updates, SMC\nsamplers, and iterative improvement to an estimate of the normalizing constant\nto reduce bias from self-normalization. In experiments with both simulated and\nreal datasets, SMC-Wake fits variational distributions that approximate the\nposterior more accurately than existing methods.\n", "date": "2024-03-15", "categories": []}
{"id": "2403.08609", "title": "On the Convergence of Locally Adaptive and Scalable Diffusion-Based\n  Sampling Methods for Deep Bayesian Neural Network Posteriors", "abstract": "  Achieving robust uncertainty quantification for deep neural networks\nrepresents an important requirement in many real-world applications of deep\nlearning such as medical imaging where it is necessary to assess the\nreliability of a neural network's prediction. Bayesian neural networks are a\npromising approach for modeling uncertainties in deep neural networks.\nUnfortunately, generating samples from the posterior distribution of neural\nnetworks is a major challenge. One significant advance in that direction would\nbe the incorporation of adaptive step sizes, similar to modern neural network\noptimizers, into Monte Carlo Markov chain sampling algorithms without\nsignificantly increasing computational demand. Over the past years, several\npapers have introduced sampling algorithms with claims that they achieve this\nproperty. However, do they indeed converge to the correct distribution? In this\npaper, we demonstrate that these methods can have a substantial bias in the\ndistribution they sample, even in the limit of vanishing step sizes and at full\nbatch size.\n", "date": "2024-03-13", "categories": []}
{"id": "2403.07282", "title": "Enhancing Transfer Learning with Flexible Nonparametric Posterior\n  Sampling", "abstract": "  Transfer learning has recently shown significant performance across various\ntasks involving deep neural networks. In these transfer learning scenarios, the\nprior distribution for downstream data becomes crucial in Bayesian model\naveraging (BMA). While previous works proposed the prior over the neural\nnetwork parameters centered around the pre-trained solution, such strategies\nhave limitations when dealing with distribution shifts between upstream and\ndownstream data. This paper introduces nonparametric transfer learning (NPTL),\na flexible posterior sampling method to address the distribution shift issue\nwithin the context of nonparametric learning. The nonparametric learning (NPL)\nmethod is a recent approach that employs a nonparametric prior for posterior\nsampling, efficiently accounting for model misspecification scenarios, which is\nsuitable for transfer learning scenarios that may involve the distribution\nshift between upstream and downstream tasks. Through extensive empirical\nvalidations, we demonstrate that our approach surpasses other baselines in BMA\nperformance.\n", "date": "2024-03-12", "categories": []}
{"id": "2305.09792", "title": "Score Operator Newton transport", "abstract": "  We propose a new approach for sampling and Bayesian computation that uses the\nscore of the target distribution to construct a transport from a given\nreference distribution to the target. Our approach is an infinite-dimensional\nNewton method, involving a linear PDE, for finding a zero of a\n``score-residual'' operator. We prove sufficient conditions for convergence to\na valid transport map. Our Newton iterates can be computed by exploiting fast\nsolvers for elliptic PDEs, resulting in new algorithms for Bayesian inference\nand other sampling tasks. We identify elementary settings where score-operator\nNewton transport achieves fast convergence while avoiding mode collapse.\n", "date": "2023-05-16", "categories": []}
{"id": "2403.01272", "title": "Can a Confident Prior Replace a Cold Posterior?", "abstract": "  Benchmark datasets used for image classification tend to have very low levels\nof label noise. When Bayesian neural networks are trained on these datasets,\nthey often underfit, misrepresenting the aleatoric uncertainty of the data. A\ncommon solution is to cool the posterior, which improves fit to the training\ndata but is challenging to interpret from a Bayesian perspective. We explore\nwhether posterior tempering can be replaced by a confidence-inducing prior\ndistribution. First, we introduce a \"DirClip\" prior that is practical to sample\nand nearly matches the performance of a cold posterior. Second, we introduce a\n\"confidence prior\" that directly approximates a cold likelihood in the limit of\ndecreasing temperature but cannot be easily sampled. Lastly, we provide several\ngeneral insights into confidence-inducing priors, such as when they might\ndiverge and how fine-tuning can mitigate numerical instability.\n", "date": "2024-03-02", "categories": []}
{"id": "2207.06154", "title": "On the Robustness of Bayesian Neural Networks to Adversarial Attacks", "abstract": "  Vulnerability to adversarial attacks is one of the principal hurdles to the\nadoption of deep learning in safety-critical applications. Despite significant\nefforts, both practical and theoretical, training deep learning models robust\nto adversarial attacks is still an open problem. In this paper, we analyse the\ngeometry of adversarial attacks in the large-data, overparameterized limit for\nBayesian Neural Networks (BNNs). We show that, in the limit, vulnerability to\ngradient-based attacks arises as a result of degeneracy in the data\ndistribution, i.e., when the data lies on a lower-dimensional submanifold of\nthe ambient space. As a direct consequence, we demonstrate that in this limit\nBNN posteriors are robust to gradient-based adversarial attacks. Crucially, we\nprove that the expected gradient of the loss with respect to the BNN posterior\ndistribution is vanishing, even when each neural network sampled from the\nposterior is vulnerable to gradient-based attacks. Experimental results on the\nMNIST, Fashion MNIST, and half moons datasets, representing the finite data\nregime, with BNNs trained with Hamiltonian Monte Carlo and Variational\nInference, support this line of arguments, showing that BNNs can display both\nhigh accuracy on clean data and robustness to both gradient-based and\ngradient-free based adversarial attacks.\n", "date": "2022-07-13", "categories": []}
{"id": "2402.18337", "title": "Probabilistic Bayesian optimal experimental design using conditional\n  normalizing flows", "abstract": "  Bayesian optimal experimental design (OED) seeks to conduct the most\ninformative experiment under budget constraints to update the prior knowledge\nof a system to its posterior from the experimental data in a Bayesian\nframework. Such problems are computationally challenging because of (1)\nexpensive and repeated evaluation of some optimality criterion that typically\ninvolves a double integration with respect to both the system parameters and\nthe experimental data, (2) suffering from the curse-of-dimensionality when the\nsystem parameters and design variables are high-dimensional, (3) the\noptimization is combinatorial and highly non-convex if the design variables are\nbinary, often leading to non-robust designs. To make the solution of the\nBayesian OED problem efficient, scalable, and robust for practical\napplications, we propose a novel joint optimization approach. This approach\nperforms simultaneous (1) training of a scalable conditional normalizing flow\n(CNF) to efficiently maximize the expected information gain (EIG) of a jointly\nlearned experimental design (2) optimization of a probabilistic formulation of\nthe binary experimental design with a Bernoulli distribution. We demonstrate\nthe performance of our proposed method for a practical MRI data acquisition\nproblem, one of the most challenging Bayesian OED problems that has\nhigh-dimensional (320 $\\times$ 320) parameters at high image resolution,\nhigh-dimensional (640 $\\times$ 386) observations, and binary mask designs to\nselect the most informative observations.\n", "date": "2024-02-28", "categories": []}
{"id": "2310.19608", "title": "On Feynman--Kac training of partial Bayesian neural networks", "abstract": "  Recently, partial Bayesian neural networks (pBNNs), which only consider a\nsubset of the parameters to be stochastic, were shown to perform competitively\nwith full Bayesian neural networks. However, pBNNs are often multi-modal in the\nlatent variable space and thus challenging to approximate with parametric\nmodels. To address this problem, we propose an efficient sampling-based\ntraining strategy, wherein the training of a pBNN is formulated as simulating a\nFeynman--Kac model. We then describe variations of sequential Monte Carlo\nsamplers that allow us to simultaneously estimate the parameters and the latent\nposterior distribution of this model at a tractable computational cost. Using\nvarious synthetic and real-world datasets we show that our proposed training\nscheme outperforms the state of the art in terms of predictive performance.\n", "date": "2023-10-30", "categories": []}
{"id": "2402.17087", "title": "A Note on Bayesian Networks with Latent Root Variables", "abstract": "  We characterise the likelihood function computed from a Bayesian network with\nlatent variables as root nodes. We show that the marginal distribution over the\nremaining, manifest, variables also factorises as a Bayesian network, which we\ncall empirical. A dataset of observations of the manifest variables allows us\nto quantify the parameters of the empirical Bayesian net. We prove that (i) the\nlikelihood of such a dataset from the original Bayesian network is dominated by\nthe global maximum of the likelihood from the empirical one; and that (ii) such\na maximum is attained if and only if the parameters of the Bayesian network are\nconsistent with those of the empirical model.\n", "date": "2024-02-26", "categories": []}
{"id": "2402.10797", "title": "BlackJAX: Composable Bayesian inference in JAX", "abstract": "  BlackJAX is a library implementing sampling and variational inference\nalgorithms commonly used in Bayesian computation. It is designed for ease of\nuse, speed, and modularity by taking a functional approach to the algorithms'\nimplementation. BlackJAX is written in Python, using JAX to compile and run\nNumpPy-like samplers and variational methods on CPUs, GPUs, and TPUs. The\nlibrary integrates well with probabilistic programming languages by working\ndirectly with the (un-normalized) target log density function. BlackJAX is\nintended as a collection of low-level, composable implementations of basic\nstatistical 'atoms' that can be combined to perform well-defined Bayesian\ninference, but also provides high-level routines for ease of use. It is\ndesigned for users who need cutting-edge methods, researchers who want to\ncreate complex sampling methods, and people who want to learn how these work.\n", "date": "2024-02-16", "categories": []}
{"id": "2402.13410", "title": "Bayesian Neural Networks with Domain Knowledge Priors", "abstract": "  Bayesian neural networks (BNNs) have recently gained popularity due to their\nability to quantify model uncertainty. However, specifying a prior for BNNs\nthat captures relevant domain knowledge is often extremely challenging. In this\nwork, we propose a framework for integrating general forms of domain knowledge\n(i.e., any knowledge that can be represented by a loss function) into a BNN\nprior through variational inference, while enabling computationally efficient\nposterior inference and sampling. Specifically, our approach results in a prior\nover neural network weights that assigns high probability mass to models that\nbetter align with our domain knowledge, leading to posterior samples that also\nexhibit this behavior. We show that BNNs using our proposed domain knowledge\npriors outperform those with standard priors (e.g., isotropic Gaussian,\nGaussian process), successfully incorporating diverse types of prior\ninformation such as fairness, physics rules, and healthcare knowledge and\nachieving better predictive performance. We also present techniques for\ntransferring the learned priors across different model architectures,\ndemonstrating their broad utility across various settings.\n", "date": "2024-02-20", "categories": []}
{"id": "2306.11380", "title": "A Bayesian Take on Gaussian Process Networks", "abstract": "  Gaussian Process Networks (GPNs) are a class of directed graphical models\nwhich employ Gaussian processes as priors for the conditional expectation of\neach variable given its parents in the network. The model allows the\ndescription of continuous joint distributions in a compact but flexible manner\nwith minimal parametric assumptions on the dependencies between variables.\nBayesian structure learning of GPNs requires computing the posterior over\ngraphs of the network and is computationally infeasible even in low dimensions.\nThis work implements Monte Carlo and Markov Chain Monte Carlo methods to sample\nfrom the posterior distribution of network structures. As such, the approach\nfollows the Bayesian paradigm, comparing models via their marginal likelihood\nand computing the posterior probability of the GPN features. Simulation studies\nshow that our method outperforms state-of-the-art algorithms in recovering the\ngraphical structure of the network and provides an accurate approximation of\nits posterior distribution.\n", "date": "2023-06-20", "categories": []}
{"id": "2402.11025", "title": "Training Bayesian Neural Networks with Sparse Subspace Variational\n  Inference", "abstract": "  Bayesian neural networks (BNNs) offer uncertainty quantification but come\nwith the downside of substantially increased training and inference costs.\nSparse BNNs have been investigated for efficient inference, typically by either\nslowly introducing sparsity throughout the training or by post-training\ncompression of dense BNNs. The dilemma of how to cut down massive training\ncosts remains, particularly given the requirement to learn about the\nuncertainty. To solve this challenge, we introduce Sparse Subspace Variational\nInference (SSVI), the first fully sparse BNN framework that maintains a\nconsistently highly sparse Bayesian model throughout the training and inference\nphases. Starting from a randomly initialized low-dimensional sparse subspace,\nour approach alternately optimizes the sparse subspace basis selection and its\nassociated parameters. While basis selection is characterized as a\nnon-differentiable problem, we approximate the optimal solution with a\nremoval-and-addition strategy, guided by novel criteria based on weight\ndistribution statistics. Our extensive experiments show that SSVI sets new\nbenchmarks in crafting sparse BNNs, achieving, for instance, a 10-20x\ncompression in model size with under 3\\% performance drop, and up to 20x FLOPs\nreduction during training compared with dense VI training. Remarkably, SSVI\nalso demonstrates enhanced robustness to hyperparameters, reducing the need for\nintricate tuning in VI and occasionally even surpassing VI-trained dense BNNs\non both accuracy and uncertainty metrics.\n", "date": "2024-02-16", "categories": []}
{"id": "2402.11973", "title": "Bayesian Active Learning for Censored Regression", "abstract": "  Bayesian active learning is based on information theoretical approaches that\nfocus on maximising the information that new observations provide to the model\nparameters. This is commonly done by maximising the Bayesian Active Learning by\nDisagreement (BALD) acquisitions function. However, we highlight that it is\nchallenging to estimate BALD when the new data points are subject to\ncensorship, where only clipped values of the targets are observed. To address\nthis, we derive the entropy and the mutual information for censored\ndistributions and derive the BALD objective for active learning in censored\nregression ($\\mathcal{C}$-BALD). We propose a novel modelling approach to\nestimate the $\\mathcal{C}$-BALD objective and use it for active learning in the\ncensored setting. Across a wide range of datasets and models, we demonstrate\nthat $\\mathcal{C}$-BALD outperforms other Bayesian active learning methods in\ncensored regression.\n", "date": "2024-02-19", "categories": []}
{"id": "2402.12292", "title": "Regularization by denoising: Bayesian model and Langevin-within-split\n  Gibbs sampling", "abstract": "  This paper introduces a Bayesian framework for image inversion by deriving a\nprobabilistic counterpart to the regularization-by-denoising (RED) paradigm. It\nadditionally implements a Monte Carlo algorithm specifically tailored for\nsampling from the resulting posterior distribution, based on an asymptotically\nexact data augmentation (AXDA). The proposed algorithm is an approximate\ninstance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo\nstep. The proposed method is applied to common imaging tasks such as\ndeblurring, inpainting and super-resolution, demonstrating its efficacy through\nextensive numerical experiments. These contributions advance Bayesian inference\nin imaging by leveraging data-driven regularization strategies within a\nprobabilistic framework.\n", "date": "2024-02-19", "categories": []}
{"id": "2304.11005", "title": "Self-Correcting Bayesian Optimization through Bayesian Active Learning", "abstract": "  Gaussian processes are the model of choice in Bayesian optimization and\nactive learning. Yet, they are highly dependent on cleverly chosen\nhyperparameters to reach their full potential, and little effort is devoted to\nfinding good hyperparameters in the literature. We demonstrate the impact of\nselecting good hyperparameters for GPs and present two acquisition functions\nthat explicitly prioritize hyperparameter learning. Statistical distance-based\nActive Learning (SAL) considers the average disagreement between samples from\nthe posterior, as measured by a statistical distance. SAL outperforms the\nstate-of-the-art in Bayesian active learning on several test functions. We then\nintroduce Self-Correcting Bayesian Optimization (SCoreBO), which extends SAL to\nperform Bayesian optimization and active learning simultaneously. SCoreBO\nlearns the model hyperparameters at improved rates compared to vanilla BO,\nwhile outperforming the latest Bayesian optimization methods on traditional\nbenchmarks. Moreover, we demonstrate the importance of self-correction on\natypical Bayesian optimization tasks.\n", "date": "2023-04-21", "categories": []}
{"id": "2402.09796", "title": "Closed-form Filtering for Non-linear Systems", "abstract": "  Sequential Bayesian Filtering aims to estimate the current state distribution\nof a Hidden Markov Model, given the past observations. The problem is\nwell-known to be intractable for most application domains, except in notable\ncases such as the tabular setting or for linear dynamical systems with gaussian\nnoise. In this work, we propose a new class of filters based on Gaussian PSD\nModels, which offer several advantages in terms of density approximation and\ncomputational efficiency. We show that filtering can be efficiently performed\nin closed form when transitions and observations are Gaussian PSD Models. When\nthe transition and observations are approximated by Gaussian PSD Models, we\nshow that our proposed estimator enjoys strong theoretical guarantees, with\nestimation error that depends on the quality of the approximation and is\nadaptive to the regularity of the transition probabilities. In particular, we\nidentify regimes in which our proposed filter attains a TV $\\epsilon$-error\nwith memory and computational complexity of $O(\\epsilon^{-1})$ and\n$O(\\epsilon^{-3/2})$ respectively, including the offline learning step, in\ncontrast to the $O(\\epsilon^{-2})$ complexity of sampling methods such as\nparticle filtering.\n", "date": "2024-02-15", "categories": []}
{"id": "2306.09686", "title": "Collapsed Inference for Bayesian Deep Learning", "abstract": "  Bayesian neural networks (BNNs) provide a formalism to quantify and calibrate\nuncertainty in deep learning. Current inference approaches for BNNs often\nresort to few-sample estimation for scalability, which can harm predictive\nperformance, while its alternatives tend to be computationally prohibitively\nexpensive. We tackle this challenge by revealing a previously unseen connection\nbetween inference on BNNs and volume computation problems. With this\nobservation, we introduce a novel collapsed inference scheme that performs\nBayesian model averaging using collapsed samples. It improves over a\nMonte-Carlo sample by limiting sampling to a subset of the network weights\nwhile pairing it with some closed-form conditional distribution over the rest.\nA collapsed sample represents uncountably many models drawn from the\napproximate posterior and thus yields higher sample efficiency. Further, we\nshow that the marginalization of a collapsed sample can be solved analytically\nand efficiently despite the non-linearity of neural networks by leveraging\nexisting volume computation solvers. Our proposed use of collapsed samples\nachieves a balance between scalability and accuracy. On various regression and\nclassification tasks, our collapsed Bayesian deep learning approach\ndemonstrates significant improvements over existing methods and sets a new\nstate of the art in terms of uncertainty estimation as well as predictive\nperformance.\n", "date": "2023-06-16", "categories": []}
{"id": "2311.04153", "title": "Kernel-, mean- and noise-marginalised Gaussian processes for exoplanet\n  transits and $H_0$ inference", "abstract": "  Using a fully Bayesian approach, Gaussian Process regression is extended to\ninclude marginalisation over the kernel choice and kernel hyperparameters. In\naddition, Bayesian model comparison via the evidence enables direct kernel\ncomparison. The calculation of the joint posterior was implemented with a\ntransdimensional sampler which simultaneously samples over the discrete kernel\nchoice and their hyperparameters by embedding these in a higher-dimensional\nspace, from which samples are taken using nested sampling. Kernel recovery and\nmean function inference were explored on synthetic data from exoplanet transit\nlight curve simulations. Subsequently, the method was extended to\nmarginalisation over mean functions and noise models and applied to the\ninference of the present-day Hubble parameter, $H_0$, from real measurements of\nthe Hubble parameter as a function of redshift, derived from the cosmologically\nmodel-independent cosmic chronometer and $\\Lambda$CDM-dependent baryon acoustic\noscillation observations. The inferred $H_0$ values from the cosmic\nchronometers, baryon acoustic oscillations and combined datasets are $H_0= 66\n\\pm 6\\, \\mathrm{km}\\,\\mathrm{s}^{-1}\\,\\mathrm{Mpc}^{-1}$, $H_0= 67 \\pm 10\\,\n\\mathrm{km}\\,\\mathrm{s}^{-1}\\,\\mathrm{Mpc}^{-1}$ and $H_0= 69 \\pm 6\\,\n\\mathrm{km}\\,\\mathrm{s}^{-1}\\,\\mathrm{Mpc}^{-1}$, respectively. The kernel\nposterior of the cosmic chronometers dataset prefers a non-stationary linear\nkernel. Finally, the datasets are shown to be not in tension with $\\ln\nR=12.17\\pm 0.02$.\n", "date": "2023-11-07", "categories": []}
{"id": "2402.06940", "title": "Efficient Incremental Belief Updates Using Weighted Virtual Observations", "abstract": "  We present an algorithmic solution to the problem of incremental belief\nupdating in the context of Monte Carlo inference in Bayesian statistical models\nrepresented by probabilistic programs. Given a model and a sample-approximated\nposterior, our solution constructs a set of weighted observations to condition\nthe model such that inference would result in the same posterior. This problem\narises e.g. in multi-level modelling, incremental inference, inference in\npresence of privacy constraints. First, a set of virtual observations is\nselected, then, observation weights are found through a computationally\nefficient optimization procedure such that the reconstructed posterior\ncoincides with or closely approximates the original posterior. We implement and\napply the solution to a number of didactic examples and case studies, showing\nefficiency and robustness of our approach. The provided reference\nimplementation is agnostic to the probabilistic programming language or the\ninference algorithm, and can be applied to most mainstream probabilistic\nprogramming environments.\n", "date": "2024-02-10", "categories": []}
{"id": "2305.18435", "title": "Statistically Efficient Bayesian Sequential Experiment Design via\n  Reinforcement Learning with Cross-Entropy Estimators", "abstract": "  Reinforcement learning can learn amortised design policies for designing\nsequences of experiments. However, current amortised methods rely on estimators\nof expected information gain (EIG) that require an exponential number of\nsamples on the magnitude of the EIG to achieve an unbiased estimation. We\npropose the use of an alternative estimator based on the cross-entropy of the\njoint model distribution and a flexible proposal distribution. This proposal\ndistribution approximates the true posterior of the model parameters given the\nexperimental history and the design policy. Our method overcomes the\nexponential-sample complexity of previous approaches and provide more accurate\nestimates of high EIG values. More importantly, it allows learning of superior\ndesign policies, and is compatible with continuous and discrete design spaces,\nnon-differentiable likelihoods and even implicit probabilistic models.\n", "date": "2023-05-29", "categories": []}
{"id": "2402.01098", "title": "Bayesian Deep Learning for Remaining Useful Life Estimation via Stein\n  Variational Gradient Descent", "abstract": "  A crucial task in predictive maintenance is estimating the remaining useful\nlife of physical systems. In the last decade, deep learning has improved\nconsiderably upon traditional model-based and statistical approaches in terms\nof predictive performance. However, in order to optimally plan maintenance\noperations, it is also important to quantify the uncertainty inherent to the\npredictions. This issue can be addressed by turning standard frequentist neural\nnetworks into Bayesian neural networks, which are naturally capable of\nproviding confidence intervals around the estimates. Several methods exist for\ntraining those models. Researchers have focused mostly on parametric\nvariational inference and sampling-based techniques, which notoriously suffer\nfrom limited approximation power and large computational burden, respectively.\nIn this work, we use Stein variational gradient descent, a recently proposed\nalgorithm for approximating intractable distributions that overcomes the\ndrawbacks of the aforementioned techniques. In particular, we show through\nexperimental studies on simulated run-to-failure turbofan engine degradation\ndata that Bayesian deep learning models trained via Stein variational gradient\ndescent consistently outperform with respect to convergence speed and\npredictive performance both the same models trained via parametric variational\ninference and their frequentist counterparts trained via backpropagation.\nFurthermore, we propose a method to enhance performance based on the\nuncertainty information provided by the Bayesian models. We release the source\ncode at https://github.com/lucadellalib/bdl-rul-svgd.\n", "date": "2024-02-02", "categories": []}
{"id": "2310.19390", "title": "Implicit Manifold Gaussian Process Regression", "abstract": "  Gaussian process regression is widely used because of its ability to provide\nwell-calibrated uncertainty estimates and handle small or sparse datasets.\nHowever, it struggles with high-dimensional data. One possible way to scale\nthis technique to higher dimensions is to leverage the implicit low-dimensional\nmanifold upon which the data actually lies, as postulated by the manifold\nhypothesis. Prior work ordinarily requires the manifold structure to be\nexplicitly provided though, i.e. given by a mesh or be known to be one of the\nwell-known manifolds like the sphere. In contrast, in this paper we propose a\nGaussian process regression technique capable of inferring implicit structure\ndirectly from data (labeled and unlabeled) in a fully differentiable way. For\nthe resulting model, we discuss its convergence to the Mat\\'ern Gaussian\nprocess on the assumed manifold. Our technique scales up to hundreds of\nthousands of data points, and may improve the predictive performance and\ncalibration of the standard Gaussian process regression in high-dimensional\nsettings.\n", "date": "2023-10-30", "categories": []}
{"id": "2401.16419", "title": "Semi-parametric Expert Bayesian Network Learning with Gaussian Processes\n  and Horseshoe Priors", "abstract": "  This paper proposes a model learning Semi-parametric relationships in an\nExpert Bayesian Network (SEBN) with linear parameter and structure constraints.\nWe use Gaussian Processes and a Horseshoe prior to introduce minimal nonlinear\ncomponents. To prioritize modifying the expert graph over adding new edges, we\noptimize differential Horseshoe scales. In real-world datasets with unknown\ntruth, we generate diverse graphs to accommodate user input, addressing\nidentifiability issues and enhancing interpretability. Evaluation on synthetic\nand UCI Liver Disorders datasets, using metrics like structural Hamming\nDistance and test likelihood, demonstrates our models outperform\nstate-of-the-art semi-parametric Bayesian Network model.\n", "date": "2024-01-29", "categories": []}
{"id": "2401.15838", "title": "Distributed Markov Chain Monte Carlo Sampling based on the Alternating\n  Direction Method of Multipliers", "abstract": "  Many machine learning applications require operating on a spatially\ndistributed dataset. Despite technological advances, privacy considerations and\ncommunication constraints may prevent gathering the entire dataset in a central\nunit. In this paper, we propose a distributed sampling scheme based on the\nalternating direction method of multipliers, which is commonly used in the\noptimization literature due to its fast convergence. In contrast to distributed\noptimization, distributed sampling allows for uncertainty quantification in\nBayesian inference tasks. We provide both theoretical guarantees of our\nalgorithm's convergence and experimental evidence of its superiority to the\nstate-of-the-art. For our theoretical results, we use convex optimization tools\nto establish a fundamental inequality on the generated local sample iterates.\nThis inequality enables us to show convergence of the distribution associated\nwith these iterates to the underlying target distribution in Wasserstein\ndistance. In simulation, we deploy our algorithm on linear and logistic\nregression tasks and illustrate its fast convergence compared to existing\ngradient-based methods.\n", "date": "2024-01-29", "categories": []}
{"id": "2306.02719", "title": "Multiple output samples per input in a single-output Gaussian process", "abstract": "  The standard Gaussian Process (GP) only considers a single output sample per\ninput in the training set. Datasets for subjective tasks, such as spoken\nlanguage assessment, may be annotated with output labels from multiple human\nraters per input. This paper proposes to generalise the GP to allow for these\nmultiple output samples in the training set, and thus make use of available\noutput uncertainty information. This differs from a multi-output GP, as all\noutput samples are from the same task here. The output density function is\nformulated to be the joint likelihood of observing all output samples, and\nlatent variables are not repeated to reduce computation cost. The test set\npredictions are inferred similarly to a standard GP, with a difference being in\nthe optimised hyper-parameters. This is evaluated on speechocean762, showing\nthat it allows the GP to compute a test set output distribution that is more\nsimilar to the collection of reference outputs from the multiple human raters.\n", "date": "2023-06-05", "categories": []}
{"id": "2401.14722", "title": "A Nonparametric Bayes Approach to Online Activity Prediction", "abstract": "  Accurately predicting the onset of specific activities within defined\ntimeframes holds significant importance in several applied contexts. In\nparticular, accurate prediction of the number of future users that will be\nexposed to an intervention is an important piece of information for\nexperimenters running online experiments (A/B tests). In this work, we propose\na novel approach to predict the number of users that will be active in a given\ntime period, as well as the temporal trajectory needed to attain a desired user\nparticipation threshold. We model user activity using a Bayesian nonparametric\napproach which allows us to capture the underlying heterogeneity in user\nengagement. We derive closed-form expressions for the number of new users\nexpected in a given period, and a simple Monte Carlo algorithm targeting the\nposterior distribution of the number of days needed to attain a desired number\nof users; the latter is important for experimental planning. We illustrate the\nperformance of our approach via several experiments on synthetic and real world\ndata, in which we show that our novel method outperforms existing competitors.\n", "date": "2024-01-26", "categories": []}
{"id": "2311.14828", "title": "Deep Latent Force Models: ODE-based Process Convolutions for Bayesian\n  Deep Learning", "abstract": "  Modelling the behaviour of highly nonlinear dynamical systems with robust\nuncertainty quantification is a challenging task which typically requires\napproaches specifically designed to address the problem at hand. We introduce a\ndomain-agnostic model to address this issue termed the deep latent force model\n(DLFM), a deep Gaussian process with physics-informed kernels at each layer,\nderived from ordinary differential equations using the framework of process\nconvolutions. Two distinct formulations of the DLFM are presented which utilise\nweight-space and variational inducing points-based Gaussian process\napproximations, both of which are amenable to doubly stochastic variational\ninference. We present empirical evidence of the capability of the DLFM to\ncapture the dynamics present in highly nonlinear real-world multi-output time\nseries data. Additionally, we find that the DLFM is capable of achieving\ncomparable performance to a range of non-physics-informed probabilistic models\non benchmark univariate regression tasks. We also empirically assess the\nnegative impact of the inducing points framework on the extrapolation\ncapabilities of LFM-based models.\n", "date": "2023-11-24", "categories": []}
{"id": "2401.13335", "title": "Full Bayesian Significance Testing for Neural Networks", "abstract": "  Significance testing aims to determine whether a proposition about the\npopulation distribution is the truth or not given observations. However,\ntraditional significance testing often needs to derive the distribution of the\ntesting statistic, failing to deal with complex nonlinear relationships. In\nthis paper, we propose to conduct Full Bayesian Significance Testing for neural\nnetworks, called \\textit{n}FBST, to overcome the limitation in relationship\ncharacterization of traditional approaches. A Bayesian neural network is\nutilized to fit the nonlinear and multi-dimensional relationships with small\nerrors and avoid hard theoretical derivation by computing the evidence value.\nBesides, \\textit{n}FBST can test not only global significance but also local\nand instance-wise significance, which previous testing methods don't focus on.\nMoreover, \\textit{n}FBST is a general framework that can be extended based on\nthe measures selected, such as Grad-\\textit{n}FBST, LRP-\\textit{n}FBST,\nDeepLIFT-\\textit{n}FBST, LIME-\\textit{n}FBST. A range of experiments on both\nsimulated and real data are conducted to show the advantages of our method.\n", "date": "2024-01-24", "categories": []}
{"id": "2401.12416", "title": "Enhancing Reliability of Neural Networks at the Edge: Inverted\n  Normalization with Stochastic Affine Transformations", "abstract": "  Bayesian Neural Networks (BayNNs) naturally provide uncertainty in their\npredictions, making them a suitable choice in safety-critical applications.\nAdditionally, their realization using memristor-based in-memory computing (IMC)\narchitectures enables them for resource-constrained edge applications. In\naddition to predictive uncertainty, however, the ability to be inherently\nrobust to noise in computation is also essential to ensure functional safety.\nIn particular, memristor-based IMCs are susceptible to various sources of\nnon-idealities such as manufacturing and runtime variations, drift, and\nfailure, which can significantly reduce inference accuracy. In this paper, we\npropose a method to inherently enhance the robustness and inference accuracy of\nBayNNs deployed in IMC architectures. To achieve this, we introduce a novel\nnormalization layer combined with stochastic affine transformations. Empirical\nresults in various benchmark datasets show a graceful degradation in inference\naccuracy, with an improvement of up to $58.11\\%$.\n", "date": "2024-01-23", "categories": []}
{"id": "2401.12418", "title": "Towards Improved Variational Inference for Deep Bayesian Models", "abstract": "  Deep learning has revolutionized the last decade, being at the forefront of\nextraordinary advances in a wide range of tasks including computer vision,\nnatural language processing, and reinforcement learning, to name but a few.\nHowever, it is well-known that deep models trained via maximum likelihood\nestimation tend to be overconfident and give poorly-calibrated predictions.\nBayesian deep learning attempts to address this by placing priors on the model\nparameters, which are then combined with a likelihood to perform posterior\ninference. Unfortunately, for deep models, the true posterior is intractable,\nforcing the user to resort to approximations. In this thesis, we explore the\nuse of variational inference (VI) as an approximation, as it is unique in\nsimultaneously approximating the posterior and providing a lower bound to the\nmarginal likelihood. If tight enough, this lower bound can be used to optimize\nhyperparameters and to facilitate model selection. However, this capacity has\nrarely been used to its full extent for Bayesian neural networks, likely\nbecause the approximate posteriors typically used in practice can lack the\nflexibility to effectively bound the marginal likelihood. We therefore explore\nthree aspects of Bayesian learning for deep models: 1) we ask whether it is\nnecessary to perform inference over as many parameters as possible, or whether\nit is reasonable to treat many of them as optimizable hyperparameters; 2) we\npropose a variational posterior that provides a unified view of inference in\nBayesian neural networks and deep Gaussian processes; 3) we demonstrate how VI\ncan be improved in certain deep Gaussian process models by analytically\nremoving symmetries from the posterior, and performing inference on Gram\nmatrices instead of features. We hope that our contributions will provide a\nstepping stone to fully realize the promises of VI in the future.\n", "date": "2024-01-23", "categories": []}
{"id": "2401.12950", "title": "Bayesian Semi-structured Subspace Inference", "abstract": "  Semi-structured regression models enable the joint modeling of interpretable\nstructured and complex unstructured feature effects. The structured model part\nis inspired by statistical models and can be used to infer the input-output\nrelationship for features of particular importance. The complex unstructured\npart defines an arbitrary deep neural network and thereby provides enough\nflexibility to achieve competitive prediction performance. While these models\ncan also account for aleatoric uncertainty, there is still a lack of work on\naccounting for epistemic uncertainty. In this paper, we address this problem by\npresenting a Bayesian approximation for semi-structured regression models using\nsubspace inference. To this end, we extend subspace inference for joint\nposterior sampling from a full parameter space for structured effects and a\nsubspace for unstructured effects. Apart from this hybrid sampling scheme, our\nmethod allows for tunable complexity of the subspace and can capture multiple\nminima in the loss landscape. Numerical experiments validate our approach's\nefficacy in recovering structured effect parameter posteriors in\nsemi-structured models and approaching the full-space posterior distribution of\nMCMC for increasing subspace dimension. Further, our approach exhibits\ncompetitive predictive performance across simulated and real-world datasets.\n", "date": "2024-01-23", "categories": []}
{"id": "2308.09460", "title": "Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling", "abstract": "  This paper presents a new accelerated proximal Markov chain Monte Carlo\nmethodology to perform Bayesian inference in imaging inverse problems with an\nunderlying convex geometry. The proposed strategy takes the form of a\nstochastic relaxed proximal-point iteration that admits two complementary\ninterpretations. For models that are smooth or regularised by Moreau-Yosida\nsmoothing, the algorithm is equivalent to an implicit midpoint discretisation\nof an overdamped Langevin diffusion targeting the posterior distribution of\ninterest. This discretisation is asymptotically unbiased for Gaussian targets\nand shown to converge in an accelerated manner for any target that is\n$\\kappa$-strongly log-concave (i.e., requiring in the order of $\\sqrt{\\kappa}$\niterations to converge, similarly to accelerated optimisation schemes),\ncomparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J.\nImaging Sciences, 13,2 (2020), pp. 905-935] which is only provably accelerated\nfor Gaussian targets and has bias. For models that are not smooth, the\nalgorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin\ndiffusion targeting a Moreau-Yosida approximation of the posterior distribution\nof interest, and hence achieves a significantly lower bias than conventional\nunadjusted Langevin strategies based on the Euler-Maruyama discretisation. For\ntargets that are $\\kappa$-strongly log-concave, the provided non-asymptotic\nconvergence analysis also identifies the optimal time step which maximizes the\nconvergence speed. The proposed methodology is demonstrated through a range of\nexperiments related to image deconvolution with Gaussian and Poisson noise,\nwith assumption-driven and data-driven convex priors. Source codes for the\nnumerical experiments of this paper are available from\nhttps://github.com/MI2G/accelerated-langevin-imla.\n", "date": "2023-08-18", "categories": []}
{"id": "2305.15349", "title": "On the Convergence of Black-Box Variational Inference", "abstract": "  We provide the first convergence guarantee for full black-box variational\ninference (BBVI), also known as Monte Carlo variational inference. While\npreliminary investigations worked on simplified versions of BBVI (e.g., bounded\ndomain, bounded support, only optimizing for the scale, and such), our setup\ndoes not need any such algorithmic modifications. Our results hold for\nlog-smooth posterior densities with and without strong log-concavity and the\nlocation-scale variational family. Also, our analysis reveals that certain\nalgorithm design choices commonly employed in practice, particularly, nonlinear\nparameterizations of the scale of the variational approximation, can result in\nsuboptimal convergence rates. Fortunately, running BBVI with proximal\nstochastic gradient descent fixes these limitations, and thus achieves the\nstrongest known convergence rate guarantees. We evaluate this theoretical\ninsight by comparing proximal SGD against other standard implementations of\nBBVI on large-scale Bayesian inference problems.\n", "date": "2023-05-24", "categories": []}
{"id": "2305.11241", "title": "Evidence Networks: simple losses for fast, amortized, neural Bayesian\n  model comparison", "abstract": "  Evidence Networks can enable Bayesian model comparison when state-of-the-art\nmethods (e.g. nested sampling) fail and even when likelihoods or priors are\nintractable or unknown. Bayesian model comparison, i.e. the computation of\nBayes factors or evidence ratios, can be cast as an optimization problem.\nThough the Bayesian interpretation of optimal classification is well-known,\nhere we change perspective and present classes of loss functions that result in\nfast, amortized neural estimators that directly estimate convenient functions\nof the Bayes factor. This mitigates numerical inaccuracies associated with\nestimating individual model probabilities. We introduce the leaky parity-odd\npower (l-POP) transform, leading to the novel ``l-POP-Exponential'' loss\nfunction. We explore neural density estimation for data probability in\ndifferent models, showing it to be less accurate and scalable than Evidence\nNetworks. Multiple real-world and synthetic examples illustrate that Evidence\nNetworks are explicitly independent of dimensionality of the parameter space\nand scale mildly with the complexity of the posterior probability density\nfunction. This simple yet powerful approach has broad implications for model\ninference tasks. As an application of Evidence Networks to real-world data we\ncompute the Bayes factor for two models with gravitational lensing data of the\nDark Energy Survey. We briefly discuss applications of our methods to other,\nrelated problems of model comparison and evaluation in implicit inference\nsettings.\n", "date": "2023-05-18", "categories": []}
{"id": "2307.06093", "title": "Online Laplace Model Selection Revisited", "abstract": "  The Laplace approximation provides a closed-form model selection objective\nfor neural networks (NN). Online variants, which optimise NN parameters jointly\nwith hyperparameters, like weight decay strength, have seen renewed interest in\nthe Bayesian deep learning community. However, these methods violate Laplace's\nmethod's critical assumption that the approximation is performed around a mode\nof the loss, calling into question their soundness. This work re-derives online\nLaplace methods, showing them to target a variational bound on a mode-corrected\nvariant of the Laplace evidence which does not make stationarity assumptions.\nOnline Laplace and its mode-corrected counterpart share stationary points where\n1. the NN parameters are a maximum a posteriori, satisfying the Laplace\nmethod's assumption, and 2. the hyperparameters maximise the Laplace evidence,\nmotivating online methods. We demonstrate that these optima are roughly\nattained in practise by online algorithms using full-batch gradient descent on\nUCI regression datasets. The optimised hyperparameters prevent overfitting and\noutperform validation-based early stopping.\n", "date": "2023-07-12", "categories": []}
{"id": "2311.03496", "title": "Asynchronous Local Computations in Distributed Bayesian Learning", "abstract": "  Due to the expanding scope of machine learning (ML) to the fields of sensor\nnetworking, cooperative robotics and many other multi-agent systems,\ndistributed deployment of inference algorithms has received a lot of attention.\nThese algorithms involve collaboratively learning unknown parameters from\ndispersed data collected by multiple agents. There are two competing aspects in\nsuch algorithms, namely, intra-agent computation and inter-agent communication.\nTraditionally, algorithms are designed to perform both synchronously. However,\ncertain circumstances need frugal use of communication channels as they are\neither unreliable, time-consuming, or resource-expensive. In this paper, we\npropose gossip-based asynchronous communication to leverage fast computations\nand reduce communication overhead simultaneously. We analyze the effects of\nmultiple (local) intra-agent computations by the active agents between\nsuccessive inter-agent communications. For local computations, Bayesian\nsampling via unadjusted Langevin algorithm (ULA) MCMC is utilized. The\ncommunication is assumed to be over a connected graph (e.g., as in\ndecentralized learning), however, the results can be extended to coordinated\ncommunication where there is a central server (e.g., federated learning). We\ntheoretically quantify the convergence rates in the process. To demonstrate the\nefficacy of the proposed algorithm, we present simulations on a toy problem as\nwell as on real world data sets to train ML models to perform classification\ntasks. We observe faster initial convergence and improved performance accuracy,\nespecially in the low data range. We achieve on average 78% and over 90%\nclassification accuracy respectively on the Gamma Telescope and mHealth data\nsets from the UCI ML repository.\n", "date": "2023-11-06", "categories": []}
{"id": "2401.02544", "title": "Hyperparameter Estimation for Sparse Bayesian Learning Models", "abstract": "  Sparse Bayesian Learning (SBL) models are extensively used in signal\nprocessing and machine learning for promoting sparsity through hierarchical\npriors. The hyperparameters in SBL models are crucial for the model's\nperformance, but they are often difficult to estimate due to the non-convexity\nand the high-dimensionality of the associated objective function. This paper\npresents a comprehensive framework for hyperparameter estimation in SBL models,\nencompassing well-known algorithms such as the expectation-maximization (EM),\nMacKay, and convex bounding (CB) algorithms. These algorithms are cohesively\ninterpreted within an alternating minimization and linearization (AML)\nparadigm, distinguished by their unique linearized surrogate functions.\nAdditionally, a novel algorithm within the AML framework is introduced, showing\nenhanced efficiency, especially under low signal noise ratios. This is further\nimproved by a new alternating minimization and quadratic approximation (AMQ)\nparadigm, which includes a proximal regularization term. The paper\nsubstantiates these advancements with thorough convergence analysis and\nnumerical experiments, demonstrating the algorithm's effectiveness in various\nnoise conditions and signal-to-noise ratios.\n", "date": "2024-01-04", "categories": []}
{"id": "2302.10975", "title": "Improved uncertainty quantification for neural networks with Bayesian\n  last layer", "abstract": "  Uncertainty quantification is an important task in machine learning - a task\nin which standardneural networks (NNs) have traditionally not excelled. This\ncan be a limitation for safety-critical applications, where uncertainty-aware\nmethods like Gaussian processes or Bayesian linear regression are often\npreferred. Bayesian neural networks are an approach to address this limitation.\nThey assume probability distributions for all parameters and yield distributed\npredictions. However, training and inference are typically intractable and\napproximations must be employed. A promising approximation is NNs with Bayesian\nlast layer (BLL). They assume distributed weights only in the linear output\nlayer and yield a normally distributed prediction. To approximate the\nintractable Bayesian neural network, point estimates of the distributed weights\nin all but the last layer should be obtained by maximizing the marginal\nlikelihood. This has previously been challenging, as the marginal likelihood is\nexpensive to evaluate in this setting. We present a reformulation of the\nlog-marginal likelihood of a NN with BLL which allows for efficient training\nusing backpropagation. Furthermore, we address the challenge of uncertainty\nquantification for extrapolation points. We provide a metric to quantify the\ndegree of extrapolation and derive a method to improve the uncertainty\nquantification for these points. Our methods are derived for the multivariate\ncase and demonstrated in a simulation study. In comparison to Bayesian linear\nregression with fixed features, and a Bayesian neural network trained with\nvariational inference, our proposed method achieves the highest log-predictive\ndensity on test data.\n", "date": "2023-02-21", "categories": []}
{"id": "2212.08123", "title": "Bayesian posterior approximation with stochastic ensembles", "abstract": "  We introduce ensembles of stochastic neural networks to approximate the\nBayesian posterior, combining stochastic methods such as dropout with deep\nensembles. The stochastic ensembles are formulated as families of distributions\nand trained to approximate the Bayesian posterior with variational inference.\nWe implement stochastic ensembles based on Monte Carlo dropout, DropConnect and\na novel non-parametric version of dropout and evaluate them on a toy problem\nand CIFAR image classification. For both tasks, we test the quality of the\nposteriors directly against Hamiltonian Monte Carlo simulations. Our results\nshow that stochastic ensembles provide more accurate posterior estimates than\nother popular baselines for Bayesian inference.\n", "date": "2022-12-15", "categories": []}
{"id": "2312.17411", "title": "Generative Posterior Networks for Approximately Bayesian Epistemic\n  Uncertainty Estimation", "abstract": "  In many real-world problems, there is a limited set of training data, but an\nabundance of unlabeled data. We propose a new method, Generative Posterior\nNetworks (GPNs), that uses unlabeled data to estimate epistemic uncertainty in\nhigh-dimensional problems. A GPN is a generative model that, given a prior\ndistribution over functions, approximates the posterior distribution directly\nby regularizing the network towards samples from the prior. We prove\ntheoretically that our method indeed approximates the Bayesian posterior and\nshow empirically that it improves epistemic uncertainty estimation and\nscalability over competing methods.\n", "date": "2023-12-29", "categories": []}
{"id": "2312.17199", "title": "Tractable Function-Space Variational Inference in Bayesian Neural\n  Networks", "abstract": "  Reliable predictive uncertainty estimation plays an important role in\nenabling the deployment of neural networks to safety-critical settings. A\npopular approach for estimating the predictive uncertainty of neural networks\nis to define a prior distribution over the network parameters, infer an\napproximate posterior distribution, and use it to make stochastic predictions.\nHowever, explicit inference over neural network parameters makes it difficult\nto incorporate meaningful prior information about the data-generating process\ninto the model. In this paper, we pursue an alternative approach. Recognizing\nthat the primary object of interest in most settings is the distribution over\nfunctions induced by the posterior distribution over neural network parameters,\nwe frame Bayesian inference in neural networks explicitly as inferring a\nposterior distribution over functions and propose a scalable function-space\nvariational inference method that allows incorporating prior information and\nresults in reliable predictive uncertainty estimates. We show that the proposed\nmethod leads to state-of-the-art uncertainty estimation and predictive\nperformance on a range of prediction tasks and demonstrate that it performs\nwell on a challenging safety-critical medical diagnosis task in which reliable\nuncertainty estimation is essential.\n", "date": "2023-12-28", "categories": []}
{"id": "2312.17210", "title": "Continual Learning via Sequential Function-Space Variational Inference", "abstract": "  Sequential Bayesian inference over predictive functions is a natural\nframework for continual learning from streams of data. However, applying it to\nneural networks has proved challenging in practice. Addressing the drawbacks of\nexisting techniques, we propose an optimization objective derived by\nformulating continual learning as sequential function-space variational\ninference. In contrast to existing methods that regularize neural network\nparameters directly, this objective allows parameters to vary widely during\ntraining, enabling better adaptation to new tasks. Compared to objectives that\ndirectly regularize neural network predictions, the proposed objective allows\nfor more flexible variational distributions and more effective regularization.\nWe demonstrate that, across a range of task sequences, neural networks trained\nvia sequential function-space variational inference achieve better predictive\naccuracy than networks trained with related methods while depending less on\nmaintaining a set of representative points from previous tasks.\n", "date": "2023-12-28", "categories": []}
{"id": "2312.15297", "title": "Make Me a BNN: A Simple Strategy for Estimating Bayesian Uncertainty\n  from Pre-trained Models", "abstract": "  Deep Neural Networks (DNNs) are powerful tools for various computer vision\ntasks, yet they often struggle with reliable uncertainty quantification - a\ncritical requirement for real-world applications. Bayesian Neural Networks\n(BNN) are equipped for uncertainty estimation but cannot scale to large DNNs\nthat are highly unstable to train. To address this challenge, we introduce the\nAdaptable Bayesian Neural Network (ABNN), a simple and scalable strategy to\nseamlessly transform DNNs into BNNs in a post-hoc manner with minimal\ncomputational and training overheads. ABNN preserves the main predictive\nproperties of DNNs while enhancing their uncertainty quantification abilities\nthrough simple BNN adaptation layers (attached to normalization layers) and a\nfew fine-tuning steps on pre-trained models. We conduct extensive experiments\nacross multiple datasets for image classification and semantic segmentation\ntasks, and our results demonstrate that ABNN achieves state-of-the-art\nperformance without the computational budget typically associated with ensemble\nmethods.\n", "date": "2023-12-23", "categories": []}
{"id": "2312.03824", "title": "nbi: the Astronomer's Package for Neural Posterior Estimation", "abstract": "  Despite the promise of Neural Posterior Estimation (NPE) methods in\nastronomy, the adaptation of NPE into the routine inference workflow has been\nslow. We identify three critical issues: the need for custom featurizer\nnetworks tailored to the observed data, the inference inexactness, and the\nunder-specification of physical forward models. To address the first two\nissues, we introduce a new framework and open-source software nbi (Neural\nBayesian Inference), which supports both amortized and sequential NPE. First,\nnbi provides built-in \"featurizer\" networks with demonstrated efficacy on\nsequential data, such as light curve and spectra, thus obviating the need for\nthis customization on the user end. Second, we introduce a modified algorithm\nSNPE-IS, which facilities asymptotically exact inference by using the surrogate\nposterior under NPE only as a proposal distribution for importance sampling.\nThese features allow nbi to be applied off-the-shelf to astronomical inference\nproblems involving light curves and spectra. We discuss how nbi may serve as an\neffective alternative to existing methods such as Nested Sampling. Our package\nis at https://github.com/kmzzhang/nbi.\n", "date": "2023-12-06", "categories": []}
{"id": "2303.17768", "title": "Scalable Bayesian Meta-Learning through Generalized Implicit Gradients", "abstract": "  Meta-learning owns unique effectiveness and swiftness in tackling emerging\ntasks with limited data. Its broad applicability is revealed by viewing it as a\nbi-level optimization problem. The resultant algorithmic viewpoint however,\nfaces scalability issues when the inner-level optimization relies on\ngradient-based iterations. Implicit differentiation has been considered to\nalleviate this challenge, but it is restricted to an isotropic Gaussian prior,\nand only favors deterministic meta-learning approaches. This work markedly\nmitigates the scalability bottleneck by cross-fertilizing the benefits of\nimplicit differentiation to probabilistic Bayesian meta-learning. The novel\nimplicit Bayesian meta-learning (iBaML) method not only broadens the scope of\nlearnable priors, but also quantifies the associated uncertainty. Furthermore,\nthe ultimate complexity is well controlled regardless of the inner-level\noptimization trajectory. Analytical error bounds are established to demonstrate\nthe precision and efficiency of the generalized implicit gradient over the\nexplicit one. Extensive numerical tests are also carried out to empirically\nvalidate the performance of the proposed method.\n", "date": "2023-03-31", "categories": []}
{"id": "2312.13484", "title": "Bayesian Transfer Learning", "abstract": "  Transfer learning is a burgeoning concept in statistical machine learning\nthat seeks to improve inference and/or predictive accuracy on a domain of\ninterest by leveraging data from related domains. While the term \"transfer\nlearning\" has garnered much recent interest, its foundational principles have\nexisted for years under various guises. Prior literature reviews in computer\nscience and electrical engineering have sought to bring these ideas into focus,\nprimarily surveying general methodologies and works from these disciplines.\nThis article highlights Bayesian approaches to transfer learning, which have\nreceived relatively limited attention despite their innate compatibility with\nthe notion of drawing upon prior knowledge to guide new learning tasks. Our\nsurvey encompasses a wide range of Bayesian transfer learning frameworks\napplicable to a variety of practical settings. We discuss how these methods\naddress the problem of finding the optimal information to transfer between\ndomains, which is a central question in transfer learning. We illustrate the\nutility of Bayesian transfer learning methods via a simulation study where we\ncompare performance against frequentist competitors.\n", "date": "2023-12-20", "categories": []}
{"id": "2309.15312", "title": "MAPTree: Beating \"Optimal\" Decision Trees with Bayesian Decision Trees", "abstract": "  Decision trees remain one of the most popular machine learning models today,\nlargely due to their out-of-the-box performance and interpretability. In this\nwork, we present a Bayesian approach to decision tree induction via maximum a\nposteriori inference of a posterior distribution over trees. We first\ndemonstrate a connection between maximum a posteriori inference of decision\ntrees and AND/OR search. Using this connection, we propose an AND/OR search\nalgorithm, dubbed MAPTree, which is able to recover the maximum a posteriori\ntree. Lastly, we demonstrate the empirical performance of the maximum a\nposteriori tree both on synthetic data and in real world settings. On 16 real\nworld datasets, MAPTree either outperforms baselines or demonstrates comparable\nperformance but with much smaller trees. On a synthetic dataset, MAPTree also\ndemonstrates greater robustness to noise and better generalization than\nexisting approaches. Finally, MAPTree recovers the maxiumum a posteriori tree\nfaster than existing sampling approaches and, in contrast with those\nalgorithms, is able to provide a certificate of optimality. The code for our\nexperiments is available at https://github.com/ThrunGroup/maptree.\n", "date": "2023-09-26", "categories": []}
{"id": "2312.05955", "title": "Learning Differentiable Particle Filter on the Fly", "abstract": "  Differentiable particle filters are an emerging class of sequential Bayesian\ninference techniques that use neural networks to construct components in state\nspace models. Existing approaches are mostly based on offline supervised\ntraining strategies. This leads to the delay of the model deployment and the\nobtained filters are susceptible to distribution shift of test-time data. In\nthis paper, we propose an online learning framework for differentiable particle\nfilters so that model parameters can be updated as data arrive. The technical\nconstraint is that there is no known ground truth state information in the\nonline inference setting. We address this by adopting an unsupervised loss to\nconstruct the online model updating procedure, which involves a sequence of\nfiltering operations for online maximum likelihood-based parameter estimation.\nWe empirically evaluate the effectiveness of the proposed method, and compare\nit with supervised learning methods in simulation settings including a\nmultivariate linear Gaussian state-space model and a simulated object tracking\nexperiment.\n", "date": "2023-12-10", "categories": []}
{"id": "2312.11169", "title": "Distributed Collapsed Gibbs Sampler for Dirichlet Process Mixture Models\n  in Federated Learning", "abstract": "  Dirichlet Process Mixture Models (DPMMs) are widely used to address\nclustering problems. Their main advantage lies in their ability to\nautomatically estimate the number of clusters during the inference process\nthrough the Bayesian non-parametric framework. However, the inference becomes\nconsiderably slow as the dataset size increases. This paper proposes a new\ndistributed Markov Chain Monte Carlo (MCMC) inference method for DPMMs (DisCGS)\nusing sufficient statistics. Our approach uses the collapsed Gibbs sampler and\nis specifically designed to work on distributed data across independent and\nheterogeneous machines, which habilitates its use in horizontal federated\nlearning. Our method achieves highly promising results and notable scalability.\nFor instance, with a dataset of 100K data points, the centralized algorithm\nrequires approximately 12 hours to complete 100 iterations while our approach\nachieves the same number of iterations in just 3 minutes, reducing the\nexecution time by a factor of 200 without compromising clustering performance.\nThe code source is publicly available at\nhttps://github.com/redakhoufache/DisCGS.\n", "date": "2023-12-18", "categories": []}
{"id": "2302.09639", "title": "An overview of differentiable particle filters for data-adaptive\n  sequential Bayesian inference", "abstract": "  By approximating posterior distributions with weighted samples, particle\nfilters (PFs) provide an efficient mechanism for solving non-linear sequential\nstate estimation problems. While the effectiveness of particle filters has been\nrecognised in various applications, their performance relies on the knowledge\nof dynamic models and measurement models, as well as the construction of\neffective proposal distributions. An emerging trend involves constructing\ncomponents of particle filters using neural networks and optimising them by\ngradient descent, and such data-adaptive particle filtering approaches are\noften called differentiable particle filters. Due to the expressiveness of\nneural networks, differentiable particle filters are a promising computational\ntool for performing inference on sequential data in complex, high-dimensional\ntasks, such as vision-based robot localisation. In this paper, we review recent\nadvances in differentiable particle filters and their applications. We place\nspecial emphasis on different design choices for key components of\ndifferentiable particle filters, including dynamic models, measurement models,\nproposal distributions, optimisation objectives, and differentiable resampling\ntechniques.\n", "date": "2023-02-19", "categories": []}
{"id": "2212.07311", "title": "Bayesian data fusion with shared priors", "abstract": "  The integration of data and knowledge from several sources is known as data\nfusion. When data is only available in a distributed fashion or when different\nsensors are used to infer a quantity of interest, data fusion becomes\nessential. In Bayesian settings, a priori information of the unknown quantities\nis available and, possibly, present among the different distributed estimators.\nWhen the local estimates are fused, the prior knowledge used to construct\nseveral local posteriors might be overused unless the fusion node accounts for\nthis and corrects it. In this paper, we analyze the effects of shared priors in\nBayesian data fusion contexts. Depending on different common fusion rules, our\nanalysis helps to understand the performance behavior as a function of the\nnumber of collaborative agents and as a consequence of different types of\npriors. The analysis is performed by using two divergences which are common in\nBayesian inference, and the generality of the results allows to analyze very\ngeneric distributions. These theoretical results are corroborated through\nexperiments in a variety of estimation and classification problems, including\nlinear and nonlinear models, and federated learning schemes.\n", "date": "2022-12-14", "categories": []}
{"id": "2312.03911", "title": "Improving Gradient-guided Nested Sampling for Posterior Inference", "abstract": "  We present a performant, general-purpose gradient-guided nested sampling\nalgorithm, ${\\tt GGNS}$, combining the state of the art in differentiable\nprogramming, Hamiltonian slice sampling, clustering, mode separation, dynamic\nnested sampling, and parallelization. This unique combination allows ${\\tt\nGGNS}$ to scale well with dimensionality and perform competitively on a variety\nof synthetic and real-world problems. We also show the potential of combining\nnested sampling with generative flow networks to obtain large amounts of\nhigh-quality samples from the posterior distribution. This combination leads to\nfaster mode discovery and more accurate estimates of the partition function.\n", "date": "2023-12-06", "categories": []}
{"id": "2208.01180", "title": "Bayesian Variable Selection in a Million Dimensions", "abstract": "  Bayesian variable selection is a powerful tool for data analysis, as it\noffers a principled method for variable selection that accounts for prior\ninformation and uncertainty. However, wider adoption of Bayesian variable\nselection has been hampered by computational challenges, especially in\ndifficult regimes with a large number of covariates P or non-conjugate\nlikelihoods. To scale to the large P regime we introduce an efficient MCMC\nscheme whose cost per iteration is sublinear in P. In addition we show how this\nscheme can be extended to generalized linear models for count data, which are\nprevalent in biology, ecology, economics, and beyond. In particular we design\nefficient algorithms for variable selection in binomial and negative binomial\nregression, which includes logistic regression as a special case. In\nexperiments we demonstrate the effectiveness of our methods, including on\ncancer and maize genomic data.\n", "date": "2022-08-02", "categories": []}
{"id": "2307.07816", "title": "Minimal Random Code Learning with Mean-KL Parameterization", "abstract": "  This paper studies the qualitative behavior and robustness of two variants of\nMinimal Random Code Learning (MIRACLE) used to compress variational Bayesian\nneural networks. MIRACLE implements a powerful, conditionally Gaussian\nvariational approximation for the weight posterior $Q_{\\mathbf{w}}$ and uses\nrelative entropy coding to compress a weight sample from the posterior using a\nGaussian coding distribution $P_{\\mathbf{w}}$. To achieve the desired\ncompression rate, $D_{\\mathrm{KL}}[Q_{\\mathbf{w}} \\Vert P_{\\mathbf{w}}]$ must\nbe constrained, which requires a computationally expensive annealing procedure\nunder the conventional mean-variance (Mean-Var) parameterization for\n$Q_{\\mathbf{w}}$. Instead, we parameterize $Q_{\\mathbf{w}}$ by its mean and KL\ndivergence from $P_{\\mathbf{w}}$ to constrain the compression cost to the\ndesired value by construction. We demonstrate that variational training with\nMean-KL parameterization converges twice as fast and maintains predictive\nperformance after compression. Furthermore, we show that Mean-KL leads to more\nmeaningful variational distributions with heavier tails and compressed weight\nsamples which are more robust to pruning.\n", "date": "2023-07-15", "categories": []}
{"id": "2312.01577", "title": "RJHMC-Tree for Exploration of the Bayesian Decision Tree Posterior", "abstract": "  Decision trees have found widespread application within the machine learning\ncommunity due to their flexibility and interpretability. This paper is directed\ntowards learning decision trees from data using a Bayesian approach, which is\nchallenging due to the potentially enormous parameter space required to span\nall tree models. Several approaches have been proposed to combat this\nchallenge, with one of the more successful being Markov chain Monte Carlo\n(MCMC) methods. The efficacy and efficiency of MCMC methods fundamentally rely\non the quality of the so-called proposals, which is the focus of this paper. In\nparticular, this paper investigates using a Hamiltonian Monte Carlo (HMC)\napproach to explore the posterior of Bayesian decision trees more efficiently\nby exploiting the geometry of the likelihood within a global update scheme. Two\nimplementations of the novel algorithm are developed and compared to existing\nmethods by testing against standard datasets in the machine learning and\nBayesian decision tree literature. HMC-based methods are shown to perform\nfavourably with respect to predictive test accuracy, acceptance rate, and tree\ncomplexity.\n", "date": "2023-12-04", "categories": []}
{"id": "2212.02003", "title": "Bayesian Learning with Information Gain Provably Bounds Risk for a\n  Robust Adversarial Defense", "abstract": "  We present a new algorithm to learn a deep neural network model robust\nagainst adversarial attacks. Previous algorithms demonstrate an adversarially\ntrained Bayesian Neural Network (BNN) provides improved robustness. We\nrecognize the adversarial learning approach for approximating the multi-modal\nposterior distribution of a Bayesian model can lead to mode collapse;\nconsequently, the model's achievements in robustness and performance are\nsub-optimal. Instead, we first propose preventing mode collapse to better\napproximate the multi-modal posterior distribution. Second, based on the\nintuition that a robust model should ignore perturbations and only consider the\ninformative content of the input, we conceptualize and formulate an information\ngain objective to measure and force the information learned from both benign\nand adversarial training instances to be similar. Importantly. we prove and\ndemonstrate that minimizing the information gain objective allows the\nadversarial risk to approach the conventional empirical risk. We believe our\nefforts provide a step toward a basis for a principled method of adversarially\ntraining BNNs. Our model demonstrate significantly improved robustness--up to\n20%--compared with adversarial training and Adv-BNN under PGD attacks with\n0.035 distortion on both CIFAR-10 and STL-10 datasets.\n", "date": "2022-12-05", "categories": []}
{"id": "2302.13095", "title": "Bayesian Neural Networks Avoid Encoding Complex and\n  Perturbation-Sensitive Concepts", "abstract": "  In this paper, we focus on mean-field variational Bayesian Neural Networks\n(BNNs) and explore the representation capacity of such BNNs by investigating\nwhich types of concepts are less likely to be encoded by the BNN. It has been\nobserved and studied that a relatively small set of interactive concepts\nusually emerge in the knowledge representation of a sufficiently-trained neural\nnetwork, and such concepts can faithfully explain the network output. Based on\nthis, our study proves that compared to standard deep neural networks (DNNs),\nit is less likely for BNNs to encode complex concepts. Experiments verify our\ntheoretical proofs. Note that the tendency to encode less complex concepts does\nnot necessarily imply weak representation power, considering that complex\nconcepts exhibit low generalization power and high adversarial vulnerability.\nThe code is available at https://github.com/sjtu-xai-lab/BNN-concepts.\n", "date": "2023-02-25", "categories": []}
{"id": "2312.00232", "title": "Uncertainty in Graph Contrastive Learning with Bayesian Neural Networks", "abstract": "  Graph contrastive learning has shown great promise when labeled data is\nscarce, but large unlabeled datasets are available. However, it often does not\ntake uncertainty estimation into account. We show that a variational Bayesian\nneural network approach can be used to improve not only the uncertainty\nestimates but also the downstream performance on semi-supervised\nnode-classification tasks. Moreover, we propose a new measure of uncertainty\nfor contrastive learning, that is based on the disagreement in likelihood due\nto different positive samples.\n", "date": "2023-11-30", "categories": []}
{"id": "2312.00373", "title": "Streaming Bayesian Modeling for predicting Fat-Tailed Customer Lifetime\n  Value", "abstract": "  We develop an online learning MCMC approach applicable for hierarchical\nbayesian models and GLMS. We also develop a fat-tailed LTV model that\ngeneralizes over several kinds of fat and thin tails. We demonstrate both\ndevelopments on commercial LTV data from a large mobile app.\n", "date": "2023-12-01", "categories": []}
{"id": "2311.18012", "title": "Bayesian Imaging for Radio Interferometry with Score-Based Priors", "abstract": "  The inverse imaging task in radio interferometry is a key limiting factor to\nretrieving Bayesian uncertainties in radio astronomy in a computationally\neffective manner. We use a score-based prior derived from optical images of\ngalaxies to recover images of protoplanetary disks from the DSHARP survey. We\ndemonstrate that our method produces plausible posterior samples despite the\nmisspecified galaxy prior. We show that our approach produces results which are\ncompetitive with existing radio interferometry imaging algorithms.\n", "date": "2023-11-29", "categories": []}
{"id": "2302.14545", "title": "Modern Bayesian Experimental Design", "abstract": "  Bayesian experimental design (BED) provides a powerful and general framework\nfor optimizing the design of experiments. However, its deployment often poses\nsubstantial computational challenges that can undermine its practical use. In\nthis review, we outline how recent advances have transformed our ability to\novercome these challenges and thus utilize BED effectively, before discussing\nsome key areas for future development in the field.\n", "date": "2023-02-28", "categories": []}
{"id": "2311.12878", "title": "Adaptive Bayesian Learning with Action and State-Dependent Signal\n  Variance", "abstract": "  This manuscript presents an advanced framework for Bayesian learning by\nincorporating action and state-dependent signal variances into decision-making\nmodels. This framework is pivotal in understanding complex data-feedback loops\nand decision-making processes in various economic systems. Through a series of\nexamples, we demonstrate the versatility of this approach in different\ncontexts, ranging from simple Bayesian updating in stable environments to\ncomplex models involving social learning and state-dependent uncertainties. The\npaper uniquely contributes to the understanding of the nuanced interplay\nbetween data, actions, outcomes, and the inherent uncertainty in economic\nmodels.\n", "date": "2023-11-20", "categories": []}
{"id": "2311.15610", "title": "Bayesian Approach to Linear Bayesian Networks", "abstract": "  This study proposes the first Bayesian approach for learning high-dimensional\nlinear Bayesian networks. The proposed approach iteratively estimates each\nelement of the topological ordering from backward and its parent using the\ninverse of a partial covariance matrix. The proposed method successfully\nrecovers the underlying structure when Bayesian regularization for the inverse\ncovariance matrix with unequal shrinkage is applied. Specifically, it shows\nthat the number of samples $n = \\Omega( d_M^2 \\log p)$ and $n = \\Omega(d_M^2\np^{2/m})$ are sufficient for the proposed algorithm to learn linear Bayesian\nnetworks with sub-Gaussian and 4m-th bounded-moment error distributions,\nrespectively, where $p$ is the number of nodes and $d_M$ is the maximum degree\nof the moralized graph. The theoretical findings are supported by extensive\nsimulation studies including real data analysis. Furthermore the proposed\nmethod is demonstrated to outperform state-of-the-art frequentist approaches,\nsuch as the BHLSM, LISTEN, and TD algorithms in synthetic data.\n", "date": "2023-11-27", "categories": []}
{"id": "2311.15990", "title": "Should We Learn Most Likely Functions or Parameters?", "abstract": "  Standard regularized training procedures correspond to maximizing a posterior\ndistribution over parameters, known as maximum a posteriori (MAP) estimation.\nHowever, model parameters are of interest only insomuch as they combine with\nthe functional form of a model to provide a function that can make good\npredictions. Moreover, the most likely parameters under the parameter posterior\ndo not generally correspond to the most likely function induced by the\nparameter posterior. In fact, we can re-parametrize a model such that any\nsetting of parameters can maximize the parameter posterior. As an alternative,\nwe investigate the benefits and drawbacks of directly estimating the most\nlikely function implied by the model and the data. We show that this procedure\nleads to pathological solutions when using neural networks and prove conditions\nunder which the procedure is well-behaved, as well as a scalable approximation.\nUnder these conditions, we find that function-space MAP estimation can lead to\nflatter minima, better generalization, and improved robustness to overfitting.\n", "date": "2023-11-27", "categories": []}
{"id": "2301.11873", "title": "A Deep Learning Method for Comparing Bayesian Hierarchical Models", "abstract": "  Bayesian model comparison (BMC) offers a principled approach for assessing\nthe relative merits of competing computational models and propagating\nuncertainty into model selection decisions. However, BMC is often intractable\nfor the popular class of hierarchical models due to their high-dimensional\nnested parameter structure. To address this intractability, we propose a deep\nlearning method for performing BMC on any set of hierarchical models which can\nbe instantiated as probabilistic programs. Since our method enables amortized\ninference, it allows efficient re-estimation of posterior model probabilities\nand fast performance validation prior to any real-data application. In a series\nof extensive validation studies, we benchmark the performance of our method\nagainst the state-of-the-art bridge sampling method and demonstrate excellent\namortized inference across all BMC settings. We then showcase our method by\ncomparing four hierarchical evidence accumulation models that have previously\nbeen deemed intractable for BMC due to partly implicit likelihoods.\nAdditionally, we demonstrate how transfer learning can be leveraged to enhance\ntraining efficiency. We provide reproducible code for all analyses and an\nopen-source implementation of our method.\n", "date": "2023-01-27", "categories": []}
{"id": "2311.14601", "title": "A Metalearned Neural Circuit for Nonparametric Bayesian Inference", "abstract": "  Most applications of machine learning to classification assume a closed set\nof balanced classes. This is at odds with the real world, where class\noccurrence statistics often follow a long-tailed power-law distribution and it\nis unlikely that all classes are seen in a single sample. Nonparametric\nBayesian models naturally capture this phenomenon, but have significant\npractical barriers to widespread adoption, namely implementation complexity and\ncomputational inefficiency. To address this, we present a method for extracting\nthe inductive bias from a nonparametric Bayesian model and transferring it to\nan artificial neural network. By simulating data with a nonparametric Bayesian\nprior, we can metalearn a sequence model that performs inference over an\nunlimited set of classes. After training, this \"neural circuit\" has distilled\nthe corresponding inductive bias and can successfully perform sequential\ninference over an open set of classes. Our experimental results show that the\nmetalearned neural circuit achieves comparable or better performance than\nparticle filter-based methods for inference in these models while being faster\nand simpler to use than methods that explicitly incorporate Bayesian\nnonparametric inference.\n", "date": "2023-11-24", "categories": []}
{"id": "2211.14605", "title": "Looking at the posterior: accuracy and uncertainty of neural-network\n  predictions", "abstract": "  Bayesian inference can quantify uncertainty in the predictions of neural\nnetworks using posterior distributions for model parameters and network output.\nBy looking at these posterior distributions, one can separate the origin of\nuncertainty into aleatoric and epistemic contributions. One goal of uncertainty\nquantification is to inform on prediction accuracy. Here we show that\nprediction accuracy depends on both epistemic and aleatoric uncertainty in an\nintricate fashion that cannot be understood in terms of marginalized\nuncertainty distributions alone. How the accuracy relates to epistemic and\naleatoric uncertainties depends not only on the model architecture, but also on\nthe properties of the dataset. We discuss the significance of these results for\nactive learning and introduce a novel acquisition function that outperforms\ncommon uncertainty-based methods. To arrive at our results, we approximated the\nposteriors using deep ensembles, for fully-connected, convolutional and\nattention-based neural networks.\n", "date": "2022-11-26", "categories": []}
{"id": "2311.00474", "title": "Diffusion models for probabilistic programming", "abstract": "  We propose Diffusion Model Variational Inference (DMVI), a novel method for\nautomated approximate inference in probabilistic programming languages (PPLs).\nDMVI utilizes diffusion models as variational approximations to the true\nposterior distribution by deriving a novel bound to the marginal likelihood\nobjective used in Bayesian modelling. DMVI is easy to implement, allows\nhassle-free inference in PPLs without the drawbacks of, e.g., variational\ninference using normalizing flows, and does not make any constraints on the\nunderlying neural network model. We evaluate DMVI on a set of common Bayesian\nmodels and show that its posterior inferences are in general more accurate than\nthose of contemporary methods used in PPLs while having a similar computational\ncost and requiring less manual tuning.\n", "date": "2023-11-01", "categories": []}
{"id": "2311.13036", "title": "Favour: FAst Variance Operator for Uncertainty Rating", "abstract": "  Bayesian Neural Networks (BNN) have emerged as a crucial approach for\ninterpreting ML predictions. By sampling from the posterior distribution, data\nscientists may estimate the uncertainty of an inference. Unfortunately many\ninference samples are often needed, the overhead of which greatly hinder BNN's\nwide adoption. To mitigate this, previous work proposed propagating the first\nand second moments of the posterior directly through the network. However, on\nits own this method is even slower than sampling, so the propagated variance\nneeds to be approximated such as assuming independence between neural nodes.\nThe resulting trade-off between quality and inference time did not match even\nplain Monte Carlo sampling.\n  Our contribution is a more principled variance propagation framework based on\n\"spiked covariance matrices\", which smoothly interpolates between quality and\ninference time. This is made possible by a new fast algorithm for updating a\ndiagonal-plus-low-rank matrix approximation under various operations. We tested\nour algorithm against sampling based MC Dropout and Variational Inference on a\nnumber of downstream uncertainty themed tasks, such as calibration and\nout-of-distribution testing. We find that Favour is as fast as performing 2-3\ninference samples, while matching the performance of 10-100 samples.\n  In summary, this work enables the use of BNN in the realm of performance\ncritical tasks where they have previously been out of reach.\n", "date": "2023-11-21", "categories": []}
{"id": "2311.13411", "title": "Bayesian inference of a new Mallows model for characterising symptom\n  sequences applied in primary progressive aphasia", "abstract": "  Machine learning models offer the potential to understand diverse datasets in\na data-driven way, powering insights into individual disease experiences and\nensuring equitable healthcare. In this study, we explore Bayesian inference for\ncharacterising symptom sequences, and the associated modelling challenges. We\nadapted the Mallows model to account for partial rankings and right-censored\ndata, employing custom MCMC fitting. Our evaluation, encompassing synthetic\ndata and a primary progressive aphasia dataset, highlights the model's efficacy\nin revealing mean orderings and estimating ranking variance. This holds the\npotential to enhance clinical comprehension of symptom occurrence. However, our\nwork encounters limitations concerning model scalability and small dataset\nsizes.\n", "date": "2023-11-22", "categories": []}
{"id": "2311.10300", "title": "Supervised structure learning", "abstract": "  This paper concerns structure learning or discovery of discrete generative\nmodels. It focuses on Bayesian model selection and the assimilation of training\ndata or content, with a special emphasis on the order in which data are\ningested. A key move - in the ensuing schemes - is to place priors on the\nselection of models, based upon expected free energy. In this setting, expected\nfree energy reduces to a constrained mutual information, where the constraints\ninherit from priors over outcomes (i.e., preferred outcomes). The resulting\nscheme is first used to perform image classification on the MNIST dataset to\nillustrate the basic idea, and then tested on a more challenging problem of\ndiscovering models with dynamics, using a simple sprite-based visual\ndisentanglement paradigm and the Tower of Hanoi (cf., blocks world) problem. In\nthese examples, generative models are constructed autodidactically to recover\n(i.e., disentangle) the factorial structure of latent states - and their\ncharacteristic paths or dynamics.\n", "date": "2023-11-17", "categories": []}
{"id": "2311.08442", "title": "Mean-field variational inference with the TAP free energy: Geometric and\n  statistical properties in linear models", "abstract": "  We study mean-field variational inference in a Bayesian linear model when the\nsample size n is comparable to the dimension p. In high dimensions, the common\napproach of minimizing a Kullback-Leibler divergence from the posterior\ndistribution, or maximizing an evidence lower bound, may deviate from the true\nposterior mean and underestimate posterior uncertainty. We study instead\nminimization of the TAP free energy, showing in a high-dimensional asymptotic\nframework that it has a local minimizer which provides a consistent estimate of\nthe posterior marginals and may be used for correctly calibrated posterior\ninference. Geometrically, we show that the landscape of the TAP free energy is\nstrongly convex in an extensive neighborhood of this local minimizer, which\nunder certain general conditions can be found by an Approximate Message Passing\n(AMP) algorithm. We then exhibit an efficient algorithm that linearly converges\nto the minimizer within this local neighborhood. In settings where it is\nconjectured that no efficient algorithm can find this local neighborhood, we\nprove analogous geometric properties for a local minimizer of the TAP free\nenergy reachable by AMP, and show that posterior inference based on this\nminimizer remains correctly calibrated.\n", "date": "2023-11-14", "categories": []}
{"id": "2302.02713", "title": "Flat Seeking Bayesian Neural Networks", "abstract": "  Bayesian Neural Networks (BNNs) provide a probabilistic interpretation for\ndeep learning models by imposing a prior distribution over model parameters and\ninferring a posterior distribution based on observed data. The model sampled\nfrom the posterior distribution can be used for providing ensemble predictions\nand quantifying prediction uncertainty. It is well-known that deep learning\nmodels with lower sharpness have better generalization ability. However,\nexisting posterior inferences are not aware of sharpness/flatness in terms of\nformulation, possibly leading to high sharpness for the models sampled from\nthem. In this paper, we develop theories, the Bayesian setting, and the\nvariational inference approach for the sharpness-aware posterior. Specifically,\nthe models sampled from our sharpness-aware posterior, and the optimal\napproximate posterior estimating this sharpness-aware posterior, have better\nflatness, hence possibly possessing higher generalization ability. We conduct\nexperiments by leveraging the sharpness-aware posterior with state-of-the-art\nBayesian Neural Networks, showing that the flat-seeking counterparts outperform\ntheir baselines in all metrics of interest.\n", "date": "2023-02-06", "categories": []}
{"id": "2304.04307", "title": "PriorCVAE: scalable MCMC parameter inference with Bayesian deep\n  generative modelling", "abstract": "  Recent advances have shown that GP priors, or their finite realisations, can\nbe encoded using deep generative models such as variational autoencoders\n(VAEs). These learned generators can serve as drop-in replacements for the\noriginal priors during MCMC inference. While this approach enables efficient\ninference, it loses information about the hyperparameters of the original\nmodels, and consequently makes inference over hyperparameters impossible and\nthe learned priors indistinct. To overcome this limitation, we condition the\nVAE on stochastic process hyperparameters. This allows the joint encoding of\nhyperparameters with GP realizations and their subsequent estimation during\ninference. Further, we demonstrate that our proposed method, PriorCVAE, is\nagnostic to the nature of the models which it approximates, and can be used,\nfor instance, to encode solutions of ODEs. It provides a practical tool for\napproximate inference and shows potential in real-life spatial and\nspatiotemporal applications.\n", "date": "2023-04-09", "categories": []}
{"id": "2310.06643", "title": "Implicit Variational Inference for High-Dimensional Posteriors", "abstract": "  In variational inference, the benefits of Bayesian models rely on accurately\ncapturing the true posterior distribution. We propose using neural samplers\nthat specify implicit distributions, which are well-suited for approximating\ncomplex multimodal and correlated posteriors in high-dimensional spaces. Our\napproach introduces novel bounds for approximate inference using implicit\ndistributions by locally linearising the neural sampler. This is distinct from\nexisting methods that rely on additional discriminator networks and unstable\nadversarial objectives. Furthermore, we present a new sampler architecture\nthat, for the first time, enables implicit distributions over tens of millions\nof latent variables, addressing computational concerns by using differentiable\nnumerical approximations. We empirically show that our method is capable of\nrecovering correlations across layers in large Bayesian neural networks, a\nproperty that is crucial for a network's performance but notoriously\nchallenging to achieve. To the best of our knowledge, no other method has been\nshown to accomplish this task for such large models. Through experiments in\ndownstream tasks, we demonstrate that our expressive posteriors outperform\nstate-of-the-art uncertainty quantification methods, validating the\neffectiveness of our training algorithm and the quality of the learned implicit\napproximation.\n", "date": "2023-10-10", "categories": []}
{"id": "2305.17058", "title": "Exact Bayesian Inference on Discrete Models via Probability Generating\n  Functions: A Probabilistic Programming Approach", "abstract": "  We present an exact Bayesian inference method for discrete statistical\nmodels, which can find exact solutions to a large class of discrete inference\nproblems, even with infinite support and continuous priors. To express such\nmodels, we introduce a probabilistic programming language that supports\ndiscrete and continuous sampling, discrete observations, affine functions,\n(stochastic) branching, and conditioning on discrete events. Our key tool is\nprobability generating functions: they provide a compact closed-form\nrepresentation of distributions that are definable by programs, thus enabling\nthe exact computation of posterior probabilities, expectation, variance, and\nhigher moments. Our inference method is provably correct and fully automated in\na tool called Genfer, which uses automatic differentiation (specifically,\nTaylor polynomials), but does not require computer algebra. Our experiments\nshow that Genfer is often faster than the existing exact inference tools PSI,\nDice, and Prodigy. On a range of real-world inference problems that none of\nthese exact tools can solve, Genfer's performance is competitive with\napproximate Monte Carlo methods, while avoiding approximation errors.\n", "date": "2023-05-26", "categories": []}
{"id": "2209.03028", "title": "Bayesian learning of feature spaces for multitasks problems", "abstract": "  This paper introduces a novel approach for multi-task regression that\nconnects Kernel Machines (KMs) and Extreme Learning Machines (ELMs) through the\nexploitation of the Random Fourier Features (RFFs) approximation of the RBF\nkernel. In this sense, one of the contributions of this paper shows that for\nthe proposed models, the KM and the ELM formulations can be regarded as two\nsides of the same coin. These proposed models, termed RFF-BLR, stand on a\nBayesian framework that simultaneously addresses two main design goals. On the\none hand, it fits multitask regressors based on KMs endowed with RBF kernels.\nOn the other hand, it enables the introduction of a common-across-tasks prior\nthat promotes multioutput sparsity in the ELM view. This Bayesian approach\nfacilitates the simultaneous consideration of both the KM and ELM perspectives\nenabling (i) the optimisation of the RBF kernel parameter $\\gamma$ within a\nprobabilistic framework, (ii) the optimisation of the model complexity, and\n(iii) an efficient transfer of knowledge across tasks. The experimental results\nshow that this framework can lead to significant performance improvements\ncompared to the state-of-the-art methods in multitask nonlinear regression.\n", "date": "2022-09-07", "categories": []}
{"id": "2311.02019", "title": "Reproducible Parameter Inference Using Bagged Posteriors", "abstract": "  Under model misspecification, it is known that Bayesian posteriors often do\nnot properly quantify uncertainty about true or pseudo-true parameters. Even\nmore fundamentally, misspecification leads to a lack of reproducibility in the\nsense that the same model will yield contradictory posteriors on independent\ndata sets from the true distribution. To define a criterion for reproducible\nuncertainty quantification under misspecification, we consider the probability\nthat two confidence sets constructed from independent data sets have nonempty\noverlap, and we establish a lower bound on this overlap probability that holds\nfor any valid confidence sets. We prove that credible sets from the standard\nposterior can strongly violate this bound, particularly in high-dimensional\nsettings (i.e., with dimension increasing with sample size), indicating that it\nis not internally coherent under misspecification. To improve reproducibility\nin an easy-to-use and widely applicable way, we propose to apply bagging to the\nBayesian posterior (\"BayesBag\"'); that is, to use the average of posterior\ndistributions conditioned on bootstrapped datasets. We motivate BayesBag from\nfirst principles based on Jeffrey conditionalization and show that the bagged\nposterior typically satisfies the overlap lower bound. Further, we prove a\nBernstein--Von Mises theorem for the bagged posterior, establishing its\nasymptotic normal distribution. We demonstrate the benefits of BayesBag via\nsimulation experiments and an application to crime rate prediction.\n", "date": "2023-11-03", "categories": []}
{"id": "2212.10649", "title": "Inversion of Bayesian Networks", "abstract": "  Variational autoencoders and Helmholtz machines use a recognition network\n(encoder) to approximate the posterior distribution of a generative model\n(decoder). In this paper we study the necessary and sufficient properties of a\nrecognition network so that it can model the true posterior distribution\nexactly. These results are derived in the general context of probabilistic\ngraphical modelling / Bayesian networks, for which the network represents a set\nof conditional independence statements. We derive both global conditions, in\nterms of d-separation, and local conditions for the recognition network to have\nthe desired qualities. It turns out that for the local conditions the property\nperfectness (for every node, all parents are joined) plays an important role.\n", "date": "2022-12-20", "categories": []}
{"id": "2305.15208", "title": "Generalized Bayesian Inference for Scientific Simulators via Amortized\n  Cost Estimation", "abstract": "  Simulation-based inference (SBI) enables amortized Bayesian inference for\nsimulators with implicit likelihoods. But when we are primarily interested in\nthe quality of predictive simulations, or when the model cannot exactly\nreproduce the observed data (i.e., is misspecified), targeting the Bayesian\nposterior may be overly restrictive. Generalized Bayesian Inference (GBI) aims\nto robustify inference for (misspecified) simulator models, replacing the\nlikelihood-function with a cost function that evaluates the goodness of\nparameters relative to data. However, GBI methods generally require running\nmultiple simulations to estimate the cost function at each parameter value\nduring inference, making the approach computationally infeasible for even\nmoderately complex simulators. Here, we propose amortized cost estimation (ACE)\nfor GBI to address this challenge: We train a neural network to approximate the\ncost function, which we define as the expected distance between simulations\nproduced by a parameter and observed data. The trained network can then be used\nwith MCMC to infer GBI posteriors for any observation without running\nadditional simulations. We show that, on several benchmark tasks, ACE\naccurately predicts cost and provides predictive simulations that are closer to\nsynthetic observations than other SBI methods, especially for misspecified\nsimulators. Finally, we apply ACE to infer parameters of the Hodgkin-Huxley\nmodel given real intracellular recordings from the Allen Cell Types Database.\nACE identifies better data-matching parameters while being an order of\nmagnitude more simulation-efficient than a standard SBI method. In summary, ACE\ncombines the strengths of SBI methods and GBI to perform robust and\nsimulation-amortized inference for scientific simulators.\n", "date": "2023-05-24", "categories": []}
{"id": "2311.00564", "title": "Online Student-$t$ Processes with an Overall-local Scale Structure for\n  Modelling Non-stationary Data", "abstract": "  Time-dependent data often exhibit characteristics, such as non-stationarity\nand heavy-tailed errors, that would be inappropriate to model with the typical\nassumptions used in popular models. Thus, more flexible approaches are required\nto be able to accommodate such issues. To this end, we propose a Bayesian\nmixture of student-$t$ processes with an overall-local scale structure for the\ncovariance. Moreover, we use a sequential Monte Carlo (SMC) sampler in order to\nperform online inference as data arrive in real-time. We demonstrate the\nsuperiority of our proposed approach compared to typical Gaussian process-based\nmodels on real-world data sets in order to prove the necessity of using\nmixtures of student-$t$ processes.\n", "date": "2023-11-01", "categories": []}
{"id": "2311.00599", "title": "Structure Learning with Adaptive Random Neighborhood Informed MCMC", "abstract": "  In this paper, we introduce a novel MCMC sampler, PARNI-DAG, for a\nfully-Bayesian approach to the problem of structure learning under\nobservational data. Under the assumption of causal sufficiency, the algorithm\nallows for approximate sampling directly from the posterior distribution on\nDirected Acyclic Graphs (DAGs). PARNI-DAG performs efficient sampling of DAGs\nvia locally informed, adaptive random neighborhood proposal that results in\nbetter mixing properties. In addition, to ensure better scalability with the\nnumber of nodes, we couple PARNI-DAG with a pre-tuning procedure of the\nsampler's parameters that exploits a skeleton graph derived through some\nconstraint-based or scoring-based algorithms. Thanks to these novel features,\nPARNI-DAG quickly converges to high-probability regions and is less likely to\nget stuck in local modes in the presence of high correlation between nodes in\nhigh-dimensional settings. After introducing the technical novelties in\nPARNI-DAG, we empirically demonstrate its mixing efficiency and accuracy in\nlearning DAG structures on a variety of experiments.\n", "date": "2023-11-01", "categories": []}
{"id": "2310.20447", "title": "Efficient Bayesian Learning Curve Extrapolation using Prior-Data Fitted\n  Networks", "abstract": "  Learning curve extrapolation aims to predict model performance in later\nepochs of training, based on the performance in earlier epochs. In this work,\nwe argue that, while the inherent uncertainty in the extrapolation of learning\ncurves warrants a Bayesian approach, existing methods are (i) overly\nrestrictive, and/or (ii) computationally expensive. We describe the first\napplication of prior-data fitted neural networks (PFNs) in this context. A PFN\nis a transformer, pre-trained on data generated from a prior, to perform\napproximate Bayesian inference in a single forward pass. We propose LC-PFN, a\nPFN trained to extrapolate 10 million artificial right-censored learning curves\ngenerated from a parametric prior proposed in prior art using MCMC. We\ndemonstrate that LC-PFN can approximate the posterior predictive distribution\nmore accurately than MCMC, while being over 10 000 times faster. We also show\nthat the same LC-PFN achieves competitive performance extrapolating a total of\n20 000 real learning curves from four learning curve benchmarks (LCBench,\nNAS-Bench-201, Taskset, and PD1) that stem from training a wide range of model\narchitectures (MLPs, CNNs, RNNs, and Transformers) on 53 different datasets\nwith varying input modalities (tabular, image, text, and protein data).\nFinally, we investigate its potential in the context of model selection and\nfind that a simple LC-PFN based predictive early stopping criterion obtains 2 -\n6x speed-ups on 45 of these datasets, at virtually no overhead.\n", "date": "2023-10-31", "categories": []}
{"id": "2305.14593", "title": "Discriminative calibration: Check Bayesian computation from simulations\n  and flexible classifier", "abstract": "  To check the accuracy of Bayesian computations, it is common to use\nrank-based simulation-based calibration (SBC). However, SBC has drawbacks: The\ntest statistic is somewhat ad-hoc, interactions are difficult to examine,\nmultiple testing is a challenge, and the resulting p-value is not a divergence\nmetric. We propose to replace the marginal rank test with a flexible\nclassification approach that learns test statistics from data. This measure\ntypically has a higher statistical power than the SBC rank test and returns an\ninterpretable divergence measure of miscalibration, computed from\nclassification accuracy. This approach can be used with different data\ngenerating processes to address likelihood-free inference or traditional\ninference methods like Markov chain Monte Carlo or variational inference. We\nillustrate an automated implementation using neural networks and\nstatistically-inspired features, and validate the method with numerical and\nreal data experiments.\n", "date": "2023-05-24", "categories": []}
{"id": "2305.19147", "title": "Conditional score-based diffusion models for Bayesian inference in\n  infinite dimensions", "abstract": "  Since their initial introduction, score-based diffusion models (SDMs) have\nbeen successfully applied to solve a variety of linear inverse problems in\nfinite-dimensional vector spaces due to their ability to efficiently\napproximate the posterior distribution. However, using SDMs for inverse\nproblems in infinite-dimensional function spaces has only been addressed\nrecently, primarily through methods that learn the unconditional score. While\nthis approach is advantageous for some inverse problems, it is mostly heuristic\nand involves numerous computationally costly forward operator evaluations\nduring posterior sampling. To address these limitations, we propose a\ntheoretically grounded method for sampling from the posterior of\ninfinite-dimensional Bayesian linear inverse problems based on amortized\nconditional SDMs. In particular, we prove that one of the most successful\napproaches for estimating the conditional score in finite dimensions - the\nconditional denoising estimator - can also be applied in infinite dimensions. A\nsignificant part of our analysis is dedicated to demonstrating that extending\ninfinite-dimensional SDMs to the conditional setting requires careful\nconsideration, as the conditional score typically blows up for small times,\ncontrarily to the unconditional score. We conclude by presenting stylized and\nlarge-scale numerical examples that validate our approach, offer additional\ninsights, and demonstrate that our method enables large-scale,\ndiscretization-invariant Bayesian inference.\n", "date": "2023-05-28", "categories": []}
{"id": "2305.19366", "title": "Joint Bayesian Inference of Graphical Structure and Parameters with a\n  Single Generative Flow Network", "abstract": "  Generative Flow Networks (GFlowNets), a class of generative models over\ndiscrete and structured sample spaces, have been previously applied to the\nproblem of inferring the marginal posterior distribution over the directed\nacyclic graph (DAG) of a Bayesian Network, given a dataset of observations.\nBased on recent advances extending this framework to non-discrete sample\nspaces, we propose in this paper to approximate the joint posterior over not\nonly the structure of a Bayesian Network, but also the parameters of its\nconditional probability distributions. We use a single GFlowNet whose sampling\npolicy follows a two-phase process: the DAG is first generated sequentially one\nedge at a time, and then the corresponding parameters are picked once the full\nstructure is known. Since the parameters are included in the posterior\ndistribution, this leaves more flexibility for the local probability models of\nthe Bayesian Network, making our approach applicable even to non-linear models\nparametrized by neural networks. We show that our method, called JSP-GFN,\noffers an accurate approximation of the joint posterior, while comparing\nfavorably against existing methods on both simulated and real data.\n", "date": "2023-05-30", "categories": []}
{"id": "2306.03783", "title": "Asymptotics of Bayesian Uncertainty Estimation in Random Features\n  Regression", "abstract": "  In this paper we compare and contrast the behavior of the posterior\npredictive distribution to the risk of the maximum a posteriori estimator for\nthe random features regression model in the overparameterized regime. We will\nfocus on the variance of the posterior predictive distribution (Bayesian model\naverage) and compare its asymptotics to that of the risk of the MAP estimator.\nIn the regime where the model dimensions grow faster than any constant multiple\nof the number of samples, asymptotic agreement between these two quantities is\ngoverned by the phase transition in the signal-to-noise ratio. They also\nasymptotically agree with each other when the number of samples grow faster\nthan any constant multiple of model dimensions. Numerical simulations\nillustrate finer distributional properties of the two quantities for finite\ndimensions. We conjecture they have Gaussian fluctuations and exhibit similar\nproperties as found by previous authors in a Gaussian sequence model, which is\nof independent theoretical interest.\n", "date": "2023-06-06", "categories": []}
{"id": "2310.17852", "title": "Function Space Bayesian Pseudocoreset for Bayesian Neural Networks", "abstract": "  A Bayesian pseudocoreset is a compact synthetic dataset summarizing essential\ninformation of a large-scale dataset and thus can be used as a proxy dataset\nfor scalable Bayesian inference. Typically, a Bayesian pseudocoreset is\nconstructed by minimizing a divergence measure between the posterior\nconditioning on the pseudocoreset and the posterior conditioning on the full\ndataset. However, evaluating the divergence can be challenging, particularly\nfor the models like deep neural networks having high-dimensional parameters. In\nthis paper, we propose a novel Bayesian pseudocoreset construction method that\noperates on a function space. Unlike previous methods, which construct and\nmatch the coreset and full data posteriors in the space of model parameters\n(weights), our method constructs variational approximations to the coreset\nposterior on a function space and matches it to the full data posterior in the\nfunction space. By working directly on the function space, our method could\nbypass several challenges that may arise when working on a weight space,\nincluding limited scalability and multi-modality issue. Through various\nexperiments, we demonstrate that the Bayesian pseudocoresets constructed from\nour method enjoys enhanced uncertainty quantification and better robustness\nacross various model architectures.\n", "date": "2023-10-27", "categories": []}
{"id": "2308.07983", "title": "Monte Carlo guided Diffusion for Bayesian linear inverse problems", "abstract": "  Ill-posed linear inverse problems arise frequently in various applications,\nfrom computational photography to medical imaging. A recent line of research\nexploits Bayesian inference with informative priors to handle the ill-posedness\nof such problems. Amongst such priors, score-based generative models (SGM) have\nrecently been successfully applied to several different inverse problems. In\nthis study, we exploit the particular structure of the prior defined by the SGM\nto define a sequence of intermediate linear inverse problems. As the noise\nlevel decreases, the posteriors of these inverse problems get closer to the\ntarget posterior of the original inverse problem. To sample from this sequence\nof posteriors, we propose the use of Sequential Monte Carlo (SMC) methods. The\nproposed algorithm, MCGDiff, is shown to be theoretically grounded and we\nprovide numerical simulations showing that it outperforms competing baselines\nwhen dealing with ill-posed inverse problems in a Bayesian setting.\n", "date": "2023-08-15", "categories": []}
{"id": "2306.12306", "title": "Beyond Deep Ensembles: A Large-Scale Evaluation of Bayesian Deep\n  Learning under Distribution Shift", "abstract": "  Bayesian deep learning (BDL) is a promising approach to achieve\nwell-calibrated predictions on distribution-shifted data. Nevertheless, there\nexists no large-scale survey that evaluates recent SOTA methods on diverse,\nrealistic, and challenging benchmark tasks in a systematic manner. To provide a\nclear picture of the current state of BDL research, we evaluate modern BDL\nalgorithms on real-world datasets from the WILDS collection containing\nchallenging classification and regression tasks, with a focus on generalization\ncapability and calibration under distribution shift. We compare the algorithms\non a wide range of large, convolutional and transformer-based neural network\narchitectures. In particular, we investigate a signed version of the expected\ncalibration error that reveals whether the methods are over- or\nunder-confident, providing further insight into the behavior of the methods.\nFurther, we provide the first systematic evaluation of BDL for fine-tuning\nlarge pre-trained models, where training from scratch is prohibitively\nexpensive. Finally, given the recent success of Deep Ensembles, we extend\npopular single-mode posterior approximations to multiple modes by the use of\nensembles. While we find that ensembling single-mode approximations generally\nimproves the generalization capability and calibration of the models by a\nsignificant margin, we also identify a failure mode of ensembles when\nfinetuning large transformer-based language models. In this setting,\nvariational inference based approaches such as last-layer Bayes By Backprop\noutperform other methods in terms of accuracy by a large margin, while modern\napproximate inference algorithms such as SWAG achieve the best calibration.\n", "date": "2023-06-21", "categories": []}
{"id": "2209.10888", "title": "Amortized Variational Inference: A Systematic Review", "abstract": "  The core principle of Variational Inference (VI) is to convert the\nstatistical inference problem of computing complex posterior probability\ndensities into a tractable optimization problem. This property enables VI to be\nfaster than several sampling-based techniques. However, the traditional VI\nalgorithm is not scalable to large data sets and is unable to readily infer\nout-of-bounds data points without re-running the optimization process. Recent\ndevelopments in the field, like stochastic-, black box-, and amortized-VI, have\nhelped address these issues. Generative modeling tasks nowadays widely make use\nof amortized VI for its efficiency and scalability, as it utilizes a\nparameterized function to learn the approximate posterior density parameters.\nIn this paper, we review the mathematical foundations of various VI techniques\nto form the basis for understanding amortized VI. Additionally, we provide an\noverview of the recent trends that address several issues of amortized VI, such\nas the amortization gap, generalization issues, inconsistent representation\nlearning, and posterior collapse. Finally, we analyze alternate divergence\nmeasures that improve VI optimization.\n", "date": "2022-09-22", "categories": []}
{"id": "2310.13975", "title": "ASBART:Accelerated Soft Bayes Additive Regression Trees", "abstract": "  Bayes additive regression trees(BART) is a nonparametric regression model\nwhich has gained wide-spread popularity in recent years due to its flexibility\nand high accuracy of estimation. Soft BART,one variation of BART,improves both\npractically and heoretically on existing Bayesian sum-of-trees models. One\nbottleneck for Soft BART is its slow speed in the long MCMC loop. Compared to\nBART,it use more than about 20 times to complete the calculation with the\ndefault setting. We proposed a variant of BART named accelerate Soft\nBART(ASBART). Simulation studies show that the new method is about 10 times\nfaster than the Soft BART with comparable accuracy. Our code is open-source and\navailable at https://github.com/richael008/XSBART.\n", "date": "2023-10-21", "categories": []}
{"id": "2210.12595", "title": "Online Probabilistic Model Identification using Adaptive Recursive MCMC", "abstract": "  Although the Bayesian paradigm offers a formal framework for estimating the\nentire probability distribution over uncertain parameters, its online\nimplementation can be challenging due to high computational costs. We suggest\nthe Adaptive Recursive Markov Chain Monte Carlo (ARMCMC) method, which\neliminates the shortcomings of conventional online techniques while computing\nthe entire probability density function of model parameters. The limitations to\nGaussian noise, the application to only linear in the parameters (LIP) systems,\nand the persistent excitation (PE) needs are some of these drawbacks. In\nARMCMC, a temporal forgetting factor (TFF)-based variable jump distribution is\nproposed. The forgetting factor can be presented adaptively using the TFF in\nmany dynamical systems as an alternative to a constant hyperparameter. By\noffering a trade-off between exploitation and exploration, the specific jump\ndistribution has been optimised towards hybrid/multi-modal systems that permit\ninferences among modes. These trade-off are adjusted based on parameter\nevolution rate. We demonstrate that ARMCMC requires fewer samples than\nconventional MCMC methods to achieve the same precision and reliability. We\ndemonstrate our approach using parameter estimation in a soft bending actuator\nand the Hunt-Crossley dynamic model, two challenging hybrid/multi-modal\nbenchmarks. Additionally, we compare our method with recursive least squares\nand the particle filter, and show that our technique has significantly more\naccurate point estimates as well as a decrease in tracking error of the value\nof interest.\n", "date": "2022-10-23", "categories": []}
{"id": "2310.13402", "title": "Calibrating Neural Simulation-Based Inference with Differentiable\n  Coverage Probability", "abstract": "  Bayesian inference allows expressing the uncertainty of posterior belief\nunder a probabilistic model given prior information and the likelihood of the\nevidence. Predominantly, the likelihood function is only implicitly established\nby a simulator posing the need for simulation-based inference (SBI). However,\nthe existing algorithms can yield overconfident posteriors (Hermans *et al.*,\n2022) defeating the whole purpose of credibility if the uncertainty\nquantification is inaccurate. We propose to include a calibration term directly\ninto the training objective of the neural model in selected amortized SBI\ntechniques. By introducing a relaxation of the classical formulation of\ncalibration error we enable end-to-end backpropagation. The proposed method is\nnot tied to any particular neural model and brings moderate computational\noverhead compared to the profits it introduces. It is directly applicable to\nexisting computational pipelines allowing reliable black-box posterior\ninference. We empirically show on six benchmark problems that the proposed\nmethod achieves competitive or better results in terms of coverage and expected\nposterior density than the previously existing approaches.\n", "date": "2023-10-20", "categories": []}
{"id": "2310.10171", "title": "On permutation symmetries in Bayesian neural network posteriors: a\n  variational perspective", "abstract": "  The elusive nature of gradient-based optimization in neural networks is tied\nto their loss landscape geometry, which is poorly understood. However recent\nwork has brought solid evidence that there is essentially no loss barrier\nbetween the local solutions of gradient descent, once accounting for\nweight-permutations that leave the network's computation unchanged. This raises\nquestions for approximate inference in Bayesian neural networks (BNNs), where\nwe are interested in marginalizing over multiple points in the loss landscape.\nIn this work, we first extend the formalism of marginalized loss barrier and\nsolution interpolation to BNNs, before proposing a matching algorithm to search\nfor linearly connected solutions. This is achieved by aligning the\ndistributions of two independent approximate Bayesian solutions with respect to\npermutation matrices. We build on the results of Ainsworth et al. (2023),\nreframing the problem as a combinatorial optimization one, using an\napproximation to the sum of bilinear assignment problem. We then experiment on\na variety of architectures and datasets, finding nearly zero marginalized loss\nbarriers for linearly connected solutions.\n", "date": "2023-10-16", "categories": []}
{"id": "2303.06171", "title": "DP-Fast MH: Private, Fast, and Accurate Metropolis-Hastings for\n  Large-Scale Bayesian Inference", "abstract": "  Bayesian inference provides a principled framework for learning from complex\ndata and reasoning under uncertainty. It has been widely applied in machine\nlearning tasks such as medical diagnosis, drug design, and policymaking. In\nthese common applications, data can be highly sensitive. Differential privacy\n(DP) offers data analysis tools with powerful worst-case privacy guarantees and\nhas been developed as the leading approach in privacy-preserving data analysis.\nIn this paper, we study Metropolis-Hastings (MH), one of the most fundamental\nMCMC methods, for large-scale Bayesian inference under differential privacy.\nWhile most existing private MCMC algorithms sacrifice accuracy and efficiency\nto obtain privacy, we provide the first exact and fast DP MH algorithm, using\nonly a minibatch of data in most iterations. We further reveal, for the first\ntime, a three-way trade-off among privacy, scalability (i.e. the batch size),\nand efficiency (i.e. the convergence rate), theoretically characterizing how\nprivacy affects the utility and computational cost in Bayesian inference. We\nempirically demonstrate the effectiveness and efficiency of our algorithm in\nvarious experiments.\n", "date": "2023-03-10", "categories": []}
{"id": "2310.08287", "title": "A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors", "abstract": "  The distribution of the weights of modern deep neural networks (DNNs) -\ncrucial for uncertainty quantification and robustness - is an eminently complex\nobject due to its extremely high dimensionality. This paper proposes one of the\nfirst large-scale explorations of the posterior distribution of deep Bayesian\nNeural Networks (BNNs), expanding its study to real-world vision tasks and\narchitectures. Specifically, we investigate the optimal approach for\napproximating the posterior, analyze the connection between posterior quality\nand uncertainty quantification, delve into the impact of modes on the\nposterior, and explore methods for visualizing the posterior. Moreover, we\nuncover weight-space symmetries as a critical aspect for understanding the\nposterior. To this extent, we develop an in-depth assessment of the impact of\nboth permutation and scaling symmetries that tend to obfuscate the Bayesian\nposterior. While the first type of transformation is known for duplicating\nmodes, we explore the relationship between the latter and L2 regularization,\nchallenging previous misconceptions. Finally, to help the community improve our\nunderstanding of the Bayesian posterior, we will shortly release the first\nlarge-scale checkpoint dataset, including thousands of real-world models and\nour codes.\n", "date": "2023-10-12", "categories": []}
{"id": "2310.05456", "title": "Ensemble-based Hybrid Optimization of Bayesian Neural Networks and\n  Traditional Machine Learning Algorithms", "abstract": "  This research introduces a novel methodology for optimizing Bayesian Neural\nNetworks (BNNs) by synergistically integrating them with traditional machine\nlearning algorithms such as Random Forests (RF), Gradient Boosting (GB), and\nSupport Vector Machines (SVM). Feature integration solidifies these results by\nemphasizing the second-order conditions for optimality, including stationarity\nand positive definiteness of the Hessian matrix. Conversely, hyperparameter\ntuning indicates a subdued impact in improving Expected Improvement (EI),\nrepresented by EI(x). Overall, the ensemble method stands out as a robust,\nalgorithmically optimized approach.\n", "date": "2023-10-09", "categories": []}
{"id": "2310.03435", "title": "Variational Inference for GARCH-family Models", "abstract": "  The Bayesian estimation of GARCH-family models has been typically addressed\nthrough Monte Carlo sampling. Variational Inference is gaining popularity and\nattention as a robust approach for Bayesian inference in complex machine\nlearning models; however, its adoption in econometrics and finance is limited.\nThis paper discusses the extent to which Variational Inference constitutes a\nreliable and feasible alternative to Monte Carlo sampling for Bayesian\ninference in GARCH-like models. Through a large-scale experiment involving the\nconstituents of the S&P 500 index, several Variational Inference optimizers, a\nvariety of volatility models, and a case study, we show that Variational\nInference is an attractive, remarkably well-calibrated, and competitive method\nfor Bayesian learning.\n", "date": "2023-10-05", "categories": []}
{"id": "2304.12541", "title": "Efficient Bayesian inference using physics-informed invertible neural\n  networks for inverse problems", "abstract": "  In this paper, we introduce an innovative approach for addressing Bayesian\ninverse problems through the utilization of physics-informed invertible neural\nnetworks (PI-INN). The PI-INN framework encompasses two sub-networks: an\ninvertible neural network (INN) and a neural basis network (NB-Net). The\nprimary role of the NB-Net lies in modeling the spatial basis functions\ncharacterizing the solution to the forward problem dictated by the underlying\npartial differential equation. Simultaneously, the INN is designed to partition\nthe parameter vector linked to the input physical field into two distinct\ncomponents: the expansion coefficients representing the forward problem\nsolution and the Gaussian latent noise. If the forward mapping is precisely\nestimated, and the statistical independence between expansion coefficients and\nlatent noise is well-maintained, the PI-INN offers a precise and efficient\ngenerative model for Bayesian inverse problems, yielding tractable posterior\ndensity estimates. As a particular physics-informed deep learning model, the\nprimary training challenge for PI-INN centers on enforcing the independence\nconstraint, which we tackle by introducing a novel independence loss based on\nestimated density. We support the efficacy and precision of the proposed PI-INN\nthrough a series of numerical experiments, including inverse kinematics,\n1-dimensional and 2-dimensional diffusion equations, and seismic traveltime\ntomography. Specifically, our experimental results showcase the superior\nperformance of the proposed independence loss in comparison to the commonly\nused but computationally demanding kernel-based maximum mean discrepancy loss.\n", "date": "2023-04-25", "categories": []}
{"id": "2309.15499", "title": "Bayesian Personalized Federated Learning with Shared and Personalized\n  Uncertainty Representations", "abstract": "  Bayesian personalized federated learning (BPFL) addresses challenges in\nexisting personalized FL (PFL). BPFL aims to quantify the uncertainty and\nheterogeneity within and across clients towards uncertainty representations by\naddressing the statistical heterogeneity of client data. In PFL, some recent\npreliminary work proposes to decompose hidden neural representations into\nshared and local components and demonstrates interesting results. However, most\nof them do not address client uncertainty and heterogeneity in FL systems,\nwhile appropriately decoupling neural representations is challenging and often\nad hoc. In this paper, we make the first attempt to introduce a general BPFL\nframework to decompose and jointly learn shared and personalized uncertainty\nrepresentations on statistically heterogeneous client data over time. A\nBayesian federated neural network BPFed instantiates BPFL by jointly learning\ncross-client shared uncertainty and client-specific personalized uncertainty\nover statistically heterogeneous and randomly participating clients. We further\ninvolve continual updating of prior distribution in BPFed to speed up the\nconvergence and avoid catastrophic forgetting. Theoretical analysis and\nguarantees are provided in addition to the experimental evaluation of BPFed\nagainst the diversified baselines.\n", "date": "2023-09-27", "categories": []}
{"id": "2306.06528", "title": "Push: Concurrent Probabilistic Programming for Bayesian Deep Learning", "abstract": "  We introduce a library called Push that takes a probabilistic programming\napproach to Bayesian deep learning (BDL). This library enables concurrent\nexecution of BDL inference algorithms on multi-GPU hardware for neural network\n(NN) models. To accomplish this, Push introduces an abstraction that represents\nan input NN as a particle. Push enables easy creation of particles so that an\ninput NN can be replicated and particles can communicate asynchronously so that\na variety of parameter updates can be expressed, including common BDL\nalgorithms. Our hope is that Push lowers the barrier to experimenting with BDL\nby streamlining the scaling of particles across GPUs. We evaluate the scaling\nbehavior of particles on single-node multi-GPU devices on vision and scientific\nmachine learning (SciML) tasks.\n", "date": "2023-06-10", "categories": []}
{"id": "2310.00941", "title": "Improved Variational Bayesian Phylogenetic Inference using Mixtures", "abstract": "  We present VBPI-Mixtures, an algorithm designed to enhance the accuracy of\nphylogenetic posterior distributions, particularly for tree-topology and\nbranch-length approximations. Despite the Variational Bayesian Phylogenetic\nInference (VBPI), a leading-edge black-box variational inference (BBVI)\nframework, achieving remarkable approximations of these distributions, the\nmultimodality of the tree-topology posterior presents a formidable challenge to\nsampling-based learning techniques such as BBVI. Advanced deep learning\nmethodologies such as normalizing flows and graph neural networks have been\nexplored to refine the branch-length posterior approximation, yet efforts to\nameliorate the posterior approximation over tree topologies have been lacking.\nOur novel VBPI-Mixtures algorithm bridges this gap by harnessing the latest\nbreakthroughs in mixture learning within the BBVI domain. As a result,\nVBPI-Mixtures is capable of capturing distributions over tree-topologies that\nVBPI fails to model. We deliver state-of-the-art performance on difficult\ndensity estimation tasks across numerous real phylogenetic datasets.\n", "date": "2023-10-02", "categories": []}
{"id": "2310.01189", "title": "If there is no underfitting, there is no Cold Posterior Effect", "abstract": "  The cold posterior effect (CPE) (Wenzel et al., 2020) in Bayesian deep\nlearning shows that, for posteriors with a temperature $T<1$, the resulting\nposterior predictive could have better performances than the Bayesian posterior\n($T=1$). As the Bayesian posterior is known to be optimal under perfect model\nspecification, many recent works have studied the presence of CPE as a model\nmisspecification problem, arising from the prior and/or from the likelihood\nfunction. In this work, we provide a more nuanced understanding of the CPE as\nwe show that misspecification leads to CPE only when the resulting Bayesian\nposterior underfits. In fact, we theoretically show that if there is no\nunderfitting, there is no CPE.\n", "date": "2023-10-02", "categories": []}
{"id": "2309.16314", "title": "A Primer on Bayesian Neural Networks: Review and Debates", "abstract": "  Neural networks have achieved remarkable performance across various problem\ndomains, but their widespread applicability is hindered by inherent limitations\nsuch as overconfidence in predictions, lack of interpretability, and\nvulnerability to adversarial attacks. To address these challenges, Bayesian\nneural networks (BNNs) have emerged as a compelling extension of conventional\nneural networks, integrating uncertainty estimation into their predictive\ncapabilities.\n  This comprehensive primer presents a systematic introduction to the\nfundamental concepts of neural networks and Bayesian inference, elucidating\ntheir synergistic integration for the development of BNNs. The target audience\ncomprises statisticians with a potential background in Bayesian methods but\nlacking deep learning expertise, as well as machine learners proficient in deep\nneural networks but with limited exposure to Bayesian statistics. We provide an\noverview of commonly employed priors, examining their impact on model behavior\nand performance. Additionally, we delve into the practical considerations\nassociated with training and inference in BNNs.\n  Furthermore, we explore advanced topics within the realm of BNN research,\nacknowledging the existence of ongoing debates and controversies. By offering\ninsights into cutting-edge developments, this primer not only equips\nresearchers and practitioners with a solid foundation in BNNs, but also\nilluminates the potential applications of this dynamic field. As a valuable\nresource, it fosters an understanding of BNNs and their promising prospects,\nfacilitating further advancements in the pursuit of knowledge and innovation.\n", "date": "2023-09-28", "categories": []}
{"id": "2309.12928", "title": "BayesDLL: Bayesian Deep Learning Library", "abstract": "  We release a new Bayesian neural network library for PyTorch for large-scale\ndeep networks. Our library implements mainstream approximate Bayesian inference\nalgorithms: variational inference, MC-dropout, stochastic-gradient MCMC, and\nLaplace approximation. The main differences from other existing Bayesian neural\nnetwork libraries are as follows: 1) Our library can deal with very large-scale\ndeep networks including Vision Transformers (ViTs). 2) We need virtually zero\ncode modifications for users (e.g., the backbone network definition codes do\nnot neet to be modified at all). 3) Our library also allows the pre-trained\nmodel weights to serve as a prior mean, which is very useful for performing\nBayesian inference with the large-scale foundation models like ViTs that are\nhard to optimise from scratch with the downstream data alone. Our code is\npublicly available at: \\url{https://github.com/SamsungLabs/BayesDLL}\\footnote{A\nmirror repository is also available at:\n\\url{https://github.com/minyoungkim21/BayesDLL}.}.\n", "date": "2023-09-22", "categories": []}
{"id": "2302.02522", "title": "Prior Density Learning in Variational Bayesian Phylogenetic Parameters\n  Inference", "abstract": "  The advances in variational inference are providing promising paths in\nBayesian estimation problems. These advances make variational phylogenetic\ninference an alternative approach to Markov Chain Monte Carlo methods for\napproximating the phylogenetic posterior. However, one of the main drawbacks of\nsuch approaches is modelling the prior through fixed distributions, which could\nbias the posterior approximation if they are distant from the current data\ndistribution. In this paper, we propose an approach and an implementation\nframework to relax the rigidity of the prior densities by learning their\nparameters using a gradient-based method and a neural network-based\nparameterization. We applied this approach for branch lengths and evolutionary\nparameters estimation under several Markov chain substitution models. The\nresults of performed simulations show that the approach is powerful in\nestimating branch lengths and evolutionary model parameters. They also show\nthat a flexible prior model could provide better results than a predefined\nprior model. Finally, the results highlight that using neural networks improves\nthe initialization of the optimization of the prior density parameters.\n", "date": "2023-02-06", "categories": []}
{"id": "2309.05292", "title": "The fine print on tempered posteriors", "abstract": "  We conduct a detailed investigation of tempered posteriors and uncover a\nnumber of crucial and previously undiscussed points. Contrary to previous\nresults, we first show that for realistic models and datasets and the tightly\ncontrolled case of the Laplace approximation to the posterior, stochasticity\ndoes not in general improve test accuracy. The coldest temperature is often\noptimal. One might think that Bayesian models with some stochasticity can at\nleast obtain improvements in terms of calibration. However, we show empirically\nthat when gains are obtained this comes at the cost of degradation in test\naccuracy. We then discuss how targeting Frequentist metrics using Bayesian\nmodels provides a simple explanation of the need for a temperature parameter\n$\\lambda$ in the optimization objective. Contrary to prior works, we finally\nshow through a PAC-Bayesian analysis that the temperature $\\lambda$ cannot be\nseen as simply fixing a misspecified prior or likelihood.\n", "date": "2023-09-11", "categories": []}
{"id": "2309.03018", "title": "Amortised Inference in Bayesian Neural Networks", "abstract": "  Meta-learning is a framework in which machine learning models train over a\nset of datasets in order to produce predictions on new datasets at test time.\nProbabilistic meta-learning has received an abundance of attention from the\nresearch community in recent years, but a problem shared by many existing\nprobabilistic meta-models is that they require a very large number of datasets\nin order to produce high-quality predictions with well-calibrated uncertainty\nestimates. In many applications, however, such quantities of data are simply\nnot available.\n  In this dissertation we present a significantly more data-efficient approach\nto probabilistic meta-learning through per-datapoint amortisation of inference\nin Bayesian neural networks, introducing the Amortised Pseudo-Observation\nVariational Inference Bayesian Neural Network (APOVI-BNN). First, we show that\nthe approximate posteriors obtained under our amortised scheme are of similar\nor better quality to those obtained through traditional variational inference,\ndespite the fact that the amortised inference is performed in a single forward\npass. We then discuss how the APOVI-BNN may be viewed as a new member of the\nneural process family, motivating the use of neural process training objectives\nfor potentially better predictive performance on complex problems as a result.\nFinally, we assess the predictive performance of the APOVI-BNN against other\nprobabilistic meta-models in both a one-dimensional regression problem and in a\nsignificantly more complex image completion setting. In both cases, when the\namount of training data is limited, our model is the best in its class.\n", "date": "2023-09-06", "categories": []}
{"id": "2309.03061", "title": "Learning Active Subspaces for Effective and Scalable Uncertainty\n  Quantification in Deep Neural Networks", "abstract": "  Bayesian inference for neural networks, or Bayesian deep learning, has the\npotential to provide well-calibrated predictions with quantified uncertainty\nand robustness. However, the main hurdle for Bayesian deep learning is its\ncomputational complexity due to the high dimensionality of the parameter space.\nIn this work, we propose a novel scheme that addresses this limitation by\nconstructing a low-dimensional subspace of the neural network\nparameters-referred to as an active subspace-by identifying the parameter\ndirections that have the most significant influence on the output of the neural\nnetwork. We demonstrate that the significantly reduced active subspace enables\neffective and scalable Bayesian inference via either Monte Carlo (MC) sampling\nmethods, otherwise computationally intractable, or variational inference.\nEmpirically, our approach provides reliable predictions with robust uncertainty\nestimates for various regression tasks.\n", "date": "2023-09-06", "categories": []}
{"id": "2308.16022", "title": "PAVI: Plate-Amortized Variational Inference", "abstract": "  Given observed data and a probabilistic generative model, Bayesian inference\nsearches for the distribution of the model's parameters that could have yielded\nthe data. Inference is challenging for large population studies where millions\nof measurements are performed over a cohort of hundreds of subjects, resulting\nin a massive parameter space. This large cardinality renders off-the-shelf\nVariational Inference (VI) computationally impractical.\n  In this work, we design structured VI families that efficiently tackle large\npopulation studies. Our main idea is to share the parameterization and learning\nacross the different i.i.d. variables in a generative model, symbolized by the\nmodel's \\textit{plates}. We name this concept \\textit{plate amortization}.\nContrary to off-the-shelf stochastic VI, which slows down inference, plate\namortization results in orders of magnitude faster to train variational\ndistributions.\n  Applied to large-scale hierarchical problems, PAVI yields expressive,\nparsimoniously parameterized VI with an affordable training time. This faster\nconvergence effectively unlocks inference in those large regimes. We illustrate\nthe practical utility of PAVI through a challenging Neuroimaging example\nfeaturing 400 million latent parameters, demonstrating a significant step\ntowards scalable and expressive Variational Inference.\n", "date": "2023-08-30", "categories": []}
{"id": "2308.13731", "title": "Learning variational autoencoders via MCMC speed measures", "abstract": "  Variational autoencoders (VAEs) are popular likelihood-based generative\nmodels which can be efficiently trained by maximizing an Evidence Lower Bound\n(ELBO). There has been much progress in improving the expressiveness of the\nvariational distribution to obtain tighter variational bounds and increased\ngenerative performance. Whilst previous work has leveraged Markov chain Monte\nCarlo (MCMC) methods for the construction of variational densities,\ngradient-based methods for adapting the proposal distributions for deep latent\nvariable models have received less attention. This work suggests an\nentropy-based adaptation for a short-run Metropolis-adjusted Langevin (MALA) or\nHamiltonian Monte Carlo (HMC) chain while optimising a tighter variational\nbound to the log-evidence. Experiments show that this approach yields higher\nheld-out log-likelihoods as well as improved generative metrics. Our implicit\nvariational density can adapt to complicated posterior geometries of latent\nhierarchical representations arising in hierarchical VAEs.\n", "date": "2023-08-26", "categories": []}
{"id": "2308.12785", "title": "Single-shot Bayesian approximation for neural networks", "abstract": "  Deep neural networks (NNs) are known for their high-prediction performances.\nHowever, NNs are prone to yield unreliable predictions when encountering\ncompletely new situations without indicating their uncertainty. Bayesian\nvariants of NNs (BNNs), such as Monte Carlo (MC) dropout BNNs, do provide\nuncertainty measures and simultaneously increase the prediction performance.\nThe only disadvantage of BNNs is their higher computation time during test time\nbecause they rely on a sampling approach. Here we present a single-shot MC\ndropout approximation that preserves the advantages of BNNs while being as fast\nas NNs. Our approach is based on moment propagation (MP) and allows to\nanalytically approximate the expected value and the variance of the MC dropout\nsignal for commonly used layers in NNs, i.e. convolution, max pooling, dense,\nsoftmax, and dropout layers. The MP approach can convert an NN into a BNN\nwithout re-training given the NN has been trained with standard dropout. We\nevaluate our approach on different benchmark datasets and a simulated toy\nexample in a classification and regression setting. We demonstrate that our\nsingle-shot MC dropout approximation resembles the point estimate and the\nuncertainty estimate of the predictive distribution that is achieved with an MC\napproach, while being fast enough for real-time deployments of BNNs. We show\nthat using part of the saved time to combine our MP approach with deep ensemble\ntechniques does further improve the uncertainty measures.\n", "date": "2023-08-24", "categories": []}
{"id": "2210.15081", "title": "Bayesian Hyperbolic Multidimensional Scaling", "abstract": "  Multidimensional scaling (MDS) is a widely used approach to representing\nhigh-dimensional, dependent data. MDS works by assigning each observation a\nlocation on a low-dimensional geometric manifold, with distance on the manifold\nrepresenting similarity. We propose a Bayesian approach to multidimensional\nscaling when the low-dimensional manifold is hyperbolic. Using hyperbolic space\nfacilitates representing tree-like structures common in many settings (e.g.\ntext or genetic data with hierarchical structure). A Bayesian approach provides\nregularization that minimizes the impact of measurement error in the observed\ndata and assesses uncertainty. We also propose a case-control likelihood\napproximation that allows for efficient sampling from the posterior\ndistribution in larger data settings, reducing computational complexity from\napproximately $O(n^2)$ to $O(n)$. We evaluate the proposed method against\nstate-of-the-art alternatives using simulations, canonical reference datasets,\nIndian village network data, and human gene expression data.\n", "date": "2022-10-26", "categories": []}
{"id": "2308.06293", "title": "Target Detection on Hyperspectral Images Using MCMC and VI Trained\n  Bayesian Neural Networks", "abstract": "  Neural networks (NN) have become almost ubiquitous with image classification,\nbut in their standard form produce point estimates, with no measure of\nconfidence. Bayesian neural networks (BNN) provide uncertainty quantification\n(UQ) for NN predictions and estimates through the posterior distribution. As NN\nare applied in more high-consequence applications, UQ is becoming a\nrequirement. BNN provide a solution to this problem by not only giving accurate\npredictions and estimates, but also an interval that includes reasonable values\nwithin a desired probability. Despite their positive attributes, BNN are\nnotoriously difficult and time consuming to train. Traditional Bayesian methods\nuse Markov Chain Monte Carlo (MCMC), but this is often brushed aside as being\ntoo slow. The most common method is variational inference (VI) due to its fast\ncomputation, but there are multiple concerns with its efficacy. We apply and\ncompare MCMC- and VI-trained BNN in the context of target detection in\nhyperspectral imagery (HSI), where materials of interest can be identified by\ntheir unique spectral signature. This is a challenging field, due to the\nnumerous permuting effects practical collection of HSI has on measured spectra.\nBoth models are trained using out-of-the-box tools on a high fidelity HSI\ntarget detection scene. Both MCMC- and VI-trained BNN perform well overall at\ntarget detection on a simulated HSI scene. This paper provides an example of\nhow to utilize the benefits of UQ, but also to increase awareness that\ndifferent training methods can give different results for the same model. If\nsufficient computational resources are available, the best approach rather than\nthe fastest or most efficient should be used, especially for high consequence\nproblems.\n", "date": "2023-08-11", "categories": []}
{"id": "2210.10780", "title": "An out-of-distribution discriminator based on Bayesian neural network\n  epistemic uncertainty", "abstract": "  Neural networks have revolutionized the field of machine learning with\nincreased predictive capability. In addition to improving the predictions of\nneural networks, there is a simultaneous demand for reliable uncertainty\nquantification on estimates made by machine learning methods such as neural\nnetworks. Bayesian neural networks (BNNs) are an important type of neural\nnetwork with built-in capability for quantifying uncertainty. This paper\ndiscusses aleatoric and epistemic uncertainty in BNNs and how they can be\ncalculated. With an example dataset of images where the goal is to identify the\namplitude of an event in the image, it is shown that epistemic uncertainty\ntends to be lower in images which are well-represented in the training dataset\nand tends to be high in images which are not well-represented. An algorithm for\nout-of-distribution (OoD) detection with BNN epistemic uncertainty is\nintroduced along with various experiments demonstrating factors influencing the\nOoD detection capability in a BNN. The OoD detection capability with epistemic\nuncertainty is shown to be comparable to the OoD detection in the discriminator\nnetwork of a generative adversarial network (GAN) with comparable network\narchitecture.\n", "date": "2022-10-18", "categories": []}
{"id": "2308.02451", "title": "Pruning a neural network using Bayesian inference", "abstract": "  Neural network pruning is a highly effective technique aimed at reducing the\ncomputational and memory demands of large neural networks. In this research\npaper, we present a novel approach to pruning neural networks utilizing\nBayesian inference, which can seamlessly integrate into the training procedure.\nOur proposed method leverages the posterior probabilities of the neural network\nprior to and following pruning, enabling the calculation of Bayes factors. The\ncalculated Bayes factors guide the iterative pruning. Through comprehensive\nevaluations conducted on multiple benchmarks, we demonstrate that our method\nachieves desired levels of sparsity while maintaining competitive accuracy.\n", "date": "2023-08-04", "categories": []}
{"id": "2308.01271", "title": "A Probabilistic Approach to Self-Supervised Learning using Cyclical\n  Stochastic Gradient MCMC", "abstract": "  In this paper we present a practical Bayesian self-supervised learning method\nwith Cyclical Stochastic Gradient Hamiltonian Monte Carlo (cSGHMC). Within this\nframework, we place a prior over the parameters of a self-supervised learning\nmodel and use cSGHMC to approximate the high dimensional and multimodal\nposterior distribution over the embeddings. By exploring an expressive\nposterior over the embeddings, Bayesian self-supervised learning produces\ninterpretable and diverse representations. Marginalizing over these\nrepresentations yields a significant gain in performance, calibration and\nout-of-distribution detection on a variety of downstream classification tasks.\nWe provide experimental results on multiple classification tasks on four\nchallenging datasets. Moreover, we demonstrate the effectiveness of the\nproposed method in out-of-distribution detection using the SVHN and CIFAR-10\ndatasets.\n", "date": "2023-08-02", "categories": []}
{"id": "2303.02859", "title": "Bayesian inference with finitely wide neural networks", "abstract": "  The analytic inference, e.g. predictive distribution being in closed form,\nmay be an appealing benefit for machine learning practitioners when they treat\nwide neural networks as Gaussian process in Bayesian setting. The realistic\nwidths, however, are finite and cause weak deviation from the Gaussianity under\nwhich partial marginalization of random variables in a model is\nstraightforward. On the basis of multivariate Edgeworth expansion, we propose a\nnon-Gaussian distribution in differential form to model a finite set of outputs\nfrom a random neural network, and derive the corresponding marginal and\nconditional properties. Thus, we are able to derive the non-Gaussian posterior\ndistribution in Bayesian regression task. In addition, in the bottlenecked deep\nneural networks, a weight space representation of deep Gaussian process, the\nnon-Gaussianity is investigated through the marginal kernel.\n", "date": "2023-03-06", "categories": []}
{"id": "2307.14628", "title": "Rapid and Scalable Bayesian AB Testing", "abstract": "  AB testing aids business operators with their decision making, and is\nconsidered the gold standard method for learning from data to improve digital\nuser experiences. However, there is usually a gap between the requirements of\npractitioners, and the constraints imposed by the statistical hypothesis\ntesting methodologies commonly used for analysis of AB tests. These include the\nlack of statistical power in multivariate designs with many factors,\ncorrelations between these factors, the need of sequential testing for early\nstopping, and the inability to pool knowledge from past tests. Here, we propose\na solution that applies hierarchical Bayesian estimation to address the above\nlimitations. In comparison to current sequential AB testing methodology, we\nincrease statistical power by exploiting correlations between factors, enabling\nsequential testing and progressive early stopping, without incurring excessive\nfalse positive risk. We also demonstrate how this methodology can be extended\nto enable the extraction of composite global learnings from past AB tests, to\naccelerate future tests. We underpin our work with a solid theoretical\nframework that articulates the value of hierarchical estimation. We demonstrate\nits utility using both numerical simulations and a large set of real-world AB\ntests. Together, these results highlight the practical value of our approach\nfor statistical inference in the technology industry.\n", "date": "2023-07-27", "categories": []}
{"id": "2207.03104", "title": "Quantum Advantage in Variational Bayes Inference", "abstract": "  Variational Bayes (VB) inference algorithm is used widely to estimate both\nthe parameters and the unobserved hidden variables in generative statistical\nmodels. The algorithm -- inspired by variational methods used in computational\nphysics -- is iterative and can get easily stuck in local minima, even when\nclassical techniques, such as deterministic annealing (DA), are used. We study\na variational Bayes (VB) inference algorithm based on a non-traditional quantum\nannealing approach -- referred to as quantum annealing variational Bayes (QAVB)\ninference -- and show that there is indeed a quantum advantage to QAVB over its\nclassical counterparts. In particular, we show that such better performance is\nrooted in key concepts from quantum mechanics: (i) the ground state of the\nHamiltonian of a quantum system -- defined from the given variational Bayes\n(VB) problem -- corresponds to an optimal solution for the minimization problem\nof the variational free energy at very low temperatures; (ii) such a ground\nstate can be achieved by a technique paralleling the quantum annealing process;\nand (iii) starting from this ground state, the optimal solution to the VB\nproblem can be achieved by increasing the heat bath temperature to unity, and\nthereby avoiding local minima introduced by spontaneous symmetry breaking\nobserved in classical physics based VB algorithms. We also show that the update\nequations of QAVB can be potentially implemented using $\\lceil \\log K \\rceil$\nqubits and $\\mathcal{O} (K)$ operations per step. Thus, QAVB can match the time\ncomplexity of existing VB algorithms, while delivering higher performance.\n", "date": "2022-07-07", "categories": []}
{"id": "2208.11389", "title": "Approximate blocked Gibbs sampling for Bayesian neural networks", "abstract": "  In this work, minibatch MCMC sampling for feedforward neural networks is made\nmore feasible. To this end, it is proposed to sample subgroups of parameters\nvia a blocked Gibbs sampling scheme. By partitioning the parameter space,\nsampling is possible irrespective of layer width. It is also possible to\nalleviate vanishing acceptance rates for increasing depth by reducing the\nproposal variance in deeper layers. Increasing the length of a non-convergent\nchain increases the predictive accuracy in classification tasks, so avoiding\nvanishing acceptance rates and consequently enabling longer chain runs have\npractical benefits. Moreover, non-convergent chain realizations aid in the\nquantification of predictive uncertainty. An open problem is how to perform\nminibatch MCMC sampling for feedforward neural networks in the presence of\naugmented data.\n", "date": "2022-08-24", "categories": []}
{"id": "2307.11954", "title": "On-Robot Bayesian Reinforcement Learning for POMDPs", "abstract": "  Robot learning is often difficult due to the expense of gathering data. The\nneed for large amounts of data can, and should, be tackled with effective\nalgorithms and leveraging expert information on robot dynamics. Bayesian\nreinforcement learning (BRL), thanks to its sample efficiency and ability to\nexploit prior knowledge, is uniquely positioned as such a solution method.\nUnfortunately, the application of BRL has been limited due to the difficulties\nof representing expert knowledge as well as solving the subsequent inference\nproblem. This paper advances BRL for robotics by proposing a specialized\nframework for physical systems. In particular, we capture this knowledge in a\nfactored representation, then demonstrate the posterior factorizes in a similar\nshape, and ultimately formalize the model in a Bayesian framework. We then\nintroduce a sample-based online solution method, based on Monte-Carlo tree\nsearch and particle filtering, specialized to solve the resulting model. This\napproach can, for example, utilize typical low-level robot simulators and\nhandle uncertainty over unknown dynamics of the environment. We empirically\ndemonstrate its efficiency by performing on-robot learning in two human-robot\ninteraction tasks with uncertainty about human behavior, achieving near-optimal\nperformance after only a handful of real-world episodes. A video of learned\npolicies is at https://youtu.be/H9xp60ngOes.\n", "date": "2023-07-22", "categories": []}
{"id": "2209.11533", "title": "A Unified Perspective on Natural Gradient Variational Inference with\n  Gaussian Mixture Models", "abstract": "  Variational inference with Gaussian mixture models (GMMs) enables learning of\nhighly tractable yet multi-modal approximations of intractable target\ndistributions with up to a few hundred dimensions. The two currently most\neffective methods for GMM-based variational inference, VIPS and iBayes-GMM,\nboth employ independent natural gradient updates for the individual components\nand their weights. We show for the first time, that their derived updates are\nequivalent, although their practical implementations and theoretical guarantees\ndiffer. We identify several design choices that distinguish both approaches,\nnamely with respect to sample selection, natural gradient estimation, stepsize\nadaptation, and whether trust regions are enforced or the number of components\nadapted. We argue that for both approaches, the quality of the learned\napproximations can heavily suffer from the respective design choices: By\nupdating the individual components using samples from the mixture model,\niBayes-GMM often fails to produce meaningful updates to low-weight components,\nand by using a zero-order method for estimating the natural gradient, VIPS\nscales badly to higher-dimensional problems. Furthermore, we show that\ninformation-geometric trust-regions (used by VIPS) are effective even when\nusing first-order natural gradient estimates, and often outperform the improved\nBayesian learning rule (iBLR) update used by iBayes-GMM. We systematically\nevaluate the effects of design choices and show that a hybrid approach\nsignificantly outperforms both prior works. Along with this work, we publish\nour highly modular and efficient implementation for natural gradient\nvariational inference with Gaussian mixture models, which supports 432\ndifferent combinations of design choices, facilitates the reproduction of all\nour experiments, and may prove valuable for the practitioner.\n", "date": "2022-09-23", "categories": []}
{"id": "2302.09921", "title": "Free-Form Variational Inference for Gaussian Process State-Space Models", "abstract": "  Gaussian process state-space models (GPSSMs) provide a principled and\nflexible approach to modeling the dynamics of a latent state, which is observed\nat discrete-time points via a likelihood model. However, inference in GPSSMs is\ncomputationally and statistically challenging due to the large number of latent\nvariables in the model and the strong temporal dependencies between them. In\nthis paper, we propose a new method for inference in Bayesian GPSSMs, which\novercomes the drawbacks of previous approaches, namely over-simplified\nassumptions, and high computational requirements. Our method is based on\nfree-form variational inference via stochastic gradient Hamiltonian Monte Carlo\nwithin the inducing-variable formalism. Furthermore, by exploiting our proposed\nvariational distribution, we provide a collapsed extension of our method where\nthe inducing variables are marginalized analytically. We also showcase results\nwhen combining our framework with particle MCMC methods. We show that, on six\nreal-world datasets, our approach can learn transition dynamics and latent\nstates more accurately than competing methods.\n", "date": "2023-02-20", "categories": []}
{"id": "2307.07849", "title": "Variational Inference with Gaussian Score Matching", "abstract": "  Variational inference (VI) is a method to approximate the computationally\nintractable posterior distributions that arise in Bayesian statistics.\nTypically, VI fits a simple parametric distribution to the target posterior by\nminimizing an appropriate objective such as the evidence lower bound (ELBO). In\nthis work, we present a new approach to VI based on the principle of score\nmatching, that if two distributions are equal then their score functions (i.e.,\ngradients of the log density) are equal at every point on their support. With\nthis, we develop score matching VI, an iterative algorithm that seeks to match\nthe scores between the variational approximation and the exact posterior. At\neach iteration, score matching VI solves an inner optimization, one that\nminimally adjusts the current variational estimate to match the scores at a\nnewly sampled value of the latent variables. We show that when the variational\nfamily is a Gaussian, this inner optimization enjoys a closed form solution,\nwhich we call Gaussian score matching VI (GSM-VI). GSM-VI is also a ``black\nbox'' variational algorithm in that it only requires a differentiable joint\ndistribution, and as such it can be applied to a wide class of models. We\ncompare GSM-VI to black box variational inference (BBVI), which has similar\nrequirements but instead optimizes the ELBO. We study how GSM-VI behaves as a\nfunction of the problem dimensionality, the condition number of the target\ncovariance matrix (when the target is Gaussian), and the degree of mismatch\nbetween the approximating and exact posterior distribution. We also study\nGSM-VI on a collection of real-world Bayesian inference problems from the\nposteriorDB database of datasets and models. In all of our studies we find that\nGSM-VI is faster than BBVI, but without sacrificing accuracy. It requires\n10-100x fewer gradient evaluations to obtain a comparable quality of\napproximation.\n", "date": "2023-07-15", "categories": []}
{"id": "2307.06055", "title": "Function-Space Regularization for Deep Bayesian Classification", "abstract": "  Bayesian deep learning approaches assume model parameters to be latent random\nvariables and infer posterior distributions to quantify uncertainty, increase\nsafety and trust, and prevent overconfident and unpredictable behavior.\nHowever, weight-space priors are model-specific, can be difficult to interpret\nand are hard to specify. Instead, we apply a Dirichlet prior in predictive\nspace and perform approximate function-space variational inference. To this\nend, we interpret conventional categorical predictions from stochastic neural\nnetwork classifiers as samples from an implicit Dirichlet distribution. By\nadapting the inference, the same function-space prior can be combined with\ndifferent models without affecting model architecture or size. We illustrate\nthe flexibility and efficacy of such a prior with toy experiments and\ndemonstrate scalability, improved uncertainty quantification and adversarial\nrobustness with large-scale image classification experiments.\n", "date": "2023-07-12", "categories": []}
{"id": "2306.16015", "title": "BayesFlow: Amortized Bayesian Workflows With Neural Networks", "abstract": "  Modern Bayesian inference involves a mixture of computational techniques for\nestimating, validating, and drawing conclusions from probabilistic models as\npart of principled workflows for data analysis. Typical problems in Bayesian\nworkflows are the approximation of intractable posterior distributions for\ndiverse model types and the comparison of competing models of the same process\nin terms of their complexity and predictive performance. This manuscript\nintroduces the Python library BayesFlow for simulation-based training of\nestablished neural network architectures for amortized data compression and\ninference. Amortized Bayesian inference, as implemented in BayesFlow, enables\nusers to train custom neural networks on model simulations and re-use these\nnetworks for any subsequent application of the models. Since the trained\nnetworks can perform inference almost instantaneously, the upfront neural\nnetwork training is quickly amortized.\n", "date": "2023-06-28", "categories": []}
{"id": "2304.14251", "title": "Variational Bayes Made Easy", "abstract": "  Variational Bayes is a popular method for approximate inference but its\nderivation can be cumbersome. To simplify the process, we give a 3-step recipe\nto identify the posterior form by explicitly looking for linearity with respect\nto expectations of well-known distributions. We can then directly write the\nupdate by simply ``reading-off'' the terms in front of those expectations. The\nrecipe makes the derivation easier, faster, shorter, and more general.\n", "date": "2023-04-27", "categories": []}
{"id": "2210.04882", "title": "Layer Ensembles", "abstract": "  Deep Ensembles, as a type of Bayesian Neural Networks, can be used to\nestimate uncertainty on the prediction of multiple neural networks by\ncollecting votes from each network and computing the difference in those\npredictions. In this paper, we introduce a method for uncertainty estimation\nthat considers a set of independent categorical distributions for each layer of\nthe network, giving many more possible samples with overlapped layers than in\nthe regular Deep Ensembles. We further introduce an optimized inference\nprocedure that reuses common layer outputs, achieving up to 19x speed up and\nreducing memory usage quadratically. We also show that the method can be\nfurther improved by ranking samples, resulting in models that require less\nmemory and time to run while achieving higher uncertainty quality than Deep\nEnsembles.\n", "date": "2022-10-10", "categories": []}
{"id": "2302.03314", "title": "Federated Variational Inference Methods for Structured Latent Variable\n  Models", "abstract": "  Federated learning methods enable model training across distributed data\nsources without data leaving their original locations and have gained\nincreasing interest in various fields. However, existing approaches are\nlimited, excluding many structured probabilistic models. We present a general\nand elegant solution based on structured variational inference, widely used in\nBayesian machine learning, adapted for the federated setting. Additionally, we\nprovide a communication-efficient variant analogous to the canonical FedAvg\nalgorithm. The proposed algorithms' effectiveness is demonstrated, and their\nperformance is compared with hierarchical Bayesian neural networks and topic\nmodels.\n", "date": "2023-02-07", "categories": []}
{"id": "2307.01417", "title": "Free energy of Bayesian Convolutional Neural Network with Skip\n  Connection", "abstract": "  Since the success of Residual Network(ResNet), many of architectures of\nConvolutional Neural Networks(CNNs) have adopted skip connection. While the\ngeneralization performance of CNN with skip connection has been explained\nwithin the framework of Ensemble Learning, the dependency on the number of\nparameters have not been revealed. In this paper, we show that Bayesian free\nenergy of Convolutional Neural Network both with and without skip connection in\nBayesian learning. The upper bound of free energy of Bayesian CNN with skip\nconnection does not depend on the oveparametrization and, the generalization\nerror of Bayesian CNN has similar property.\n", "date": "2023-07-04", "categories": []}
{"id": "2306.17795", "title": "Hierarchical Bayesian Regression for Multi-Location Sales Transaction\n  Forecasting", "abstract": "  The features in many prediction models naturally take the form of a\nhierarchy. The lower levels represent individuals or events. These units group\nnaturally into locations and intervals or other aggregates, often at multiple\nlevels. Levels of groupings may intersect and join, much as relational database\ntables do. Besides representing the structure of the data, predictive features\nin hierarchical models can be assigned to their proper levels. Such models lend\nthemselves to hierarchical Bayes solution methods that ``share'' results of\ninference between groups by generalizing over the case of individual models for\neach group versus one model that aggregates all groups into one.\n  In this paper we show our work-in-progress applying a hierarchical Bayesian\nmodel to forecast purchases throughout the day at store franchises, with\ngroupings over locations and days of the week. We demonstrate using the\n\\textsf{stan} package on individual sales transaction data collected over the\ncourse of a year. We show how this solves the dilemma of having limited data\nand hence modest accuracy for each day and location, while being able to scale\nto a large number of locations with improved accuracy.\n", "date": "2023-06-30", "categories": []}
{"id": "2306.16923", "title": "NAUTILUS: boosting Bayesian importance nested sampling with deep\n  learning", "abstract": "  We introduce a novel approach to boost the efficiency of the importance\nnested sampling (INS) technique for Bayesian posterior and evidence estimation\nusing deep learning. Unlike rejection-based sampling methods such as vanilla\nnested sampling (NS) or Markov chain Monte Carlo (MCMC) algorithms, importance\nsampling techniques can use all likelihood evaluations for posterior and\nevidence estimation. However, for efficient importance sampling, one needs\nproposal distributions that closely mimic the posterior distributions. We show\nhow to combine INS with deep learning via neural network regression to\naccomplish this task. We also introduce NAUTILUS, a reference open-source\nPython implementation of this technique for Bayesian posterior and evidence\nestimation. We compare NAUTILUS against popular NS and MCMC packages, including\nEMCEE, DYNESTY, ULTRANEST and POCOMC, on a variety of challenging synthetic\nproblems and real-world applications in exoplanet detection, galaxy SED fitting\nand cosmology. In all applications, the sampling efficiency of NAUTILUS is\nsubstantially higher than that of all other samplers, often by more than an\norder of magnitude. Simultaneously, NAUTILUS delivers highly accurate results\nand needs fewer likelihood evaluations than all other samplers tested. We also\nshow that NAUTILUS has good scaling with the dimensionality of the likelihood\nand is easily parallelizable to many CPUs.\n", "date": "2023-06-29", "categories": []}
{"id": "2303.09519", "title": "PyVBMC: Efficient Bayesian inference in Python", "abstract": "  PyVBMC is a Python implementation of the Variational Bayesian Monte Carlo\n(VBMC) algorithm for posterior and model inference for black-box computational\nmodels (Acerbi, 2018, 2020). VBMC is an approximate inference method designed\nfor efficient parameter estimation and model assessment when model evaluations\nare mildly-to-very expensive (e.g., a second or more) and/or noisy.\nSpecifically, VBMC computes:\n  - a flexible (non-Gaussian) approximate posterior distribution of the model\nparameters, from which statistics and posterior samples can be easily\nextracted;\n  - an approximation of the model evidence or marginal likelihood, a metric\nused for Bayesian model selection.\n  PyVBMC can be applied to any computational or statistical model with up to\nroughly 10-15 continuous parameters, with the only requirement that the user\ncan provide a Python function that computes the target log likelihood of the\nmodel, or an approximation thereof (e.g., an estimate of the likelihood\nobtained via simulation or Monte Carlo methods). PyVBMC is particularly\neffective when the model takes more than about a second per evaluation, with\ndramatic speed-ups of 1-2 orders of magnitude when compared to traditional\napproximate inference methods.\n  Extensive benchmarks on both artificial test problems and a large number of\nreal models from the computational sciences, particularly computational and\ncognitive neuroscience, show that VBMC generally - and often vastly -\noutperforms alternative methods for sample-efficient Bayesian inference, and is\napplicable to both exact and simulator-based models (Acerbi, 2018, 2019, 2020).\nPyVBMC brings this state-of-the-art inference algorithm to Python, along with\nan easy-to-use Pythonic interface for running the algorithm and manipulating\nand visualizing its results.\n", "date": "2023-03-16", "categories": []}
{"id": "2305.19535", "title": "Low-rank extended Kalman filtering for online learning of neural\n  networks from streaming data", "abstract": "  We propose an efficient online approximate Bayesian inference algorithm for\nestimating the parameters of a nonlinear function from a potentially\nnon-stationary data stream. The method is based on the extended Kalman filter\n(EKF), but uses a novel low-rank plus diagonal decomposition of the posterior\nprecision matrix, which gives a cost per step which is linear in the number of\nmodel parameters. In contrast to methods based on stochastic variational\ninference, our method is fully deterministic, and does not require step-size\ntuning. We show experimentally that this results in much faster (more sample\nefficient) learning, which results in more rapid adaptation to changing\ndistributions, and faster accumulation of reward when used as part of a\ncontextual bandit algorithm.\n", "date": "2023-05-31", "categories": []}
{"id": "2210.12928", "title": "GFlowOut: Dropout with Generative Flow Networks", "abstract": "  Bayesian Inference offers principled tools to tackle many critical problems\nwith modern neural networks such as poor calibration and generalization, and\ndata inefficiency. However, scaling Bayesian inference to large architectures\nis challenging and requires restrictive approximations. Monte Carlo Dropout has\nbeen widely used as a relatively cheap way for approximate Inference and to\nestimate uncertainty with deep neural networks. Traditionally, the dropout mask\nis sampled independently from a fixed distribution. Recent works show that the\ndropout mask can be viewed as a latent variable, which can be inferred with\nvariational inference. These methods face two important challenges: (a) the\nposterior distribution over masks can be highly multi-modal which can be\ndifficult to approximate with standard variational inference and (b) it is not\ntrivial to fully utilize sample-dependent information and correlation among\ndropout masks to improve posterior estimation. In this work, we propose\nGFlowOut to address these issues. GFlowOut leverages the recently proposed\nprobabilistic framework of Generative Flow Networks (GFlowNets) to learn the\nposterior distribution over dropout masks. We empirically demonstrate that\nGFlowOut results in predictive distributions that generalize better to\nout-of-distribution data, and provide uncertainty estimates which lead to\nbetter performance in downstream tasks.\n", "date": "2022-10-24", "categories": []}
{"id": "2301.03991", "title": "Constraining cosmological parameters from N-body simulations with\n  Variational Bayesian Neural Networks", "abstract": "  Methods based on Deep Learning have recently been applied on astrophysical\nparameter recovery thanks to their ability to capture information from complex\ndata. One of these methods is the approximate Bayesian Neural Networks (BNNs)\nwhich have demonstrated to yield consistent posterior distribution into the\nparameter space, helpful for uncertainty quantification. However, as any modern\nneural networks, they tend to produce overly confident uncertainty estimates\nand can introduce bias when BNNs are applied to data. In this work, we\nimplement multiplicative normalizing flows (MNFs), a family of approximate\nposteriors for the parameters of BNNs with the purpose of enhancing the\nflexibility of the variational posterior distribution, to extract $\\Omega_m$,\n$h$, and $\\sigma_8$ from the QUIJOTE simulations. We have compared this method\nwith respect to the standard BNNs, and the flipout estimator. We found that\nMNFs combined with BNNs outperform the other models obtaining predictive\nperformance with almost one order of magnitude larger that standard BNNs,\n$\\sigma_8$ extracted with high accuracy ($r^2=0.99$), and precise uncertainty\nestimates. The latter implies that MNFs provide more realistic predictive\ndistribution closer to the true posterior mitigating the bias introduced by the\nvariational approximation and allowing to work with well-calibrated networks.\n", "date": "2023-01-09", "categories": []}
{"id": "2302.09125", "title": "JANA: Jointly Amortized Neural Approximation of Complex Bayesian Models", "abstract": "  This work proposes ``jointly amortized neural approximation'' (JANA) of\nintractable likelihood functions and posterior densities arising in Bayesian\nsurrogate modeling and simulation-based inference. We train three complementary\nnetworks in an end-to-end fashion: 1) a summary network to compress individual\ndata points, sets, or time series into informative embedding vectors; 2) a\nposterior network to learn an amortized approximate posterior; and 3) a\nlikelihood network to learn an amortized approximate likelihood. Their\ninteraction opens a new route to amortized marginal likelihood and posterior\npredictive estimation -- two important ingredients of Bayesian workflows that\nare often too expensive for standard methods. We benchmark the fidelity of JANA\non a variety of simulation models against state-of-the-art Bayesian methods and\npropose a powerful and interpretable diagnostic for joint calibration. In\naddition, we investigate the ability of recurrent likelihood networks to\nemulate complex time series models without resorting to hand-crafted summary\nstatistics.\n", "date": "2023-02-17", "categories": []}
{"id": "2306.09702", "title": "A Hierarchical Bayesian Model for Deep Few-Shot Meta Learning", "abstract": "  We propose a novel hierarchical Bayesian model for learning with a large\n(possibly infinite) number of tasks/episodes, which suits well the few-shot\nmeta learning problem. We consider episode-wise random variables to model\nepisode-specific target generative processes, where these local random\nvariables are governed by a higher-level global random variate. The global\nvariable helps memorize the important information from historic episodes while\ncontrolling how much the model needs to be adapted to new episodes in a\nprincipled Bayesian manner. Within our model framework, the prediction on a\nnovel episode/task can be seen as a Bayesian inference problem. However, a main\nobstacle in learning with a large/infinite number of local random variables in\nonline nature, is that one is not allowed to store the posterior distribution\nof the current local random variable for frequent future updates, typical in\nconventional variational inference. We need to be able to treat each local\nvariable as a one-time iterate in the optimization. We propose a\nNormal-Inverse-Wishart model, for which we show that this one-time iterate\noptimization becomes feasible due to the approximate closed-form solutions for\nthe local posterior distributions. The resulting algorithm is more attractive\nthan the MAML in that it is not required to maintain computational graphs for\nthe whole gradient optimization steps per episode. Our approach is also\ndifferent from existing Bayesian meta learning methods in that unlike dealing\nwith a single random variable for the whole episodes, our approach has a\nhierarchical structure that allows one-time episodic optimization, desirable\nfor principled Bayesian learning with many/infinite tasks. The code is\navailable at \\url{https://github.com/minyoungkim21/niwmeta}.\n", "date": "2023-06-16", "categories": []}
{"id": "2306.08352", "title": "Bayesian Non-linear Latent Variable Modeling via Random Fourier Features", "abstract": "  The Gaussian process latent variable model (GPLVM) is a popular probabilistic\nmethod used for nonlinear dimension reduction, matrix factorization, and\nstate-space modeling. Inference for GPLVMs is computationally tractable only\nwhen the data likelihood is Gaussian. Moreover, inference for GPLVMs has\ntypically been restricted to obtaining maximum a posteriori point estimates,\nwhich can lead to overfitting, or variational approximations, which\nmischaracterize the posterior uncertainty. Here, we present a method to perform\nMarkov chain Monte Carlo (MCMC) inference for generalized Bayesian nonlinear\nlatent variable modeling. The crucial insight necessary to generalize GPLVMs to\narbitrary observation models is that we approximate the kernel function in the\nGaussian process mappings with random Fourier features; this allows us to\ncompute the gradient of the posterior in closed form with respect to the latent\nvariables. We show that we can generalize GPLVMs to non-Gaussian observations,\nsuch as Poisson, negative binomial, and multinomial distributions, using our\nrandom feature latent variable model (RFLVM). Our generalized RFLVMs perform on\npar with state-of-the-art latent variable models on a wide range of\napplications, including motion capture, images, and text data for the purpose\nof estimating the latent structure and imputing the missing data of these\ncomplex data sets.\n", "date": "2023-06-14", "categories": []}
{"id": "2306.07060", "title": "Prediction Algorithms Achieving Bayesian Decision Theoretical Optimality\n  Based on Decision Trees as Data Observation Processes", "abstract": "  In the field of decision trees, most previous studies have difficulty\nensuring the statistical optimality of a prediction of new data and suffer from\noverfitting because trees are usually used only to represent prediction\nfunctions to be constructed from given data. In contrast, some studies,\nincluding this paper, used the trees to represent stochastic data observation\nprocesses behind given data. Moreover, they derived the statistically optimal\nprediction, which is robust against overfitting, based on the Bayesian decision\ntheory by assuming a prior distribution for the trees. However, these studies\nstill have a problem in computing this Bayes optimal prediction because it\ninvolves an infeasible summation for all division patterns of a feature space,\nwhich is represented by the trees and some parameters. In particular, an open\nproblem is a summation with respect to combinations of division axes, i.e., the\nassignment of features to inner nodes of the tree. We solve this by a Markov\nchain Monte Carlo method, whose step size is adaptively tuned according to a\nposterior distribution for the trees.\n", "date": "2023-06-12", "categories": []}
{"id": "2306.07158", "title": "Riemannian Laplace approximations for Bayesian neural networks", "abstract": "  Bayesian neural networks often approximate the weight-posterior with a\nGaussian distribution. However, practical posteriors are often, even locally,\nhighly non-Gaussian, and empirical performance deteriorates. We propose a\nsimple parametric approximate posterior that adapts to the shape of the true\nposterior through a Riemannian metric that is determined by the log-posterior\ngradient. We develop a Riemannian Laplace approximation where samples naturally\nfall into weight-regions with low negative log-posterior. We show that these\nsamples can be drawn by solving a system of ordinary differential equations,\nwhich can be done efficiently by leveraging the structure of the Riemannian\nmetric and automatic differentiation. Empirically, we demonstrate that our\napproach consistently improves over the conventional Laplace approximation\nacross tasks. We further show that, unlike the conventional Laplace\napproximation, our method is not overly sensitive to the choice of prior, which\nalleviates a practical pitfall of current approaches.\n", "date": "2023-06-12", "categories": []}
{"id": "2306.03968", "title": "Stochastic Marginal Likelihood Gradients using Neural Tangent Kernels", "abstract": "  Selecting hyperparameters in deep learning greatly impacts its effectiveness\nbut requires manual effort and expertise. Recent works show that Bayesian model\nselection with Laplace approximations can allow to optimize such\nhyperparameters just like standard neural network parameters using gradients\nand on the training data. However, estimating a single hyperparameter gradient\nrequires a pass through the entire dataset, limiting the scalability of such\nalgorithms. In this work, we overcome this issue by introducing lower bounds to\nthe linearized Laplace approximation of the marginal likelihood. In contrast to\nprevious estimators, these bounds are amenable to stochastic-gradient-based\noptimization and allow to trade off estimation accuracy against computational\ncomplexity. We derive them using the function-space form of the linearized\nLaplace, which can be estimated using the neural tangent kernel.\nExperimentally, we show that the estimators can significantly accelerate\ngradient-based hyperparameter optimization.\n", "date": "2023-06-06", "categories": []}
{"id": "2211.06714", "title": "Bayesian Learning of Coupled Biogeochemical-Physical Models", "abstract": "  Predictive dynamical models for marine ecosystems are used for a variety of\nneeds. Due to sparse measurements and limited understanding of the myriad of\nocean processes, there is however significant uncertainty. There is model\nuncertainty in the parameter values, functional forms with diverse\nparameterizations, level of complexity needed, and thus in the state fields. We\ndevelop a Bayesian model learning methodology that allows interpolation in the\nspace of candidate models and discovery of new models from noisy, sparse, and\nindirect observations, all while estimating state fields and parameter values,\nas well as the joint PDFs of all learned quantities. We address the challenges\nof high-dimensional and multidisciplinary dynamics governed by PDEs by using\nstate augmentation and the computationally efficient GMM-DO filter. Our\ninnovations include stochastic formulation and complexity parameters to unify\ncandidate models into a single general model as well as stochastic expansion\nparameters within piecewise function approximations to generate dense candidate\nmodel spaces. These innovations allow handling many compatible and embedded\ncandidate models, possibly none of which are accurate, and learning elusive\nunknown functional forms. Our new methodology is generalizable, interpretable,\nand extrapolates out of the space of models to discover new ones. We perform a\nseries of twin experiments based on flows past a ridge coupled with\nthree-to-five component ecosystem models, including flows with chaotic\nadvection. The probabilities of known, uncertain, and unknown model\nformulations, and of state fields and parameters, are updated jointly using\nBayes' law. Non-Gaussian statistics, ambiguity, and biases are captured. The\nparameter values and model formulations that best explain the data are\nidentified. When observations are sufficiently informative, model complexity\nand functions are discovered.\n", "date": "2022-11-12", "categories": []}
{"id": "2302.00564", "title": "Automatically Marginalized MCMC in Probabilistic Programming", "abstract": "  Hamiltonian Monte Carlo (HMC) is a powerful algorithm to sample latent\nvariables from Bayesian models. The advent of probabilistic programming\nlanguages (PPLs) frees users from writing inference algorithms and lets users\nfocus on modeling. However, many models are difficult for HMC to solve\ndirectly, and often require tricks like model reparameterization. We are\nmotivated by the fact that many of those models could be simplified by\nmarginalization. We propose to use automatic marginalization as part of the\nsampling process using HMC in a graphical model extracted from a PPL, which\nsubstantially improves sampling from real-world hierarchical models.\n", "date": "2023-02-01", "categories": []}
{"id": "2301.11294", "title": "Coin Sampling: Gradient-Based Bayesian Inference without Learning Rates", "abstract": "  In recent years, particle-based variational inference (ParVI) methods such as\nStein variational gradient descent (SVGD) have grown in popularity as scalable\nmethods for Bayesian inference. Unfortunately, the properties of such methods\ninvariably depend on hyperparameters such as the learning rate, which must be\ncarefully tuned by the practitioner in order to ensure convergence to the\ntarget measure at a suitable rate. In this paper, we introduce a suite of new\nparticle-based methods for scalable Bayesian inference based on coin betting,\nwhich are entirely learning-rate free. We illustrate the performance of our\napproach on a range of numerical examples, including several high-dimensional\nmodels and datasets, demonstrating comparable performance to other ParVI\nalgorithms with no need to tune a learning rate.\n", "date": "2023-01-26", "categories": []}
{"id": "2306.00361", "title": "Sharded Bayesian Additive Regression Trees", "abstract": "  In this paper we develop the randomized Sharded Bayesian Additive Regression\nTrees (SBT) model. We introduce a randomization auxiliary variable and a\nsharding tree to decide partitioning of data, and fit each partition component\nto a sub-model using Bayesian Additive Regression Tree (BART). By observing\nthat the optimal design of a sharding tree can determine optimal sharding for\nsub-models on a product space, we introduce an intersection tree structure to\ncompletely specify both the sharding and modeling using only tree structures.\nIn addition to experiments, we also derive the theoretical optimal weights for\nminimizing posterior contractions and prove the worst-case complexity of SBT.\n", "date": "2023-06-01", "categories": []}
{"id": "2305.19350", "title": "Non-convex Bayesian Learning via Stochastic Gradient Markov Chain Monte\n  Carlo", "abstract": "  The rise of artificial intelligence (AI) hinges on the efficient training of\nmodern deep neural networks (DNNs) for non-convex optimization and uncertainty\nquantification, which boils down to a non-convex Bayesian learning problem. A\nstandard tool to handle the problem is Langevin Monte Carlo, which proposes to\napproximate the posterior distribution with theoretical guarantees. In this\nthesis, we start with the replica exchange Langevin Monte Carlo (also known as\nparallel tempering), which proposes appropriate swaps between exploration and\nexploitation to achieve accelerations. However, the na\\\"ive extension of swaps\nto big data problems leads to a large bias, and bias-corrected swaps are\nrequired. Such a mechanism leads to few effective swaps and insignificant\naccelerations. To alleviate this issue, we first propose a control variates\nmethod to reduce the variance of noisy energy estimators and show a potential\nto accelerate the exponential convergence. We also present the population-chain\nreplica exchange based on non-reversibility and obtain an optimal round-trip\nrate for deep learning. In the second part of the thesis, we study scalable\ndynamic importance sampling algorithms based on stochastic approximation.\nTraditional dynamic importance sampling algorithms have achieved success,\nhowever, the lack of scalability has greatly limited their extensions to big\ndata. To handle this scalability issue, we resolve the vanishing gradient\nproblem and propose two dynamic importance sampling algorithms. Theoretically,\nwe establish the stability condition for the underlying ordinary differential\nequation (ODE) system and guarantee the asymptotic convergence of the latent\nvariable to the desired fixed point. Interestingly, such a result still holds\ngiven non-convex energy landscapes.\n", "date": "2023-05-30", "categories": []}
{"id": "2305.18488", "title": "A Bayesian sparse factor model with adaptive posterior concentration", "abstract": "  In this paper, we propose a new Bayesian inference method for a\nhigh-dimensional sparse factor model that allows both the factor dimensionality\nand the sparse structure of the loading matrix to be inferred. The novelty is\nto introduce a certain dependence between the sparsity level and the factor\ndimensionality, which leads to adaptive posterior concentration while keeping\ncomputational tractability. We show that the posterior distribution\nasymptotically concentrates on the true factor dimensionality, and more\nimportantly, this posterior consistency is adaptive to the sparsity level of\nthe true loading matrix and the noise variance. We also prove that the proposed\nBayesian model attains the optimal detection rate of the factor dimensionality\nin a more general situation than those found in the literature. Moreover, we\nobtain a near-optimal posterior concentration rate of the covariance matrix.\nNumerical studies are conducted and show the superiority of the proposed method\ncompared with other competitors.\n", "date": "2023-05-29", "categories": []}
{"id": "2305.18774", "title": "Bayesian Decision Trees Inspired from Evolutionary Algorithms", "abstract": "  Bayesian Decision Trees (DTs) are generally considered a more advanced and\naccurate model than a regular Decision Tree (DT) because they can handle\ncomplex and uncertain data. Existing work on Bayesian DTs uses Markov Chain\nMonte Carlo (MCMC) with an accept-reject mechanism and sample using naive\nproposals to proceed to the next iteration, which can be slow because of the\nburn-in time needed. We can reduce the burn-in period by proposing a more\nsophisticated way of sampling or by designing a different numerical Bayesian\napproach. In this paper, we propose a replacement of the MCMC with an\ninherently parallel algorithm, the Sequential Monte Carlo (SMC), and a more\neffective sampling strategy inspired by the Evolutionary Algorithms (EA).\nExperiments show that SMC combined with the EA can produce more accurate\nresults compared to MCMC in 100 times fewer iterations.\n", "date": "2023-05-30", "categories": []}
{"id": "2301.13303", "title": "Variational sparse inverse Cholesky approximation for latent Gaussian\n  processes via double Kullback-Leibler minimization", "abstract": "  To achieve scalable and accurate inference for latent Gaussian processes, we\npropose a variational approximation based on a family of Gaussian distributions\nwhose covariance matrices have sparse inverse Cholesky (SIC) factors. We\ncombine this variational approximation of the posterior with a similar and\nefficient SIC-restricted Kullback-Leibler-optimal approximation of the prior.\nWe then focus on a particular SIC ordering and nearest-neighbor-based sparsity\npattern resulting in highly accurate prior and posterior approximations. For\nthis setting, our variational approximation can be computed via stochastic\ngradient descent in polylogarithmic time per iteration. We provide numerical\ncomparisons showing that the proposed double-Kullback-Leibler-optimal\nGaussian-process approximation (DKLGP) can sometimes be vastly more accurate\nfor stationary kernels than alternative approaches such as inducing-point and\nmean-field approximations at similar computational complexity.\n", "date": "2023-01-30", "categories": []}
{"id": "2305.14454", "title": "An Improved Variational Approximate Posterior for the Deep Wishart\n  Process", "abstract": "  Deep kernel processes are a recently introduced class of deep Bayesian models\nthat have the flexibility of neural networks, but work entirely with Gram\nmatrices. They operate by alternately sampling a Gram matrix from a\ndistribution over positive semi-definite matrices, and applying a deterministic\ntransformation. When the distribution is chosen to be Wishart, the model is\ncalled a deep Wishart process (DWP). This particular model is of interest\nbecause its prior is equivalent to a deep Gaussian process (DGP) prior, but at\nthe same time it is invariant to rotational symmetries, leading to a simpler\nposterior distribution. Practical inference in the DWP was made possible in\nrecent work (\"A variational approximate posterior for the deep Wishart process\"\nOber and Aitchison 2021a) where the authors used a generalisation of the\nBartlett decomposition of the Wishart distribution as the variational\napproximate posterior. However, predictive performance in that paper was less\nimpressive than one might expect, with the DWP only beating a DGP on a few of\nthe UCI datasets used for comparison. In this paper, we show that further\ngeneralising their distribution to allow linear combinations of rows and\ncolumns in the Bartlett decomposition results in better predictive performance,\nwhile incurring negligible additional computation cost.\n", "date": "2023-05-23", "categories": []}
{"id": "2305.14765", "title": "Masked Bayesian Neural Networks : Theoretical Guarantee and its\n  Posterior Inference", "abstract": "  Bayesian approaches for learning deep neural networks (BNN) have been\nreceived much attention and successfully applied to various applications.\nParticularly, BNNs have the merit of having better generalization ability as\nwell as better uncertainty quantification. For the success of BNN, search an\nappropriate architecture of the neural networks is an important task, and\nvarious algorithms to find good sparse neural networks have been proposed. In\nthis paper, we propose a new node-sparse BNN model which has good theoretical\nproperties and is computationally feasible. We prove that the posterior\nconcentration rate to the true model is near minimax optimal and adaptive to\nthe smoothness of the true model. In particular the adaptiveness is the first\nof its kind for node-sparse BNNs. In addition, we develop a novel MCMC\nalgorithm which makes the Bayesian inference of the node-sparse BNN model\nfeasible in practice.\n", "date": "2023-05-24", "categories": []}
{"id": "2305.15340", "title": "Bayesian calibration of differentiable agent-based models", "abstract": "  Agent-based modelling (ABMing) is a powerful and intuitive approach to\nmodelling complex systems; however, the intractability of ABMs' likelihood\nfunctions and the non-differentiability of the mathematical operations\ncomprising these models present a challenge to their use in the real world.\nThese difficulties have in turn generated research on approximate Bayesian\ninference methods for ABMs and on constructing differentiable approximations to\narbitrary ABMs, but little work has been directed towards designing approximate\nBayesian inference techniques for the specific case of differentiable ABMs. In\nthis work, we aim to address this gap and discuss how generalised variational\ninference procedures may be employed to provide misspecification-robust\nBayesian parameter inferences for differentiable ABMs. We demonstrate with\nexperiments on a differentiable ABM of the COVID-19 pandemic that our approach\ncan result in accurate inferences, and discuss avenues for future work.\n", "date": "2023-05-24", "categories": []}
{"id": "2206.00853", "title": "Masked Bayesian Neural Networks : Computation and Optimality", "abstract": "  As data size and computing power increase, the architectures of deep neural\nnetworks (DNNs) have been getting more complex and huge, and thus there is a\ngrowing need to simplify such complex and huge DNNs. In this paper, we propose\na novel sparse Bayesian neural network (BNN) which searches a good DNN with an\nappropriate complexity. We employ the masking variables at each node which can\nturn off some nodes according to the posterior distribution to yield a nodewise\nsparse DNN. We devise a prior distribution such that the posterior distribution\nhas theoretical optimalities (i.e. minimax optimality and adaptiveness), and\ndevelop an efficient MCMC algorithm. By analyzing several benchmark datasets,\nwe illustrate that the proposed BNN performs well compared to other existing\nmethods in the sense that it discovers well condensed DNN architectures with\nsimilar prediction accuracy and uncertainty quantification compared to large\nDNNs.\n", "date": "2022-06-02", "categories": []}
{"id": "2305.11022", "title": "Massively Parallel Reweighted Wake-Sleep", "abstract": "  Reweighted wake-sleep (RWS) is a machine learning method for performing\nBayesian inference in a very general class of models. RWS draws $K$ samples\nfrom an underlying approximate posterior, then uses importance weighting to\nprovide a better estimate of the true posterior. RWS then updates its\napproximate posterior towards the importance-weighted estimate of the true\nposterior. However, recent work [Chattergee and Diaconis, 2018] indicates that\nthe number of samples required for effective importance weighting is\nexponential in the number of latent variables. Attaining such a large number of\nimportance samples is intractable in all but the smallest models. Here, we\ndevelop massively parallel RWS, which circumvents this issue by drawing $K$\nsamples of all $n$ latent variables, and individually reasoning about all $K^n$\npossible combinations of samples. While reasoning about $K^n$ combinations\nmight seem intractable, the required computations can be performed in\npolynomial time by exploiting conditional independencies in the generative\nmodel. We show considerable improvements over standard \"global\" RWS, which\ndraws $K$ samples from the full joint.\n", "date": "2023-05-18", "categories": []}
{"id": "2211.17158", "title": "Proximal Residual Flows for Bayesian Inverse Problems", "abstract": "  Normalizing flows are a powerful tool for generative modelling, density\nestimation and posterior reconstruction in Bayesian inverse problems. In this\npaper, we introduce proximal residual flows, a new architecture of normalizing\nflows. Based on the fact, that proximal neural networks are by definition\naveraged operators, we ensure invertibility of certain residual blocks.\nMoreover, we extend the architecture to conditional proximal residual flows for\nposterior reconstruction within Bayesian inverse problems. We demonstrate the\nperformance of proximal residual flows on numerical examples.\n", "date": "2022-11-30", "categories": []}
{"id": "2303.13930", "title": "Particle Mean Field Variational Bayes", "abstract": "  The Mean Field Variational Bayes (MFVB) method is one of the most\ncomputationally efficient techniques for Bayesian inference. However, its use\nhas been restricted to models with conjugate priors or those that require\nanalytical calculations. This paper proposes a novel particle-based MFVB\napproach that greatly expands the applicability of the MFVB method. We\nestablish the theoretical basis of the new method by leveraging the connection\nbetween Wasserstein gradient flows and Langevin diffusion dynamics, and\ndemonstrate the effectiveness of this approach using Bayesian logistic\nregression, stochastic volatility, and deep neural networks.\n", "date": "2023-03-24", "categories": []}
{"id": "2212.14457", "title": "Bayesian Interpolation with Deep Linear Networks", "abstract": "  Characterizing how neural network depth, width, and dataset size jointly\nimpact model quality is a central problem in deep learning theory. We give here\na complete solution in the special case of linear networks with output\ndimension one trained using zero noise Bayesian inference with Gaussian weight\npriors and mean squared error as a negative log-likelihood. For any training\ndataset, network depth, and hidden layer widths, we find non-asymptotic\nexpressions for the predictive posterior and Bayesian model evidence in terms\nof Meijer-G functions, a class of meromorphic special functions of a single\ncomplex variable. Through novel asymptotic expansions of these Meijer-G\nfunctions, a rich new picture of the joint role of depth, width, and dataset\nsize emerges. We show that linear networks make provably optimal predictions at\ninfinite depth: the posterior of infinitely deep linear networks with\ndata-agnostic priors is the same as that of shallow networks with\nevidence-maximizing data-dependent priors. This yields a principled reason to\nprefer deeper networks when priors are forced to be data-agnostic. Moreover, we\nshow that with data-agnostic priors, Bayesian model evidence in wide linear\nnetworks is maximized at infinite depth, elucidating the salutary role of\nincreased depth for model selection. Underpinning our results is a novel\nemergent notion of effective depth, given by the number of hidden layers times\nthe number of data points divided by the network width; this determines the\nstructure of the posterior in the large-data limit.\n", "date": "2022-12-29", "categories": []}
{"id": "2305.08733", "title": "Refining Amortized Posterior Approximations using Gradient-Based Summary\n  Statistics", "abstract": "  We present an iterative framework to improve the amortized approximations of\nposterior distributions in the context of Bayesian inverse problems, which is\ninspired by loop-unrolled gradient descent methods and is theoretically\ngrounded in maximally informative summary statistics. Amortized variational\ninference is restricted by the expressive power of the chosen variational\ndistribution and the availability of training data in the form of joint data\nand parameter samples, which often lead to approximation errors such as the\namortization gap. To address this issue, we propose an iterative framework that\nrefines the current amortized posterior approximation at each step. Our\napproach involves alternating between two steps: (1) constructing a training\ndataset consisting of pairs of summarized data residuals and parameters, where\nthe summarized data residual is generated using a gradient-based summary\nstatistic, and (2) training a conditional generative model -- a normalizing\nflow in our examples -- on this dataset to obtain a probabilistic update of the\nunknown parameter. This procedure leads to iterative refinement of the\namortized posterior approximations without the need for extra training data. We\nvalidate our method in a controlled setting by applying it to a stylized\nproblem, and observe improved posterior approximations with each iteration.\nAdditionally, we showcase the capability of our method in tackling\nrealistically sized problems by applying it to transcranial ultrasound, a\nhigh-dimensional, nonlinear inverse problem governed by wave physics, and\nobserve enhanced posterior quality through better image reconstruction with the\nposterior mean.\n", "date": "2023-05-15", "categories": []}
{"id": "2302.04759", "title": "Robust and Scalable Bayesian Online Changepoint Detection", "abstract": "  This paper proposes an online, provably robust, and scalable Bayesian\napproach for changepoint detection. The resulting algorithm has key advantages\nover previous work: it provides provable robustness by leveraging the\ngeneralised Bayesian perspective, and also addresses the scalability issues of\nprevious attempts. Specifically, the proposed generalised Bayesian formalism\nleads to conjugate posteriors whose parameters are available in closed form by\nleveraging diffusion score matching. The resulting algorithm is exact, can be\nupdated through simple algebra, and is more than 10 times faster than its\nclosest competitor.\n", "date": "2023-02-09", "categories": []}
{"id": "2305.07334", "title": "Locking and Quacking: Stacking Bayesian model predictions by log-pooling\n  and superposition", "abstract": "  Combining predictions from different models is a central problem in Bayesian\ninference and machine learning more broadly. Currently, these predictive\ndistributions are almost exclusively combined using linear mixtures such as\nBayesian model averaging, Bayesian stacking, and mixture of experts. Such\nlinear mixtures impose idiosyncrasies that might be undesirable for some\napplications, such as multi-modality. While there exist alternative strategies\n(e.g. geometric bridge or superposition), optimising their parameters usually\ninvolves computing an intractable normalising constant repeatedly. We present\ntwo novel Bayesian model combination tools. These are generalisations of model\nstacking, but combine posterior densities by log-linear pooling (locking) and\nquantum superposition (quacking). To optimise model weights while avoiding the\nburden of normalising constants, we investigate the Hyvarinen score of the\ncombined posterior predictions. We demonstrate locking with an illustrative\nexample and discuss its practical application with importance sampling.\n", "date": "2023-05-12", "categories": []}
{"id": "2305.03395", "title": "Sparsifying Bayesian neural networks with latent binary variables and\n  normalizing flows", "abstract": "  Artificial neural networks (ANNs) are powerful machine learning methods used\nin many modern applications such as facial recognition, machine translation,\nand cancer diagnostics. A common issue with ANNs is that they usually have\nmillions or billions of trainable parameters, and therefore tend to overfit to\nthe training data. This is especially problematic in applications where it is\nimportant to have reliable uncertainty estimates. Bayesian neural networks\n(BNN) can improve on this, since they incorporate parameter uncertainty. In\naddition, latent binary Bayesian neural networks (LBBNN) also take into account\nstructural uncertainty by allowing the weights to be turned on or off, enabling\ninference in the joint space of weights and structures. In this paper, we will\nconsider two extensions to the LBBNN method: Firstly, by using the local\nreparametrization trick (LRT) to sample the hidden units directly, we get a\nmore computationally efficient algorithm. More importantly, by using\nnormalizing flows on the variational posterior distribution of the LBBNN\nparameters, the network learns a more flexible variational posterior\ndistribution than the mean field Gaussian. Experimental results show that this\nimproves predictive power compared to the LBBNN method, while also obtaining\nmore sparse networks. We perform two simulation studies. In the first study, we\nconsider variable selection in a logistic regression setting, where the more\nflexible variational distribution leads to improved results. In the second\nstudy, we compare predictive uncertainty based on data generated from\ntwo-dimensional Gaussian distributions. Here, we argue that our Bayesian\nmethods lead to more realistic estimates of predictive uncertainty.\n", "date": "2023-05-05", "categories": []}
{"id": "2206.08201", "title": "Learning Physics between Digital Twins with Low-Fidelity Models and\n  Physics-Informed Gaussian Processes", "abstract": "  A digital twin is a computer model that represents an individual, for\nexample, a component, a patient or a process. In many situations, we want to\ngain knowledge about an individual from its data while incorporating imperfect\nphysical knowledge and also learn from data from other individuals. In this\npaper, we introduce a fully Bayesian methodology for learning between digital\ntwins in a setting where the physical parameters of each individual are of\ninterest. A model discrepancy term is incorporated in the model formulation of\neach personalized model to account for the missing physics of the low-fidelity\nmodel. To allow sharing of information between individuals, we introduce a\nBayesian Hierarchical modelling framework where the individual models are\nconnected through a new level in the hierarchy. Our methodology is demonstrated\nin two case studies, a toy example previously used in the literature extended\nto more individuals and a cardiovascular model relevant for the treatment of\nHypertension. The case studies show that 1) models not accounting for imperfect\nphysical models are biased and over-confident, 2) the models accounting for\nimperfect physical models are more uncertain but cover the truth, 3) the models\nlearning between digital twins have less uncertainty than the corresponding\nindependent individual models, but are not over-confident.\n", "date": "2022-06-16", "categories": []}
{"id": "2305.00934", "title": "Variational Inference for Bayesian Neural Networks under Model and\n  Parameter Uncertainty", "abstract": "  Bayesian neural networks (BNNs) have recently regained a significant amount\nof attention in the deep learning community due to the development of scalable\napproximate Bayesian inference techniques. There are several advantages of\nusing a Bayesian approach: Parameter and prediction uncertainties become easily\navailable, facilitating rigorous statistical analysis. Furthermore, prior\nknowledge can be incorporated. However, so far, there have been no scalable\ntechniques capable of combining both structural and parameter uncertainty. In\nthis paper, we apply the concept of model uncertainty as a framework for\nstructural learning in BNNs and hence make inference in the joint space of\nstructures/models and parameters. Moreover, we suggest an adaptation of a\nscalable variational inference approach with reparametrization of marginal\ninclusion probabilities to incorporate the model space constraints.\nExperimental results on a range of benchmark datasets show that we obtain\ncomparable accuracy results with the competing models, but based on methods\nthat are much more sparse than ordinary BNNs.\n", "date": "2023-05-01", "categories": []}
{"id": "2304.13267", "title": "Bayesian Federated Learning: A Survey", "abstract": "  Federated learning (FL) demonstrates its advantages in integrating\ndistributed infrastructure, communication, computing and learning in a\nprivacy-preserving manner. However, the robustness and capabilities of existing\nFL methods are challenged by limited and dynamic data and conditions,\ncomplexities including heterogeneities and uncertainties, and analytical\nexplainability. Bayesian federated learning (BFL) has emerged as a promising\napproach to address these issues. This survey presents a critical overview of\nBFL, including its basic concepts, its relations to Bayesian learning in the\ncontext of FL, and a taxonomy of BFL from both Bayesian and federated\nperspectives. We categorize and discuss client- and server-side and FL-based\nBFL methods and their pros and cons. The limitations of the existing BFL\nmethods and the future directions of BFL research further address the intricate\nrequirements of real-life FL applications.\n", "date": "2023-04-26", "categories": []}
{"id": "2304.11251", "title": "Machine Learning and the Future of Bayesian Computation", "abstract": "  Bayesian models are a powerful tool for studying complex data, allowing the\nanalyst to encode rich hierarchical dependencies and leverage prior\ninformation. Most importantly, they facilitate a complete characterization of\nuncertainty through the posterior distribution. Practical posterior computation\nis commonly performed via MCMC, which can be computationally infeasible for\nhigh dimensional models with many observations. In this article we discuss the\npotential to improve posterior computation using ideas from machine learning.\nConcrete future directions are explored in vignettes on normalizing flows,\nBayesian coresets, distributed Bayesian inference, and variational inference.\n", "date": "2023-04-21", "categories": []}
{"id": "2304.11134", "title": "Plug-and-Play split Gibbs sampler: embedding deep generative priors in\n  Bayesian inference", "abstract": "  This paper introduces a stochastic plug-and-play (PnP) sampling algorithm\nthat leverages variable splitting to efficiently sample from a posterior\ndistribution. The algorithm based on split Gibbs sampling (SGS) draws\ninspiration from the alternating direction method of multipliers (ADMM). It\ndivides the challenging task of posterior sampling into two simpler sampling\nproblems. The first problem depends on the likelihood function, while the\nsecond is interpreted as a Bayesian denoising problem that can be readily\ncarried out by a deep generative model. Specifically, for an illustrative\npurpose, the proposed method is implemented in this paper using\nstate-of-the-art diffusion-based generative models. Akin to its deterministic\nPnP-based counterparts, the proposed method exhibits the great advantage of not\nrequiring an explicit choice of the prior distribution, which is rather encoded\ninto a pre-trained generative model. However, unlike optimization methods\n(e.g., PnP-ADMM) which generally provide only point estimates, the proposed\napproach allows conventional Bayesian estimators to be accompanied by\nconfidence intervals at a reasonable additional computational cost. Experiments\non commonly studied image processing problems illustrate the efficiency of the\nproposed sampling strategy. Its performance is compared to recent\nstate-of-the-art optimization and sampling methods.\n", "date": "2023-04-21", "categories": []}
{"id": "2304.04455", "title": "Bayesian optimization for sparse neural networks with trainable\n  activation functions", "abstract": "  In the literature on deep neural networks, there is considerable interest in\ndeveloping activation functions that can enhance neural network performance. In\nrecent years, there has been renewed scientific interest in proposing\nactivation functions that can be trained throughout the learning process, as\nthey appear to improve network performance, especially by reducing overfitting.\nIn this paper, we propose a trainable activation function whose parameters need\nto be estimated. A fully Bayesian model is developed to automatically estimate\nfrom the learning data both the model weights and activation function\nparameters. An MCMC-based optimization scheme is developed to build the\ninference. The proposed method aims to solve the aforementioned problems and\nimprove convergence time by using an efficient sampling scheme that guarantees\nconvergence to the global maximum. The proposed scheme is tested on three\ndatasets with three different CNNs. Promising results demonstrate the\nusefulness of our proposed approach in improving model accuracy due to the\nproposed activation function and Bayesian estimation of the parameters.\n", "date": "2023-04-10", "categories": []}
{"id": "2304.08151", "title": "Prediction-Oriented Bayesian Active Learning", "abstract": "  Information-theoretic approaches to active learning have traditionally\nfocused on maximising the information gathered about the model parameters, most\ncommonly by optimising the BALD score. We highlight that this can be suboptimal\nfrom the perspective of predictive performance. For example, BALD lacks a\nnotion of an input distribution and so is prone to prioritise data of limited\nrelevance. To address this we propose the expected predictive information gain\n(EPIG), an acquisition function that measures information gain in the space of\npredictions rather than parameters. We find that using EPIG leads to stronger\npredictive performance compared with BALD across a range of datasets and\nmodels, and thus provides an appealing drop-in replacement.\n", "date": "2023-04-17", "categories": []}
{"id": "2210.00993", "title": "Efficient Bayes Inference in Neural Networks through Adaptive Importance\n  Sampling", "abstract": "  Bayesian neural networks (BNNs) have received an increased interest in the\nlast years. In BNNs, a complete posterior distribution of the unknown weight\nand bias parameters of the network is produced during the training stage. This\nprobabilistic estimation offers several advantages with respect to point-wise\nestimates, in particular, the ability to provide uncertainty quantification\nwhen predicting new data. This feature inherent to the Bayesian paradigm, is\nuseful in countless machine learning applications. It is particularly appealing\nin areas where decision-making has a crucial impact, such as medical healthcare\nor autonomous driving. The main challenge of BNNs is the computational cost of\nthe training procedure since Bayesian techniques often face a severe curse of\ndimensionality. Adaptive importance sampling (AIS) is one of the most prominent\nMonte Carlo methodologies benefiting from sounded convergence guarantees and\nease for adaptation. This work aims to show that AIS constitutes a successful\napproach for designing BNNs. More precisely, we propose a novel algorithm\nPMCnet that includes an efficient adaptation mechanism, exploiting geometric\ninformation on the complex (often multimodal) posterior distribution. Numerical\nresults illustrate the excellent performance and the improved exploration\ncapabilities of the proposed method for both shallow and deep neural networks.\n", "date": "2022-10-03", "categories": []}
{"id": "2304.06349", "title": "Neural State-Space Models: Empirical Evaluation of Uncertainty\n  Quantification", "abstract": "  Effective quantification of uncertainty is an essential and still missing\nstep towards a greater adoption of deep-learning approaches in different\napplications, including mission-critical ones. In particular, investigations on\nthe predictive uncertainty of deep-learning models describing non-linear\ndynamical systems are very limited to date. This paper is aimed at filling this\ngap and presents preliminary results on uncertainty quantification for system\nidentification with neural state-space models. We frame the learning problem in\na Bayesian probabilistic setting and obtain posterior distributions for the\nneural network's weights and outputs through approximate inference techniques.\nBased on the posterior, we construct credible intervals on the outputs and\ndefine a surprise index which can effectively diagnose usage of the model in a\npotentially dangerous out-of-distribution regime, where predictions cannot be\ntrusted.\n", "date": "2023-04-13", "categories": []}
{"id": "2304.05398", "title": "Forward-backward Gaussian variational inference via JKO in the\n  Bures-Wasserstein Space", "abstract": "  Variational inference (VI) seeks to approximate a target distribution $\\pi$\nby an element of a tractable family of distributions. Of key interest in\nstatistics and machine learning is Gaussian VI, which approximates $\\pi$ by\nminimizing the Kullback-Leibler (KL) divergence to $\\pi$ over the space of\nGaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian\nVariational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach\nexploits the composite structure of the KL divergence, which can be written as\nthe sum of a smooth term (the potential) and a non-smooth term (the entropy)\nover the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein\ndistance. For our proposed algorithm, we obtain state-of-the-art convergence\nguarantees when $\\pi$ is log-smooth and log-concave, as well as the first\nconvergence guarantees to first-order stationary solutions when $\\pi$ is only\nlog-smooth.\n", "date": "2023-04-10", "categories": []}
{"id": "2304.04824", "title": "Gradient-based Uncertainty Attribution for Explainable Bayesian Deep\n  Learning", "abstract": "  Predictions made by deep learning models are prone to data perturbations,\nadversarial attacks, and out-of-distribution inputs. To build a trusted AI\nsystem, it is therefore critical to accurately quantify the prediction\nuncertainties. While current efforts focus on improving uncertainty\nquantification accuracy and efficiency, there is a need to identify uncertainty\nsources and take actions to mitigate their effects on predictions. Therefore,\nwe propose to develop explainable and actionable Bayesian deep learning methods\nto not only perform accurate uncertainty quantification but also explain the\nuncertainties, identify their sources, and propose strategies to mitigate the\nuncertainty impacts. Specifically, we introduce a gradient-based uncertainty\nattribution method to identify the most problematic regions of the input that\ncontribute to the prediction uncertainty. Compared to existing methods, the\nproposed UA-Backprop has competitive accuracy, relaxed assumptions, and high\nefficiency. Moreover, we propose an uncertainty mitigation strategy that\nleverages the attribution results as attention to further improve the model\nperformance. Both qualitative and quantitative evaluations are conducted to\ndemonstrate the effectiveness of our proposed methods.\n", "date": "2023-04-10", "categories": []}
{"id": "2206.04734", "title": "Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel\n  Recombination", "abstract": "  Calculation of Bayesian posteriors and model evidences typically requires\nnumerical integration. Bayesian quadrature (BQ), a surrogate-model-based\napproach to numerical integration, is capable of superb sample efficiency, but\nits lack of parallelisation has hindered its practical applications. In this\nwork, we propose a parallelised (batch) BQ method, employing techniques from\nkernel quadrature, that possesses an empirically exponential convergence rate.\nAdditionally, just as with Nested Sampling, our method permits simultaneous\ninference of both posteriors and model evidence. Samples from our BQ surrogate\nmodel are re-selected to give a sparse set of samples, via a kernel\nrecombination algorithm, requiring negligible additional time to increase the\nbatch size. Empirically, we find that our approach significantly outperforms\nthe sampling efficiency of both state-of-the-art BQ techniques and Nested\nSampling in various real-world datasets, including lithium-ion battery\nanalytics.\n", "date": "2022-06-09", "categories": []}
{"id": "2304.02899", "title": "Variable-Complexity Weighted-Tempered Gibbs Samplers for Bayesian\n  Variable Selection", "abstract": "  Subset weighted-Tempered Gibbs Sampler (wTGS) has been recently introduced by\nJankowiak to reduce the computation complexity per MCMC iteration in\nhigh-dimensional applications where the exact calculation of the posterior\ninclusion probabilities (PIP) is not essential. However, the Rao-Backwellized\nestimator associated with this sampler has a high variance as the ratio between\nthe signal dimension and the number of conditional PIP estimations is large. In\nthis paper, we design a new subset weighted-Tempered Gibbs Sampler (wTGS) where\nthe expected number of computations of conditional PIPs per MCMC iteration can\nbe much smaller than the signal dimension. Different from the subset wTGS and\nwTGS, our sampler has a variable complexity per MCMC iteration. We provide an\nupper bound on the variance of an associated Rao-Blackwellized estimator for\nthis sampler at a finite number of iterations, $T$, and show that the variance\nis $O\\big(\\big(\\frac{P}{S}\\big)^2 \\frac{\\log T}{T}\\big)$ for a given dataset\nwhere $S$ is the expected number of conditional PIP computations per MCMC\niteration. Experiments show that our Rao-Blackwellized estimator can have a\nsmaller variance than its counterpart associated with the subset wTGS.\n", "date": "2023-04-06", "categories": []}
{"id": "2304.02902", "title": "Towards Efficient MCMC Sampling in Bayesian Neural Networks by\n  Exploiting Symmetry", "abstract": "  Bayesian inference in deep neural networks is challenging due to the\nhigh-dimensional, strongly multi-modal parameter posterior density landscape.\nMarkov chain Monte Carlo approaches asymptotically recover the true posterior\nbut are considered prohibitively expensive for large modern architectures.\nLocal methods, which have emerged as a popular alternative, focus on specific\nparameter regions that can be approximated by functions with tractable\nintegrals. While these often yield satisfactory empirical results, they fail,\nby definition, to account for the multi-modality of the parameter posterior. In\nthis work, we argue that the dilemma between exact-but-unaffordable and\ncheap-but-inexact approaches can be mitigated by exploiting symmetries in the\nposterior landscape. Such symmetries, induced by neuron interchangeability and\ncertain activation functions, manifest in different parameter values leading to\nthe same functional output value. We show theoretically that the posterior\npredictive density in Bayesian neural networks can be restricted to a\nsymmetry-free parameter reference set. By further deriving an upper bound on\nthe number of Monte Carlo chains required to capture the functional diversity,\nwe propose a straightforward approach for feasible Bayesian inference. Our\nexperiments suggest that efficient sampling is indeed possible, opening up a\npromising path to accurate uncertainty quantification in deep learning.\n", "date": "2023-04-06", "categories": []}
{"id": "2211.02218", "title": "Fully Bayesian inference for latent variable Gaussian process models", "abstract": "  Real engineering and scientific applications often involve one or more\nqualitative inputs. Standard Gaussian processes (GPs), however, cannot directly\naccommodate qualitative inputs. The recently introduced latent variable\nGaussian process (LVGP) overcomes this issue by first mapping each qualitative\nfactor to underlying latent variables (LVs), and then uses any standard GP\ncovariance function over these LVs. The LVs are estimated similarly to the\nother GP hyperparameters through maximum likelihood estimation, and then\nplugged into the prediction expressions. However, this plug-in approach will\nnot account for uncertainty in estimation of the LVs, which can be significant\nespecially with limited training data. In this work, we develop a fully\nBayesian approach for the LVGP model and for visualizing the effects of the\nqualitative inputs via their LVs. We also develop approximations for scaling up\nLVGPs and fully Bayesian inference for the LVGP hyperparameters. We conduct\nnumerical studies comparing plug-in inference against fully Bayesian inference\nover a few engineering models and material design applications. In contrast to\nprevious studies on standard GP modeling that have largely concluded that a\nfully Bayesian treatment offers limited improvements, our results show that for\nLVGP modeling it offers significant improvements in prediction accuracy and\nuncertainty quantification over the plug-in approach.\n", "date": "2022-11-04", "categories": []}
{"id": "2303.10085", "title": "Robust probabilistic inference via a constrained transport metric", "abstract": "  Flexible Bayesian models are typically constructed using limits of large\nparametric models with a multitude of parameters that are often\nuninterpretable. In this article, we offer a novel alternative by constructing\nan exponentially tilted empirical likelihood carefully designed to concentrate\nnear a parametric family of distributions of choice with respect to a novel\nvariant of the Wasserstein metric, which is then combined with a prior\ndistribution on model parameters to obtain a robustified posterior. The\nproposed approach finds applications in a wide variety of robust inference\nproblems, where we intend to perform inference on the parameters associated\nwith the centering distribution in presence of outliers. Our proposed transport\nmetric enjoys great computational simplicity, exploiting the Sinkhorn\nregularization for discrete optimal transport problems, and being inherently\nparallelizable. We demonstrate superior performance of our methodology when\ncompared against state-of-the-art robust Bayesian inference methods. We also\ndemonstrate equivalence of our approach with a nonparametric Bayesian\nformulation under a suitable asymptotic framework, testifying to its\nflexibility. The constrained entropy maximization that sits at the heart of our\nlikelihood formulation finds its utility beyond robust Bayesian inference; an\nillustration is provided in a trustworthy machine learning application.\n", "date": "2023-03-17", "categories": []}
{"id": "2210.04994", "title": "Sampling-based inference for large linear models, with application to\n  linearised Laplace", "abstract": "  Large-scale linear models are ubiquitous throughout machine learning, with\ncontemporary application as surrogate models for neural network uncertainty\nquantification; that is, the linearised Laplace method. Alas, the computational\ncost associated with Bayesian linear models constrains this method's\napplication to small networks, small output spaces and small datasets. We\naddress this limitation by introducing a scalable sample-based Bayesian\ninference method for conjugate Gaussian multi-output linear models, together\nwith a matching method for hyperparameter (regularisation) selection.\nFurthermore, we use a classic feature normalisation method (the g-prior) to\nresolve a previously highlighted pathology of the linearised Laplace method.\nTogether, these contributions allow us to perform linearised neural network\ninference with ResNet-18 on CIFAR100 (11M parameters, 100 outputs x 50k\ndatapoints), with ResNet-50 on Imagenet (50M parameters, 1000 outputs x 1.2M\ndatapoints) and with a U-Net on a high-resolution tomographic reconstruction\ntask (2M parameters, 251k output~dimensions).\n", "date": "2022-10-10", "categories": []}
{"id": "2303.08230", "title": "Bayesian Beta-Bernoulli Process Sparse Coding with Deep Neural Networks", "abstract": "  Several approximate inference methods have been proposed for deep discrete\nlatent variable models. However, non-parametric methods which have previously\nbeen successfully employed for classical sparse coding models have largely been\nunexplored in the context of deep models. We propose a non-parametric iterative\nalgorithm for learning discrete latent representations in such deep models.\nAdditionally, to learn scale invariant discrete features, we propose local data\nscaling variables. Lastly, to encourage sparsity in our representations, we\npropose a Beta-Bernoulli process prior on the latent factors. We evaluate our\nspare coding model coupled with different likelihood models. We evaluate our\nmethod across datasets with varying characteristics and compare our results to\ncurrent amortized approximate inference methods.\n", "date": "2023-03-14", "categories": []}
{"id": "2303.00573", "title": "Dimension-reduced KRnet maps for high-dimensional Bayesian inverse\n  problems", "abstract": "  We present a dimension-reduced KRnet map approach (DR-KRnet) for\nhigh-dimensional Bayesian inverse problems, which is based on an explicit\nconstruction of a map that pushes forward the prior measure to the posterior\nmeasure in the latent space. Our approach consists of two main components:\ndata-driven VAE prior and density approximation of the posterior of the latent\nvariable. In reality, it may not be trivial to initialize a prior distribution\nthat is consistent with available prior data; in other words, the complex prior\ninformation is often beyond simple hand-crafted priors. We employ variational\nautoencoder (VAE) to approximate the underlying distribution of the prior\ndataset, which is achieved through a latent variable and a decoder. Using the\ndecoder provided by the VAE prior, we reformulate the problem in a\nlow-dimensional latent space. In particular, we seek an invertible transport\nmap given by KRnet to approximate the posterior distribution of the latent\nvariable. Moreover, an efficient physics-constrained surrogate model without\nany labeled data is constructed to reduce the computational cost of solving\nboth forward and adjoint problems involved in likelihood computation. With\nnumerical experiments, we demonstrate the accuracy and efficiency of DR-KRnet\nfor high-dimensional Bayesian inverse problems.\n", "date": "2023-03-01", "categories": []}
{"id": "2303.04397", "title": "The Lie-Group Bayesian Learning Rule", "abstract": "  The Bayesian Learning Rule provides a framework for generic algorithm design\nbut can be difficult to use for three reasons. First, it requires a specific\nparameterization of exponential family. Second, it uses gradients which can be\ndifficult to compute. Third, its update may not always stay on the manifold. We\naddress these difficulties by proposing an extension based on Lie-groups where\nposteriors are parametrized through transformations of an arbitrary base\ndistribution and updated via the group's exponential map. This simplifies all\nthree difficulties for many cases, providing flexible parametrizations through\ngroup's action, simple gradient computation through reparameterization, and\nupdates that always stay on the manifold. We use the new learning rule to\nderive a new algorithm for deep learning with desirable biologically-plausible\nattributes to learn sparse features. Our work opens a new frontier for the\ndesign of new algorithms by exploiting Lie-group structures.\n", "date": "2023-03-08", "categories": []}
{"id": "2303.03701", "title": "Variational Inference for Neyman-Scott Processes", "abstract": "  Neyman-Scott processes (NSPs) have been applied across a range of fields to\nmodel points or temporal events with a hierarchy of clusters. Markov chain\nMonte Carlo (MCMC) is typically used for posterior sampling in the model.\nHowever, MCMC's mixing time can cause the resulting inference to be slow, and\nthereby slow down model learning and prediction. We develop the first\nvariational inference (VI) algorithm for NSPs, and give two examples of\nsuitable variational posterior point process distributions. Our method\nminimizes the inclusive Kullback-Leibler (KL) divergence for VI to obtain the\nvariational parameters. We generate samples from the approximate posterior\npoint processes much faster than MCMC, as we can directly estimate the\napproximate posterior point processes without any MCMC steps or gradient\ndescent. We include synthetic and real-world data experiments that demonstrate\nour VI algorithm achieves better prediction performance than MCMC when\ncomputational time is limited.\n", "date": "2023-03-07", "categories": []}
{"id": "2208.08068", "title": "Quantum Bayesian Computation", "abstract": "  Quantum Bayesian Computation (QBC) is an emerging field that levers the\ncomputational gains available from quantum computers to provide an exponential\nspeed-up in Bayesian computation. Our paper adds to the literature in two ways.\nFirst, we show how von Neumann quantum measurement can be used to simulate\nmachine learning algorithms such as Markov chain Monte Carlo (MCMC) and Deep\nLearning (DL) that are fundamental to Bayesian learning. Second, we describe\ndata encoding methods needed to implement quantum machine learning including\nthe counterparts to traditional feature extraction and kernel embeddings\nmethods. Our goal then is to show how to apply quantum algorithms directly to\nstatistical machine learning problems. On the theoretical side, we provide\nquantum versions of high dimensional regression, Gaussian processes (Q-GP) and\nstochastic gradient descent (Q-SGD). On the empirical side, we apply a Quantum\nFFT model to Chicago housing data. Finally, we conclude with directions for\nfuture research.\n", "date": "2022-08-17", "categories": []}
{"id": "2303.01540", "title": "Variational EP with Probabilistic Backpropagation for Bayesian Neural\n  Networks", "abstract": "  I propose a novel approach for nonlinear Logistic regression using a\ntwo-layer neural network (NN) model structure with hierarchical priors on the\nnetwork weights. I present a hybrid of expectation propagation called\nVariational Expectation Propagation approach (VEP) for approximate integration\nover the posterior distribution of the weights, the hierarchical scale\nparameters of the priors and zeta. Using a factorized posterior approximation I\nderive a computationally efficient algorithm, whose complexity scales similarly\nto an ensemble of independent sparse logistic models. The approach can be\nextended beyond standard activation functions and NN model structures to form\nflexible nonlinear binary predictors from multiple sparse linear models. I\nconsider a hierarchical Bayesian model with logistic regression likelihood and\na Gaussian prior distribution over the parameters called weights and\nhyperparameters. I work in the perspective of E step and M step for computing\nthe approximating posterior and updating the parameters using the computed\nposterior respectively.\n", "date": "2023-03-02", "categories": []}
{"id": "2207.08435", "title": "Robust Simulation-Based Inference in Cosmology with Bayesian Neural\n  Networks", "abstract": "  Simulation-based inference (SBI) is rapidly establishing itself as a standard\nmachine learning technique for analyzing data in cosmological surveys. Despite\ncontinual improvements to the quality of density estimation by learned models,\napplications of such techniques to real data are entirely reliant on the\ngeneralization power of neural networks far outside the training distribution,\nwhich is mostly unconstrained. Due to the imperfections in scientist-created\nsimulations, and the large computational expense of generating all possible\nparameter combinations, SBI methods in cosmology are vulnerable to such\ngeneralization issues. Here, we discuss the effects of both issues, and show\nhow using a Bayesian neural network framework for training SBI can mitigate\nbiases, and result in more reliable inference outside the training set. We\nintroduce cosmoSWAG, the first application of Stochastic Weight Averaging to\ncosmology, and apply it to SBI trained for inference on the cosmic microwave\nbackground.\n", "date": "2022-07-18", "categories": []}
{"id": "2302.14247", "title": "Sequential edge detection using joint hierarchical Bayesian learning", "abstract": "  This paper introduces a new sparse Bayesian learning (SBL) algorithm that\njointly recovers a temporal sequence of edge maps from noisy and under-sampled\nFourier data. The new method is cast in a Bayesian framework and uses a prior\nthat simultaneously incorporates intra-image information to promote sparsity in\neach individual edge map with inter-image information to promote similarities\nin any unchanged regions. By treating both the edges as well as the similarity\nbetween adjacent images as random variables, there is no need to separately\nform regions of change. Thus we avoid both additional computational cost as\nwell as any information loss resulting from pre-processing the image. Our\nnumerical examples demonstrate that our new method compares favorably with more\nstandard SBL approaches.\n", "date": "2023-02-28", "categories": []}
{"id": "2302.12419", "title": "A Targeted Accuracy Diagnostic for Variational Approximations", "abstract": "  Variational Inference (VI) is an attractive alternative to Markov Chain Monte\nCarlo (MCMC) due to its computational efficiency in the case of large datasets\nand/or complex models with high-dimensional parameters. However, evaluating the\naccuracy of variational approximations remains a challenge. Existing methods\ncharacterize the quality of the whole variational distribution, which is almost\nalways poor in realistic applications, even if specific posterior functionals\nsuch as the component-wise means or variances are accurate. Hence, these\ndiagnostics are of practical value only in limited circumstances. To address\nthis issue, we propose the TArgeted Diagnostic for Distribution Approximation\nAccuracy (TADDAA), which uses many short parallel MCMC chains to obtain lower\nbounds on the error of each posterior functional of interest. We also develop a\nreliability check for TADDAA to determine when the lower bounds should not be\ntrusted. Numerical experiments validate the practical utility and computational\nefficiency of our approach on a range of synthetic distributions and real-data\nexamples, including sparse logistic regression and Bayesian neural network\nmodels.\n", "date": "2023-02-24", "categories": []}
{"id": "2206.12361", "title": "Robustness to corruption in pre-trained Bayesian neural networks", "abstract": "  We develop ShiftMatch, a new training-data-dependent likelihood for\nrobustness to corruption in Bayesian neural networks (BNNs). ShiftMatch is\ninspired by the training-data-dependent \"EmpCov\" priors from Izmailov et al.\n(2021a), and efficiently matches test-time spatial correlations to those at\ntraining time. Critically, ShiftMatch is designed to leave the neural network's\ntraining time likelihood unchanged, allowing it to use publicly available\nsamples from pre-trained BNNs. Using pre-trained HMC samples, ShiftMatch gives\nstrong performance improvements on CIFAR-10-C, outperforms EmpCov priors\n(though ShiftMatch uses extra information from a minibatch of corrupted test\npoints), and is perhaps the first Bayesian method capable of convincingly\noutperforming plain deep ensembles.\n", "date": "2022-06-24", "categories": []}
{"id": "2302.12130", "title": "Bayesian Structure Scores for Probabilistic Circuits", "abstract": "  Probabilistic circuits (PCs) are a prominent representation of probability\ndistributions with tractable inference. While parameter learning in PCs is\nrigorously studied, structure learning is often more based on heuristics than\non principled objectives. In this paper, we develop Bayesian structure scores\nfor deterministic PCs, i.e., the structure likelihood with parameters\nmarginalized out, which are well known as rigorous objectives for structure\nlearning in probabilistic graphical models. When used within a greedy cutset\nalgorithm, our scores effectively protect against overfitting and yield a fast\nand almost hyper-parameter-free structure learner, distinguishing it from\nprevious approaches. In experiments, we achieve good trade-offs between\ntraining time and model fit in terms of log-likelihood. Moreover, the\nprincipled nature of Bayesian scores unlocks PCs for accommodating frameworks\nsuch as structural expectation-maximization.\n", "date": "2023-02-23", "categories": []}
{"id": "2302.10706", "title": "Variational Boosted Soft Trees", "abstract": "  Gradient boosting machines (GBMs) based on decision trees consistently\ndemonstrate state-of-the-art results on regression and classification tasks\nwith tabular data, often outperforming deep neural networks. However, these\nmodels do not provide well-calibrated predictive uncertainties, which prevents\ntheir use for decision making in high-risk applications. The Bayesian treatment\nis known to improve predictive uncertainty calibration, but previously proposed\nBayesian GBM methods are either computationally expensive, or resort to crude\napproximations. Variational inference is often used to implement Bayesian\nneural networks, but is difficult to apply to GBMs, because the decision trees\nused as weak learners are non-differentiable. In this paper, we propose to\nimplement Bayesian GBMs using variational inference with soft decision trees, a\nfully differentiable alternative to standard decision trees introduced by Irsoy\net al. Our experiments demonstrate that variational soft trees and variational\nsoft GBMs provide useful uncertainty estimates, while retaining good predictive\nperformance. The proposed models show higher test likelihoods when compared to\nthe state-of-the-art Bayesian GBMs in 7/10 tabular regression datasets and\nimproved out-of-distribution detection in 5/10 datasets.\n", "date": "2023-02-21", "categories": []}
{"id": "2210.07278", "title": "Meta-Uncertainty in Bayesian Model Comparison", "abstract": "  Bayesian model comparison (BMC) offers a principled probabilistic approach to\nstudy and rank competing models. In standard BMC, we construct a discrete\nprobability distribution over the set of possible models, conditional on the\nobserved data of interest. These posterior model probabilities (PMPs) are\nmeasures of uncertainty, but -- when derived from a finite number of\nobservations -- are also uncertain themselves. In this paper, we conceptualize\ndistinct levels of uncertainty which arise in BMC. We explore a fully\nprobabilistic framework for quantifying meta-uncertainty, resulting in an\napplied method to enhance any BMC workflow. Drawing on both Bayesian and\nfrequentist techniques, we represent the uncertainty over the uncertain PMPs\nvia meta-models which combine simulated and observed data into a predictive\ndistribution for PMPs on new data. We demonstrate the utility of the proposed\nmethod in the context of conjugate Bayesian regression, likelihood-based\ninference with Markov chain Monte Carlo, and simulation-based inference with\nneural networks.\n", "date": "2022-10-13", "categories": []}
{"id": "2206.06462", "title": "Quasi-Bayesian Nonparametric Density Estimation via Autoregressive\n  Predictive Updates", "abstract": "  Bayesian methods are a popular choice for statistical inference in small-data\nregimes due to the regularization effect induced by the prior. In the context\nof density estimation, the standard nonparametric Bayesian approach is to\ntarget the posterior predictive of the Dirichlet process mixture model. In\ngeneral, direct estimation of the posterior predictive is intractable and so\nmethods typically resort to approximating the posterior distribution as an\nintermediate step. The recent development of quasi-Bayesian predictive copula\nupdates, however, has made it possible to perform tractable predictive density\nestimation without the need for posterior approximation. Although these\nestimators are computationally appealing, they tend to struggle on non-smooth\ndata distributions. This is due to the comparatively restrictive form of the\nlikelihood models from which the proposed copula updates were derived. To\naddress this shortcoming, we consider a Bayesian nonparametric model with an\nautoregressive likelihood decomposition and a Gaussian process prior. While the\npredictive update of such a model is typically intractable, we derive a\nquasi-Bayesian predictive update that achieves state-of-the-art results in\nsmall-data regimes.\n", "date": "2022-06-13", "categories": []}
{"id": "2211.06291", "title": "Do Bayesian Neural Networks Need To Be Fully Stochastic?", "abstract": "  We investigate the benefit of treating all the parameters in a Bayesian\nneural network stochastically and find compelling theoretical and empirical\nevidence that this standard construction may be unnecessary. To this end, we\nprove that expressive predictive distributions require only small amounts of\nstochasticity. In particular, partially stochastic networks with only $n$\nstochastic biases are universal probabilistic predictors for $n$-dimensional\npredictive problems. In empirical investigations, we find no systematic benefit\nof full stochasticity across four different inference modalities and eight\ndatasets; partially stochastic networks can match and sometimes even outperform\nfully stochastic networks, despite their reduced memory costs.\n", "date": "2022-11-11", "categories": []}
{"id": "2206.06720", "title": "Deep Variational Implicit Processes", "abstract": "  Implicit processes (IPs) are a generalization of Gaussian processes (GPs).\nIPs may lack a closed-form expression but are easy to sample from. Examples\ninclude, among others, Bayesian neural networks or neural samplers. IPs can be\nused as priors over functions, resulting in flexible models with\nwell-calibrated prediction uncertainty estimates. Methods based on IPs usually\ncarry out function-space approximate inference, which overcomes some of the\ndifficulties of parameter-space approximate inference. Nevertheless, the\napproximations employed often limit the expressiveness of the final model,\nresulting, e.g., in a Gaussian predictive distribution, which can be\nrestrictive. We propose here a multi-layer generalization of IPs called the\nDeep Variational Implicit process (DVIP). This generalization is similar to\nthat of deep GPs over GPs, but it is more flexible due to the use of IPs as the\nprior distribution over the latent functions. We describe a scalable\nvariational inference algorithm for training DVIP and show that it outperforms\nprevious IP-based methods and also deep GPs. We support these claims via\nextensive regression and classification experiments. We also evaluate DVIP on\nlarge datasets with up to several million data instances to illustrate its good\nscalability and performance.\n", "date": "2022-06-14", "categories": []}
{"id": "2301.05490", "title": "Scalable Batch Acquisition for Deep Bayesian Active Learning", "abstract": "  In deep active learning, it is especially important to choose multiple\nexamples to markup at each step to work efficiently, especially on large\ndatasets. At the same time, existing solutions to this problem in the Bayesian\nsetup, such as BatchBALD, have significant limitations in selecting a large\nnumber of examples, associated with the exponential complexity of computing\nmutual information for joint random variables. We, therefore, present the Large\nBatchBALD algorithm, which gives a well-grounded approximation to the BatchBALD\nmethod that aims to achieve comparable quality while being more computationally\nefficient. We provide a complexity analysis of the algorithm, showing a\nreduction in computation time, especially for large batches. Furthermore, we\npresent an extensive set of experimental results on image and text data, both\non toy datasets and larger ones such as CIFAR-100.\n", "date": "2023-01-13", "categories": []}
{"id": "2302.07407", "title": "Bayesian Decision Trees via Tractable Priors and Probabilistic\n  Context-Free Grammars", "abstract": "  Decision Trees are some of the most popular machine learning models today due\nto their out-of-the-box performance and interpretability. Often, Decision Trees\nmodels are constructed greedily in a top-down fashion via heuristic search\ncriteria, such as Gini impurity or entropy. However, trees constructed in this\nmanner are sensitive to minor fluctuations in training data and are prone to\noverfitting. In contrast, Bayesian approaches to tree construction formulate\nthe selection process as a posterior inference problem; such approaches are\nmore stable and provide greater theoretical guarantees. However, generating\nBayesian Decision Trees usually requires sampling from complex, multimodal\nposterior distributions. Current Markov Chain Monte Carlo-based approaches for\nsampling Bayesian Decision Trees are prone to mode collapse and long mixing\ntimes, which makes them impractical. In this paper, we propose a new criterion\nfor training Bayesian Decision Trees. Our criterion gives rise to BCART-PCFG,\nwhich can efficiently sample decision trees from a posterior distribution\nacross trees given the data and find the maximum a posteriori (MAP) tree.\nLearning the posterior and training the sampler can be done in time that is\npolynomial in the dataset size. Once the posterior has been learned, trees can\nbe sampled efficiently (linearly in the number of nodes). At the core of our\nmethod is a reduction of sampling the posterior to sampling a derivation from a\nprobabilistic context-free grammar. We find that trees sampled via BCART-PCFG\nperform comparable to or better than greedily-constructed Decision Trees in\nclassification accuracy on several datasets. Additionally, the trees sampled\nvia BCART-PCFG are significantly smaller -- sometimes by as much as 20x.\n", "date": "2023-02-15", "categories": []}
{"id": "2206.06779", "title": "Benchmarking Bayesian neural networks and evaluation metrics for\n  regression tasks", "abstract": "  Due to the growing adoption of deep neural networks in many fields of science\nand engineering, modeling and estimating their uncertainties has become of\nprimary importance. Despite the growing literature about uncertainty\nquantification in deep learning, the quality of the uncertainty estimates\nremains an open question. In this work, we assess for the first time the\nperformance of several approximation methods for Bayesian neural networks on\nregression tasks by evaluating the quality of the confidence regions with\nseveral coverage metrics. The selected algorithms are also compared in terms of\npredictivity, kernelized Stein discrepancy and maximum mean discrepancy with\nrespect to a reference posterior in both weight and function space. Our\nfindings show that (i) some algorithms have excellent predictive performance\nbut tend to largely over or underestimate uncertainties (ii) it is possible to\nachieve good accuracy and a given target coverage with finely tuned\nhyperparameters and (iii) the promising kernel Stein discrepancy cannot be\nexclusively relied on to assess the posterior approximation. As a by-product of\nthis benchmark, we also compute and visualize the similarity of all algorithms\nand corresponding hyperparameters: interestingly we identify a few clusters of\nalgorithms with similar behavior in weight space, giving new insights on how\nthey explore the posterior distribution.\n", "date": "2022-06-08", "categories": []}
{"id": "2206.05630", "title": "Mathematical Theory of Bayesian Statistics for Unknown Information\n  Source", "abstract": "  In statistical inference, uncertainty is unknown and all models are wrong.\nThat is to say, a person who makes a statistical model and a prior distribution\nis simultaneously aware that both are fictional candidates. To study such\ncases, statistical measures have been constructed, such as cross validation,\ninformation criteria, and marginal likelihood, however, their mathematical\nproperties have not yet been completely clarified when statistical models are\nunder- and over- parametrized.\n  We introduce a place of mathematical theory of Bayesian statistics for\nunknown uncertainty, which clarifies general properties of cross validation,\ninformation criteria, and marginal likelihood, even if an unknown\ndata-generating process is unrealizable by a model or even if the posterior\ndistribution cannot be approximated by any normal distribution. Hence it gives\na helpful standpoint for a person who cannot believe in any specific model and\nprior.\n  This paper consists of three parts. The first is a new result, whereas the\nsecond and third are well-known previous results with new experiments. We show\nthere exists a more precise estimator of the generalization loss than\nleave-one-out cross validation, there exists a more accurate approximation of\nmarginal likelihood than BIC, and the optimal hyperparameters for\ngeneralization loss and marginal likelihood are different.\n", "date": "2022-06-11", "categories": []}
{"id": "2301.05708", "title": "A domain-decomposed VAE method for Bayesian inverse problems", "abstract": "  Bayesian inverse problems are often computationally challenging when the\nforward model is governed by complex partial differential equations (PDEs).\nThis is typically caused by expensive forward model evaluations and\nhigh-dimensional parameterization of priors. This paper proposes a\ndomain-decomposed variational auto-encoder Markov chain Monte Carlo\n(DD-VAE-MCMC) method to tackle these challenges simultaneously. Through\npartitioning the global physical domain into small subdomains, the proposed\nmethod first constructs local deterministic generative models based on local\nhistorical data, which provide efficient local prior representations. Gaussian\nprocess models with active learning address the domain decomposition interface\nconditions. Then inversions are conducted on each subdomain independently in\nparallel and in low-dimensional latent parameter spaces. The local inference\nsolutions are post-processed through the Poisson image blending procedure to\nresult in an efficient global inference result. Numerical examples are provided\nto demonstrate the performance of the proposed method.\n", "date": "2023-01-09", "categories": []}
{"id": "2211.11865", "title": "Bayesian Learning for Neural Networks: an algorithmic survey", "abstract": "  The last decade witnessed a growing interest in Bayesian learning. Yet, the\ntechnicality of the topic and the multitude of ingredients involved therein,\nbesides the complexity of turning theory into practical implementations, limit\nthe use of the Bayesian learning paradigm, preventing its widespread adoption\nacross different fields and applications. This self-contained survey engages\nand introduces readers to the principles and algorithms of Bayesian Learning\nfor Neural Networks. It provides an introduction to the topic from an\naccessible, practical-algorithmic perspective. Upon providing a general\nintroduction to Bayesian Neural Networks, we discuss and present both standard\nand recent approaches for Bayesian inference, with an emphasis on solutions\nrelying on Variational Inference and the use of Natural gradients. We also\ndiscuss the use of manifold optimization as a state-of-the-art approach to\nBayesian learning. We examine the characteristic properties of all the\ndiscussed methods, and provide pseudo-codes for their implementation, paying\nattention to practical aspects, such as the computation of the gradients.\n", "date": "2022-11-21", "categories": []}
{"id": "2301.13401", "title": "Classified as unknown: A novel Bayesian neural network", "abstract": "  We establish estimations for the parameters of the output distribution for\nthe softmax activation function using the probit function. As an application,\nwe develop a new efficient Bayesian learning algorithm for fully connected\nneural networks, where training and predictions are performed within the\nBayesian inference framework in closed-form. This approach allows sequential\nlearning and requires no computationally expensive gradient calculation and\nMonte Carlo sampling. Our work generalizes the Bayesian algorithm for a single\nperceptron for binary classification in \\cite{H} to multi-layer perceptrons for\nmulti-class classification.\n", "date": "2023-01-31", "categories": []}
{"id": "2210.12236", "title": "Uncertain Evidence in Probabilistic Models and Stochastic Simulators", "abstract": "  We consider the problem of performing Bayesian inference in probabilistic\nmodels where observations are accompanied by uncertainty, referred to as\n\"uncertain evidence.\" We explore how to interpret uncertain evidence, and by\nextension the importance of proper interpretation as it pertains to inference\nabout latent variables. We consider a recently-proposed method \"distributional\nevidence\" as well as revisit two older methods: Jeffrey's rule and virtual\nevidence. We devise guidelines on how to account for uncertain evidence and we\nprovide new insights, particularly regarding consistency. To showcase the\nimpact of different interpretations of the same uncertain evidence, we carry\nout experiments in which one interpretation is defined as \"correct.\" We then\ncompare inference results from each different interpretation illustrating the\nimportance of careful consideration of uncertain evidence.\n", "date": "2022-10-21", "categories": []}
{"id": "2301.11276", "title": "BayesSpeech: A Bayesian Transformer Network for Automatic Speech\n  Recognition", "abstract": "  Recent developments using End-to-End Deep Learning models have been shown to\nhave near or better performance than state of the art Recurrent Neural Networks\n(RNNs) on Automatic Speech Recognition tasks. These models tend to be lighter\nweight and require less training time than traditional RNN-based approaches.\nHowever, these models take frequentist approach to weight training. In theory,\nnetwork weights are drawn from a latent, intractable probability distribution.\nWe introduce BayesSpeech for end-to-end Automatic Speech Recognition.\nBayesSpeech is a Bayesian Transformer Network where these intractable\nposteriors are learned through variational inference and the local\nreparameterization trick without recurrence. We show how the introduction of\nvariance in the weights leads to faster training time and near state-of-the-art\nperformance on LibriSpeech-960.\n", "date": "2023-01-16", "categories": []}
{"id": "2207.01234", "title": "Incorporating functional summary information in Bayesian neural networks\n  using a Dirichlet process likelihood approach", "abstract": "  Bayesian neural networks (BNNs) can account for both aleatoric and epistemic\nuncertainty. However, in BNNs the priors are often specified over the weights\nwhich rarely reflects true prior knowledge in large and complex neural network\narchitectures. We present a simple approach to incorporate prior knowledge in\nBNNs based on external summary information about the predicted classification\nprobabilities for a given dataset. The available summary information is\nincorporated as augmented data and modeled with a Dirichlet process, and we\nderive the corresponding \\emph{Summary Evidence Lower BOund}. The approach is\nfounded on Bayesian principles, and all hyperparameters have a proper\nprobabilistic interpretation. We show how the method can inform the model about\ntask difficulty and class imbalance. Extensive experiments show that, with\nnegligible computational overhead, our method parallels and in many cases\noutperforms popular alternatives in accuracy, uncertainty calibration, and\nrobustness against corruptions with both balanced and imbalanced data.\n", "date": "2022-07-04", "categories": []}
{"id": "2301.09090", "title": "Parallel Approaches to Accelerate Bayesian Decision Trees", "abstract": "  Markov Chain Monte Carlo (MCMC) is a well-established family of algorithms\nprimarily used in Bayesian statistics to sample from a target distribution when\ndirect sampling is challenging. Existing work on Bayesian decision trees uses\nMCMC. Unfortunately, this can be slow, especially when considering large\nvolumes of data. It is hard to parallelise the accept-reject component of the\nMCMC. None-the-less, we propose two methods for exploiting parallelism in the\nMCMC: in the first, we replace the MCMC with another numerical Bayesian\napproach, the Sequential Monte Carlo (SMC) sampler, which has the appealing\nproperty that it is an inherently parallel algorithm; in the second, we\nconsider data partitioning. Both methods use multi-core processing with a\nHighPerformance Computing (HPC) resource. We test the two methods in various\nstudy settings to determine which method is the most beneficial for each test\ncase. Experiments show that data partitioning has limited utility in the\nsettings we consider and that the use of the SMC sampler can improve run-time\n(compared to the sequential implementation) by up to a factor of 343.\n", "date": "2023-01-22", "categories": []}
{"id": "2212.14715", "title": "Bayesian statistical learning using density operators", "abstract": "  This short study reformulates the statistical Bayesian learning problem using\na quantum mechanics framework. Density operators representing ensembles of pure\nstates of sample wave functions are used in place probability densities. We\nshow that such representation allows to formulate the statistical Bayesian\nlearning problem in different coordinate systems on the sample space. We\nfurther show that such representation allows to learn projections of density\noperators using a kernel trick. In particular, the study highlights that\ndecomposing wave functions rather than probability densities, as it is done in\nkernel embedding, allows to preserve the nature of probability operators.\nResults are illustrated with a simple example using discrete orthogonal wavelet\ntransform of density operators.\n", "date": "2022-12-28", "categories": []}
{"id": "2301.05297", "title": "Towards Dependable Autonomous Systems Based on Bayesian Deep Learning\n  Components", "abstract": "  As autonomous systems increasingly rely on Deep Neural Networks (DNN) to\nimplement the navigation pipeline functions, uncertainty estimation methods\nhave become paramount for estimating confidence in DNN predictions. Bayesian\nDeep Learning (BDL) offers a principled approach to model uncertainties in\nDNNs. However, in DNN-based systems, not all the components use uncertainty\nestimation methods and typically ignore the uncertainty propagation between\nthem. This paper provides a method that considers the uncertainty and the\ninteraction between BDL components to capture the overall system uncertainty.\nWe study the effect of uncertainty propagation in a BDL-based system for\nautonomous aerial navigation. Experiments show that our approach allows us to\ncapture useful uncertainty estimates while slightly improving the system's\nperformance in its final task. In addition, we discuss the benefits,\nchallenges, and implications of adopting BDL to build dependable autonomous\nsystems.\n", "date": "2023-01-12", "categories": []}
{"id": "2301.00032", "title": "Bayesian Learning for Dynamic Inference", "abstract": "  The traditional statistical inference is static, in the sense that the\nestimate of the quantity of interest does not affect the future evolution of\nthe quantity. In some sequential estimation problems however, the future values\nof the quantity to be estimated depend on the estimate of its current value.\nThis type of estimation problems has been formulated as the dynamic inference\nproblem. In this work, we formulate the Bayesian learning problem for dynamic\ninference, where the unknown quantity-generation model is assumed to be\nrandomly drawn according to a random model parameter. We derive the optimal\nBayesian learning rules, both offline and online, to minimize the inference\nloss. Moreover, learning for dynamic inference can serve as a meta problem,\nsuch that all familiar machine learning problems, including supervised\nlearning, imitation learning and reinforcement learning, can be cast as its\nspecial cases or variants. Gaining a good understanding of this unifying meta\nproblem thus sheds light on a broad spectrum of machine learning problems as\nwell.\n", "date": "2022-12-30", "categories": []}
{"id": "2211.09008", "title": "Normalizing Flows for Hierarchical Bayesian Analysis: A Gravitational\n  Wave Population Study", "abstract": "  We propose parameterizing the population distribution of the gravitational\nwave population modeling framework (Hierarchical Bayesian Analysis) with a\nnormalizing flow. We first demonstrate the merit of this method on illustrative\nexperiments and then analyze four parameters of the latest LIGO/Virgo data\nrelease: primary mass, secondary mass, redshift, and effective spin. Our\nresults show that despite the small and notoriously noisy dataset, the\nposterior predictive distributions (assuming a prior over the parameters of the\nflow) of the observed gravitational wave population recover structure that\nagrees with robust previous phenomenological modeling results while being less\nsusceptible to biases introduced by less flexible models. Therefore, the method\nforms a promising flexible, reliable replacement for population inference\ndistributions, even when data is highly noisy.\n", "date": "2022-11-15", "categories": []}
{"id": "2210.10964", "title": "Uncertainty Disentanglement with Non-stationary Heteroscedastic Gaussian\n  Processes for Active Learning", "abstract": "  Gaussian processes are Bayesian non-parametric models used in many areas. In\nthis work, we propose a Non-stationary Heteroscedastic Gaussian process model\nwhich can be learned with gradient-based techniques. We demonstrate the\ninterpretability of the proposed model by separating the overall uncertainty\ninto aleatoric (irreducible) and epistemic (model) uncertainty. We illustrate\nthe usability of derived epistemic uncertainty on active learning problems. We\ndemonstrate the efficacy of our model with various ablations on multiple\ndatasets.\n", "date": "2022-10-20", "categories": []}
{"id": "2212.06965", "title": "Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian\n  Physics-Informed Neural Networks", "abstract": "  Physics-Informed Neural Networks (PINNs) are gaining popularity as a method\nfor solving differential equations. While being more feasible in some contexts\nthan the classical numerical techniques, PINNs still lack credibility. A remedy\nfor that can be found in Uncertainty Quantification (UQ) which is just\nbeginning to emerge in the context of PINNs. Assessing how well the trained\nPINN complies with imposed differential equation is the key to tackling\nuncertainty, yet there is lack of comprehensive methodology for this task. We\npropose a framework for UQ in Bayesian PINNs (B-PINNs) that incorporates the\ndiscrepancy between the B-PINN solution and the unknown true solution. We\nexploit recent results on error bounds for PINNs on linear dynamical systems\nand demonstrate the predictive uncertainty on a class of linear ODEs.\n", "date": "2022-12-14", "categories": []}
{"id": "2206.08900", "title": "Adapting the Linearised Laplace Model Evidence for Modern Deep Learning", "abstract": "  The linearised Laplace method for estimating model uncertainty has received\nrenewed attention in the Bayesian deep learning community. The method provides\nreliable error bars and admits a closed-form expression for the model evidence,\nallowing for scalable selection of model hyperparameters. In this work, we\nexamine the assumptions behind this method, particularly in conjunction with\nmodel selection. We show that these interact poorly with some now-standard\ntools of deep learning--stochastic approximation methods and normalisation\nlayers--and make recommendations for how to better adapt this classic method to\nthe modern setting. We provide theoretical support for our recommendations and\nvalidate them empirically on MLPs, classic CNNs, residual networks with and\nwithout normalisation layers, generative autoencoders and transformers.\n", "date": "2022-06-17", "categories": []}
{"id": "2212.04259", "title": "Fast Parallel Bayesian Network Structure Learning", "abstract": "  Bayesian networks (BNs) are a widely used graphical model in machine learning\nfor representing knowledge with uncertainty. The mainstream BN structure\nlearning methods require performing a large number of conditional independence\n(CI) tests. The learning process is very time-consuming, especially for\nhigh-dimensional problems, which hinders the adoption of BNs to more\napplications. Existing works attempt to accelerate the learning process with\nparallelism, but face issues including load unbalancing, costly atomic\noperations and dominant parallel overhead. In this paper, we propose a fast\nsolution named Fast-BNS on multi-core CPUs to enhance the efficiency of the BN\nstructure learning. Fast-BNS is powered by a series of efficiency optimizations\nincluding (i) designing a dynamic work pool to monitor the processing of edges\nand to better schedule the workloads among threads, (ii) grouping the CI tests\nof the edges with the same endpoints to reduce the number of unnecessary CI\ntests, (iii) using a cache-friendly data storage to improve the memory\nefficiency, and (iv) generating the conditioning sets on-the-fly to avoid extra\nmemory consumption. A comprehensive experimental study shows that the\nsequential version of Fast-BNS is up to 50 times faster than its counterpart,\nand the parallel version of Fast-BNS achieves 4.8 to 24.5 times speedup over\nthe state-of-the-art multi-threaded solution. Moreover, Fast-BNS has a good\nscalability to the network size as well as sample size. Fast-BNS source code is\nfreely available at https://github.com/jjiantong/FastBN.\n", "date": "2022-12-08", "categories": []}
{"id": "2212.03103", "title": "A Comprehensively Improved Hybrid Algorithm for Learning Bayesian\n  Networks: Multiple Compound Memory Erasing", "abstract": "  Using a Bayesian network to analyze the causal relationship between nodes is\na hot spot. The existing network learning algorithms are mainly\nconstraint-based and score-based network generation methods. The\nconstraint-based method is mainly the application of conditional independence\n(CI) tests, but the inaccuracy of CI tests in the case of high dimensionality\nand small samples has always been a problem for the constraint-based method.\nThe score-based method uses the scoring function and search strategy to find\nthe optimal candidate network structure, but the search space increases too\nmuch with the increase of the number of nodes, and the learning efficiency is\nvery low. This paper presents a new hybrid algorithm, MCME (multiple compound\nmemory erasing). This method retains the advantages of the first two methods,\nsolves the shortcomings of the above CI tests, and makes innovations in the\nscoring function in the direction discrimination stage. A large number of\nexperiments show that MCME has better or similar performance than some existing\nalgorithms.\n", "date": "2022-12-05", "categories": []}
{"id": "2211.00729", "title": "A Bayesian Framework on Asymmetric Mixture of Factor Analyser", "abstract": "  Mixture of factor analyzer (MFA) model is an efficient model for the analysis\nof high dimensional data through which the factor-analyzer technique based on\nthe covariance matrices reducing the number of free parameters. The model also\nprovides an important methodology to determine latent groups in data. There are\nseveral pieces of research to extend the model based on the asymmetrical and/or\nwith outlier datasets with some known computational limitations that have been\nexamined in frequentist cases. In this paper, an MFA model with a rich and\nflexible class of skew normal (unrestricted) generalized hyperbolic (called\nSUNGH) distributions along with a Bayesian structure with several computational\nbenefits have been introduced. The SUNGH family provides considerable\nflexibility to model skewness in different directions as well as allowing for\nheavy tailed data. There are several desirable properties in the structure of\nthe SUNGH family, including, an analytically flexible density which leads to\neasing up the computation applied for the estimation of parameters. Considering\nfactor analysis models, the SUNGH family also allows for skewness and heavy\ntails for both the error component and factor scores. In the present study, the\nadvantages of using this family of distributions have been discussed and the\nsuitable efficiency of the introduced MFA model using real data examples and\nsimulation has been demonstrated.\n", "date": "2022-11-01", "categories": []}
{"id": "2211.15860", "title": "Bayesian Experimental Design for Symbolic Discovery", "abstract": "  This study concerns the formulation and application of Bayesian optimal\nexperimental design to symbolic discovery, which is the inference from\nobservational data of predictive models taking general functional forms. We\napply constrained first-order methods to optimize an appropriate selection\ncriterion, using Hamiltonian Monte Carlo to sample from the prior. A step for\ncomputing the predictive distribution, involving convolution, is computed via\neither numerical integration, or via fast transform methods.\n", "date": "2022-11-29", "categories": []}
{"id": "2211.09184", "title": "An Empirical Analysis of the Advantages of Finite- v.s. Infinite-Width\n  Bayesian Neural Networks", "abstract": "  Comparing Bayesian neural networks (BNNs) with different widths is\nchallenging because, as the width increases, multiple model properties change\nsimultaneously, and, inference in the finite-width case is intractable. In this\nwork, we empirically compare finite- and infinite-width BNNs, and provide\nquantitative and qualitative explanations for their performance difference. We\nfind that when the model is mis-specified, increasing width can hurt BNN\nperformance. In these cases, we provide evidence that finite-width BNNs\ngeneralize better partially due to the properties of their frequency spectrum\nthat allows them to adapt under model mismatch.\n", "date": "2022-11-16", "categories": []}
{"id": "2206.00241", "title": "Asymptotic Properties for Bayesian Neural Network in Besov Space", "abstract": "  Neural networks have shown great predictive power when dealing with various\nunstructured data such as images and natural languages. The Bayesian neural\nnetwork captures the uncertainty of prediction by putting a prior distribution\nfor the parameter of the model and computing the posterior distribution. In\nthis paper, we show that the Bayesian neural network using spike-and-slab prior\nhas consistency with nearly minimax convergence rate when the true regression\nfunction is in the Besov space. Even when the smoothness of the regression\nfunction is unknown the same posterior convergence rate holds and thus the\nspike-and-slab prior is adaptive to the smoothness of the regression function.\nWe also consider the shrinkage prior, which is more feasible than other priors,\nand show that it has the same convergence rate. In other words, we propose a\npractical Bayesian neural network with guaranteed asymptotic properties.\n", "date": "2022-06-01", "categories": []}
{"id": "2211.13402", "title": "MP-GELU Bayesian Neural Networks: Moment Propagation by GELU\n  Nonlinearity", "abstract": "  Bayesian neural networks (BNNs) have been an important framework in the study\nof uncertainty quantification. Deterministic variational inference, one of the\ninference methods, utilizes moment propagation to compute the predictive\ndistributions and objective functions. Unfortunately, deriving the moments\nrequires computationally expensive Taylor expansion in nonlinear functions,\nsuch as a rectified linear unit (ReLU) or a sigmoid function. Therefore, a new\nnonlinear function that realizes faster moment propagation than conventional\nfunctions is required. In this paper, we propose a novel nonlinear function\nnamed moment propagating-Gaussian error linear unit (MP-GELU) that enables the\nfast derivation of first and second moments in BNNs. MP-GELU enables the\nanalytical computation of moments by applying nonlinearity to the input\nstatistics, thereby reducing the computationally expensive calculations\nrequired for nonlinear functions. In empirical experiments on regression tasks,\nwe observed that the proposed MP-GELU provides higher prediction accuracy and\nbetter quality of uncertainty with faster execution than those of ReLU-based\nBNNs.\n", "date": "2022-11-24", "categories": []}
{"id": "2211.10837", "title": "Non-reversible Parallel Tempering for Deep Posterior Approximation", "abstract": "  Parallel tempering (PT), also known as replica exchange, is the go-to\nworkhorse for simulations of multi-modal distributions. The key to the success\nof PT is to adopt efficient swap schemes. The popular deterministic even-odd\n(DEO) scheme exploits the non-reversibility property and has successfully\nreduced the communication cost from $O(P^2)$ to $O(P)$ given sufficiently many\n$P$ chains. However, such an innovation largely disappears in big data due to\nthe limited chains and few bias-corrected swaps. To handle this issue, we\ngeneralize the DEO scheme to promote non-reversibility and propose a few\nsolutions to tackle the underlying bias caused by the geometric stopping time.\nNotably, in big data scenarios, we obtain an appealing communication cost\n$O(P\\log P)$ based on the optimal window size. In addition, we also adopt\nstochastic gradient descent (SGD) with large and constant learning rates as\nexploration kernels. Such a user-friendly nature enables us to conduct\napproximation tasks for complex posteriors without much tuning costs.\n", "date": "2022-11-20", "categories": []}
{"id": "2206.03743", "title": "Using Mixed-Effects Models to Learn Bayesian Networks from Related Data\n  Sets", "abstract": "  We commonly assume that data are a homogeneous set of observations when\nlearning the structure of Bayesian networks. However, they often comprise\ndifferent data sets that are related but not homogeneous because they have been\ncollected in different ways or from different populations.\n  In our previous work (Azzimonti, Corani and Scutari, 2021), we proposed a\nclosed-form Bayesian Hierarchical Dirichlet score for discrete data that pools\ninformation across related data sets to learn a single encompassing network\nstructure, while taking into account the differences in their probabilistic\nstructures. In this paper, we provide an analogous solution for learning a\nBayesian network from continuous data using mixed-effects models to pool\ninformation across the related data sets. We study its structural, parametric,\npredictive and classification accuracy and we show that it outperforms both\nconditional Gaussian Bayesian networks (that do not perform any pooling) and\nclassical Gaussian Bayesian networks (that disregard the heterogeneous nature\nof the data). The improvement is marked for low sample sizes and for unbalanced\ndata sets.\n", "date": "2022-06-08", "categories": []}
{"id": "2211.08393", "title": "On the Performance of Direct Loss Minimization for Bayesian Neural\n  Networks", "abstract": "  Direct Loss Minimization (DLM) has been proposed as a pseudo-Bayesian method\nmotivated as regularized loss minimization. Compared to variational inference,\nit replaces the loss term in the evidence lower bound (ELBO) with the\npredictive log loss, which is the same loss function used in evaluation. A\nnumber of theoretical and empirical results in prior work suggest that DLM can\nsignificantly improve over ELBO optimization for some models. However, as we\npoint out in this paper, this is not the case for Bayesian neural networks\n(BNNs). The paper explores the practical performance of DLM for BNN, the\nreasons for its failure and its relationship to optimizing the ELBO, uncovering\nsome interesting facts about both algorithms.\n", "date": "2022-11-15", "categories": []}
{"id": "2211.06139", "title": "Understanding Approximation for Bayesian Inference in Neural Networks", "abstract": "  Bayesian inference has theoretical attractions as a principled framework for\nreasoning about beliefs. However, the motivations of Bayesian inference which\nclaim it to be the only 'rational' kind of reasoning do not apply in practice.\nThey create a binary split in which all approximate inference is equally\n'irrational'. Instead, we should ask ourselves how to define a spectrum of\nmore- and less-rational reasoning that explains why we might prefer one\nBayesian approximation to another. I explore approximate inference in Bayesian\nneural networks and consider the unintended interactions between the\nprobabilistic model, approximating distribution, optimization algorithm, and\ndataset. The complexity of these interactions highlights the difficulty of any\nstrategy for evaluating Bayesian approximations which focuses entirely on the\nmethod, outside the context of specific datasets and decision-problems. For\ngiven applications, the expected utility of the approximate posterior can\nmeasure inference quality. To assess a model's ability to incorporate different\nparts of the Bayesian framework we can identify desirable characteristic\nbehaviours of Bayesian reasoning and pick decision-problems that make heavy use\nof those behaviours. Here, we use continual learning (testing the ability to\nupdate sequentially) and active learning (testing the ability to represent\ncredence). But existing continual and active learning set-ups pose challenges\nthat have nothing to do with posterior quality which can distort their ability\nto evaluate Bayesian approximations. These unrelated challenges can be removed\nor reduced, allowing better evaluation of approximate inference methods.\n", "date": "2022-11-11", "categories": []}
{"id": "2211.06260", "title": "Towards Improved Learning in Gaussian Processes: The Best of Two Worlds", "abstract": "  Gaussian process training decomposes into inference of the (approximate)\nposterior and learning of the hyperparameters. For non-Gaussian (non-conjugate)\nlikelihoods, two common choices for approximate inference are Expectation\nPropagation (EP) and Variational Inference (VI), which have complementary\nstrengths and weaknesses. While VI's lower bound to the marginal likelihood is\na suitable objective for inferring the approximate posterior, it does not\nautomatically imply it is a good learning objective for hyperparameter\noptimization. We design a hybrid training procedure where the inference\nleverages conjugate-computation VI and the learning uses an EP-like marginal\nlikelihood approximation. We empirically demonstrate on binary classification\nthat this provides a good learning objective and generalizes better.\n", "date": "2022-11-11", "categories": []}
{"id": "2211.04847", "title": "Hyper-Parameter Auto-Tuning for Sparse Bayesian Learning", "abstract": "  Choosing the values of hyper-parameters in sparse Bayesian learning (SBL) can\nsignificantly impact performance. However, the hyper-parameters are normally\ntuned manually, which is often a difficult task. Most recently, effective\nautomatic hyper-parameter tuning was achieved by using an empirical auto-tuner.\nIn this work, we address the issue of hyper-parameter auto-tuning using neural\nnetwork (NN)-based learning. Inspired by the empirical auto-tuner, we design\nand learn a NN-based auto-tuner, and show that considerable improvement in\nconvergence rate and recovery performance can be achieved.\n", "date": "2022-11-09", "categories": []}
{"id": "2211.03248", "title": "Sparse Horseshoe Estimation via Expectation-Maximisation", "abstract": "  The horseshoe prior is known to possess many desirable properties for\nBayesian estimation of sparse parameter vectors, yet its density function lacks\nan analytic form. As such, it is challenging to find a closed-form solution for\nthe posterior mode. Conventional horseshoe estimators use the posterior mean to\nestimate the parameters, but these estimates are not sparse. We propose a novel\nexpectation-maximisation (EM) procedure for computing the MAP estimates of the\nparameters in the case of the standard linear model. A particular strength of\nour approach is that the M-step depends only on the form of the prior and it is\nindependent of the form of the likelihood. We introduce several simple\nmodifications of this EM procedure that allow for straightforward extension to\ngeneralised linear models. In experiments performed on simulated and real data,\nour approach performs comparable, or superior to, state-of-the-art sparse\nestimation methods in terms of statistical performance and computational cost.\n", "date": "2022-11-07", "categories": []}
{"id": "2211.02476", "title": "Sparse Gaussian Process Hyperparameters: Optimize or Integrate?", "abstract": "  The kernel function and its hyperparameters are the central model selection\nchoice in a Gaussian proces (Rasmussen and Williams, 2006). Typically, the\nhyperparameters of the kernel are chosen by maximising the marginal likelihood,\nan approach known as Type-II maximum likelihood (ML-II). However, ML-II does\nnot account for hyperparameter uncertainty, and it is well-known that this can\nlead to severely biased estimates and an underestimation of predictive\nuncertainty. While there are several works which employ a fully Bayesian\ncharacterisation of GPs, relatively few propose such approaches for the sparse\nGPs paradigm. In this work we propose an algorithm for sparse Gaussian process\nregression which leverages MCMC to sample from the hyperparameter posterior\nwithin the variational inducing point framework of Titsias (2009). This work is\nclosely related to Hensman et al. (2015b) but side-steps the need to sample the\ninducing points, thereby significantly improving sampling efficiency in the\nGaussian likelihood case. We compare this scheme against natural baselines in\nliterature along with stochastic variational GPs (SVGPs) along with an\nextensive computational analysis.\n", "date": "2022-11-04", "categories": []}
{"id": "2211.01518", "title": "Bayesian Counterfactual Mean Embeddings and Off-Policy Evaluation", "abstract": "  The counterfactual distribution models the effect of the treatment in the\nuntreated group. While most of the work focuses on the expected values of the\ntreatment effect, one may be interested in the whole counterfactual\ndistribution or other quantities associated to it. Building on the framework of\nBayesian conditional mean embeddings, we propose a Bayesian approach for\nmodeling the counterfactual distribution, which leads to quantifying the\nepistemic uncertainty about the distribution. The framework naturally extends\nto the setting where one observes multiple treatment effects (e.g. an\nintermediate effect after an interim period, and an ultimate treatment effect\nwhich is of main interest) and allows for additionally modelling uncertainty\nabout the relationship of these effects. For such goal, we present three novel\nBayesian methods to estimate the expectation of the ultimate treatment effect,\nwhen only noisy samples of the dependence between intermediate and ultimate\neffects are provided. These methods differ on the source of uncertainty\nconsidered and allow for combining two sources of data. Moreover, we generalize\nthese ideas to the off-policy evaluation framework, which can be seen as an\nextension of the counterfactual estimation problem. We empirically explore the\ncalibration of the algorithms in two different experimental settings which\nrequire data fusion, and illustrate the value of considering the uncertainty\nstemming from the two sources of data.\n", "date": "2022-11-02", "categories": []}
{"id": "2208.13723", "title": "Bayesian Continual Learning via Spiking Neural Networks", "abstract": "  Among the main features of biological intelligence are energy efficiency,\ncapacity for continual adaptation, and risk management via uncertainty\nquantification. Neuromorphic engineering has been thus far mostly driven by the\ngoal of implementing energy-efficient machines that take inspiration from the\ntime-based computing paradigm of biological brains. In this paper, we take\nsteps towards the design of neuromorphic systems that are capable of adaptation\nto changing learning tasks, while producing well-calibrated uncertainty\nquantification estimates. To this end, we derive online learning rules for\nspiking neural networks (SNNs) within a Bayesian continual learning framework.\nIn it, each synaptic weight is represented by parameters that quantify the\ncurrent epistemic uncertainty resulting from prior knowledge and observed data.\nThe proposed online rules update the distribution parameters in a streaming\nfashion as data are observed. We instantiate the proposed approach for both\nreal-valued and binary synaptic weights. Experimental results using Intel's\nLava platform show the merits of Bayesian over frequentist learning in terms of\ncapacity for adaptation and uncertainty quantification.\n", "date": "2022-08-29", "categories": []}
{"id": "2211.00100", "title": "Federated Averaging Langevin Dynamics: Toward a unified theory and new\n  algorithms", "abstract": "  This paper focuses on Bayesian inference in a federated learning context\n(FL). While several distributed MCMC algorithms have been proposed, few\nconsider the specific limitations of FL such as communication bottlenecks and\nstatistical heterogeneity. Recently, Federated Averaging Langevin Dynamics\n(FALD) was introduced, which extends the Federated Averaging algorithm to\nBayesian inference. We obtain a novel tight non-asymptotic upper bound on the\nWasserstein distance to the global posterior for FALD. This bound highlights\nthe effects of statistical heterogeneity, which causes a drift in the local\nupdates that negatively impacts convergence. We propose a new algorithm\nVR-FALD* that uses control variates to correct the client drift. We establish\nnon-asymptotic bounds showing that VR-FALD* is not affected by statistical\nheterogeneity. Finally, we illustrate our results on several FL benchmarks for\nBayesian inference.\n", "date": "2022-10-31", "categories": []}
{"id": "2208.10461", "title": "Scale invariant process regression: Towards Bayesian ML with minimal\n  assumptions", "abstract": "  Current methods for regularization in machine learning require quite specific\nmodel assumptions (e.g. a kernel shape) that are not derived from prior\nknowledge about the application, but must be imposed merely to make the method\nwork. We show in this paper that regularization can indeed be achieved by\nassuming nothing but invariance principles (w.r.t. scaling, translation, and\nrotation of input and output space) and the degree of differentiability of the\ntrue function.\n  Concretely, we derive a novel (non-Gaussian) stochastic process from the\nabove minimal assumptions, and we present a corresponding Bayesian inference\nmethod for regression. The mean posterior turns out to be a polyharmonic\nspline, and the posterior process is a mixture of t-processes.\n  Compared with Gaussian process regression, the proposed method shows equal\nperformance and has the advantages of being (i) less arbitrary (no choice of\nkernel) (ii) potentially faster (no kernel parameter optimization), and (iii)\nhaving better extrapolation behavior.\n  We believe that the proposed theory has central importance for the conceptual\nfoundations of regularization and machine learning and also has great potential\nto enable practical advances in ML areas beyond regression.\n", "date": "2022-08-22", "categories": []}
{"id": "2210.16176", "title": "A Novel Sparse Bayesian Learning and Its Application to Fault Diagnosis\n  for Multistation Assembly Systems", "abstract": "  This paper addresses the problem of fault diagnosis in multistation assembly\nsystems. Fault diagnosis is to identify process faults that cause the excessive\ndimensional variation of the product using dimensional measurements. For such\nproblems, the challenge is solving an underdetermined system caused by a common\nphenomenon in practice; namely, the number of measurements is less than that of\nthe process errors. To address this challenge, this paper attempts to solve the\nfollowing two problems: (1) how to utilize the temporal correlation in the time\nseries data of each process error and (2) how to apply prior knowledge\nregarding which process errors are more likely to be process faults. A novel\nsparse Bayesian learning method is proposed to achieve the above objectives.\nThe method consists of three hierarchical layers. The first layer has\nparameterized prior distribution that exploits the temporal correlation of each\nprocess error. Furthermore, the second and third layers achieve the prior\ndistribution representing the prior knowledge of process faults. Then, these\nprior distributions are updated with the likelihood function of the measurement\nsamples from the process, resulting in the accurate posterior distribution of\nprocess faults from an underdetermined system. Since posterior distributions of\nprocess faults are intractable, this paper derives approximate posterior\ndistributions via Variational Bayes inference. Numerical and simulation case\nstudies using an actual autobody assembly process are performed to demonstrate\nthe effectiveness of the proposed method.\n", "date": "2022-10-28", "categories": []}
{"id": "2208.06120", "title": "Bayesian Inference with Latent Hamiltonian Neural Networks", "abstract": "  When sampling for Bayesian inference, one popular approach is to use\nHamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS)\nwhich automatically decides the end time of the Hamiltonian trajectory.\nHowever, HMC and NUTS can require numerous numerical gradients of the target\ndensity, and can prove slow in practice. We propose Hamiltonian neural networks\n(HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained,\nHNNs do not require numerical gradients of the target density during sampling.\nMoreover, they satisfy important properties such as perfect time reversibility\nand Hamiltonian conservation, making them well-suited for use within HMC and\nNUTS because stationarity can be shown. We also propose an HNN extension called\nlatent HNNs (L-HNNs), which are capable of predicting latent variable outputs.\nCompared to HNNs, L-HNNs offer improved expressivity and reduced integration\nerrors. Finally, we employ L-HNNs in NUTS with an online error monitoring\nscheme to prevent sample degeneracy in regions of low probability density. We\ndemonstrate L-HNNs in NUTS with online error monitoring on several examples\ninvolving complex, heavy-tailed, and high-local-curvature probability\ndensities. Overall, L-HNNs in NUTS with online error monitoring satisfactorily\ninferred these probability densities. Compared to traditional NUTS, L-HNNs in\nNUTS with online error monitoring required 1--2 orders of magnitude fewer\nnumerical gradients of the target density and improved the effective sample\nsize (ESS) per gradient by an order of magnitude.\n", "date": "2022-08-12", "categories": []}
{"id": "2210.12363", "title": "Bayesian Convolutional Deep Sets with Task-Dependent Stationary Prior", "abstract": "  Convolutional deep sets are the architecture of a deep neural network (DNN)\nthat can model stationary stochastic process. This architecture uses the kernel\nsmoother and the DNN to construct the translation equivariant functional\nrepresentations, and thus reflects the inductive bias of the stationarity into\nDNN. However, since this architecture employs the kernel smoother known as the\nnon-parametric model, it may produce ambiguous representations when the number\nof data points is not given sufficiently. To remedy this issue, we introduce\nBayesian convolutional deep sets that construct the random translation\nequivariant functional representations with stationary prior. Furthermore, we\npresent how to impose the task-dependent prior for each dataset because a\nwrongly imposed prior forms an even worse representation than that of the\nkernel smoother. We validate the proposed architecture and its training on\nvarious experiments with time-series and image datasets.\n", "date": "2022-10-22", "categories": []}
{"id": "2210.12642", "title": "Accelerated Linearized Laplace Approximation for Bayesian Deep Learning", "abstract": "  Laplace approximation (LA) and its linearized variant (LLA) enable effortless\nadaptation of pretrained deep neural networks to Bayesian neural networks. The\ngeneralized Gauss-Newton (GGN) approximation is typically introduced to improve\ntheir tractability. However, LA and LLA are still confronted with non-trivial\ninefficiency issues and should rely on Kronecker-factored, diagonal, or even\nlast-layer approximate GGN matrices in practical use. These approximations are\nlikely to harm the fidelity of learning outcomes. To tackle this issue,\ninspired by the connections between LLA and neural tangent kernels (NTKs), we\ndevelop a Nystrom approximation to NTKs to accelerate LLA. Our method benefits\nfrom the capability of popular deep learning libraries for forward mode\nautomatic differentiation, and enjoys reassuring theoretical guarantees.\nExtensive studies reflect the merits of the proposed method in aspects of both\nscalability and performance. Our method can even scale up to architectures like\nvision transformers. We also offer valuable ablation studies to diagnose our\nmethod. Code is available at \\url{https://github.com/thudzj/ELLA}.\n", "date": "2022-10-23", "categories": []}
{"id": "2210.12957", "title": "On the optimization and pruning for Bayesian deep learning", "abstract": "  The goal of Bayesian deep learning is to provide uncertainty quantification\nvia the posterior distribution. However, exact inference over the weight space\nis computationally intractable due to the ultra-high dimensions of the neural\nnetwork. Variational inference (VI) is a promising approach, but naive\napplication on weight space does not scale well and often underperform on\npredictive accuracy. In this paper, we propose a new adaptive variational\nBayesian algorithm to train neural networks on weight space that achieves high\npredictive accuracy. By showing that there is an equivalence to Stochastic\nGradient Hamiltonian Monte Carlo(SGHMC) with preconditioning matrix, we then\npropose an MCMC within EM algorithm, which incorporates the spike-and-slab\nprior to capture the sparsity of the neural network. The EM-MCMC algorithm\nallows us to perform optimization and model pruning within one-shot. We\nevaluate our methods on CIFAR-10, CIFAR-100 and ImageNet datasets, and\ndemonstrate that our dense model can reach the state-of-the-art performance and\nour sparse model perform very well compared to previously proposed pruning\nschemes.\n", "date": "2022-10-24", "categories": []}
{"id": "2210.13232", "title": "A Complete Recipe for Bayesian Knowledge Transfer: Object Tracking", "abstract": "  The problem of sequentially transferring from a source object track and a\nmodel to another Bayesian filter has become ubiquitous. Due to the lack of a\nstructural model that can capture the dependence among different models, the\ntransfer may not be fully specified. In this paper, we introduce a novel\nBayesian model that accounts for the model-jump from which the object can\nchoose a model and follow. We aim to track the trajectory of the object while\nsequentially transferring from the source object to the target object. The main\nidea is to impute the dynamical model while tracking the object and estimating\nthe state parameters of the moving object according to discretized dynamic\nsystems. We demonstrate this procedure can handle the model mismatch as it\nsequentially corrects the predictive model. Particularly, for a fixed number of\nmotion models, the object can learn what motion to follow at each time step. We\nemploy a prior model for each model and then adaptively correct for changing\none model to another to robustly estimate object trajectory under various\nmotions. More concretely, we propose a robust Bayesian recipe to handle the\nmodel-jump and then integrate it with a Markov chain Monte Carlo (MCMC)\napproach to sample from the posterior distribution. We demonstrate through\nexperiments the advantage of accounting for model-jump in our proposed method\nfor knowledge transfer between learning tasks in Bayesian transfer learning.\n", "date": "2022-10-20", "categories": []}
{"id": "2210.11737", "title": "Bayesian deep learning framework for uncertainty quantification in high\n  dimensions", "abstract": "  We develop a novel deep learning method for uncertainty quantification in\nstochastic partial differential equations based on Bayesian neural network\n(BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior\ndistribution of the parameters in deep neural networks by performing Bayesian\ninference on the network parameters. The posterior distribution is efficiently\nsampled using HMC to quantify uncertainties in the system. Several numerical\nexamples are shown for both forward and inverse problems in high dimension to\ndemonstrate the effectiveness of the proposed method for uncertainty\nquantification. These also show promising results that the computational cost\nis almost independent of the dimension of the problem demonstrating the\npotential of the method for tackling the so-called curse of dimensionality.\n", "date": "2022-10-21", "categories": []}
{"id": "2210.03283", "title": "Design Amortization for Bayesian Optimal Experimental Design", "abstract": "  Bayesian optimal experimental design is a sub-field of statistics focused on\ndeveloping methods to make efficient use of experimental resources. Any\npotential design is evaluated in terms of a utility function, such as the\n(theoretically well-justified) expected information gain (EIG); unfortunately\nhowever, under most circumstances the EIG is intractable to evaluate. In this\nwork we build off of successful variational approaches, which optimize a\nparameterized variational model with respect to bounds on the EIG. Past work\nfocused on learning a new variational model from scratch for each new design\nconsidered. Here we present a novel neural architecture that allows\nexperimenters to optimize a single variational model that can estimate the EIG\nfor potentially infinitely many designs. To further improve computational\nefficiency, we also propose to train the variational model on a significantly\ncheaper-to-evaluate lower bound, and show empirically that the resulting model\nprovides an excellent guide for more accurate, but expensive to evaluate bounds\non the EIG. We demonstrate the effectiveness of our technique on generalized\nlinear models, a class of statistical models that is widely used in the\nanalysis of controlled experiments. Experiments show that our method is able to\ngreatly improve accuracy over existing approximation strategies, and achieve\nthese results with far better sample efficiency.\n", "date": "2022-10-07", "categories": []}
{"id": "2210.10973", "title": "Scalable Bayesian Transformed Gaussian Processes", "abstract": "  The Bayesian transformed Gaussian process (BTG) model, proposed by Kedem and\nOliviera, is a fully Bayesian counterpart to the warped Gaussian process (WGP)\nand marginalizes out a joint prior over input warping and kernel\nhyperparameters. This fully Bayesian treatment of hyperparameters often\nprovides more accurate regression estimates and superior uncertainty\npropagation, but is prohibitively expensive. The BTG posterior predictive\ndistribution, itself estimated through high-dimensional integration, must be\ninverted in order to perform model prediction. To make the Bayesian approach\npractical and comparable in speed to maximum-likelihood estimation (MLE), we\npropose principled and fast techniques for computing with BTG. Our framework\nuses doubly sparse quadrature rules, tight quantile bounds, and rank-one matrix\nalgebra to enable both fast model prediction and model selection. These\nscalable methods allow us to regress over higher-dimensional datasets and apply\nBTG with layered transformations that greatly improve its expressibility. We\ndemonstrate that BTG achieves superior empirical performance over MLE-based\nmodels.\n", "date": "2022-10-20", "categories": []}
{"id": "2210.11385", "title": "On Representations of Mean-Field Variational Inference", "abstract": "  The mean field variational inference (MFVI) formulation restricts the general\nBayesian inference problem to the subspace of product measures. We present a\nframework to analyze MFVI algorithms, which is inspired by a similar\ndevelopment for general variational Bayesian formulations. Our approach enables\nthe MFVI problem to be represented in three different manners: a gradient flow\non Wasserstein space, a system of Fokker-Planck-like equations and a diffusion\nprocess. Rigorous guarantees are established to show that a time-discretized\nimplementation of the coordinate ascent variational inference algorithm in the\nproduct Wasserstein space of measures yields a gradient flow in the limit. A\nsimilar result is obtained for their associated densities, with the limit being\ngiven by a quasi-linear partial differential equation. A popular class of\npractical algorithms falls in this framework, which provides tools to establish\nconvergence. We hope this framework could be used to guarantee convergence of\nalgorithms in a variety of approaches, old and new, to solve variational\ninference problems.\n", "date": "2022-10-20", "categories": []}
{"id": "2210.10526", "title": "Propagating Variational Model Uncertainty for Bioacoustic Call Label\n  Smoothing", "abstract": "  We focus on using the predictive uncertainty signal calculated by Bayesian\nneural networks to guide learning in the self-same task the model is being\ntrained on. Not opting for costly Monte Carlo sampling of weights, we propagate\nthe approximate hidden variance in an end-to-end manner, throughout a\nvariational Bayesian adaptation of a ResNet with attention and\nsqueeze-and-excitation blocks, in order to identify data samples that should\ncontribute less into the loss value calculation. We, thus, propose\nuncertainty-aware, data-specific label smoothing, where the smoothing\nprobability is dependent on this epistemic uncertainty. We show that, through\nthe explicit usage of the epistemic uncertainty in the loss calculation, the\nvariational model is led to improved predictive and calibration performance.\nThis core machine learning methodology is exemplified at wildlife call\ndetection, from audio recordings made via passive acoustic monitoring equipment\nin the animals' natural habitats, with the future goal of automating large\nscale annotation in a trustworthy manner.\n", "date": "2022-10-19", "categories": []}
{"id": "2210.09352", "title": "A Mixing Time Lower Bound for a Simplified Version of BART", "abstract": "  Bayesian Additive Regression Trees (BART) is a popular Bayesian\nnon-parametric regression algorithm. The posterior is a distribution over sums\nof decision trees, and predictions are made by averaging approximate samples\nfrom the posterior.\n  The combination of strong predictive performance and the ability to provide\nuncertainty measures has led BART to be commonly used in the social sciences,\nbiostatistics, and causal inference.\n  BART uses Markov Chain Monte Carlo (MCMC) to obtain approximate posterior\nsamples over a parameterized space of sums of trees, but it has often been\nobserved that the chains are slow to mix.\n  In this paper, we provide the first lower bound on the mixing time for a\nsimplified version of BART in which we reduce the sum to a single tree and use\na subset of the possible moves for the MCMC proposal distribution. Our lower\nbound for the mixing time grows exponentially with the number of data points.\n  Inspired by this new connection between the mixing time and the number of\ndata points, we perform rigorous simulations on BART. We show qualitatively\nthat BART's mixing time increases with the number of data points.\n  The slow mixing time of the simplified BART suggests a large variation\nbetween different runs of the simplified BART algorithm and a similar large\nvariation is known for BART in the literature. This large variation could\nresult in a lack of stability in the models, predictions, and posterior\nintervals obtained from the BART MCMC samples.\n  Our lower bound and simulations suggest increasing the number of chains with\nthe number of data points.\n", "date": "2022-10-17", "categories": []}
{"id": "2210.08608", "title": "Posterior Regularized Bayesian Neural Network Incorporating Soft and\n  Hard Knowledge Constraints", "abstract": "  Neural Networks (NNs) have been widely {used in supervised learning} due to\ntheir ability to model complex nonlinear patterns, often presented in\nhigh-dimensional data such as images and text. However, traditional NNs often\nlack the ability for uncertainty quantification. Bayesian NNs (BNNS) could help\nmeasure the uncertainty by considering the distributions of the NN model\nparameters. Besides, domain knowledge is commonly available and could improve\nthe performance of BNNs if it can be appropriately incorporated. In this work,\nwe propose a novel Posterior-Regularized Bayesian Neural Network (PR-BNN) model\nby incorporating different types of knowledge constraints, such as the soft and\nhard constraints, as a posterior regularization term. Furthermore, we propose\nto combine the augmented Lagrangian method and the existing BNN solvers for\nefficient inference. The experiments in simulation and two case studies about\naviation landing prediction and solar energy output prediction have shown the\nknowledge constraints and the performance improvement of the proposed model\nover traditional BNNs without the constraints.\n", "date": "2022-10-16", "categories": []}
{"id": "2210.06205", "title": "On Divergence Measures for Bayesian Pseudocoresets", "abstract": "  A Bayesian pseudocoreset is a small synthetic dataset for which the posterior\nover parameters approximates that of the original dataset. While promising, the\nscalability of Bayesian pseudocoresets is not yet validated in realistic\nproblems such as image classification with deep neural networks. On the other\nhand, dataset distillation methods similarly construct a small dataset such\nthat the optimization using the synthetic dataset converges to a solution with\nperformance competitive with optimization using full data. Although dataset\ndistillation has been empirically verified in large-scale settings, the\nframework is restricted to point estimates, and their adaptation to Bayesian\ninference has not been explored. This paper casts two representative dataset\ndistillation algorithms as approximations to methods for constructing\npseudocoresets by minimizing specific divergence measures: reverse KL\ndivergence and Wasserstein distance. Furthermore, we provide a unifying view of\nsuch divergence measures in Bayesian pseudocoreset construction. Finally, we\npropose a novel Bayesian pseudocoreset algorithm based on minimizing forward KL\ndivergence. Our empirical results demonstrate that the pseudocoresets\nconstructed from these methods reflect the true posterior even in\nhigh-dimensional Bayesian inference problems.\n", "date": "2022-10-12", "categories": []}
{"id": "2206.01606", "title": "Excess risk analysis for epistemic uncertainty with application to\n  variational inference", "abstract": "  Bayesian deep learning plays an important role especially for its ability\nevaluating epistemic uncertainty (EU). Due to computational complexity issues,\napproximation methods such as variational inference (VI) have been used in\npractice to obtain posterior distributions and their generalization abilities\nhave been analyzed extensively, for example, by PAC-Bayesian theory; however,\nlittle analysis exists on EU, although many numerical experiments have been\nconducted on it. In this study, we analyze the EU of supervised learning in\napproximate Bayesian inference by focusing on its excess risk. First, we\ntheoretically show the novel relations between generalization error and the\nwidely used EU measurements, such as the variance and mutual information of\npredictive distribution, and derive their convergence behaviors. Next, we\nclarify how the objective function of VI regularizes the EU. With this\nanalysis, we propose a new objective function for VI that directly controls the\nprediction performance and the EU based on the PAC-Bayesian theory. Numerical\nexperiments show that our algorithm significantly improves the EU evaluation\nover the existing VI methods.\n", "date": "2022-06-02", "categories": []}
{"id": "2209.07157", "title": "On the detrimental effect of invariances in the likelihood for\n  variational inference", "abstract": "  Variational Bayesian posterior inference often requires simplifying\napproximations such as mean-field parametrisation to ensure tractability.\nHowever, prior work has associated the variational mean-field approximation for\nBayesian neural networks with underfitting in the case of small datasets or\nlarge model sizes. In this work, we show that invariances in the likelihood\nfunction of over-parametrised models contribute to this phenomenon because\nthese invariances complicate the structure of the posterior by introducing\ndiscrete and/or continuous modes which cannot be well approximated by Gaussian\nmean-field distributions. In particular, we show that the mean-field\napproximation has an additional gap in the evidence lower bound compared to a\npurpose-built posterior that takes into account the known invariances.\nImportantly, this invariance gap is not constant; it vanishes as the\napproximation reverts to the prior. We proceed by first considering translation\ninvariances in a linear model with a single data point in detail. We show that,\nwhile the true posterior can be constructed from a mean-field parametrisation,\nthis is achieved only if the objective function takes into account the\ninvariance gap. Then, we transfer our analysis of the linear model to neural\nnetworks. Our analysis provides a framework for future work to explore\nsolutions to the invariance problem.\n", "date": "2022-09-15", "categories": []}
{"id": "2209.15208", "title": "Scale-invariant Bayesian Neural Networks with Connectivity Tangent\n  Kernel", "abstract": "  Explaining generalizations and preventing over-confident predictions are\ncentral goals of studies on the loss landscape of neural networks. Flatness,\ndefined as loss invariability on perturbations of a pre-trained solution, is\nwidely accepted as a predictor of generalization in this context. However, the\nproblem that flatness and generalization bounds can be changed arbitrarily\naccording to the scale of a parameter was pointed out, and previous studies\npartially solved the problem with restrictions: Counter-intuitively, their\ngeneralization bounds were still variant for the function-preserving parameter\nscaling transformation or limited only to an impractical network structure. As\na more fundamental solution, we propose new prior and posterior distributions\ninvariant to scaling transformations by \\textit{decomposing} the scale and\nconnectivity of parameters, thereby allowing the resulting generalization bound\nto describe the generalizability of a broad class of networks with the more\npractical class of transformations such as weight decay with batch\nnormalization. We also show that the above issue adversely affects the\nuncertainty calibration of Laplace approximation and propose a solution using\nour invariant posterior. We empirically demonstrate our posterior provides\neffective flatness and calibration measures with low complexity in such a\npractical parameter transformation case, supporting its practical effectiveness\nin line with our rationale.\n", "date": "2022-09-30", "categories": []}
{"id": "2209.14594", "title": "Bayesian Neural Network Versus Ex-Post Calibration For Prediction\n  Uncertainty", "abstract": "  Probabilistic predictions from neural networks which account for predictive\nuncertainty during classification is crucial in many real-world and high-impact\ndecision making settings. However, in practice most datasets are trained on\nnon-probabilistic neural networks which by default do not capture this inherent\nuncertainty. This well-known problem has led to the development of post-hoc\ncalibration procedures, such as Platt scaling (logistic), isotonic and beta\ncalibration, which transforms the scores into well calibrated empirical\nprobabilities. A plausible alternative to the calibration approach is to use\nBayesian neural networks, which directly models a predictive distribution.\nAlthough they have been applied to images and text datasets, they have seen\nlimited adoption in the tabular and small data regime. In this paper, we\ndemonstrate that Bayesian neural networks yields competitive performance when\ncompared to calibrated neural networks and conduct experiments across a wide\narray of datasets.\n", "date": "2022-09-29", "categories": []}
{"id": "2209.13096", "title": "BayesNetCNN: incorporating uncertainty in neural networks for\n  image-based classification tasks", "abstract": "  The willingness to trust predictions formulated by automatic algorithms is\nkey in a vast number of domains. However, a vast number of deep architectures\nare only able to formulate predictions without an associated uncertainty. In\nthis paper, we propose a method to convert a standard neural network into a\nBayesian neural network and estimate the variability of predictions by sampling\ndifferent networks similar to the original one at each forward pass. We couple\nour methods with a tunable rejection-based approach that employs only the\nfraction of the dataset that the model is able to classify with an uncertainty\nbelow a user-set threshold. We test our model in a large cohort of brain images\nfrom Alzheimer's Disease patients, where we tackle discrimination of patients\nfrom healthy controls based on morphometric images only. We demonstrate how\ncombining the estimated uncertainty with a rejection-based approach increases\nclassification accuracy from 0.86 to 0.95 while retaining 75% of the test set.\nIn addition, the model can select cases to be recommended for manual evaluation\nbased on excessive uncertainty. We believe that being able to estimate the\nuncertainty of a prediction, along with tools that can modulate the behavior of\nthe network to a degree of confidence that the user is informed about (and\ncomfortable with) can represent a crucial step in the direction of user\ncompliance and easier integration of deep learning tools into everyday tasks\ncurrently performed by human operators.\n", "date": "2022-09-27", "categories": []}
{"id": "2209.11875", "title": "Tighter Variational Bounds are Not Necessarily Better. A Research Report\n  on Implementation, Ablation Study, and Extensions", "abstract": "  This report explains, implements and extends the works presented in \"Tighter\nVariational Bounds are Not Necessarily Better\" (T Rainforth et al., 2018). We\nprovide theoretical and empirical evidence that increasing the number of\nimportance samples $K$ in the importance weighted autoencoder (IWAE) (Burda et\nal., 2016) degrades the signal-to-noise ratio (SNR) of the gradient estimator\nin the inference network and thereby affecting the full learning process. In\nother words, even though increasing $K$ decreases the standard deviation of the\ngradients, it also reduces the magnitude of the true gradient faster, thereby\nincreasing the relative variance of the gradient updates. Extensive experiments\nare performed to understand the importance of $K$. These experiments suggest\nthat tighter variational bounds are beneficial for the generative network,\nwhereas looser bounds are preferable for the inference network. With these\ninsights, three methods are implemented and studied: the partially importance\nweighted autoencoder (PIWAE), the multiply importance weighted autoencoder\n(MIWAE) and the combination importance weighted autoencoder (CIWAE). Each of\nthese three methods entails IWAE as a special case but employs the importance\nweights in different ways to ensure a higher SNR of the gradient estimators. In\nour research study and analysis, the efficacy of these algorithms is tested on\nmultiple datasets such as MNIST and Omniglot. Finally, we demonstrate that the\nthree presented IWAE variations are able to generate approximate posterior\ndistributions that are much closer to the true posterior distribution than for\nthe IWAE, while matching the performance of the IWAE generative network or\npotentially outperforming it in the case of PIWAE.\n", "date": "2022-09-23", "categories": []}
{"id": "2209.10512", "title": "Improved Marginal Unbiased Score Expansion (MUSE) via Implicit\n  Differentiation", "abstract": "  We apply the technique of implicit differentiation to boost performance,\nreduce numerical error, and remove required user-tuning in the Marginal\nUnbiased Score Expansion (MUSE) algorithm for hierarchical Bayesian inference.\nWe demonstrate these improvements on three representative inference problems:\n1) an extended Neal's funnel 2) Bayesian neural networks, and 3) probabilistic\nprincipal component analysis. On our particular test cases, MUSE with implicit\ndifferentiation is faster than Hamiltonian Monte Carlo by factors of 155, 397,\nand 5, respectively, or factors of 65, 278, and 1 without implicit\ndifferentiation, and yields good approximate marginal posteriors. The Julia and\nPython MUSE packages have been updated to use implicit differentiation, and can\nsolve problems defined by hand or with any of a number of popular probabilistic\nprogramming languages and automatic differentiation backends.\n", "date": "2022-09-21", "categories": []}
{"id": "2209.09349", "title": "Physics-Informed Machine Learning of Dynamical Systems for Efficient\n  Bayesian Inference", "abstract": "  Although the no-u-turn sampler (NUTS) is a widely adopted method for\nperforming Bayesian inference, it requires numerous posterior gradients which\ncan be expensive to compute in practice. Recently, there has been a significant\ninterest in physics-based machine learning of dynamical (or Hamiltonian)\nsystems and Hamiltonian neural networks (HNNs) is a noteworthy architecture.\nBut these types of architectures have not been applied to solve Bayesian\ninference problems efficiently. We propose the use of HNNs for performing\nBayesian inference efficiently without requiring numerous posterior gradients.\nWe introduce latent variable outputs to HNNs (L-HNNs) for improved expressivity\nand reduced integration errors. We integrate L-HNNs in NUTS and further propose\nan online error monitoring scheme to prevent sampling degeneracy in regions\nwhere L-HNNs may have little training data. We demonstrate L-HNNs in NUTS with\nonline error monitoring considering several complex high-dimensional posterior\ndensities and compare its performance to NUTS.\n", "date": "2022-09-19", "categories": []}
{"id": "2209.07267", "title": "Compressed Particle-Based Federated Bayesian Learning and Unlearning", "abstract": "  Conventional frequentist FL schemes are known to yield overconfident\ndecisions. Bayesian FL addresses this issue by allowing agents to process and\nexchange uncertainty information encoded in distributions over the model\nparameters. However, this comes at the cost of a larger per-iteration\ncommunication overhead. This letter investigates whether Bayesian FL can still\nprovide advantages in terms of calibration when constraining communication\nbandwidth. We present compressed particle-based Bayesian FL protocols for FL\nand federated \"unlearning\" that apply quantization and sparsification across\nmultiple particles. The experimental results confirm that the benefits of\nBayesian FL are robust to bandwidth constraints.\n", "date": "2022-09-14", "categories": []}
{"id": "2209.02188", "title": "Bayesian Neural Network Inference via Implicit Models and the Posterior\n  Predictive Distribution", "abstract": "  We propose a novel approach to perform approximate Bayesian inference in\ncomplex models such as Bayesian neural networks. The approach is more scalable\nto large data than Markov Chain Monte Carlo, it embraces more expressive models\nthan Variational Inference, and it does not rely on adversarial training (or\ndensity ratio estimation). We adopt the recent approach of constructing two\nmodels: (1) a primary model, tasked with performing regression or\nclassification; and (2) a secondary, expressive (e.g. implicit) model that\ndefines an approximate posterior distribution over the parameters of the\nprimary model. However, we optimise the parameters of the posterior model via\ngradient descent according to a Monte Carlo estimate of the posterior\npredictive distribution -- which is our only approximation (other than the\nposterior model). Only a likelihood needs to be specified, which can take\nvarious forms such as loss functions and synthetic likelihoods, thus providing\na form of a likelihood-free approach. Furthermore, we formulate the approach\nsuch that the posterior samples can either be independent of, or conditionally\ndependent upon the inputs to the primary model. The latter approach is shown to\nbe capable of increasing the apparent complexity of the primary model. We see\nthis being useful in applications such as surrogate and physics-based models.\nTo promote how the Bayesian paradigm offers more than just uncertainty\nquantification, we demonstrate: uncertainty quantification, multi-modality, as\nwell as an application with a recent deep forecasting neural network\narchitecture.\n", "date": "2022-09-06", "categories": []}
{"id": "2208.13259", "title": "Bayesian Neural Network Language Modeling for Speech Recognition", "abstract": "  State-of-the-art neural network language models (NNLMs) represented by long\nshort term memory recurrent neural networks (LSTM-RNNs) and Transformers are\nbecoming highly complex. They are prone to overfitting and poor generalization\nwhen given limited training data. To this end, an overarching full Bayesian\nlearning framework encompassing three methods is proposed in this paper to\naccount for the underlying uncertainty in LSTM-RNN and Transformer LMs. The\nuncertainty over their model parameters, choice of neural activations and\nhidden output representations are modeled using Bayesian, Gaussian Process and\nvariational LSTM-RNN or Transformer LMs respectively. Efficient inference\napproaches were used to automatically select the optimal network internal\ncomponents to be Bayesian learned using neural architecture search. A minimal\nnumber of Monte Carlo parameter samples as low as one was also used. These\nallow the computational costs incurred in Bayesian NNLM training and evaluation\nto be minimized. Experiments are conducted on two tasks: AMI meeting\ntranscription and Oxford-BBC LipReading Sentences 2 (LRS2) overlapped speech\nrecognition using state-of-the-art LF-MMI trained factored TDNN systems\nfeaturing data augmentation, speaker adaptation and audio-visual multi-channel\nbeamforming for overlapped speech. Consistent performance improvements over the\nbaseline LSTM-RNN and Transformer LMs with point estimated model parameters and\ndrop-out regularization were obtained across both tasks in terms of perplexity\nand word error rate (WER). In particular, on the LRS2 data, statistically\nsignificant WER reductions up to 1.3% and 1.2% absolute (12.1% and 11.3%\nrelative) were obtained over the baseline LSTM-RNN and Transformer LMs\nrespectively after model combination between Bayesian NNLMs and their\nrespective baselines.\n", "date": "2022-08-28", "categories": []}
{"id": "2208.13605", "title": "Approach of variable clustering and compression for learning large\n  Bayesian networks", "abstract": "  This paper describes a new approach for learning structures of large Bayesian\nnetworks based on blocks resulting from feature space clustering. This\nclustering is obtained using normalized mutual information. And the subsequent\naggregation of blocks is done using classical learning methods except that they\nare input with compressed information about combinations of feature values for\neach block. Validation of this approach is done for Hill-Climbing as a graph\nenumeration algorithm for two score functions: BIC and MI. In this way,\npotentially parallelizable block learning can be implemented even for those\nscore functions that are considered unsuitable for parallelizable learning. The\nadvantage of the approach is evaluated in terms of speed of work as well as the\naccuracy of the found structures.\n", "date": "2022-08-29", "categories": []}
{"id": "2208.10312", "title": "BigBraveBN: algorithm of structural learning for bayesian networks with\n  a large number of nodes", "abstract": "  Learning a Bayesian network is an NP-hard problem and with an increase in the\nnumber of nodes, classical algorithms for learning the structure of Bayesian\nnetworks become inefficient. In recent years, some methods and algorithms for\nlearning Bayesian networks with a high number of nodes (more than 50) were\ndeveloped. But these solutions have their disadvantages, for instance, they\nonly operate one type of data (discrete or continuous) or their algorithm has\nbeen created to meet a specific nature of data (medical, social, etc.). The\narticle presents a BigBraveBN algorithm for learning large Bayesian Networks\nwith a high number of nodes (over 100). The algorithm utilizes the Brave\ncoefficient that measures the mutual occurrence of instances in several groups.\nTo form these groups, we use the method of nearest neighbours based on the\nMutual information (MI) measure. In the experimental part of the article, we\ncompare the performance of BigBraveBN to other existing solutions on multiple\ndata sets both discrete and continuous. The experimental part also represents\ntests on real data. The aforementioned experimental results demonstrate the\nefficiency of the BigBraveBN algorithm in structure learning of Bayesian\nNetworks.\n", "date": "2022-08-22", "categories": []}
{"id": "2207.10673", "title": "Correcting Model Bias with Sparse Implicit Processes", "abstract": "  Model selection in machine learning (ML) is a crucial part of the Bayesian\nlearning procedure. Model choice may impose strong biases on the resulting\npredictions, which can hinder the performance of methods such as Bayesian\nneural networks and neural samplers. On the other hand, newly proposed\napproaches for Bayesian ML exploit features of approximate inference in\nfunction space with implicit stochastic processes (a generalization of Gaussian\nprocesses). The approach of Sparse Implicit Processes (SIP) is particularly\nsuccessful in this regard, since it is fully trainable and achieves flexible\npredictions. Here, we expand on the original experiments to show that SIP is\ncapable of correcting model bias when the data generating mechanism differs\nstrongly from the one implied by the model. We use synthetic datasets to show\nthat SIP is capable of providing predictive distributions that reflect the data\nbetter than the exact predictions of the initial, but wrongly assumed model.\n", "date": "2022-07-21", "categories": []}
{"id": "2208.04221", "title": "Uncertain Bayesian Networks: Learning from Incomplete Data", "abstract": "  When the historical data are limited, the conditional probabilities\nassociated with the nodes of Bayesian networks are uncertain and can be\nempirically estimated. Second order estimation methods provide a framework for\nboth estimating the probabilities and quantifying the uncertainty in these\nestimates. We refer to these cases as uncer tain or second-order Bayesian\nnetworks. When such data are complete, i.e., all variable values are observed\nfor each instantiation, the conditional probabilities are known to be\nDirichlet-distributed. This paper improves the current state-of-the-art\napproaches for handling uncertain Bayesian networks by enabling them to learn\ndistributions for their parameters, i.e., conditional probabilities, with\nincomplete data. We extensively evaluate various methods to learn the posterior\nof the parameters through the desired and empirically derived strength of\nconfidence bounds for various queries.\n", "date": "2022-08-08", "categories": []}
{"id": "2207.13167", "title": "One Simple Trick to Fix Your Bayesian Neural Network", "abstract": "  One of the most popular estimation methods in Bayesian neural networks (BNN)\nis mean-field variational inference (MFVI). In this work, we show that neural\nnetworks with ReLU activation function induce posteriors, that are hard to fit\nwith MFVI. We provide a theoretical justification for this phenomenon, study it\nempirically, and report the results of a series of experiments to investigate\nthe effect of activation function on the calibration of BNNs. We find that\nusing Leaky ReLU activations leads to more Gaussian-like weight posteriors and\nachieves a lower expected calibration error (ECE) than its ReLU-based\ncounterpart.\n", "date": "2022-07-26", "categories": []}
{"id": "2207.12688", "title": "Single MCMC Chain Parallelisation on Decision Trees", "abstract": "  Decision trees are highly famous in machine learning and usually acquire\nstate-of-the-art performance. Despite that, well-known variants like CART, ID3,\nrandom forest, and boosted trees miss a probabilistic version that encodes\nprior assumptions about tree structures and shares statistical strength between\nnode parameters. Existing work on Bayesian decision trees depend on Markov\nChain Monte Carlo (MCMC), which can be computationally slow, especially on high\ndimensional data and expensive proposals. In this study, we propose a method to\nparallelise a single MCMC decision tree chain on an average laptop or personal\ncomputer that enables us to reduce its run-time through multi-core processing\nwhile the results are statistically identical to conventional sequential\nimplementation. We also calculate the theoretical and practical reduction in\nrun time, which can be obtained utilising our method on multi-processor\narchitectures. Experiments showed that we could achieve 18 times faster running\ntime provided that the serial and the parallel implementation are statistically\nidentical.\n", "date": "2022-07-26", "categories": []}
{"id": "2207.08200", "title": "Uncertainty Calibration in Bayesian Neural Networks via Distance-Aware\n  Priors", "abstract": "  As we move away from the data, the predictive uncertainty should increase,\nsince a great variety of explanations are consistent with the little available\ninformation. We introduce Distance-Aware Prior (DAP) calibration, a method to\ncorrect overconfidence of Bayesian deep learning models outside of the training\ndomain. We define DAPs as prior distributions over the model parameters that\ndepend on the inputs through a measure of their distance from the training set.\nDAP calibration is agnostic to the posterior inference method, and it can be\nperformed as a post-processing step. We demonstrate its effectiveness against\nseveral baselines in a variety of classification and regression problems,\nincluding benchmarks designed to test the quality of predictive distributions\naway from the data.\n", "date": "2022-07-17", "categories": []}
{"id": "2207.08365", "title": "CausNet : Generational orderings based search for optimal Bayesian\n  networks via dynamic programming with parent set constraints", "abstract": "  Finding a globally optimal Bayesian Network using exhaustive search is a\nproblem with super-exponential complexity, which severely restricts the number\nof variables that it can work for. We implement a dynamic programming based\nalgorithm with built-in dimensionality reduction and parent set identification.\nThis reduces the search space drastically and can be applied to\nlarge-dimensional data. We use what we call generational orderings based search\nfor optimal networks, which is a novel way to efficiently search the space of\npossible networks given the possible parent sets. The algorithm supports both\ncontinuous and categorical data, and categorical as well as survival outcomes.\nWe demonstrate the efficacy of our algorithm on both synthetic and real data.\nIn simulations, our algorithm performs better than three state-of-art\nalgorithms that are currently used extensively. We then apply it to an Ovarian\nCancer gene expression dataset with 513 genes and a survival outcome. Our\nalgorithm is able to find an optimal network describing the disease pathway\nconsisting of 6 genes leading to the outcome node in a few minutes on a basic\ncomputer. Our generational orderings based search for optimal networks, is both\nefficient and highly scalable approach to finding optimal Bayesian Networks,\nthat can be applied to 1000s of variables. Using specifiable parameters -\ncorrelation, FDR cutoffs, and in-degree - one can increase or decrease the\nnumber of nodes and density of the networks. Availability of two scoring\noption-BIC and Bge-and implementation of survival outcomes and mixed data types\nmakes our algorithm very suitable for many types of high dimensional biomedical\ndata to find disease pathways.\n", "date": "2022-07-18", "categories": []}
{"id": "2207.03859", "title": "Variational Inference of overparameterized Bayesian Neural Networks: a\n  theoretical and empirical study", "abstract": "  This paper studies the Variational Inference (VI) used for training Bayesian\nNeural Networks (BNN) in the overparameterized regime, i.e., when the number of\nneurons tends to infinity. More specifically, we consider overparameterized\ntwo-layer BNN and point out a critical issue in the mean-field VI training.\nThis problem arises from the decomposition of the lower bound on the evidence\n(ELBO) into two terms: one corresponding to the likelihood function of the\nmodel and the second to the Kullback-Leibler (KL) divergence between the prior\ndistribution and the variational posterior. In particular, we show both\ntheoretically and empirically that there is a trade-off between these two terms\nin the overparameterized regime only when the KL is appropriately re-scaled\nwith respect to the ratio between the the number of observations and neurons.\nWe also illustrate our theoretical results with numerical experiments that\nhighlight the critical choice of this ratio.\n", "date": "2022-07-08", "categories": []}
{"id": "2206.11173", "title": "Cold Posteriors through PAC-Bayes", "abstract": "  We investigate the cold posterior effect through the lens of PAC-Bayes\ngeneralization bounds. We argue that in the non-asymptotic setting, when the\nnumber of training samples is (relatively) small, discussions of the cold\nposterior effect should take into account that approximate Bayesian inference\ndoes not readily provide guarantees of performance on out-of-sample data.\nInstead, out-of-sample error is better described through a generalization\nbound. In this context, we explore the connections between the ELBO objective\nfrom variational inference and the PAC-Bayes objectives. We note that, while\nthe ELBO and PAC-Bayes objectives are similar, the latter objectives naturally\ncontain a temperature parameter $\\lambda$ which is not restricted to be\n$\\lambda=1$. For both regression and classification tasks, in the case of\nisotropic Laplace approximations to the posterior, we show how this\nPAC-Bayesian interpretation of the temperature parameter captures the cold\nposterior effect.\n", "date": "2022-06-22", "categories": []}
{"id": "2206.07795", "title": "On Calibrated Model Uncertainty in Deep Learning", "abstract": "  Estimated uncertainty by approximate posteriors in Bayesian neural networks\nare prone to miscalibration, which leads to overconfident predictions in\ncritical tasks that have a clear asymmetric cost or significant losses. Here,\nwe extend the approximate inference for the loss-calibrated Bayesian framework\nto dropweights based Bayesian neural networks by maximising expected utility\nover a model posterior to calibrate uncertainty in deep learning. Furthermore,\nwe show that decisions informed by loss-calibrated uncertainty can improve\ndiagnostic performance to a greater extent than straightforward alternatives.\nWe propose Maximum Uncertainty Calibration Error (MUCE) as a metric to measure\ncalibrated confidence, in addition to its prediction especially for high-risk\napplications, where the goal is to minimise the worst-case deviation between\nerror and estimated uncertainty. In experiments, we show the correlation\nbetween error in prediction and estimated uncertainty by interpreting\nWasserstein distance as the accuracy of prediction. We evaluated the\neffectiveness of our approach to detecting Covid-19 from X-Ray images.\nExperimental results show that our method reduces miscalibration considerably,\nwithout impacting the models accuracy and improves reliability of\ncomputer-based diagnostics.\n", "date": "2022-06-15", "categories": []}
{"id": "2206.07673", "title": "Wide Bayesian neural networks have a simple weight posterior: theory and\n  accelerated sampling", "abstract": "  We introduce repriorisation, a data-dependent reparameterisation which\ntransforms a Bayesian neural network (BNN) posterior to a distribution whose KL\ndivergence to the BNN prior vanishes as layer widths grow. The repriorisation\nmap acts directly on parameters, and its analytic simplicity complements the\nknown neural network Gaussian process (NNGP) behaviour of wide BNNs in function\nspace. Exploiting the repriorisation, we develop a Markov chain Monte Carlo\n(MCMC) posterior sampling algorithm which mixes faster the wider the BNN. This\ncontrasts with the typically poor performance of MCMC in high dimensions. We\nobserve up to 50x higher effective sample size relative to no reparametrisation\nfor both fully-connected and residual networks. Improvements are achieved at\nall widths, with the margin between reparametrised and standard BNNs growing\nwith layer width.\n", "date": "2022-06-15", "categories": []}
{"id": "2206.05643", "title": "Density Regression and Uncertainty Quantification with Bayesian Deep\n  Noise Neural Networks", "abstract": "  Deep neural network (DNN) models have achieved state-of-the-art predictive\naccuracy in a wide range of supervised learning applications. However,\naccurately quantifying the uncertainty in DNN predictions remains a challenging\ntask. For continuous outcome variables, an even more difficult problem is to\nestimate the predictive density function, which not only provides a natural\nquantification of the predictive uncertainty, but also fully captures the\nrandom variation in the outcome. In this work, we propose the Bayesian Deep\nNoise Neural Network (B-DeepNoise), which generalizes standard Bayesian DNNs by\nextending the random noise variable from the output layer to all hidden layers.\nThe latent random noise equips B-DeepNoise with the flexibility to approximate\nhighly complex predictive distributions and accurately quantify predictive\nuncertainty. For posterior computation, the unique structure of B-DeepNoise\nleads to a closed-form Gibbs sampling algorithm that iteratively simulates from\nthe posterior full conditional distributions of the model parameters,\ncircumventing computationally intensive Metropolis-Hastings methods. A\ntheoretical analysis of B-DeepNoise establishes a recursive representation of\nthe predictive distribution and decomposes the predictive variance with respect\nto the latent parameters. We evaluate B-DeepNoise against existing methods on\nbenchmark regression datasets, demonstrating its superior performance in terms\nof prediction accuracy, uncertainty quantification accuracy, and uncertainty\nquantification efficiency. To illustrate our method's usefulness in scientific\nstudies, we apply B-DeepNoise to predict general intelligence from neuroimaging\nfeatures in the Adolescent Brain Cognitive Development (ABCD) project.\n", "date": "2022-06-12", "categories": []}
{"id": "2206.06357", "title": "Federated Bayesian Neural Regression: A Scalable Global Federated\n  Gaussian Process", "abstract": "  In typical scenarios where the Federated Learning (FL) framework applies, it\nis common for clients to have insufficient training data to produce an accurate\nmodel. Thus, models that provide not only point estimations, but also some\nnotion of confidence are beneficial. Gaussian Process (GP) is a powerful\nBayesian model that comes with naturally well-calibrated variance estimations.\nHowever, it is challenging to learn a stand-alone global GP since merging local\nkernels leads to privacy leakage. To preserve privacy, previous works that\nconsider federated GPs avoid learning a global model by focusing on the\npersonalized setting or learning an ensemble of local models. We present\nFederated Bayesian Neural Regression (FedBNR), an algorithm that learns a\nscalable stand-alone global federated GP that respects clients' privacy. We\nincorporate deep kernel learning and random features for scalability by\ndefining a unifying random kernel. We show this random kernel can recover any\nstationary kernel and many non-stationary kernels. We then derive a principled\napproach of learning a global predictive model as if all client data is\ncentralized. We also learn global kernels with knowledge distillation methods\nfor non-identically and independently distributed (non-i.i.d.) clients.\nExperiments are conducted on real-world regression datasets and show\nstatistically significant improvements compared to other federated GP models.\n", "date": "2022-06-13", "categories": []}
{"id": "2206.02435", "title": "Tackling covariate shift with node-based Bayesian neural networks", "abstract": "  Bayesian neural networks (BNNs) promise improved generalization under\ncovariate shift by providing principled probabilistic representations of\nepistemic uncertainty. However, weight-based BNNs often struggle with high\ncomputational complexity of large-scale architectures and datasets. Node-based\nBNNs have recently been introduced as scalable alternatives, which induce\nepistemic uncertainty by multiplying each hidden node with latent random\nvariables, while learning a point-estimate of the weights. In this paper, we\ninterpret these latent noise variables as implicit representations of simple\nand domain-agnostic data perturbations during training, producing BNNs that\nperform well under covariate shift due to input corruptions. We observe that\nthe diversity of the implicit corruptions depends on the entropy of the latent\nvariables, and propose a straightforward approach to increase the entropy of\nthese variables during training. We evaluate the method on out-of-distribution\nimage classification benchmarks, and show improved uncertainty estimation of\nnode-based BNNs under covariate shift due to input perturbations. As a side\neffect, the method also provides robustness against noisy training labels.\n", "date": "2022-06-06", "categories": []}
{"id": "2206.05111", "title": "PAVI: Plate-Amortized Variational Inference", "abstract": "  Given some observed data and a probabilistic generative model, Bayesian\ninference aims at obtaining the distribution of a model's latent parameters\nthat could have yielded the data. This task is challenging for large population\nstudies where thousands of measurements are performed over a cohort of hundreds\nof subjects, resulting in a massive latent parameter space. This large\ncardinality renders off-the-shelf Variational Inference (VI) computationally\nimpractical. In this work, we design structured VI families that can\nefficiently tackle large population studies. To this end, our main idea is to\nshare the parameterization and learning across the different i.i.d. variables\nin a generative model -symbolized by the model's plates. We name this concept\nplate amortization, and illustrate the powerful synergies it entitles,\nresulting in expressive, parsimoniously parameterized and orders of magnitude\nfaster to train large scale hierarchical variational distributions. We\nillustrate the practical utility of PAVI through a challenging Neuroimaging\nexample featuring a million latent parameters, demonstrating a significant step\ntowards scalable and expressive Variational Inference.\n", "date": "2022-06-10", "categories": []}
{"id": "2206.02266", "title": "Information Threshold, Bayesian Inference and Decision-Making", "abstract": "  We define the information threshold as the point of maximum curvature in the\nprior vs. posterior Bayesian curve, both of which are described as a function\nof the true positive and negative rates of the classification system in\nquestion. The nature of the threshold is such that for sufficiently adequate\nbinary classification systems, retrieving excess information beyond the\nthreshold does not significantly alter the reliability of our classification\nassessment. We hereby introduce the \"marital status thought experiment\" to\nillustrate this idea and report a previously undefined mathematical\nrelationship between the Bayesian prior and posterior, which may have\nsignificant philosophical and epistemological implications in decision theory.\nWhere the prior probability is a scalar between 0 and 1 given by $\\phi$ and the\nposterior is a scalar between 0 and 1 given by $\\rho$, then at the information\nthreshold, $\\phi_e$:\n  $\\phi_e + \\rho_e = 1$\n  Otherwise stated, given some degree of prior belief, we may assert its\npersuasiveness when sufficient quality evidence yields a posterior so that\ntheir combined sum equals 1. Retrieving further evidence beyond this point does\nnot significantly improve the posterior probability, and may serve as a\nbenchmark for confidence in decision-making.\n", "date": "2022-06-05", "categories": []}
{"id": "2606.21080", "title": "Bayesian Model Averaging under Predictor Redundancy via Density-Ratio Posterior Compression", "abstract": "Bayesian model averaging in support-indexed regression induces a posterior distribution over active predictor supports. Under predictor redundancy, posterior mass can spread across many nearly interchangeable supports, making exact-support summaries unstable or hard to interpret even when prediction is stable. We study how to report an already fitted Bayesian model averaging posterior without changing the Bayesian target. A report uses hard or soft regions of support space, and its compressed reporting law is compared with the reference posterior through an explicit density ratio. This ratio gives computable total-variation and Kullback--Leibler distortion, bounds for bounded predictive summaries, retained-mass diagnostics, and fallback-weight diagnostics. The framework covers fixed hard regions, metric-ball regions, posterior-cluster regions, and pooled-pruned region dictionaries. We prove exact error formulas and validation bounds for these region reports, and give conditions under which a few regions can replace a long list of individual supports. In simulations, our region reports often give shorter and clearer summaries while preserving the main posterior information, and the density-ratio diagnostics show when too much information has been lost.", "date": "2026-06-19", "categories": []}
{"id": "2606.21099", "title": "ShuffleFlow: Scalable Posterior Inference for Bayesian Inverse Imaging", "abstract": "Variational inference (VI) is a powerful method for principled posterior inference for scientific inverse imaging. VI learns the posterior distribution, often with a flow-based network, which can cheaply generate posterior samples upon optimization, and can flexibly incorporate score-based or classic priors. However, its application to large-scale image reconstruction is severely hindered by the poor scalability of the flow-based networks. In this work, we introduce ShuffleFlow, a scalable VI framework to address this challenge. Our method breaks down the problem into three parts: a pixel-unshuffling-based image coordinate sampler, a neural field as feature encoder, and a conditional normalizing flow (CNF) as posterior estimator. Specifically, our framework partitions an image into a stack of sub-images with pixel-unshuffling and uses a shared CNF to model the joint distribution of the sub-image stack. We condition the CNF on the output of a neural field, which embeds feature vectors corresponding to pixel-unshuffling sample locations to capture spatial structures, and share the flow's latent variable across the channels to model their correlations. We demonstrate our method's effectiveness and efficiency on both linear and nonlinear imaging inverse problems, and show its ability to more rapidly generate a high-sample-count posterior than diffusion samplers.", "date": "2026-06-19", "categories": []}
{"id": "2606.21789", "title": "Bayesian three-dimensional seismic travel-time tomography for active- and passive-source seismic data using physics-informed neural network", "abstract": "Accurate 3D seismic velocity modeling through seismic travel-time tomography using both active- and passive-source data provides critical underpinning models for seismicity monitoring and hazard assessment. Because travel-time tomography is an inherently ill-posed inverse problem, UQ of the estimated models using Bayesian methods is also important for reliable downstream interpretations and analyses. However, Bayesian inference for 3D tomography based on conventional grid-based representations faces the ``curse of dimensionality'' and severe computational bottlenecks. Consequently, rigorous Bayesian UQ for margin-wide 3D travel-time tomography has remained largely unexplored. In this study, we propose a meshless 3D Bayesian travel-time tomography method that combines PINNs with a neural representation of the velocity structure, enabling tractable and data-efficient Bayesian inference through function-space particle-based variational inference. To efficiently integrate passive-source data into the Bayesian estimation of the velocity structure, we conduct analytical marginalization treating uncertain source parameters as nuisance parameters, with passive-source relocation carried out in post-processing. We validated the capability of our approach for 3D problems through synthetic experiments. Furthermore, we applied the method to a real-world dataset from marine active-source surveys and natural earthquakes off the Kii Peninsula, Nankai Trough. Our probabilistic 3D ensemble successfully resolves key geological features and provides data-consistent uncertainty maps. The posterior mean hypocenters shifted mainly in the vertical direction by 10-15 km, consistent with a previous relocation result. Finally, the neural representation drastically reduces storage requirements for the entire ensemble velocity model, highlighting the scalability and data efficiency of the proposed framework.", "date": "2026-06-19", "categories": []}
{"id": "2606.22188", "title": "Bayesian Adaptation Gym: A Benchmark for the Bayesian Low-Rank Adaptation of Multi-Modal Language Models", "abstract": "Large multi-modal language models are increasingly deployed in high-stakes domains, making well-calibrated uncertainty essential. Traditional Bayesian methods approximate posteriors over all model weights, which becomes intractable for modern large models. For this reason, recent work instead considers Bayesian low-rank adaptation to enable tractable posterior approximation. Due to a lack of a standardized benchmark to evaluate these approaches, it remains unclear where these methods provide meaningful benefits. To fill this gap, we introduce Bayesian Adaptation Gym (BAG), a benchmark for the Bayesian adaptation of multi-modal language models. BAG provides reference implementations of classic Bayesian baselines and state-of-the-art adaptation methods, along with a multi-modal dataset and task suite designed to probe calibration, robustness under distribution shift, and decision-making under uncertainty via active learning. Using BAG, we conduct and report extensive experiments across model sizes, datasets, and tasks to highlight the successes and failures of current Bayesian adaptation approaches. To enable further research, BAG is fully open source: https://github.com/SRI-CSL/BayesAdapt.", "date": "2026-06-20", "categories": []}
{"id": "2606.22200", "title": "Neural Conjugate Aggregation: Identifiable Unsupervised Multi-Sensor Regression under Heterogeneous Sensor Bias", "abstract": "We study regression-based data fusion under uncertainty, where multiple noisy and biased measurement sources are available but ground-truth labels are absent during training. This setting arises in sensor networks, simulation ensembles, and scientific monitoring systems where supervision is costly or infeasible. We propose the Neural Conjugate Aggregation Model (NCAM), a hierarchical Bayesian framework that combines neural networks with conjugate Gaussian inference for unsupervised multi-source fusion. NCAM learns source-specific bias and reliability conditioned on contextual covariates, yielding an analytically tractable posterior over a latent target variable with decomposed epistemic and aleatoric uncertainty. Structural non-identifiability is resolved through sensor anchoring and variance regularization, enabling stable and interpretable posterior aggregation. To complement Bayesian uncertainty with finite-sample guarantees, we integrate locally adaptive Monte Carlo conformal prediction, producing heteroscedastic prediction intervals with coverage guarantees under exchangeability assumptions. Experiments on synthetic and real-world air-quality datasets demonstrate improved predictive accuracy and well-calibrated uncertainty compared to unsupervised baselines, including mean aggregation, probabilistic PCA, and Kalman filtering.", "date": "2026-06-20", "categories": []}
{"id": "2606.22601", "title": "Scalable Bayesian Additive Models for Stellar Flare Detection via Amortized Gaussian Process Inference and Hidden Markov Models", "abstract": "Gaussian Processes (GPs) are a powerful tool for Bayesian time-series modeling, yet their cubic computational cost remains a severe barrier for application to long, high-cadence datasets in astronomy. While specialized scalable solvers like Celerite elegantly reduce this scaling to linear time, repeatedly evaluating the exact likelihood during iterative Bayesian sampling is a bottleneck for developing more complex models, like hierarchical or additive models in which Celerite is only one component. To make this inference computationally tractable, we introduce a generative surrogate framework. By utilizing a Variational Autoencoder (VAE) to learn a compressed representation of the Celerite prior, we map highly correlated stochastic dependencies into a low-dimensional, isotropic manifold. This transition completely bypasses exact covariance operations, shifting the computational burden to a rapid neural network forward pass. Through an extensive simulation study, we show that the generative surrogate accurately reproduces the structural fidelity of exact physical kernels like Celerite. Finally, we demonstrate embedding our VAE approximation into an additive model that combines Celerite and a hidden Markov model (HMM) for stellar flare detection in time series data of stars. We evaluate the joint VAE+HMM architecture against the exact Celerite+HMM framework on empirical astrophysical time series and demonstrate that the proposed methodology achieves significant reductions in computational time, enabling the rigorous, large-scale characterization of stellar flares across massive data archives.", "date": "2026-06-21", "categories": []}
{"id": "2606.23662", "title": "Action-BED: Task-Driven Bayesian Experimental Design with Singly Intractable Objectives", "abstract": "Bayesian experimental design (BED) has traditionally been based on maximising expected uncertainty reductions from prior to posterior. A major shortfall of this approach is that it leads to doubly intractable objectives that are difficult to optimise, while customising them to particular downstream tasks of interest can also be difficult. Following first principles decision theory, we demonstrate that BED can alternatively be formulated in terms of an expected future loss (EFL) on downstream actions, providing a simple and naturally task-driven framework. Critically, we then show that all such EFLs can be rearranged into singly intractable objectives that can be jointly optimised with respect to both the design policy and a downstream action policy using stochastic gradients, an approach we refer to as ACTION-BED. This formulation further sidesteps the need for any explicit posterior or marginal likelihood estimation and is naturally implicit, requiring only the ability to sample from the joint model over model parameters and data, and evaluate the downstream loss function. It thus allows design policies to be learned more effectively, efficiently, and simply than existing methods, while providing easy customisation to different downstream tasks and losses.", "date": "2026-06-22", "categories": []}
{"id": "2606.25745", "title": "Gaussian Mean Field Variational Inference can Overestimate Predictive Variance", "abstract": "Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian Linear Regression (BLR), we show that this characterization is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on in distribution data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.", "date": "2026-06-24", "categories": []}
{"id": "2606.26273", "title": "Equivariance and Augmentation for Bayesian Neural Networks", "abstract": "Symmetries are important for many deep learning tasks, ranging from applications in the sciences to medical imaging. However, there is an ongoing debate about whether to impose symmetry constraints on the neural network architecture (yielding equivariant neural networks) or learn them from augmented training data. Although equivariant networks are well-studied theoretically, much less is known about data augmentation, since analyzing augmentation requires control over the training dynamics. Inspired by recent results that show that augmented infinite deep ensembles are exactly equivariant, we study data augmentation for Bayesian neural networks (BNNs) trained with variational inference. We focus on variational distributions in the exponential family and derive conditions under which exact equivariance is reached. We furthermore obtain bounds on the equivariance error and introduce three novel symmetrization techniques which boost the effect of data augmentation in this setting. We conduct extensive numerical experiments which show that one of our symmetrization methods (orbit expansion) outperforms the baseline in both equivariance and overall performance. Our code is available at github.com/dmw1998/augment-BNNs", "date": "2026-06-24", "categories": []}
{"id": "2606.27090", "title": "Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference", "abstract": "Global objectives, such as KL divergence and ELBO, are widely used in Bayesian inference for measuring distributional discrepancy. This paper studies their local-mass behaviour that is not directly captured by such objectives. We introduce and use two mathematical tools: (1) Mass Index for recording the polynomial and logarithmic decay scales of local mass, and (2) regularised extended KL (RE-KL), a set-localised divergence that can be formulated in the presence of singular components. Mass Indices help characterise how Bayesian updating changes local mass: (1) power-log likelihood factors shift it explicitly, and (2) parameter-dependent supports, or their smooth softenings, may change the local scale through the amount of mass that remains near the parameter value. Using local RE-KL, we prove absolute, relative, and directional inequalities for comparing local small-ball masses under the two KL directions. Together, these results provide a local theoretical account of local mass behaviour. Experiments provide controlled illustrations of the local behaviour. Code is available at https://github.com/Forsythia0604/Local-Mass-Framework.", "date": "2026-06-25", "categories": []}
{"id": "2606.27286", "title": "Simulation-based inference for rapid Bayesian parameter estimation in epidemiological models: a comparison with MCMC", "abstract": "Mechanistic epidemiological models are widely used to support infectious disease forecasting and public-health decision making. Bayesian calibration of such models is commonly performed using Markov chain Monte Carlo (MCMC), which can become computationally expensive for high-dimensional nonlinear systems and repeated near-real-time analyses. Here, we investigate simulation-based inference (SBI) using neural posterior estimation as a scalable alternative for Bayesian calibration of a mechanistic SECIR epidemiological model using COVID-19 intensive care unit (ICU) occupancy data from Germany during 2020. We compared SBI and MCMC across multiple epidemic phases using both 31-day inference windows and a substantially more challenging 201-day reconstruction problem involving multiple transmission change points. Posterior agreement was evaluated quantitatively using Wasserstein distances and Kullback-Leibler divergences together with posterior predictive checks. Across the 31-day windows, SBI recovered posterior distributions in strong agreement with MCMC while accurately reproducing observed ICU trajectories. In the 201-day setting, SBI preserved the dominant posterior structure despite increased uncertainty. SBI, by combining CPU and GPU resources, substantially reduced computational runtime compared with MCMC, which was restricted to running on CPUs. Whereas MCMC required approximately 1000 seconds for the 31-day inference problems, SBI achieved comparable posterior and predictive performance in approximately 60-70 seconds on a single GPU. For the 201-day inference problem, SBI required an average of 157 seconds, while the MCMC runs took over 19,000 seconds. Our results demonstrate that SBI provides a rapid and computationally efficient framework for Bayesian calibration of mechanistic epidemiological models, supporting repeated near-real-time inference and rapid outbreak analysis.", "date": "2026-06-25", "categories": []}
{"id": "2606.29203", "title": "Bayesian Best-Arm Identification with Abstention: A Polynomial-to-Exponential Phase Transition", "abstract": "We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation. Subject to an abstention budget $\\alpha$, we analyze the probability of undetected error--the risk of recommending a suboptimal arm without abstaining. Our central finding is that abstention induces a phase transition: without abstention, the error probability decays polynomially in the sampling budget $T$; in contrast, introducing any small positive abstention budget shifts this to an exponential decay. For Gaussian priors and rewards, in the regime $T\\to\\infty$ followed by $\\alpha\\downarrow0$, we establish exact matching information-theoretic lower bounds and algorithmic upper bounds on the optimal error exponent, which takes the form $\\exp(-\\frac{\\alpha^{2}T}{8\\kappa_{\\nu}^{2}})$. The hardness parameter $\\kappa_{\\nu}$ represents the prior density of the top-two gap at zero, highlighting that nearly tied instances drive the fundamental error. We introduce an adaptive algorithm, PGWS, that successfully achieves this optimal exponent by expending its abstention budget on statistically ambiguous instances. We further demonstrate that this polynomial-to-exponential improvement is exclusively a Bayesian phenomenon--in the frequentist setting, abstention only affects lower-order exponent terms. We also extend our results beyond the Gaussian model.", "date": "2026-06-28", "categories": []}
{"id": "2606.29516", "title": "A Mathematical Optimization Approach for Expert-Informed Bayesian Best Subset Selection", "abstract": "A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.", "date": "2026-06-28", "categories": []}
{"id": "2606.30440", "title": "Transformer Architectures as Complete Bayes Processes: A Formal Proof in the Measure-Theoretic Kernel Framework", "abstract": "We present a complete formal proof that transformer architectures, when their internal update mechanisms satisfy a Bayes joint-distribution condition, implement exact Bayesian posterior inference. Working within the measure-theoretic kernel framework, we define a hierarchy of abstractions -- from the core Bayesian transformer, through semantic transformers with explicit update kernels, to full transformer blocks with QKV/attention/residual/MLP pipelines, and finally multilayer stacks -- and prove at each level that the Bayes joint semantics implies the update kernel equals the posterior almost everywhere. For the block-level architecture, we derive the explicit Bayes formula through Radon-Nikodym differentiation and prove its normalization. We additionally prove that the softmax attention mechanism induces a valid probability distribution over keys, establishing the bridge between the abstract kernel framework and concrete attention implementations. The framework makes no architectural assumptions beyond the Markov kernel structure and exposes explicit conditions under which a transformer block is provably Bayesian. In essence, when this joint distribution condition is satisfied, the forward computation of a Transformer is formally equivalent to a rigorous Bayesian posterior update.", "date": "2026-06-29", "categories": []}
{"id": "2606.31063", "title": "Dynamic Gaussian Processes and the Vanilla-SPDE Exchange", "abstract": "Gaussian process inference is often limited by cubic computational costs, a challenge that becomes more pronounced in spatio-temporal settings where posterior inference is required over dense grids. While state-space SPDE formulations enable linear complexity in time, exact inference remains cubic in space and deteriorates further when observation locations are disjoint from the prediction locations, which inflates the number of considered spatial points. To address this, we propose the Vanilla-SPDE Exchange, which exploits an equivalence between the standard and SPDE formulations of GP inference to construct a hybrid scheme with improved computational cost. We demonstrate these gains through complexity analysis and numerical experiments.", "date": "2026-06-30", "categories": []}
{"id": "2606.31284", "title": "Sequential sparse Gaussian process quantile regression", "abstract": "Quantile regression aims to estimate the conditional quantiles of a response variable from observed data. In a Bayesian setting, Gaussian process quantile regression provides uncertainty quantification but faces significant computational challenges due to the nonconjugacy of the asymmetric Laplace likelihood and the cost of posterior inference. We develop a sparse Gaussian process framework in which the quantile function is represented through a reduced set of inducing variables and posterior inference is performed using a Laplace approximation. A decomposition of the predictive uncertainty into conditional-prior and posterior-induced variance components is then exploited to drive two complementary adaptive mechanisms: inducing-input infilling and data acquisition. These mechanisms are combined within a sequential algorithm that allocates computational effort toward the dominant source of predictive uncertainty and adaptively controls model complexity. Numerical experiments on benchmark problems demonstrate the accuracy of the Laplace approximation, the benefits of variance-based inducing-input placement, and the effectiveness of the proposed sequential enrichment strategy compared with predefined data-acquisition strategies.", "date": "2026-06-30", "categories": []}