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2024/Conditional Information Bottleneck Approach for Time Series Imputation/7ceca7f2-abdb-40c9-b357-bdec02d221bb_origin.pdf

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2024/Conditional Information Bottleneck Approach for Time Series Imputation/full.md

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+# CONDITIONAL INFORMATION BOTTLENECK APPROACH FOR TIME SERIES IMPUTATION
+
+MinGyu Choi
+
+Massachusetts Institute of Technology, USA chemgyu@mit.edu
+
+Changhee Lee
+
+Chung-Ang University, Korea  
+changheelede@cau.ac.kr
+
+# ABSTRACT
+
+Time series imputation presents a significant challenge because it requires capturing the underlying temporal dynamics from partially observed time series data. Among the recent successes of imputation methods based on generative models, the information bottleneck (IB) framework offers a well-suited theoretical foundation for multiple imputations, allowing us to account for the uncertainty associated with the imputed values. However, direct application of IB framework to time series data without considering their temporal context can lead to a substantial loss of temporal dependencies, which, in turn, can degrade the overall imputation performance. To address such a challenge, we propose a novel conditional information bottleneck (CIB) approach for time series imputation, which aims to mitigate the potentially negative consequences of the regularization constraint by focusing on reducing the redundant information conditioned on the temporal context. We provide a theoretical analysis of its effect by adapting variational decomposition. We use the resulting insight and propose a novel deep learning method that can approximately achieve the proposed CIB objective for time series imputation as a combination of evidence lower bound and novel temporal kernel-enhanced contrastive optimization. Our experiments, conducted on multiple real-world datasets, consistently demonstrate that our method significantly improves imputation performance (including both interpolation and extrapolation), and also enhances prediction performance based on the imputed values.
+
+# 1 INTRODUCTION
+
+Multivariate time series data often includes missing features, with diverse missing ratios and patterns depending on distinct sampling periods or measurement strategies (Johnson et al., 2016). Since these missing features can significantly impair the performance of downstream tasks and comprehension of the temporal dynamics, time series imputation, which aims to reconstruct the missing features, has become a pivotal and pervasive topic across numerous practical domains, including healthcare, environmental science, and various other fields. What makes time series imputation challenging is that an imputation method must satisfy two essential requirements: i) it must account for underlying temporal dependencies, and ii) it should allow for multiple imputations to facilitate uncertainty quantification for real-world decision-making.
+
+Generative models, particularly variational autoencoders (VAEs) (Kingma & Welling, 2014), have been employed in the context of multiple imputation tasks due to their capability to generate samples in a probabilistic manner. VAE-based imputation methods primarily focus on defining the evidence lower bound, where the reconstruction error is computed only over the observed part of the incomplete data (Sohn et al., 2015; Nazabal et al., 2020). These methods can be naturally interpreted under the information bottleneck (IB) principle (Tishby & Zaslavsky, 2015), providing an information-theoretic understanding of what constitutes an imputation-relevant representation. This understanding is based on the fundamental trade-off between maintaining a concise representation (i.e., regularization) and preserving good representation power (i.e., reconstruction) (Voloshynovskiy et al., 2019).
+
+However, a direct application of the IB principle to time series imputation struggles with capturing the underlying temporal dependencies. Our motivating examples in Figure 1(B) show that imputation methods trained with the conventional IB framework lose a significant amount of information about temporal dynamics relevant for imputing missing values. In this paper, we theoretically analyze that
+
+the overly strict regularization in the conventional IB may force the encoder to rely solely on the observed features at a particular time point, rather than learning the underlying temporal dependencies from the remaining observations from other time steps. To overcome such an issue, we propose a novel conditional information bottleneck (CIB) framework for time series imputation. Our framework adopts the reconstruction-regularization structure of the IB principle while preserving temporal information through conditional regularization, allowing us to circumvent the strict regularization constraints of the conventional IB. Throughout the experiments conducted on multiple real-world datasets including image sequences, weather measurements, and electrical health records, our proposed method consistently outperforms the state-of-the-art imputation methods with respect to both imputation performance and prediction performance based on the imputed values.
+
+# 2 PRELIMINARIES: INFORMATION BOTTLENECK APPROACH TO IMPUTATION
+
+In this section, we first formally describe the information bottleneck (IB) principle (Tishby & Zaslavsky, 2015; Alemi et al., 2017), which provides an information-theoretic understanding of what a task-relevant representation is in terms of the fundamental trade-off between having a concise representation and good representative power. Then, we present a generative model for imputing missing features under the IB principle.
+
+Let  $\mathbf{X}$  and  $\mathbf{Y}$  be random variables for the input feature and the target label, respectively. The IB principle aims to find the bottleneck random variable  $\mathbf{Z}$  that compresses the information in  $\mathbf{X}$  while keeping the information relevant for predicting  $\mathbf{Y}$  as the following (Tishby & Zaslavsky, 2015),
+
+$$
+\min  _ {\phi , \theta} I _ {\phi} (\mathbf {Z}; \mathbf {X}) - \beta I _ {\theta} (\mathbf {Y}; \mathbf {Z}) \tag {1}
+$$
+
+where  $\beta \in \mathbb{R}$  is a Lagrangian multiplier that balances the two mutual information terms, and  $\phi$  and  $\theta$  correspond to learnable parameters that define probabilistic mappings  $q_{\phi}(\mathbf{Z}|\mathbf{X})$  and  $q_{\theta}(\mathbf{Y}|\mathbf{Z})$ , respectively. The core motivation of (1) is to find the optimal distribution of latent representation  $\mathbf{Z}$  and the corresponding inference model parameters  $\phi$  that removes label-irrelevant information from  $\mathbf{X}$  while preserving the information about the class label  $\mathbf{Y}$ .
+
+This offers an information-theoretic perspective on generative model-based imputation methods which generate missing observations from the observed features.
+
+Definition 1. (Imputation) Let  $\mathbf{X}^o$  and  $\mathbf{X}^m$  be random variables for the partially observed features and missing features of  $\mathbf{X}$ , respectively, such that  $\mathbf{X} = \mathbf{X}^o \cup \mathbf{X}^m$ . Then, we define imputation as an unsupervised IB as follows:
+
+$$
+\min  _ {\phi , \theta} I _ {\phi} (\mathbf {Z}; \mathbf {X} ^ {o}) - \beta I _ {\theta} (\mathbf {X}; \mathbf {Z}) \tag {2}
+$$
+
+where  $\beta \in \mathbb{R}_{\geq}$  is a Lagrangian multiplier, and  $\phi$  and  $\theta$  correspond to learnable parameters that define probabilistic mappings  $q_{\phi}(\mathbf{Z}|\mathbf{X}^{o})$  and  $q_{\theta}(\mathbf{X}|\mathbf{Z})$ , respectively.
+
+The above definition in (2) aims at finding the distribution of latent representation  $\mathbf{Z}$  and the corresponding parameters  $\phi$  that preserves the core information for accurately reconstructing the (complete) original input  $\mathbf{X}$  while suppressing redundant information from its incomplete observation,  $\mathbf{X}^o$ .
+
+# 3 METHOD
+
+# 3.1 PROBLEM FORMULATION
+
+We consider a general temporal dynamics setting in which each instance of a (discrete) time series input comprises a sequence of measurements (i.e., observations), denoted as  $\mathbf{x}_{1:T} \stackrel{\mathrm{def}}{=} [\mathbf{x}_1, \ldots, \mathbf{x}_T]$ , collected during the time interval  $[0, \tau_T)$ . Here,  $\mathbf{x}_t \in \mathbb{R}^d$  is the complete input vector measured at time  $t \in [\tau_{t-1}, \tau_t)$  and is a realization of a random variable  $\mathbf{X}_t$ . However, in practice, time series data often contains missing features with arbitrary patterns such that  $x_t^l$  is not observed during  $[\tau_{t-1}, \tau_t)$  for any feature  $l \in \{1, \dots, d\}$  at any time step  $t \in \{1, \dots, T\}$ . This phenomenon is particularly common in domains such as healthcare (Johnson et al., 2016) where each feature may
+
+![](images/b2e7b92ba8664e930bea6f3de383be53982d1a8ddc95dc4b926d663c42a3f20d.jpg)  
+Figure 1: (A) Conceptual illustration of the IB and CIB principles. By conditioning regularization on the remaining input time steps, the latent representation can better preserve the underlying temporal dependency. (B) Motivating experimental results on interpolation (left) and extrapolation (right). Because features in a single time step are completely missing, a model must collect information from other time steps. The conventional IB approach (HI-VAE) shows deteriorating performance in both cases. Another IB approach (GP-VAE) using a Gaussian process prior demonstrates enhanced performance for interpolation but often significantly loses time series characteristics for extrapolation (i.e., the writing style is corrupted). The CIB approach (Ours) exhibits improved imputation performance for both cases. Complete quantitative results are available in Table 1.
+
+have a distinct sampling period or when non-uniform sampling strategies are employed. To denote missing observations, we partition the input vector  $\mathbf{x}_t$  at each time step into observed features  $\mathbf{x}_t^o$  and missing features  $\mathbf{x}_t^m$ , such that  $\mathbf{x}_t = \mathbf{x}_t^o \cup \mathbf{x}_t^m$ .
+
+Objective. Our aim is to reconstruct the complete time series input  $\mathbf{x}_{1:T}$  by filling in the missing features from the observed time series input  $\mathbf{x}_{1:T}^{o}$ . Formally, we seek to generate  $\mathbf{x}_t^m$  from the conditional distribution  $p(\mathbf{X}_t^m|\mathbf{X}_{1:T}^o)$ . By modeling the conditional distribution  $p(\mathbf{X}_t^m|\mathbf{X}_{1:T}^o)$  instead of using a deterministic mapping, we can generate multiple imputations, allowing us to capture the uncertainty associated with the imputed values.
+
+What makes this problem challenging is that we must account for the underlying temporal dynamics represented by  $\mathbf{x}_{1:T}^{o}$  when imputing missing features  $\mathbf{x}_t^m$  for  $t\in \{1,\dots ,T\}$ . We can straightforwardly apply the unsupervised IB described in (2) to obtain latent representations  $\mathbf{Z}_t$  by discarding information from the observed time series input  $\mathbf{X}_{1:T}^{o}$  that is redundant for reconstructing  $\mathbf{X}_t$ . Formally, this can be achieved by minimizing  $I_{\phi}(\mathbf{Z}_t;\bar{\mathbf{X}}_{1:T}^{o}) - \beta I_{\theta}(\mathbf{X}_t;\mathbf{Z}_t)$  with a comprehensive encoder (e.g., RNN or Transformer) capable of effectively modeling the temporal dependencies within the observed time series observations, i.e.,  $q_{\phi}(\mathbf{Z}_t|\mathbf{X}_{1:T}^{o})$ . However, enforcing such strict regularization constraints on the encoder may lead to a significant loss of information regarding the temporal context that can be achieved by observations at different time steps, which we denote as  $\mathbf{X}_{\backslash t}^{o}\stackrel {\mathrm{def}}{=}(\mathbf{X}_{\tau}^{o}:\tau \in \{1,\ldots ,T\} \setminus t)$ . This may cause the imputation of  $\mathbf{X}_t^m$  at time step  $t$  to heavily rely on the observed features at that particular time point, i.e.,  $\mathbf{X}_t^o$ , rather than being able to learn from temporal dependencies present in other observations, i.e.,  $\mathbf{X}_{\backslash t}^{o}$  (as shown in Figure 1(B)).
+
+To tackle this issue, we alleviate the potentially negative consequences of the regularization constraint by directing our attention to the redundant information of the observed input at time step  $t$  when it is conditioned on its temporal context represented by the remaining observed time series  $\mathbf{X}_{\backslash t}^{o}$ . This offers a novel information-theoretic rationale for time series imputation, as defined below:
+
+Definition 2. (Time Series Imputation) Let  $\mathbf{X}_t^o$  and  $\mathbf{X}_t^m$  be random variables for the partially observed features and missing features of  $\mathbf{X}_t$  at time step  $t$ . Then, given the observed time series input  $\mathbf{X}_{1:T}^o$ , we define time series imputation at time step  $t$  as an unsupervised CIB as follows:
+
+$$
+\min  _ {\phi , \theta} \underbrace {I _ {\phi} \left(\mathbf {Z} _ {t} ; \mathbf {X} _ {t} ^ {o} \mid \mathbf {X} _ {\backslash t} ^ {o}\right)} _ {\text {C o n d i t i o n a l R e g u l a r i z a t i o n}} - \underbrace {\beta I _ {\theta} \left(\mathbf {X} _ {t} ; \mathbf {Z} _ {t}\right)} _ {\text {R e c o n s t r u c t i o n}} \tag {3}
+$$
+
+where  $\mathbf{X}_{\backslash t}^{o}$  represents the random variables for the remaining input observations, excluding  $\mathbf{X}_t^o$
+
+By conditioning on  $\mathbf{X}_{\backslash t}^{o}$ , (3) guides us to find latent representations  $\mathbf{Z}_t$  and the corresponding inference model parameter  $\phi$  which encompasses all retrievable information from the entire observed input time series  $\mathbf{X}_{1:T}^{o}$  (reconstruction), while discarding information that is redundant for capturing  $\mathbf{X}_t^m$  given the available temporal context from the remaining observed time series  $\mathbf{X}_{\backslash t}^{o}$  (conditional regularization). Overall, the proposed objective in (3) enables us to more effectively utilize information
+
+from  $\mathbf{X}_{\backslash t}^{o}$  for computing  $\mathbf{X}_t^m$  compared to other IB-related alternatives, whose conceptual illustration can be seen in Figure 1(A).
+
+# 3.2 DEEP VARIATIONAL CONDITIONAL INFORMATION BOTTLENECK ON TIME SERIES
+
+In this subsection, we will transform our objective (3) into a learnable form by utilizing variational decomposition. (3) is represented as a combination of the traditional ELBO with mutual information along the time axis, which can be approximately achieved by minimizing the contrastive loss.
+
+# 3.2.1 MAXIMIZING RECONSTRUCTION:  $\min_{\phi, \theta} -I(\mathbf{X}_t; \mathbf{Z}_t)$
+
+Following the derivations introduced in (Voloshynovskiy et al., 2019), we can find a lower bound of the reconstruction term as the following:
+
+$$
+\begin{array}{l} I _ {\theta} (\mathbf {X} _ {t}; \mathbf {Z} _ {t}) = \mathcal {H} (\mathbf {x} _ {t}) + D _ {\mathrm {K L}} (p (\mathbf {x} _ {t} | \mathbf {z} _ {t}) | | p _ {\theta} (\mathbf {x} _ {t} | \mathbf {z} _ {t})) + \mathbb {E} _ {\mathbf {x} _ {1: T} ^ {o} \sim p _ {\mathrm {d a t a}}} \left[ \mathbb {E} _ {\mathbf {z} _ {t} \sim q _ {\phi} (\mathbf {z} _ {t} | \mathbf {x} _ {1: T} ^ {o})} \left[ \log p _ {\theta} (\mathbf {x} _ {t} | \mathbf {z} _ {t}) \right] \right] \\ \geq \mathbb {E} _ {\mathbf {x} _ {1: T} ^ {o} \sim p _ {\mathrm {d a t a}}} \left[ \mathbb {E} _ {\mathbf {z} _ {t} \sim q _ {\phi} (\mathbf {z} _ {t} | \mathbf {x} _ {1: T} ^ {o})} \left[ \log p _ {\theta} \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right) \right] \right] \stackrel {\text {d e f}} {=} - \mathcal {L} _ {\phi , \theta} ^ {1} \tag {4} \\ \end{array}
+$$
+
+where  $\mathcal{H}(\cdot)$  is the entropy and the last inequality holds due to the non-negativity of entropy and KL-divergence. Here, we introduce a feature estimator, denoted as  $p_{\theta}(\mathbf{X}_t|\mathbf{Z}_t)$ , as a variational approximation of  $p(\mathbf{X}_t|\mathbf{Z}_t)$ . We model the feature estimator as an isotropic Gaussian, i.e.,  $p_{\theta}(\mathbf{X}_t|\mathbf{Z}_t) = \mathcal{N}(\mu_{\theta}(\mathbf{Z}_t),\mathrm{diag}(\sigma_{\theta}(\mathbf{Z}_t)))$  where  $\mu_{\theta}(\cdot)$  and  $\sigma_{\theta}(\cdot)$  are implemented by neural networks parameterized by  $\theta$ .
+
+In many practical scenarios, the ground-truth values for missing features are unknown during training. Thus, to accurately learn the reconstruction process given the latent representation of the observed time series, we apply (4) only to the features observed at each time point, similar to the approach in (Nazabal et al., 2020).
+
+Discussion on the Conditional Reconstruction. One might question why the reconstruction term is not conditioned on  $\mathbf{X}_{\backslash t}^{o}$ , as given by an alternative form of the CIB, i.e.,  $\min_{\phi,\theta} I_{\phi}(\mathbf{Z}_t; \mathbf{X}_t^o | \mathbf{X}_{\backslash t}^o) - \beta I(\mathbf{X}_t; \mathbf{Z}_t | \mathbf{X}_{\backslash t}^o)$ . Applying the chain rule of mutual information decomposes the conditional reconstruction as the follows:  $I(\mathbf{X}_t; \mathbf{Z}_t | \mathbf{X}_{\backslash t}^o) = I(\mathbf{X}_t; \mathbf{Z}_t, \mathbf{X}_{\backslash t}^o) - I(\mathbf{X}_t; \mathbf{X}_{\backslash t}^o)$ . It turns out that the first term can be bounded by a mathematically equivalent expression as shown in (4), suggesting that this term encourages mitigating constraints on temporal context for reconstruction (see Appendix A.1). However, minimizing the second term attempts to eliminate information about the target  $\mathbf{X}_t$  at time point  $t$ , which can be achieved from the observation other than time point  $t$ , i.e.,  $\mathbf{X}_{\backslash t}$ . This contradicts the goal of time series imputation where we aim to capture temporal context from the remaining observed time steps. Our empirical results also support that minimizing  $I(\mathbf{X}_t; \mathbf{X}_{\backslash t}^o)$  deteriorates the model performance (see Appendix A.2 for derivation, B.2 for experimental results).<sup>3</sup>
+
+# 3.2.2 MINIMIZING CONDITIONAL REGULARIZATION:  $\min_{\phi, \theta} I_{\phi}(\mathbf{Z}_t; \mathbf{X}_t^o | \mathbf{X}_{\backslash t}^o)$ .
+
+We employ the chain rule for mutual information on the conditional regularization term as follows:
+
+$$
+\min  _ {\phi , \theta} I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {t} ^ {o} \mid \mathbf {X} _ {\backslash t} ^ {o}\right) = \min  _ {\phi , \theta} I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {1: T} ^ {o}\right) - I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}\right). \tag {5}
+$$
+
+It is worth highlighting that the application of the chain rule decomposes the conditional regularization into two components: (i) minimizing the information between the latent representation  $\mathbf{Z}_t$  and the entire observed time series input  $\mathbf{X}_{1:T}^o$  that encourages the latent representation to be concise, while (ii) maximizing the information from  $\mathbf{X}_{\backslash t}^o$  to capture the underlying temporal dynamics provided by the observations at the remaining time steps. This prevents a significant loss of temporal context in the IB and, in turn, enhances the utilization of temporal dependencies from the remaining time steps.
+
+Minimizing  $I(\mathbf{Z}_t; \mathbf{X}_{1:T}^o)$ . The first term in (5) can be bounded as follows (see Appendix A.3):
+
+$$
+I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {1: T} ^ {o}\right) \leq \mathbb {E} _ {\mathbf {x} _ {1: T} ^ {o} \sim p _ {\mathrm {d a t a}}} \left[ D _ {\mathrm {K L}} \left(q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1: T} ^ {o}\right) | | p (\mathbf {z} _ {t})\right) \right] \stackrel {\text {d e f}} {=} \mathcal {L} _ {\phi} ^ {2} \tag {6}
+$$
+
+where we utilize the unit isotropic Gaussian as the prior distribution, i.e.,  $p(\mathbf{Z}_t) = \mathcal{N}(\mathbf{0},I)$ . We model the stochastic encoder as a multivariate Gaussian distribution defined as  $q_{\phi}(\mathbf{Z}_t|\mathbf{X}_{1:T}^o) = \mathcal{N}(\mu_{\phi}(\mathbf{X}_{1:T}^o),\mathrm{diag}(\sigma_{\phi}(\mathbf{X}_{1:T}^o))),$  where  $\mu_{\phi}(\cdot)$  and  $\sigma_{\phi}(\cdot)$  are implemented as neural networks parameterized by  $\phi$ . This explains why the (unconditional) IB struggles with modeling temporal dynamics, as discussed in Section 3.1. That is, (6) forces the encoder mappings that depend on time series inputs to converge to the unit Gaussian, imposing overly strict regularization. This hinders capturing of temporal dependencies present in other observations, motivating us to explicitly capture temporal dynamics by maximizing  $I(\mathbf{Z}_t;\mathbf{X}_{\backslash t}^o)$  rather than relying solely on the reconstruction signal in (4).
+
+Maximizing  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$ . To bound the second term in (5), we adopt the InfoNCE minimization from the contrastive learning on latent representations that approximately achieves maximizing the corresponding mutual information (Oord et al., 2018; Tian et al., 2020). Suppose we have the latent representation at time step  $t$  given an observed time series  $\mathbf{x}_{1:T}^o$  as a reference, i.e.,  $\mathbf{z}_t \sim q_{\phi}(\mathbf{Z}_t | \mathbf{x}_{1:T}^o)$ . Since our aim is to maximize information from  $\mathbf{X}_{\backslash t}^o$ , we intentionally make missing observations at time step  $t$  from the reference, and employ latent representations of time steps other than  $t$  as positive pairs. For negative pairs, we consider latent representations at different time steps from other time series observations. Finally, we define our novel contrastive learning loss with cosine similarity of latent representations along the time axis as follows (see Appendix A.4):
+
+$$
+I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}\right) \geq \mathbb {E} _ {\mathbf {x} _ {1: T} ^ {o} \sim p _ {\mathrm {d a t a}}} \left[ \log \left(\frac {\sum_ {t ^ {\prime} \in \{1 , \dots , T \} \backslash t} \exp \left(\mathbf {z} _ {t} ^ {T} \tilde {\mathbf {z}} _ {t ^ {\prime}} / \tau\right)}{\sum_ {\mathbf {x} _ {1 : T} ^ {-} \in \mathcal {X} _ {1 : T} ^ {-}} \sum_ {t ^ {\prime} \in \{1 , \dots , T \},} \exp \left(\mathbf {z} _ {t} ^ {T} \mathbf {z} _ {t ^ {\prime}} ^ {-} / \tau\right)}\right) \right] \stackrel {\text {d e f}} {=} - \mathcal {L} _ {\phi} ^ {3} \quad (7)
+$$
+
+where  $\tau$  is the temperature parameter. Here,  $\tilde{\mathbf{z}}_{t'} \sim q_{\phi}(\tilde{\mathbf{Z}}_{t'}|\mathbf{x}_{t'}^{o})$  denotes the positive pair obtained by masking the reference time series, such that  $\mathbf{x}_{t'}^{o}$  is created by replacing  $\mathbf{x}_t^o$  with zeros from  $\mathbf{x}_{1:T}^{o}$ . We regard such positive pairs as augmentations of a given time series since latent representations with missing values at time step  $t$  share task-relevant information about the underlying temporal dynamics of a given time series. We denote  $\mathcal{X}_{1:T}^{-}$  a set of negative samples comprising other time series in the same mini-batch, where  $\mathbf{x}_{1:T}^{-}$  indicates an observed time series from  $\mathcal{X}_{1:T}^{-}$ . This makes our encoder capture time series-level semantics – such as underlying disease progression patterns that can be distinguished from others – by pushing these samples from the reference. Such an attribution is necessary for reconstructing missing values (and associated downstream tasks in the experiments) specific to the input time series. Please refer to Appendix C for implementation details.
+
+# 3.2.3 OPTIMIZATION
+
+Now, we introduce a novel imputation method, which we refer to as Time-series Imputation using Conditional Information Bottleneck (TimeCIB), that consists of the stochastic encoder,  $q_{\phi}$ , and the feature estimator,  $p_{\theta}$ , introduced above. Please see Figure C1 for a schematic illustration of our framework. Overall, we optimize our method based on the following objective by combining all the loss functions that allows us to approximately achieve time series imputation defined in (3):
+
+$$
+\min  _ {\phi , \theta} \quad \beta \mathcal {L} _ {\phi , \theta} ^ {1} + \mathcal {L} _ {\phi} ^ {2} + \gamma \mathcal {L} _ {\phi} ^ {3} \tag {8}
+$$
+
+where  $\gamma \in \mathbb{R}_{\geq 0}$  is a balancing coefficient that trades off the impact of  $\mathcal{L}_\phi^3$ . We provide sensitivity analysis on  $\beta, \gamma$  in Appendix B.4.
+
+# 3.3 INTRODUCING INDUCTIVE BIAS ABOUT TEMPORAL DYNAMICS
+
+Now, we illustrate how we can inject inductive bias about the underlying temporal dynamics by employing temporal kernels to further improve the expressive power of TimeCIB.
+
+The alignment of the latent representation (Wang & Isola, 2020), attained through contrastive learning based on (7), renders the similarity between latent representations at two adjacent time points indistinguishable from the similarity between those at two distant time points. This phenomenon appears to contradict real-world temporal dynamics, such as gradually deteriorating or periodic behavior of disease progression patterns. To address this, we employ conditional alignment (Dufumier et al., 2021) that introduces inductive bias about the underlying temporal dynamics with temporal kernels as the following:
+
+$$
+I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}\right) \geq \mathbb {E} _ {\mathbf {x} _ {1: T} ^ {o} \sim p _ {\mathrm {d a t a}}} \left[ \log \left(\frac {\sum_ {t ^ {\prime} \in \{1 , \dots , T \} \backslash t} c _ {t , t ^ {\prime}} \exp \left(\mathbf {z} _ {t} ^ {T} \tilde {\mathbf {z}} _ {t ^ {\prime}} / \tau\right)}{\sum_ {\mathbf {x} _ {1 : T} ^ {-} \in \mathcal {X} _ {1 : T} ^ {-}} \sum_ {t ^ {\prime} \in \{1 , \dots , T \},}} \exp \left(\mathbf {z} _ {t} ^ {T} \mathbf {z} _ {t ^ {\prime}} ^ {-} / \tau\right)}\right) \right] \stackrel {\text {d e f}} {=} - \mathcal {L} _ {\phi} ^ {3 ^ {\prime}} \tag {9}
+$$
+
+where  $c_{t,t'} \in \mathbb{R}$  is a kernel coefficient as a function of two time points  $t$  and  $t'$ .
+
+Incorporating prior knowledge of underlying similarity into contrastive learning is not a novel concept. Several previous works have leveraged supervisory information to adapt contrastive learning, including semantic similarities for text classification (Suresh & Ong, 2021), angle similarity for gaze estimation (Wang et al., 2022), and Gaussian priors for time series and video representation learning (Tonekaboni et al., 2021; Chen et al., 2022). In this paper, we evaluate the following two representative temporal kernels to evaluate our method on data with different temporal behaviors and will leave the choice of the kernel as a hyperparameter:
+
+$$
+c _ {\text {c a u c h y}} \left(\tau , \tau^ {\prime}\right) = \sigma^ {2} \left(1 + \frac {\left(\tau - \tau^ {\prime}\right) ^ {2}}{l ^ {2}}\right) ^ {- 1}, \quad c _ {\text {p e r i o d i c}} \left(\tau , \tau^ {\prime}\right) = \sigma^ {2} \exp \left(- \frac {2 \sin^ {2} \left(\pi \left(\tau - \tau^ {\prime}\right) / p\right)}{l ^ {2}}\right) \tag {10}
+$$
+
+- Cauchy Smoothing. Under the assumption that two nearby time points should be more similar than those far away, we smooth the latent representations of time series by assigning higher weights to nearby time points when pulling the representations, utilizing the Cauchy kernel defined as  $c_{\mathrm{cauchy}}(\tau, \tau')$  in (10) which is the mixture of infinite RBF kernels with different time scales (Rasmussen, 2004). This is a generalized form that reduces to uniform weights as in (7), i.e.,  $l = \infty$  gives  $c(\tau, \tau') = \sigma^2$ .  
+- Periodic Smoothing. Unfortunately, Cauchy smoothing may not be appropriate when the underlying temporal dynamics exhibit periodic behavior (e.g., seasonality). To incorporate our domain knowledge about periodic time series data, we utilize the exponentiated sine-squared kernel given as  $c_{\mathrm{periodic}}(\tau, \tau')$  in (10) where  $l \in \mathbb{R}$  corresponds to the length scale and  $p \in \mathbb{R}$  reflects the periodicity.
+
+# 4 RELATED WORKS
+
+Time Series Imputation: Predictive Methods. Earlier works on time series imputation have been proposed for single imputation utilizing predictive models. M-RNN (Yoon et al., 2018b) and BRITS (Cao et al., 2018) are representative RNN-based methods that predict missing observations by employing bidirectional RNNs to capture both past and future temporal dependencies. Inspired by the recent success of transformers in modeling time series data over conventional RNN architectures, recent predictive methods for time series imputation employ a self-attention mechanism to enhance imputation performance (Bansal et al., 2021; Du et al., 2023; Shan et al., 2023). However, these methods cannot provide multiple imputations and, therefore, fail to incorporate the uncertainty associated with imputed values.
+
+Time Series Imputation: Generative Methods. Two main strands of generative models – VAEs (Kingma & Welling, 2014) and generative adversarial networks (GANs) (Makhzani et al., 2015) – have been introduced for multiple imputation due to their ability to stochastically generate samples. Here, we describe VAE-based methods. See Appendix D.3 for more related works.
+
+VAE-based imputation methods primarily focus on defining the evidence lower bound, where the reconstruction error is computed only over the observed part of the incomplete data while missing values are filled with arbitrary values (e.g., zeros) during inference (Sohn et al., 2015; Nazabal et al., 2020). Fortuin et al. (2020) proposed GP-VAE which adopts a similar approach to efficiently handle incomplete (missing) data in a temporal setting by assuming that the latent representation of input time series evolves smoothly over time according to a Gaussian process (GP). While introducing the GP prior improves the ability to capture the underlying temporal dynamics, GP-VAE still cannot capture shared temporal structures across time series data, as it employs an independent GP prior for each time series. More recently, L-VAE (Ramchandran et al., 2021) and its conditional extension (Ramchandran et al., 2022) further improve the GP prior of GP-VAE by utilizing auxiliary covariates information. We focus our comparison on VAE-based models since these models can be information-theoretically interpreted as optimizing the IB(Voloshynovskiy et al., 2019).
+
+It is worth clarifying that TimeCIB can be distinguished from GP-VAE in terms of how they achieve time series imputation and leverage temporal kernels. Under the IB framework, GP-VAE models the smooth temporal evolution of latent variables by replacing the traditional unit Gaussian prior with a GP prior specified by temporal kernels. However, TimeCIB is motivated by the inherent limitation of the IB in discarding temporal information (see Section 3.1) and proposes a novel CIB principle that alleviates the strict regularization of IB. Temporal kernels are optionally adopted to introduce an inductive bias to the underlying temporal dynamics.
+
+Information Bottleneck with Conditional Information. Several works have tailored the IB principle (Tishby & Zaslavsky, 2015; Alemi et al., 2017) by introducing conditional reconstruction or applying conditional regularization to extract information in alignment with their specific objectives. (Gondek & Hofmann, 2003) proposed conditional reconstruction to discover a new meaningful set of clusters that is orthogonal to the known class labels. (Fischer, 2020; Tezuka & Namekawa, 2021) introduced conditional regularization for supervised learning, which minimizes only redundant information given label information, thereby preventing the loss of label-related information due to overly strict regularization in the conventional IB principle. More recently, (Lee et al., 2023) utilized both conditional reconstruction and regularization to discover a label-related core subgraph from a pair of two molecular graphs.
+
+From this perspective, our work aligns with conditional regularization approaches. While previous works have primarily focused on mitigating regularization concerning target label information, our method aims to alleviate overly strict regularization that can hinder the learning of underlying temporal dynamics. Moreover, to the best of our knowledge, this is the first work that presents an information-theoretic definition for time series imputation and proposes a novel conditional IB that can effectively preserve temporal dynamics for better imputation.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+Evaluation Metrics. We evaluate the imputation performance from two perspectives: i) Imputation performance which measures feature-wise (pixel-wise) reconstruction. Specifically, we assess the negative log-likelihood (NLL) and mean squared error (MSE) of the imputed values on artificially missing features. ii) Prediction performance, which indirectly measures how well the imputed values preserve task-relevant information, which is a crucial aspect of imputation methods in practice. Following the experimental setup in (Fortuin et al., 2020) and (Yoon et al., 2019), we train separate classifiers or predictors with imputed values to predict the target labels. Then, we evaluate the area under the receiver operating characteristic (AUROC) for classification tasks and the MSE of the forecast (ForecastMSE) for forecasting tasks to measure the discriminative and predictive performance of imputation methods, respectively.
+
+**Baseline Models.** We focus our comparison on VAE-based models since these models can be interpreted under the IB principle as suggested in (Voloshynovskiy et al., 2019). Moreover, these multiple imputation methods can provide uncertainty of the imputed values, which is often crucial to support decision-making processes such as clinical interventions in healthcare. Hence, for baseline models, we compare our proposed method with the following: i) GP-VAE (Fortuin et al., 2020) which utilizes the Gaussian process (GP) prior to model time dependency, ii) HI-VAE (Nazabal et al., 2020) and iii) VAE (Kingma & Welling, 2014), both of which use an autoencoder architecture and are capable of imputing values at each time step. In addition to the above baselines, we compare with cutting-edge predictive methods: an RNN-based method, BRITS (Cao et al., 2018), and a transformer-based methods, SAITS (Du et al., 2023). Moreover, we also compare with state-of-the-art generative imputation methods, attention-based autoencoder approach, mTANs (Shukla & Marlin, 2021), and diffusion-based approach, CSDI (Tashiro et al., 2021). For a fair comparison, the magnitudes of the number of parameters are set to be equivalent among the evaluated methods. More detailed explanations about baseline models are provided in Appendix D.
+
+Table 1: Imputation and prediction performance on the image sequence datasets.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">HealingMNIST (missing with MNAR pattern)</td><td colspan="2">RotatedMNIST (interpolation &amp; extrapolation)</td></tr><tr><td>NLL(↓)</td><td>MSE(↓)</td><td>AUROC(↑)</td><td>NLL(↓)</td><td>MSE(↓)</td></tr><tr><td>No Imp.</td><td>-</td><td>0.293 ± 0.000</td><td>0.920 ± 0.000</td><td>-</td><td>0.133 ± 0.000</td></tr><tr><td>Mean Imp.</td><td>-</td><td>0.168 ± 0.000</td><td>0.938 ± 0.000</td><td>-</td><td>0.085 ± 0.000</td></tr><tr><td>Forward Imp.</td><td>-</td><td>0.177 ± 0.000</td><td>0.946 ± 0.000</td><td>-</td><td>0.080 ± 0.000</td></tr><tr><td>VAE</td><td>0.480 ± 0.002</td><td>0.232 ± 0.000</td><td>0.922 ± 0.000</td><td>1.773 ± 0.127</td><td>0.133 ± 0.000</td></tr><tr><td>HI-VAE</td><td>0.290 ± 0.001</td><td>0.134 ± 0.003</td><td>0.962 ± 0.001</td><td>0.207 ± 0.007</td><td>0.087 ± 0.001</td></tr><tr><td>GP-VAE</td><td>0.261 ± 0.001</td><td>0.114 ± 0.002</td><td>0.960 ± 0.002</td><td>0.190 ± 0.001</td><td>0.080 ± 0.004</td></tr><tr><td>Ours(Uniform)</td><td>0.204 ± 0.002</td><td>0.090 ± 0.001</td><td>0.967 ± 0.001</td><td>0.184 ± 0.001</td><td>0.077 ± 0.001</td></tr><tr><td>Ours(Cauchy)</td><td>0.202 ± 0.004</td><td>0.088 ± 0.002</td><td>0.967 ± 0.000</td><td>0.184 ± 0.001</td><td>0.076 ± 0.002</td></tr></table>
+
+![](images/6e4a5a9480b264bf555b71e7e6bab7850b1bf7edd047b7d1ba934a9c9f4a5d04.jpg)  
+(a) Robustness on missing patterns.
+
+![](images/3699bafc433c4b8316e903da0305dd6bb9fc2b1b911640ac36afcebb44c48498.jpg)  
+(b) Robustness on missing ratios.  
+Figure 2: Robustness analysis for missing patterns and missing ratios on HealingMNIST.
+
+# 5.2 MAIN RESULTS
+
+Imputation on image sequences. To evaluate imputation performance on diverse missing scenarios, we assess imputation performance on two MNIST sequence benchmarks with various missing patterns. HealingMNIST (Krishnan et al., 2015) has approximately  $60\%$  of missing pixels under a missing-not-at-random (MNAR) pattern on every time step, where the missing probability of white pixels is twice larger than that of black pixels. Given that the model is not provided with information about the underlying missing mechanism, this task is particularly challenging, yet it mirrors many practical scenarios. For example, in healthcare, patients with depression are more likely to refuse answers about the severity of their condition (Gliklich et al., 2014). RotatedMNIST (Ramchandran et al., 2021) evaluates performance on interpolation and extrapolation, where all features at an arbitrary time step are completely missing. This makes imputation more challenging since the model must reconstruct all the missing values at a given time step solely based on the temporal dependency. Table 1 demonstrates that TimeCIB provides state-of-the-art imputation and prediction performance on both datasets, and the application of the Cauchy kernel (10) can further improve the performance.
+
+We further evaluate the robustness of our model on HealingMNIST with four additional missing patterns with  $60\%$  of missing ratio (Figure B2a) and that with different missing ratios ranging from  $10\%$  to  $90\%$  with the random missing pattern (Figure B2b). To assess the robustness of missing patterns, we employ Random, Spatial (i.e., neighboring pixels have correlated missing probabilities), and Temporal+/- (i.e., positive/negative temporal correlation). The imputation performance of our proposed method is most robust on diverse missing patterns and missing ratios. Please refer to Appendix B for more experimental details and results.
+
+Imputation for weather forecasting. Weather forecasting is one of the representative fields where we can observe diverse scales of seasonality – such as daily, weekly, monthly, or yearly basis – which is one of the most important characteristics of time series data. In this experiment, we focused on two weather forecasting datasets - Beijing Air Quality (Zhang et al., 2017) and US Local<sup>5</sup> whose time series measurements are collected every hour. Our model is capable of using the prior knowledge on the periodicity of the target data, by applying conditional alignment with temporal kernels (Section 3.3); we assume daily periodicity by setting  $p = 24$ . Inspired by the experiments outlined in (Yoon et al., 2019), we also evaluate the utility of imputation methods on forecasting by assessing the prediction performance (ForecastMSE) of a separately trained LSTM using time series data with imputed values. As shown in Table 2, TimeCIB outperforms the benchmarks on both weather forecasting datasets in terms of both imputation and prediction performance. The
+
+Table 2: Imputation and prediction performance on the weather forecasting datasets.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">Beijing (T=24)</td><td colspan="3">US Local (T=168)</td></tr><tr><td>NLL(↓)</td><td>MSE(↓)</td><td>ForecastMSE(↓)</td><td>NLL(↓)</td><td>MSE(↓)</td><td>ForecastMSE(↓)</td></tr><tr><td>No Imp.</td><td>-</td><td>1.015 ± 0.000</td><td>0.539 ± 0.000</td><td>-</td><td>1.113 ± 0.000</td><td>0.610 ± 0.000</td></tr><tr><td>Mean Imp.</td><td>-</td><td>0.460 ± 0.000</td><td>0.517 ± 0.000</td><td>-</td><td>0.509 ± 0.000</td><td>0.432 ± 0.000</td></tr><tr><td>Forward Imp.</td><td>-</td><td>0.399 ± 0.000</td><td>0.502 ± 0.000</td><td>-</td><td>0.391 ± 0.000</td><td>0.401 ± 0.000</td></tr><tr><td>BRITS</td><td>-</td><td>0.396 ± 0.002</td><td>0.490 ± 0.005</td><td>-</td><td>0.384 ± 0.001</td><td>0.398 ± 0.027</td></tr><tr><td>SAITS</td><td>-</td><td>0.283 ± 0.013</td><td>0.450 ± 0.005</td><td>-</td><td>0.275 ± 0.002</td><td>0.350 ± 0.067</td></tr><tr><td>mTANs</td><td>*</td><td>0.287 ± 0.005</td><td>0.436 ± 0.005</td><td>*</td><td>0.268 ± 0.018</td><td>0.337 ± 0.033</td></tr><tr><td>CSDI(n=5)</td><td>*</td><td>0.287 ± 0.003</td><td>0.423 ± 0.003</td><td>*</td><td>0.378 ± 0.001</td><td>0.364 ± 0.036</td></tr><tr><td>CSDI(n=25)</td><td>*</td><td>0.270 ± 0.001</td><td>0.423 ± 0.006</td><td>*</td><td>0.340 ± 0.000</td><td>0.347 ± 0.036</td></tr><tr><td>VAE</td><td>1.427 ± 0.001</td><td>1.016 ± 0.002</td><td>0.524 ± 0.006</td><td>1.462 ± 0.002</td><td>1.086 ± 0.004</td><td>0.467 ± 0.034</td></tr><tr><td>HI-VAE</td><td>1.081 ± 0.003</td><td>0.321 ± 0.008</td><td>0.464 ± 0.008</td><td>1.078 ± 0.005</td><td>0.317 ± 0.010</td><td>0.380 ± 0.060</td></tr><tr><td>GP-VAE</td><td>1.077 ± 0.006</td><td>0.316 ± 0.011</td><td>0.463 ± 0.008</td><td>1.078 ± 0.005</td><td>0.318 ± 0.010</td><td>0.385 ± 0.051</td></tr><tr><td>Ours(Uniform)</td><td>1.063 ± 0.001</td><td>0.291 ± 0.004</td><td>0.445 ± 0.003</td><td>1.052 ± 0.001</td><td>0.265 ± 0.002</td><td>0.351 ± 0.060</td></tr><tr><td>Ours(Periodic)</td><td>1.060 ± 0.002</td><td>0.283 ± 0.004</td><td>0.443 ± 0.004</td><td>1.049 ± 0.002</td><td>0.260 ± 0.003</td><td>0.327 ± 0.022</td></tr></table>
+
+![](images/796f617ed8fbd4207492973167ecaf2d56539f922c818addb9775b27c00650d6.jpg)  
+(a) Beijing
+
+![](images/208cbe63c0afc29f06062a5021fdd5b5426c598522eea9a0ad1d7e450b1a807c.jpg)  
+(b) US Local  
+Figure 3: Comparison of the imputed values for examples in (a) Beijing, (b) US Local, and (c) Physi- oment2012 datasets, highlighting that TimeCIB provides more accurate imputations by considering temporal dependencies. Dots and crosses are observed and missing ground-truth values, respectively.
+
+![](images/58e089c5c59d1f9684fe252b2e3cc51d7456b1f4b4252b0ada421b0eda15ba64.jpg)  
+(c) Physionet2012
+
+performance of our method is further enhanced when equipped with a temporal periodic kernel (10), highlighting our model's ability to incorporate the correct inductive bias.
+
+Imputation for electrical health records. Time series imputation is of special importance in healthcare where each feature may have a distinct sampling period and strategies. In this context, we evaluate imputation methods on Physionet2012 - Mortality Prediction Challenge (Silva et al., 2012), which aims to predict in-hospital mortality of intensive care unit (ICU) patients from 48 hours of records with roughly  $80\%$  of missing features. Furthermore, we conduct additional evaluations to assess whether the imputation methods preserve the critical characteristics of a given time series - i.e., whether a patient's status is
+
+deteriorating or not – after replacing the missing features with imputed values. Table 3 shows that TimeCIB provides imputation performance comparable to the best benchmark while outperforming the VAE-based methods by a great margin. Furthermore, it achieves the best classification performance, successfully capturing information about the temporal dynamics of patients' status. Note that while the mean imputation provides better imputation performance, the imputed values drastically lose the crucial information for discriminating patient's status.
+
+Table 3: Imputation and prediction performance on the clinical dataset.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">Physionet2012 (mortality prediction)</td></tr><tr><td>NLL(↓)</td><td>MSE(↓)</td><td>AUROC(↑)</td></tr><tr><td>No Imp.</td><td>-</td><td>0.962 ± 0.000</td><td>0.692 ± 0.000</td></tr><tr><td>Mean Imp.</td><td>-</td><td>0.511 ± 0.000</td><td>0.703 ± 0.000</td></tr><tr><td>Forward Imp.</td><td>-</td><td>0.613 ± 0.000</td><td>0.710 ± 0.000</td></tr><tr><td>BRITS</td><td>-</td><td>0.529 ± 0.004</td><td>0.700 ± 0.005</td></tr><tr><td>SAITS</td><td>-</td><td>0.501 ± 0.024</td><td>0.713 ± 0.007</td></tr><tr><td>mTANs</td><td>*</td><td>0.499 ± 0.008</td><td>0.721 ± 0.004</td></tr><tr><td>CSDI(n=5)</td><td>*</td><td>0.548 ± 0.014</td><td>0.705 ± 0.005</td></tr><tr><td>CSDI(n=25)</td><td>*</td><td>0.478 ± 0.002</td><td>0.683 ± 0.033</td></tr><tr><td>VAE</td><td>1.400 ± 0.000</td><td>0.962 ± 0.000</td><td>0.691 ± 0.001</td></tr><tr><td>HI-VAE</td><td>1.345 ± 0.009</td><td>0.852 ± 0.018</td><td>0.696 ± 0.004</td></tr><tr><td>GP-VAE</td><td>1.227 ± 0.007</td><td>0.616 ± 0.013</td><td>0.730 ± 0.006</td></tr><tr><td>Ours(Uniform)</td><td>1.183 ± 0.007</td><td>0.528 ± 0.014</td><td>0.744 ± 0.009</td></tr><tr><td>Ours(Cauchy)</td><td>1.179 ± 0.006</td><td>0.521 ± 0.012</td><td>0.744 ± 0.009</td></tr></table>
+
+# 6 CONCLUSION
+
+In this paper, we have presented TimeCIB, a novel information-theoretic approach for time series imputation. While inheriting the multiple imputation and uncertainty measurement properties of the IB, TimeCIB addresses the limitation of the IB principle in capturing underlying temporal dynamics by replacing conventional regularization with conditional regularization. Our variational decomposition showed that CIB could be approximated by optimizing the evidence lower bound (ELBO) and the contrastive objective. We also demonstrated that introducing inductive bias based on temporal kernels can further enhance expressive power, acting as a form of conditional alignment. Our empirical results on image sequences, weather forecasting, and electrical health records prove that TimeCIB is effective in a wide range of practical cases.
+
+# ACKNOWLEDGMENTS
+
+We thank anonymous reviewers for many insightful comments and suggestions. CL was supported through the IITP grant funded by the Korea government(MSIT) (No. 2021-0-01341, AI Graduate School Program, CAU).
+
+# CODEAVAILABILITY
+
+Codebase used in this paper is available at https://github.com/Chemgyu/TimeCIB.
+
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+
+# A DERIVATIONS
+
+# A.1 VARIATIONAL APPROXIMATION OF FIRST TERM IN CONDITIONAL RECONSTRUCTION.
+
+Here, we show that the first term of the conditional reconstruction  $I(\mathbf{X}_t; \mathbf{Z}_t, \mathbf{X}_{\backslash t}^o)$  can be approximated as in (4) based on the following derivation:
+
+$$
+\begin{array}{l} I (\mathbf {X} _ {t}; \mathbf {Z} _ {t}, \mathbf {X} _ {\backslash t} ^ {o}) = \mathbb {E} _ {p (\mathbf {x} _ {t}, \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o})} \left[ \log \frac {p (\mathbf {x} _ {t} | \mathbf {z} _ {t} , \mathbf {x} _ {\backslash t} ^ {o})}{p (\mathbf {x} _ {t})} \right] \\ = \mathbb {E} _ {p (\mathbf {x} _ {t}, \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o})} \left[ \log p (\mathbf {x} _ {t} | \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}) \right] - \mathbb {E} _ {p (\mathbf {x} _ {t})} \left[ \log p (\mathbf {x} _ {t}) \right] \\ = \int_ {\mathbf {x} _ {t}} \int_ {\mathbf {z} _ {t}} \int_ {\mathbf {x} _ {\backslash t} ^ {o}} p \left(\mathbf {x} _ {t}, \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}\right) \log p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}\right) d \mathbf {x} _ {t} d \mathbf {z} _ {t} d \mathbf {x} _ {\backslash t} ^ {o} + \mathcal {H} (\mathbf {x} _ {t}) \tag {11} \\ \geq \int_ {\mathbf {x} _ {t}} \int_ {\mathbf {z} _ {t}} \int_ {\mathbf {x} _ {\backslash t} ^ {o}} p (\mathbf {x} _ {t}, \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}) \log p (\mathbf {x} _ {t} | \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}) d \mathbf {x} _ {t} d \mathbf {z} _ {t} d \mathbf {x} _ {\backslash t} ^ {o} \\ \end{array}
+$$
+
+The last inequality holds because of the non-negativity of entropy.
+
+Note that  $p(\mathbf{X}_t,\mathbf{Z}_t,\mathbf{X}_t^o)$  can be marginalized by introducing  $\mathbf{X}_t^o$
+
+$$
+p \left(\mathbf {X} _ {t}, \mathbf {Z} _ {t}, \mathbf {X} _ {\backslash t} ^ {o}\right) = \int_ {\mathbf {X} _ {t} ^ {o}} p \left(\mathbf {X} _ {t}, \mathbf {X} _ {1: T} ^ {o}\right) q _ {\phi} \left(\mathbf {Z} _ {t} \mid \mathbf {X} _ {1: T} ^ {o}\right) d \mathbf {X} _ {t} ^ {o} \tag {12}
+$$
+
+Then, putting (12) into the (11) gives
+
+$$
+\begin{array}{l} I \left(\mathbf {X} _ {t}; \mathbf {Z} _ {t}, \mathbf {X} _ {\backslash t} ^ {o}\right) \geq \int_ {\mathbf {x} _ {t}} \int_ {\mathbf {z} _ {t}} \int_ {\mathbf {x} _ {1: T} ^ {o}} p \left(\mathbf {x} _ {t}, \mathbf {x} _ {1: T} ^ {o}\right) q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1: T} ^ {o}\right) \log p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}\right) d \mathbf {x} _ {t} d \mathbf {z} _ {t} d \mathbf {x} _ {1: T} ^ {o} \\ = \mathbb {E} _ {p (\mathbf {x} _ {t}, \mathbf {x} _ {1: T} ^ {o})} \left[ \mathbb {E} _ {q _ {\phi} (\mathbf {z} _ {t} | \mathbf {x} _ {1: T} ^ {o})} \left[ \log p (\mathbf {x} _ {t} | \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}) \right] \right] \\ = \mathbb {E} _ {p (\mathbf {x} _ {t}, \mathbf {x} _ {1: T} ^ {o})} \left[ \mathbb {E} _ {q _ {\phi} (\mathbf {z} _ {t} | \mathbf {x} _ {1: T} ^ {o})} \left[ \log p (\mathbf {x} _ {t} | \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}) \frac {p _ {\theta} (\mathbf {x} _ {t} | \mathbf {z} _ {t})}{p _ {\theta} (\mathbf {x} _ {t} | \mathbf {z} _ {t})} \right] \right] \\ = \mathbb {E} _ {p \left(\mathbf {x} _ {t}, \mathbf {x} _ {1: T} ^ {o}\right)} \left[ \mathbb {E} _ {q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1: T} ^ {o}\right)} \left[ \log p _ {\theta} \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right) \right] \right] + \mathbb {E} _ {p \left(\mathbf {x} _ {t}, \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}\right)} \left[ \log \frac {p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t} , \mathbf {x} _ {\backslash t} ^ {o}\right)}{p _ {\theta} \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right)} \right] \tag {13} \\ \end{array}
+$$
+
+The last term can be expressed in a form of the KL-divergence:
+
+$$
+\begin{array}{l} \mathbb {E} _ {p \left(\mathbf {x} _ {t}, \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {\circ}\right)} \left[ \log \frac {p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t} , \mathbf {x} _ {\backslash t} ^ {\circ}\right)}{p _ {\theta} \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right)} \right] = \mathbb {E} _ {p \left(\mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {\circ}\right)} \left[ \mathbb {E} _ {p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}, \mathbf {x} _ {\backslash t} ^ {\circ}\right)} \left[ \log \frac {p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t} , \mathbf {x} _ {\backslash t} ^ {\circ}\right)}{p _ {\theta} \left(\mathbf {x} _ {t} \vert \mathbf {z} _ {t}\right)} \right] \right] \tag {14} \\ = \mathbb {E} _ {p (\mathbf {z} _ {t}, \mathbf {x} _ {t})} \left[ D _ {\mathrm {K L}} \left(p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}, \mathbf {x} _ {t} ^ {o}\right) | | p _ {\theta} \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right)\right) \right] \\ \end{array}
+$$
+
+Due to the non-negativity of the KL-divergence, plugging (14) into (13) gives (4) as
+
+$$
+I \left(\mathbf {X} _ {t}; \mathbf {Z} _ {t}, \mathbf {X} _ {\backslash t} ^ {o}\right) \geq \mathbb {E} _ {p \left(\mathbf {X} _ {t}, \mathbf {X} _ {1: T} ^ {o}\right)} \left[ \mathbb {E} _ {q _ {\phi} \left(\mathbf {Z} _ {t} \mid \mathbf {X} _ {1: T} ^ {o}\right)} \left[ \log p _ {\theta} \left(\mathbf {X} _ {t} \mid \mathbf {Z} _ {t}\right) \right] \right]. \tag {15}
+$$
+
+A.2 VARIATIONAL APPROXIMATION OF SECOND TERM IN CONDITIONAL RECONSTRUCTION. Analogous to the derivation in (Alemi et al., 2017),  $I(\mathbf{X}_t; \mathbf{X}_{\backslash t}^o)$  is upper bounded as follows:
+
+$$
+\begin{array}{l} I (\mathbf {X} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}) = \mathbb {E} _ {p (\mathbf {x} _ {t}, \mathbf {x} _ {\backslash t} ^ {o})} \left[ \log \frac {p (\mathbf {x} _ {t} | \mathbf {x} _ {\backslash t} ^ {o})}{p (\mathbf {x} _ {t})} \right] \\ = \mathbb {E} _ {p (\mathbf {x} _ {t}, \mathbf {x} _ {\backslash t} ^ {o})} \left[ \log \frac {p (\mathbf {x} _ {t} | \mathbf {x} _ {\backslash t} ^ {o}) q (\mathbf {x} _ {t})}{p (\mathbf {x} _ {t}) q (\mathbf {x} _ {t})} \right] \\ = \mathbb {E} _ {p \left(\mathbf {x} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}\right)} \left[ \log \frac {p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {\backslash t} ^ {o}\right)}{q \left(\mathbf {x} _ {t}\right)} \right] - \mathbb {E} _ {p \left(\mathbf {x} _ {t}, \mathbf {x} _ {\backslash t} ^ {o}\right)} \left[ \log \frac {p \left(\mathbf {x} _ {t}\right)}{q \left(\mathbf {x} _ {t}\right)} \right] \tag {16} \\ = \mathbb {E} _ {p \left(\mathbf {x} _ {\backslash t} ^ {o}\right)} \left[ D _ {\mathrm {K L}} \left(p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {\backslash t} ^ {o}\right) \mid \mid q \left(\mathbf {x} _ {t}\right)\right) \right] - D _ {\mathrm {K L}} \left(p \left(\mathbf {x} _ {t}\right) \mid \mid q \left(\mathbf {x} _ {t}\right)\right) \\ \leq \mathbb {E} _ {p \left(\mathbf {x} _ {t} ^ {\circ}\right)} \left[ D _ {\mathrm {K L}} \left(p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {\backslash t} ^ {o}\right) \mid | q \left(\mathbf {x} _ {t}\right)\right) \right] \\ \end{array}
+$$
+
+where we set  $q(\mathbf{X}_t) = \mathcal{N}(0,I)$  (Alemi et al., 2017; Lee et al., 2023). Specifically, in our implementation, we deterministically encode  $\tilde{\mathbf{Z}}_t$  by taking  $\mu_{\phi}(\mathbf{X}_{\backslash t}^{o})$  and then calculate the KL-divergence between the decoder distribution  $p_{\theta}(\mathbf{X}_t|\tilde{\mathbf{Z}}_t)$  with  $q(\mathbf{X}_t)$ .
+
+A.3 VARIATIONAL APPROXIMATION OF FIRST TERM IN CONDITIONAL REGULARIZATION.
+
+$$
+\begin{array}{l} I (\mathbf {Z} _ {t}; \mathbf {X} _ {1: T} ^ {o}) = \mathbb {E} _ {q _ {\phi} (\mathbf {z} _ {t}, \mathbf {x} _ {1: T} ^ {o})} \left[ \log \frac {q _ {\phi} (\mathbf {x} _ {1 : T} ^ {o} , \mathbf {z} _ {t})}{q _ {\phi} (\mathbf {z} _ {t}) p _ {\mathrm {d a t a}} (\mathbf {x} _ {1 : T} ^ {o})} \right] \\ = \mathbb {E} _ {q _ {\phi} (\mathbf {z} _ {t}, \mathbf {x} _ {1: T} ^ {o})} \left[ \log \frac {q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1 : T} ^ {o}\right)}{q _ {\phi} (\mathbf {z} _ {t})} \frac {p (\mathbf {z} _ {t})}{p (\mathbf {z} _ {t})} \right] (17) \\ = \mathbb {E} _ {q _ {\phi} (\mathbf {z} _ {t}, \mathbf {x} _ {1: T} ^ {o})} \left[ \log \frac {q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1 : T} ^ {o}\right)}{p (\mathbf {z} _ {t})} \right] + \mathbb {E} _ {q _ {\phi} \left(\mathbf {z} _ {t}, \mathbf {x} _ {1: T} ^ {o}\right)} \left[ \log \frac {p (\mathbf {z} _ {t})}{q _ {\phi} (\mathbf {z} _ {t})} \right] (17) \\ = \mathbb {E} _ {p _ {\text {d a t a}} (\mathbf {x} _ {1: T} ^ {o})} \left[ D _ {\mathrm {K L}} \left(q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1: T} ^ {o}\right) \mid \mid p (\mathbf {z} _ {t})\right) \right] - D _ {\mathrm {K L}} \left(q _ {\phi} (\mathbf {z} _ {t}) \mid \mid p (\mathbf {z} _ {t})\right) \\ \leq \mathbb {E} _ {p _ {\text {d a t a}} (\mathbf {x} _ {1: T} ^ {o})} \left[ D _ {\mathrm {K L}} \left(q _ {\phi} \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {1: T} ^ {o}\right) | | p (\mathbf {z} _ {t})\right) \right] \\ \end{array}
+$$
+
+The last inequality holds because of the non-negativity of KL-divergence.
+
+# A.4 CONTRASTIVE APPROXIMATION OF SECOND TERM IN CONDITIONAL REGULARIZATION.
+
+In Section 3.2.2, we optimize  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$  by approximate the mutual information into a contrastive form, which is similar to (Oord et al., 2018; Tian et al., 2020).
+
+$$
+\begin{array}{l} I (\mathbf {Z} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}) = - \mathbb {E} _ {X} \log \left[ \frac {p (\mathbf {X} _ {\backslash t} ^ {o})}{p (\mathbf {X} _ {\backslash t} ^ {o} | \mathbf {Z} _ {t})} \right] \\ = - \mathbb {E} _ {X} \log \left[ \frac {p \left(\mathbf {X} _ {\backslash t} ^ {o}\right)}{p \left(\mathbf {X} _ {\backslash t} ^ {o} \mid \mathbf {Z} _ {t}\right)} N \right] + \log (N) \\ \geq - \mathbb {E} _ {X} \log \left[ \frac {p \left(\mathbf {X} _ {\backslash t} ^ {o}\right)}{p \left(\mathbf {X} _ {\backslash t} ^ {o} \mid \mathbf {Z} _ {t}\right)} N \right] \\ \geq - \mathbb {E} _ {X} \log \left[ 1 + \frac {p \left(\mathbf {X} _ {\backslash t} ^ {o}\right)}{p \left(\mathbf {X} _ {\backslash t} ^ {o} \mid \mathbf {Z} _ {t}\right)} (N - 1) \right] \\ = - \mathbb {E} _ {X} \log \left[ 1 + \frac {p \left(\mathbf {X} _ {\backslash t} ^ {o}\right)}{p \left(\mathbf {X} _ {\backslash t} ^ {o} \mid \mathbf {Z} _ {t}\right)} (N - 1) \mathbb {E} _ {\mathbf {X} _ {\backslash t} ^ {o, j}} \frac {p \left(\mathbf {X} _ {\backslash t} ^ {o , j} \mid \mathbf {Z} _ {t}\right)}{\mathbf {X} _ {\backslash t} ^ {o , j}} \right] \tag {18} \\ \approx - \mathbb {E} _ {X} \log \left[ 1 + \frac {p \left(\mathbf {X} _ {\backslash t} ^ {o}\right)}{p \left(\mathbf {X} _ {\backslash t} ^ {o} \mid \mathbf {Z} _ {t}\right)} \sum_ {\mathbf {X} _ {\backslash t} ^ {-} \in \mathcal {X} ^ {-}} \frac {p \left(\mathbf {X} _ {\backslash t} ^ {-} \mid \mathbf {Z} _ {t}\right)}{p \left(\mathbf {X} _ {\backslash t} ^ {-}\right)} \right] \\ = \mathbb {E} _ {X} \log \left[ \frac {\frac {p (\mathbf {X} _ {\backslash t} ^ {o} | \mathbf {Z} _ {t})}{p (\mathbf {X} _ {\backslash t} ^ {o})}}{\frac {p (\mathbf {X} _ {\backslash t} ^ {o} | \mathbf {Z} _ {t})}{p (\mathbf {X} _ {\backslash t} ^ {o})} + \sum_ {\mathbf {X} _ {\backslash t} ^ {-} \in \mathcal {X} ^ {-}} \frac {p (\mathbf {X} _ {\backslash t} ^ {-} | \mathbf {Z} _ {t})}{p (\mathbf {X} _ {\backslash t} ^ {-})}} \right] \\ \approx \mathbb {E} _ {X} \log \left[ \frac {\frac {p (\mathbf {X} _ {\backslash t} ^ {o} | \mathbf {Z} _ {t})}{p (\mathbf {X} _ {\backslash t} ^ {o})}}{\sum_ {\mathbf {X} _ {1 : T} ^ {-} \in \mathcal {X} ^ {-}} \frac {p (\mathbf {X} _ {1 : T} ^ {-} | \mathbf {Z} _ {t})}{p (\mathbf {X} _ {1 : T} ^ {-})}} \right] \\ = \mathbb {E} _ {X} \log \left[ \frac {f (\mathbf {Z} _ {t} , \mathbf {X} _ {\backslash t} ^ {o})}{\sum_ {\mathbf {X} _ {1 : T} ^ {-} \in \mathcal {X} ^ {-}} f (\mathbf {Z} _ {t} , \mathbf {X} _ {1 : T} ^ {-})} \right] \\ \end{array}
+$$
+
+There remain two design choices: i) selection of  $\mathcal{X}^{-}$  and ii) formulation of function  $f$ . For i), we use a mini-batch approach that  $\mathbf{X}_{1:T}^{-}$  are chosen from other time series inputs in the same mini-batch. For ii), we adopt the average of cosine similarities along the time axis. Specifically, we define the function  $f$  using an alternative representations  $\tilde{\mathbf{Z}}_{t'}$  which is obtained by inputting the masked input  $q_{\phi}(\tilde{\mathbf{Z}}_{t'}|\mathbf{X}_{\backslash t}^{o})$ :
+
+$$
+f \left(\mathbf {Z} _ {t}, \mathbf {X} _ {\backslash t} ^ {o}\right) = \sum_ {t ^ {\prime} \in \{1: T \} \backslash t} f \left(\mathbf {Z} _ {t}, \tilde {\mathbf {Z}} _ {t ^ {\prime}}\right) = \sum_ {t ^ {\prime} \in \{1: T \} \backslash t} \exp \left(\mathbf {Z} _ {t} ^ {T} \tilde {\mathbf {Z}} _ {t ^ {\prime}} / \tau\right) \tag {19}
+$$
+
+where  $\tau \in \Re$  is a temperature hyperparameter.
+
+Alternatively, we can also define the function for negative samples in a similar way.
+
+$$
+f \left(\mathbf {Z} _ {t}, \mathbf {X} _ {1: T} ^ {-}\right) = \sum_ {t ^ {\prime} \in \{1: T \}} f \left(\mathbf {Z} _ {t}, \mathbf {Z} _ {t ^ {\prime}} ^ {-}\right) = \sum_ {t ^ {\prime} \in \{1: T \}} \exp \left(\mathbf {Z} _ {t} ^ {T} \mathbf {Z} _ {t ^ {\prime}} ^ {-} / \tau\right) \tag {20}
+$$
+
+Then  $I(\mathbf{Z}_t;\mathbf{X}_{\backslash t}^o)$  is lower bounded by
+
+$$
+I \left(\mathbf {Z} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}\right) \geq \log \left[ \frac {\sum_ {t ^ {\prime} \in \{1 : T \} \backslash t} \exp \left(\mathbf {Z} _ {t} ^ {T} \tilde {\mathbf {Z}} _ {t ^ {\prime}} / \tau\right)}{\sum_ {\mathbf {X} _ {1 : T} ^ {-} \in \mathcal {X} _ {1 : T} ^ {-}} \sum_ {t ^ {\prime} \in \{1 : T \}} \exp \left(\mathbf {Z} _ {t} ^ {T} \mathbf {Z} _ {t ^ {\prime}} ^ {-} / \tau\right)} \right] \tag {21}
+$$
+
+We can observe that  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$  is lower bounded by the similar form of NT-Xent objective function in (Sohn, 2016) and specifically in-version of the supervised contrastive loss in (Khosla et al., 2020).
+
+# B ADDITIONAL RESULTS
+
+# B.1 MORE ON ROBUSTNESS
+
+Missing Patterns. To assess the robustness of our model on diverse missing patterns, we tested our model on five different types of missingness generated by the missing mechanism outlined in (Fortuin et al., 2020). Here, we describe the data-generating process of each missing pattern as the following: Spatial: Define a spatial Gaussian process with RBF kernel, and draw the missingness patterns by sampling from this process. Temporal+: Define a temporal Gaussian process with RBF kernel for each feature dimension, and draw from this process. Temporal-: Implement with determinantal point process for each feature dimension.
+
+Table B1: Robustness on missing patterns, HealingMNIST (NLL)  
+
+<table><tr><td>Pattern</td><td>Mean imp.</td><td>Forward imp.</td><td>VAE</td><td>HI-VAE</td><td>GP-VAE</td><td>Ours</td></tr><tr><td>Random</td><td>-</td><td>-</td><td>0.3360 ± 0.0010</td><td>0.2218 ± 0.0008</td><td>0.1957 ± 0.0006</td><td>0.0887 ± 0.0010</td></tr><tr><td>Spatial</td><td>-</td><td>-</td><td>0.4802 ± 0.0016</td><td>0.2259 ± 0.0010</td><td>0.1779 ± 0.0007</td><td>0.1509 ± 0.0038</td></tr><tr><td>Temporal-</td><td>-</td><td>-</td><td>0.1940 ± 0.0006</td><td>0.1349 ± 0.0005</td><td>0.1175 ± 0.0004</td><td>0.0879 ± 0.0004</td></tr><tr><td>Temporal+</td><td>-</td><td>-</td><td>0.1918 ± 0.0006</td><td>0.1332 ± 0.0005</td><td>0.1206 ± 0.0004</td><td>0.0927 ± 0.0006</td></tr><tr><td>MNAR</td><td>-</td><td>-</td><td>0.4798 ± 0.0016</td><td>0.2896 ± 0.0010</td><td>0.2606 ± 0.0008</td><td>0.2017 ± 0.0041</td></tr></table>
+
+Table B2: Robustness on missing patterns, HealingMNIST (MSE)  
+
+<table><tr><td>Pattern</td><td>Mean imp.</td><td>Forward imp.</td><td>VAE</td><td>HI-VAE</td><td>GP-VAE</td><td>Ours</td></tr><tr><td>Random</td><td>0.069 ± 0.000</td><td>0.099 ± 0.000</td><td>0.100 ± 0.000</td><td>0.046 ± 0.001</td><td>0.036 ± 0.000</td><td>0.036 ± 0.000</td></tr><tr><td>Spatial</td><td>0.090 ± 0.000</td><td>0.099 ± 0.000</td><td>0.122 ± 0.000</td><td>0.097 ± 0.000</td><td>0.071 ± 0.001</td><td>0.059 ± 0.000</td></tr><tr><td>Temporal-</td><td>0.086 ± 0.000</td><td>0.093 ± 0.000</td><td>0.101 ± 0.000</td><td>0.046 ± 0.001</td><td>0.037 ± 0.001</td><td>0.036 ± 0.000</td></tr><tr><td>Temporal+</td><td>0.107 ± 0.000</td><td>0.117 ± 0.000</td><td>0.101 ± 0.000</td><td>0.047 ± 0.001</td><td>0.038 ± 0.001</td><td>0.037 ± 0.000</td></tr><tr><td>MNAR</td><td>0.168 ± 0.000</td><td>0.177 ± 0.000</td><td>0.232 ± 0.000</td><td>0.134 ± 0.003</td><td>0.114 ± 0.002</td><td>0.088 ± 0.002</td></tr></table>
+
+Missing Ratio. Analogous to (Fortuin et al., 2020), we validate the robustness of our model on a variety of different missing ratios using random missing patterns, ranging from  $10\%$  to  $90\%$ .
+
+Table B3: Robustness on random missingness, HealingMNIST (NLL)  
+
+<table><tr><td>Missing</td><td>VAE</td><td>HI-VAE</td><td>GP-VAE</td><td>Ours</td></tr><tr><td>10 %</td><td>0.1319 ± 0.0039</td><td>0.0783 ± 0.0022</td><td>0.0695 ± 0.0010</td><td>0.0666 ± 0.0004</td></tr><tr><td>20 %</td><td>0.1506 ± 0.0054</td><td>0.0844 ± 0.0009</td><td>0.0744 ± 0.0009</td><td>0.0688 ± 0.0004</td></tr><tr><td>30 %</td><td>0.1637 ± 0.0042</td><td>0.0874 ± 0.0057</td><td>0.0799 ± 0.0008</td><td>0.0734 ± 0.0005</td></tr><tr><td>40 %</td><td>0.1950 ± 0.0061</td><td>0.0974 ± 0.0102</td><td>0.0864 ± 0.0007</td><td>0.0782 ± 0.0003</td></tr><tr><td>50 %</td><td>0.2164 ± 0.0046</td><td>0.1172 ± 0.0127</td><td>0.0937 ± 0.0012</td><td>0.0849 ± 0.0005</td></tr><tr><td>60 %</td><td>0.2634 ± 0.0108</td><td>0.1264 ± 0.0070</td><td>0.1040 ± 0.0008</td><td>0.0944 ± 0.0001</td></tr><tr><td>70 %</td><td>0.2800 ± 0.0051</td><td>0.1489 ± 0.0034</td><td>0.1197 ± 0.0003</td><td>0.1088 ± 0.0002</td></tr><tr><td>80 %</td><td>0.2928 ± 0.0058</td><td>0.1698 ± 0.0177</td><td>0.1408 ± 0.0002</td><td>0.1317 ± 0.0009</td></tr><tr><td>90 %</td><td>0.3071 ± 0.0054</td><td>0.2290 ± 0.0042</td><td>0.1852 ± 0.0003</td><td>0.1691 ± 0.0006</td></tr></table>
+
+Table B4: Robustness on random missingness, HealingMNIST (MSE)  
+
+<table><tr><td>Missing</td><td>VAE</td><td>HI-VAE</td><td>GP-VAE</td><td>Ours</td></tr><tr><td>10 %</td><td>0.0582 ± 0.0023</td><td>0.0336 ± 0.0009</td><td>0.0295 ± 0.0004</td><td>0.0274 ± 0.0008</td></tr><tr><td>20 %</td><td>0.0700 ± 0.0030</td><td>0.0367 ± 0.0000</td><td>0.0311 ± 0.0004</td><td>0.0283 ± 0.0001</td></tr><tr><td>30 %</td><td>0.0857 ± 0.0033</td><td>0.0373 ± 0.0029</td><td>0.0336 ± 0.0007</td><td>0.0305 ± 0.0002</td></tr><tr><td>40 %</td><td>0.1185 ± 0.0033</td><td>0.0420 ± 0.0051</td><td>0.0362 ± 0.0000</td><td>0.0328 ± 0.0001</td></tr><tr><td>50 %</td><td>0.1317 ± 0.0006</td><td>0.0517 ± 0.0063</td><td>0.0390 ± 0.0005</td><td>0.0356 ± 0.0001</td></tr><tr><td>60 %</td><td>0.1328 ± 0.0000</td><td>0.0561 ± 0.0036</td><td>0.0436 ± 0.0001</td><td>0.0402 ± 0.0000</td></tr><tr><td>70 %</td><td>0.1328 ± 0.0000</td><td>0.0681 ± 0.0018</td><td>0.0505 ± 0.0001</td><td>0.0470 ± 0.0000</td></tr><tr><td>80 %</td><td>0.1328 ± 0.0000</td><td>0.0787 ± 0.0108</td><td>0.0594 ± 0.0001</td><td>0.0584 ± 0.0008</td></tr><tr><td>90 %</td><td>0.1328 ± 0.0000</td><td>0.1088 ± 0.0076</td><td>0.0826 ± 0.0009</td><td>0.0755 ± 0.0002</td></tr></table>
+
+# B.2 ABLATION STUDY ON THE CONDITIONAL RECONSTRUCTION.
+
+As derived in Section 3.2, the conditional reconstruction term is given as the follows:  $I(\mathbf{X}_t;\mathbf{Z}_t|\mathbf{X}_{\backslash t}^o) = I(\mathbf{X}_t;\mathbf{Z}_t,\mathbf{X}_{\backslash t}^o) - I(\mathbf{X}_t;\mathbf{X}_{\backslash t}^o)$ . To support our claim that the effect of minimizing  $I(\mathbf{X}_t;\mathbf{X}_{\backslash t}^o)$  deteriorates the model performance, we introduce an auxiliary weight  $\omega$  and make a weighted version of the conditional reconstruction term as
+
+$$
+\tilde {I} \left(\mathbf {X} _ {t}; \mathbf {Z} _ {t} \mid \mathbf {X} _ {\backslash t} ^ {o}\right) = I \left(\mathbf {X} _ {t}; \mathbf {Z} _ {t}, \mathbf {X} _ {\backslash t} ^ {o}\right) - \omega I \left(\mathbf {X} _ {t}; \mathbf {X} _ {\backslash t} ^ {o}\right). \tag {22}
+$$
+
+Then, we can see the effect of  $I(\mathbf{X}_t; \mathbf{X}_{\backslash t}^o)$  by varying the value of  $\omega$  from 0 to 1 by assessing the imputation performance on the Physionet2012 dataset. Figure B2 shows that, starting from  $\omega = 0$  which is equivalent to the CIB objective in (8), focusing more on  $I(\mathbf{X}_t; \mathbf{X}_{\backslash t}^o)$  gradually deteriorates both the imputation performance (MSE) and the classification performance (AUROC).
+
+![](images/758846764d5345aca2b113b1288759f1a84b9a1cccef47cdacf70b014f39e484.jpg)  
+Figure B1: The effect of  $\omega$  in the weighted conditional reconstruction in (22).
+
+# B.3 QUANTITATIVE ASSESSMENT ON  $I(\mathbf{Z}_t;\mathbf{X}_{\backslash t}^o)$
+
+In addition to the qualitative assessment on  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$  shown in Figure 1 and empirical performances in Section 5, we compare the approximated quantity on  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$  using (9) to assert that our CIB objective indeed preserves more information. Please note that, as derived in (A.4), the lower bound is given by  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o) \geq \log N - \mathcal{L}_{\phi}^3$ , which depends on the batch size  $N$  and the objective  $\mathcal{L}_{\phi}^3$ . Table B5 shows the values of  $\mathcal{L}_{\phi}^3$ , indicating that a lower value corresponds to more preserved information. This result demonstrates that our CIB objective preserves more information in  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$  based on the approximation.
+
+Table B5: Quantitative results on  $I(\mathbf{Z}_t; \mathbf{X}_{\backslash t}^o)$ . Lower value indicates more preservation of information.  
+
+<table><tr><td>Dataset</td><td>VAE</td><td>HI-VAE</td><td>GP-VAE</td><td>Ours(Uniform)</td><td>Ours(Kernel)</td></tr><tr><td>RotatedMNIST</td><td>3.619 ± 0.005</td><td>3.577 ± 0.008</td><td>3.435 ± 0.001</td><td>3.428 ± 0.006</td><td>3.418 ± 0.002</td></tr><tr><td>Beijing</td><td>3.950 ± 0.015</td><td>3.822 ± 0.012</td><td>3.762 ± 0.012</td><td>3.700 ± 0.008</td><td>3.686 ± 0.006</td></tr></table>
+
+# B.4 SENSITIVITY ANALYSIS
+
+Here, we provide the sensitivity analysis on  $\beta$  and  $\gamma$  in (8). Before presenting results, we rewrite our objective as follows to analyze the effects of individual regularization terms given fixed reconstruction,
+
+$$
+\min  _ {\phi , \theta} \mathcal {L} _ {\phi , \theta} ^ {1} + \beta^ {\prime} \mathcal {L} _ {\phi} ^ {2} + \gamma^ {\prime} \mathcal {L} _ {\phi} ^ {3} \tag {23}
+$$
+
+where  $\beta' = 1 / \beta$  and  $\gamma' = \gamma / \beta$ . Now, (23) shares the same notational convention used in  $\beta$ -VAE (Higgins et al., 2017), where we can systematically assess the sensitivity of TimeCIB on the two regularization objectives.
+
+![](images/394a48823f0e268dfe4d232653ebbc2f166bde63dd38b0e3eaf97c0be9f1811b.jpg)  
+(a) Sensitivity on  $\beta^{\prime}$
+
+![](images/3ab68080d2afa01caedd8190b30de146319eae967fc1c6d554ba5bc0bda743c7.jpg)  
+(b) Sensitivity on  $\gamma^\prime$  
+Figure B2: Sensitivity analysis on the regularization objectives,  $\beta^\prime$  and  $\gamma^\prime$ .  $\gamma^\prime = 0.5$  for (a), and  $\beta^{\prime} = 0.1$  for (b).
+
+Sensitivity on  $\beta'$ .  $\beta'$  is related to the regularization trade-off that low- $\beta'$  (weak regularization) can cause lack of timeseries-wise information and eventually overfitting on observed values, while high- $\beta'$  (strong regularization) can cause loss of timepoint-wise information since all distribution would become a unit Gaussian, as well-studied by previous works.
+
+Sensitivity on  $\gamma^\prime$ . Since  $\gamma^\prime$  also works as a regularization term, sensitivity on  $\gamma^\prime$  is similar to  $\beta^\prime$ , but the cause of behaviour is slightly different. For low- $\gamma^\prime$  (weak regularization), there is no driving force on conserving temporal dynamics (see Section 3.1), thereby cause loss of temporal dynamics and would eventually become HI-VAE (Nazabal et al., 2020) when  $\gamma^\prime = 0$ . On the other hand, high- $\gamma^\prime$  (strong regularization) cause loss of timepoint-wise information, since  $L_{\phi}^{3}$  will map all  $\mathbf{z}_t$  from the same time series into a single point, which means that we cannot distinguish individual timepoints. Figure B2 illustrates quantitative and qualitative results on the HealingMNIST.
+
+**Optimization Strategy.** Simultaneous optimization of  $\beta'$  and  $\gamma'$  could be difficult. However, motivated from the above sensitivity analysis, one can first set  $\gamma' = 0$  and find optimal  $\beta'$  (which is HI-VAE). Second-stage parameter search within the close range of order of magnitude of  $\beta'$  could help finding optimal  $\gamma'$ .
+
+# C IMPLEMENTATION DETAILS
+
+Here, we provide implementation details, including illustration of overall frameworks, algorithms, and hyperparameters. The overall framework is illustrated in Figure C1.
+
+Figure C1: Illustration of TimeCIB framework. (A) Conceptual illustration TimeCIB framework. CIB maximizes  $I_{\theta}(\mathbf{x}_t;\mathbf{z}_t)$ , while minimizing  $I_{\phi}(\mathbf{z}_t;\mathbf{x}_t^o |\mathbf{x}_{\backslash t}^o)$ . (B) Implementation of optimizing  $I_{\phi}(\mathbf{z}_t;\mathbf{x}_{\backslash t}^o)$ .  
+![](images/43bfa926c1f439881ba823062ab28c149fa6730bfd54252d47998ae4d0788064.jpg)  
+Notations:  $\mathbf{X}_t$ $\mathbf{x}_t^o$ $\mathbf{x}_t^m$  Z,  $\mathbf{x}_t^{\mathrm{masked}}$
+
+![](images/ba38f3686d7eb7b5a71ce86950a2d05e1f03f15963c7e533a0c2154bccceab08.jpg)
+
+# C.1 TRAINING AND INFERENCE.
+
+At training stage, input time sequences are first fed into the stochastic encoder  $q_{\phi}(\mathbf{Z}_t|\mathbf{X}_{1:T}^o) = \mathcal{N}(\mu_{\phi}(\mathbf{X}_{1:T}^o),\mathrm{diag}(\sigma_{\phi}(\mathbf{X}_{1:T}^o)))$  where we use a 2 layers of bidirectional LSTM. For image dataset, we used CNN preprocessor before the encoder module. We then sample  $\mathbf{z}_t$  from resulting latent distribution, and decoder  $p_{\theta}(\mathbf{X}_t|\mathbf{Z}_t) = \mathcal{N}(\mu_{\theta}(\mathbf{z}_t),\sigma_{\theta}I)$  reconstructs the complete observation from the latent vector. In addition to previous conventional ELBO procedures, we also encodes  $\mathbf{X}_{\backslash t}^{o}$ , which are generated by masking  $\mathbf{X}_t^o$  with zeros. These masked input series are also mapped by the same encoder and latent vectors are obtained. Consequently, we use two sets of time series representations:  $\mathbf{z}_{1:T}$  and  $\tilde{\mathbf{z}}_{1:T}$  for calculating (7). For training efficiency, we optimized (7) stochastically, which is, we randomly sampled  $t$  at every epoch and minimized contrastive loss on that time step. At single prediction, we use the mean of the latent distribution  $\mu_{\phi}(\mathbf{X}_{1:T}^{o})$  as the corresponding latent representation and also reconstruct the observation from the mean of the decoder distribution  $\mu_{\theta}(\mathbf{z}_t)$ . All experiments were conducted using a 48GB NVIDIA RTX A6000.
+
+# C.2 ALGORITHM
+
+We provide the pseudo-algorithm of TimeCIB as below:
+
+Algorithm 1 Conditional Information Bottleneck on Time Series  
+input batch size  $B$  time length  $T$    
+for sampled minibatch  $\{\mathbf{X}_{1:T}^{o}\}_{1}^{B}$  do sample  $t\in \{1:T\}$  generate masked input  $\{\mathbf{X}_{\backslash t}^{o}\}_{1}^{B}$    
+for  $k\in \{1,\dots ,B\}$  in parallel do for  $t^{\prime}\in \{1:T\}$  in parallel do encode and sample  $\mathbf{Z}_{t'}^k\sim q_\phi (\mathbf{Z}_{t'}^k |\mathbf{X}_{1:T}^{o,k}) = \mathcal{N}(\mu_\phi (\mathbf{X}_{1:T}^{o,k})\sigma_\phi (\mathbf{X}_{1:T}^{o,k})I)$  encode and sample  $\tilde{\mathbf{Z}}_{t'}^k\sim q_\phi (\tilde{\mathbf{Z}}_{t'}^k |\mathbf{X}_{\backslash t'}^{o,k}) = \mathcal{N}(\mu_\phi (\mathbf{X}_{\backslash t'}^{o,k}),\sigma_\phi (\mathbf{X}_{\backslash t'}^{o,k})I)$  decode  $p_{\theta}(\mathbf{X}_{t'}^k |\mathbf{Z}_{t'}) = \mathcal{N}(\mu_{\theta}(\mathbf{Z}_{t'}),\mathrm{diag}(\sigma_{\theta}(\mathbf{Z}_{t'}))$ $\mathcal{L}_{\phi ,\theta}^{1} = \mathcal{L}_{\phi ,\theta}^{1} - \log (p_{\theta}(\mathbf{X}_{t'}^{k,o}|\mathbf{Z}_{t'}))$ $\mathcal{L}_\phi^2 = \mathcal{L}_\phi^2 +KL[q_\phi (\tilde{\mathbf{Z}}_{t'}^k |\mathbf{X}_{\backslash t'}^{o,k})||\mathcal{N}(\mathbf{0},I)]$  end for end for   
+for  $k\in \{1,\ldots ,B\}$  in parallel do  $\mathcal{L}_\phi^3 = \mathcal{L}_\phi^3 +$  Contrastive Loss  $(\mathbf{Z}_{t'}^k,\tilde{\mathbf{Z}}_{\backslash t'}^k,\mathbf{Z}_{1:T}^{k})(9)$    
+end for  $\begin{array}{r}\mathcal{L} = \beta \mathcal{L}_{\phi ,\theta}^{1} + \mathcal{L}_{\phi}^{2} + \gamma \mathcal{L}_{\phi}^{3}\\ \text{update}\phi ,\theta \text{to minimize}\mathcal{L}\\ \text{end for} \end{array}$
+
+# C.3 HYPERPARAMETER SPECIFICATION.
+
+We provide hyperparameter settings used for our experiments.
+
+Table C1: Hyperparameter specifications.  
+
+<table><tr><td></td><td>Hidden Dim</td><td>Batch Size</td><td>Epochs</td><td>Learning Rate</td><td>Temperature</td><td>Kernel parameter</td></tr><tr><td>HealingMNIST</td><td>128</td><td>64</td><td>30</td><td>1e-3</td><td>1.0</td><td>2.0</td></tr><tr><td>RotatedMNIST</td><td>128</td><td>64</td><td>30</td><td>1e-3</td><td>1.0</td><td>2.0</td></tr><tr><td>Beijing</td><td>128</td><td>64</td><td>100</td><td>1e-3</td><td>1.0</td><td>4.0</td></tr><tr><td>US Local</td><td>64</td><td>16</td><td>20</td><td>1e-4</td><td>1.0</td><td>2.0</td></tr><tr><td>Physionet2012</td><td>16</td><td>256</td><td>50</td><td>1e-3</td><td>1.0</td><td>32</td></tr></table>
+
+For a fair comparison, the magnitudes of the number of parameters are set to be equivalent among the evaluated deep learning methods.
+
+Table C2: Number of learnable parameters.  
+
+<table><tr><td></td><td>Ours</td><td>VAE</td><td>HI-VAE</td><td>GP-VAE</td><td>BRITS</td><td>SAITS</td><td>mTANs</td><td>CSDI</td></tr><tr><td>HealingMNIST</td><td>1.2M</td><td>1.2M</td><td>1.2M</td><td>1.2M</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>RotatedMNIST</td><td>1.2M</td><td>1.2M</td><td>1.2M</td><td>1.2M</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Beijing</td><td>368K</td><td>368K</td><td>368K</td><td>368K</td><td>343K</td><td>362K</td><td>398K</td><td>375K</td></tr><tr><td>US Local</td><td>68K</td><td>68K</td><td>68K</td><td>68K</td><td>51K</td><td>87K</td><td>63K</td><td>62K</td></tr><tr><td>Physionet2012</td><td>21K</td><td>21K</td><td>21K</td><td>21K</td><td>24K</td><td>27K</td><td>29K</td><td>33K</td></tr></table>
+
+# D DATASET, BASELINES, AND MORE RELATED WORKS
+
+# D.1 DATASET STATISTICS
+
+We provide baseline statistics of the benchmark datasets.
+
+Table D1: Data Statistics.  
+
+<table><tr><td></td><td># Samples</td><td>Len (T)</td><td>Feature Dim</td><td># Classes</td><td>Missing Ratio (ori/art)</td></tr><tr><td>HealingMNIST</td><td>50000/10000/10000</td><td>10</td><td>28 × 28 × 1</td><td>10</td><td>- /60%</td></tr><tr><td>RotatedMNIST</td><td>50000/10000/10000</td><td>10</td><td>28 × 28 × 1</td><td>10</td><td>- /60%</td></tr><tr><td>Beijing</td><td>851/304/306</td><td>24</td><td>132</td><td>-</td><td>2%/60%</td></tr><tr><td>USLocal</td><td>847/298/298</td><td>168</td><td>11</td><td>-</td><td>- /60%</td></tr><tr><td>Physionet2012</td><td>3997/3993/3997</td><td>48</td><td>35</td><td>2</td><td>80%/ 2%</td></tr></table>
+
+# D.2 BASELINE MODELS
+
+HI-VAE (Nazabal et al., 2020) HI-VAE proposes a general generative framework that extends the VAE structure to be suitable for fitting incomplete heterogeneous data. Nazabal et al. (2020) proposed likelihood models for diverse data – real-valued, positive real-valued, interval, categorical, ordinal, and count data ��� and also expanded the conventional VAE to handle missing features. In this paper, we followed their approach to calculate reconstruction only to the observed features.
+
+GP-VAE (Fortuin et al., 2020) GP-VAE expands information bottleneck on time series imputation, by using Gaussian process prior in the latent space. To use the Gaussian process, GP-VAE's inference model encodes the input with respect to each latent dimension,  $q(\mathbf{z}_{1:T,j}|\mathbf{x}_{1:T}^o) = \mathcal{N}(m_j,\Lambda_j^{-1})$ , where  $j$  indexes the dimensions in the latent space. The latent prior is formalized by  $\mathbf{z}_t\sim \mathcal{GP}(m_z(\cdot),k_z(\cdot ,\cdot))$  where  $m$  and  $k$  are the mean and covariance functions, respectively. GP-VAE also utilizes the Cauchy kernel as a covariance function to capture temporal dynamics.
+
+# D.3 MORE RELATED WORKS
+
+GAN-based imputation methods generate imputed values adversarially, assisted by a masked reconstruction loss, with the goal of making the generated samples closely resemble the original ones based on their observed values and indistinguishable from the original incomplete data by a discriminator. GAIN (Yoon et al., 2018a) uses a random masking and hinting mechanism to aid the discriminator in better distinguishing real and generated samples, thereby improving imputation performance. GRUI-GAN (Luo et al., 2018) and its extension  $E^2$ GAN (Luo et al., 2019) adopted the GAN framework into a temporal setting by enhancing the GRU cell to account for temporal dynamics, considering time lag influences related to observations and missing values.
+
+# E TABLE OF NOTATIONS
+
+# Notations
+
+<table><tr><td>X1:T</td><td>Complete data features of one time series</td></tr><tr><td>X1o:T</td><td>Observed features of one time series</td></tr><tr><td>X1m:T</td><td>Missing features of one time series</td></tr><tr><td>X\t</td><td>Complete data features of one time series, except the time t</td></tr><tr><td>Xo:t</td><td>Observed features of one time series, except the time t</td></tr><tr><td>Xm:t</td><td>Missing features of one time series, expt the time t</td></tr><tr><td>Xt</td><td>Complete data features at time t</td></tr><tr><td>Xo</td><td>Observed features at time t</td></tr><tr><td>Xm</td><td>Missing features at time t</td></tr><tr><td>Zt</td><td>Representation features at time t, generated from Xo1:T</td></tr><tr><td>Zt</td><td>Representation features at time t, generated from Xo\t</td></tr><tr><td>φ</td><td>Set of parameters of an encoder</td></tr><tr><td>θ</td><td>Set of parameters of a decoder</td></tr></table>

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+# CONDITIONAL INSTRUMENTAL VARIABLE REGRESSION WITH REPRESENTATION LEARNING FOR CAUSAL INFERENCE
+
+Debo Cheng  $^{1*}$ , Ziqi Xu  $^{1,2*}$ , Jiuyong Li  $^{1}$ , Lin Liu  $^{1}$ , Jixue Liu  $^{1}$  & Thuc Duy Le  $^{1}$ $^{1}$ University of South Australia  $^{1}$  Data61, CSIRO  $^{2}$   
+{firstname.lastname} @unisa.edu.au  $^{1}$ , ziqi.xu@data61.csiro.au
+
+# ABSTRACT
+
+This paper studies the challenging problem of estimating causal effects from observational data, in the presence of unobserved confounders. The two-stage least square (TSLS) method and its variants with a standard instrumental variable (IV) are commonly used to eliminate confounding bias, including the bias caused by unobserved confounders, but they rely on the linearity assumption. Besides, the strict condition of unconfounded instruments posed on a standard IV is too strong to be practical. To address these challenging and practical problems of the standard IV method (linearity assumption and the strict condition), in this paper, we use a conditional IV (CIV) to relax the unconfounded instrument condition of standard IV and propose a non-linear CIV regression with Confounding Balancing Representation Learning, CBRL.CIV, for jointly eliminating the confounding bias from unobserved confounders and balancing the observed confounders, without the linearity assumption. We theoretically demonstrate the soundness of CBRL.CIV. Extensive experiments on synthetic and two real-world datasets show the competitive performance of CBRL.CIV against state-of-the-art IV-based estimators and superiority in dealing with the non-linear situation.
+
+# 1 INTRODUCTION
+
+Causal inference is a fundamental problem in various fields such as economics, epidemiology, and medicine (Hernán & Robins, 2010; Imbens & Rubin, 2015). Estimating treatment (i.e., causal) effects of a given treatment on an outcome of interest from observational data plays an important role in causal inference. Nevertheless, estimating causal effects from observational data suffers from the confounding bias caused by confounders<sup>1</sup>, and unobserved confounders make causal effect estimation more challenging. When there is an unobserved confounder affecting both the treatment and outcome variables, as illustrated by the causal DAG<sup>2</sup> in Fig. 1 (a), the endogeneity problem<sup>3</sup> (Antonakis et al., 2010; Cheng et al., 2024) arises, and in this situation, it is well-known that the causal effect of the treatment on the outcome is non-identifiable (Pearl, 2009; Shpitser & Pearl, 2006).
+
+Instrumental variable (IV) is a powerful approach to addressing the endogeneity problem and identifying causal effects in the presence of unobserved confounders, but under strong assumptions (Anderson & Rubin, 1949; Hernán & Robins, 2006). Typically, the IV approach requires a valid standard IV (denoted as  $S$ ) that satisfies the following conditions: (i)  $S$  affects the treatment  $W$  (a.k.a. the relevance condition), (ii) the causal effect of  $S$  on the outcome  $Y$  is only through  $W$  (a.k.a. the exclusion condition), and (iii)  $S$  and  $Y$  are not confounded by measured or unmeasured variables (a.k.a. the unconfounded instrument condition). For instance, the variable  $S$  in the causal DAG of Fig. 1 (b) is a standard IV that satisfies the three conditions. Under the linearity assumption, the two-
+
+![](images/17735dc22af57cb67fffd93dd4c590480b6ce017f90aff4b129d6da2bee7d8d3.jpg)  
+(a)
+
+![](images/11d2b08422eeba573e79b15fb555febb2a547f1ec27577a77cda8c6808c93ceb.jpg)  
+(b)  
+Figure 1: An illustration presents four scenarios: (a) unobserved confounders, (b) standard IV, (c) the imbalance issue with standard IV, and (d) the imbalance issue with CIV. Variables include treatment  $W$ , outcome  $Y$ , the set of observed confounders  $\mathbf{C}$ , the set of unobserved confounders  $\mathbf{U}$ , and IV  $S$ . Directed edges indicate causal relationships and dotted lines represent statistical associations. In (a), the causal effect of  $W$  on  $Y$  is non-identifiable due to  $\mathbf{U}$  affecting both  $W$  and  $Y$ ; In (b), the causal effect of  $W$  on  $Y$  can be estimated by using the IV  $S$ ; In (c), the dependence of  $\mathbf{U}$  and  $\mathbf{C}$  affecting the potential for confounding and distribution imbalance in causal effect estimation, addressed in Wu et al. (2022); In (d),  $S$  is a conditional IV (CIV). In this case,  $\mathbf{C}$  induces bias across the different groups as specified by the different values of  $S$ , potentially biasing the estimated causal effect of  $W$  on  $Y$ . This paper addresses this problem.
+
+![](images/33d848138b632c53ea0bd9f5569ba3268801f06118b173884461716e991098e0.jpg)  
+(c)
+
+![](images/ca42cba544925c553a9ce1d115b93c26368835c1675b32a00272c2f824f21e9f.jpg)  
+(d)
+
+stage least squares (TSLS) method is often employed to estimate the causal effects in the presence of unobserved confounders using a given standard IV (Angrist & Imbens, 1995; Imbens, 2014).
+
+In many real-world scenarios, non-linear variable relationships combined with the effect of unobserved confounders make estimating causal effects from observational data more challenging. Recently, some methods were proposed to address the challenging case with non-linear relationships when a standard IV is used, such as DeepIV (Hartford et al., 2017), kernelIV regression (Singh et al., 2019), double machine learning IV (DMLIV) method (Syrgkanis et al., 2019), etc. These methods are built on two-stage regression models. Using Fig. 1 (b) as an example, the first stage of such a two-stage approach is to fit a non-linear regression model for the conditional distribution of the treatment  $W$  conditioning on the standard IV  $S$  and the set of measured pretreatment variables  $\mathbf{C}$ , i.e., the estimated treatment  $\hat{W}$  is obtained based on  $P(W \mid S, \mathbf{C})$ . The second stage is to fit a non-linear regression model on  $Y$  conditioning on  $\mathbf{C}$  and  $\hat{W}$  obtained in the first stage.
+
+However, due to the spurious associations between  $\mathbf{U}$  and  $\mathbf{C}$  (as illustrated in Fig. 1 (c)), the distributions of  $\mathbf{C}$  in the treated and control groups may be imbalanced (Shalit et al., 2017), which can result in biased estimations in the non-linear outcome regression model (Abadie, 2003), especially when the outcome regression model is misspecified. To address the imbalance issue in the outcome regression model, Wu et al. Wu et al. (2022) proposed a confounder balancing IV regression (CBIV) algorithm, and demonstrated the effectiveness of the confounding balancing representation learning approach. However, CBIV ignores the confounding bias caused by the imbalance distribution of  $\mathbf{C}$  across different groups specified by the different values of  $S$  in the first stage.
+
+To address the imbalance problems and the practical issues of the standard IV approach (linearity assumption and the strict condition), we consider conditional IV (CIV) in this work. CIV relaxes the third condition of the standard IV, and allows a variable to be a valid IV by conditioning on a set of measured variables (Brito & Pearl, 2002; van der Zander et al., 2015). In real-wolrd applications, a CIV is less restrictive than a standard IV and is easier to find. For example, we aim to estimate the effect of a treatment  $(W)$  on the recovery from a disease  $(Y)$ .  $W$  and  $Y$  are affected by unobserved confounders  $\mathbf{U}$ , such as the socioeconomic status of patients. The variable of clinical practice guidelines  $(S)$  for treatment decision  $W$  is a CIV with the conditioning set  $(\mathbf{C})$ , which includes factors such as age, health condition, and an individual's propensity to seek medical assistance. This example can be represented by the causal DAG in Fig. 1 (d). In contrast, a standard IV affecting only  $W$  but not  $Y$  via other paths is difficult to find in the real-world.
+
+A two-stage regression is commonly used to obtain an estimation of the causal effect by an IV or a CIV approach. There are two risks of biases in the estimation using a two-stage solution in Fig. 1 (d). Firstly, the distribution imbalance of  $\mathbf{C}$  between the treated and control groups leads to a bias in the outcome regression model. Secondly, unobserved confounders could cause a distribution imbalance of  $\mathbf{C}$  between the different groups specified by different values of  $S$  as  $\mathbf{C}$  leading to a bias in the treatment regression model. Current solutions (Hartford et al., 2017; Wu et al., 2022) consider a standard IV and use a regression model conditioning on  $\mathbf{C}$ , and they are not applicable to a CIV case since they do not mitigate the confounding bias between  $S$  and  $W$ .
+
+To enhance the practical use of the IV approach, specifically to address the double imbalance problem caused by both observed and unobserved confounders in non-linear cases when using a CIV, we propose CBRL.CIV, a novel Confounding Balancing Representation Learning method for nonlinear CIV regression. We further demonstrate its theoretical soundness and practical usefulness. By integrating CIV with confounding balance, CBRL.CIV emerges as a comprehensive solution that adeptly navigates the intricacies of confounder effects in non-linear models, offering a significant advancement in causal inference. We summarise the main contributions of our work as follows.
+
+- We study the problem of causal effect estimation from data with unobserved confounders in a non-linear setting and design a novel and practical CIV method to solve the problem. To the best of our knowledge, this is the first work that considers the imbalance issues in the applications of CIVs when there are finite samples.  
+- We propose a novel representation learning method, CBRL.CIV for balancing the distributions and removing the confounding bias simultaneously. We theoretically show the soundness of the proposed CBRL.CIV method. Specifically, CBRL.CIV utilises two balance terms to enhance generalisation across different groups specified by different values of  $S$  in the treatment network, and different groups specified by different values of  $\hat{W}$  in the outcome regression network.  
+- Extensive experiments are conducted on synthetic and real-world datasets and the results demonstrate the effectiveness of CBRL.CIV in average causal effect estimation.
+
+# 2 PRELIMINARY
+
+We introduce some important notations and definitions used in the paper. Let  $W \in \{1,0\}$  denote a binary treatment variable where  $W = 1$  represents treated and  $W = 0$  untreated,  $Y$  the outcome variable of interest, and  $\mathbf{U}$  a set of unobserved confounders. Let  $\mathbf{X}$  denote the set of pretreatment variables, namely, all variables in  $\mathbf{X}$  are measured before  $W$  is measured. Finally, we assume that the complete data are independent and identically distributed realizations of  $(W,Y,\mathbf{U},\mathbf{X})$ , and the observed data is  $\mathcal{O} = (W,Y,\mathbf{X})$ .
+
+We aim at estimating the average causal effect of  $W$  on  $Y$  from  $\mathcal{O}$  with the presence of unobserved confounders  $\mathbf{U}$ . In the observational data  $\mathcal{O}$ , for each individual  $i$ , the potential outcomes are denoted as  $Y_{i}(W = 1)$  and  $Y_{i}(W = 0)$  corresponding to  $W$  (Imbens & Rubin, 2015; Rosenbaum & Rubin, 1983). Note that only one of the potential outcomes can be observed for a given individual, which is known as the fundamental problem in causal inference (Imbens & Rubin, 2015).
+
+Furthermore, we use a causal DAG  $\mathcal{G} = (\mathbf{V},\mathbf{E})$  to represent the underlying mechanism of the studied system, where  $\mathbf{V}$  represents the set of nodes (variables) of the DAG, and  $\mathbf{E} \subseteq \mathbf{V} \times \mathbf{V}$  is the set of directed edges (Pearl, 2009; Spirtes et al., 2000). For any  $X_{i}, X_{j} \in \mathbf{V}$  and  $i \neq j$ , the directed edge  $X_{i} \rightarrow X_{j}$  in the causal DAG  $\mathcal{G}$  indicates that  $X_{i}$  is a direct cause (parent) of  $X_{j}$ , and  $X_{j}$  is a direct effect (child) of  $X_{i}$ . A path from  $X_{i}$  to  $X_{j}$  is a directed path or causal path when all edges along it are directed towards  $X_{j}$ . Additionally, on a causal path from  $X_{i}$  to  $X_{j}$ ,  $X_{i}$  is an ancestor of  $X_{j}$ , and conversely,  $X_{j}$  is a descendant of  $X_{i}$ . Using a causal DAG, a CIV (Definition 7.4.1 on Page 248 (Pearl, 2009)) can be defined as:
+
+Definition 1 (Conditional IV). Let  $\mathcal{G} = (\mathbf{V},\mathbf{E})$  be a causal DAG with  $\mathbf{V} = \mathbf{X}\cup \mathbf{U}\cup \{W,Y\}$ , a variable  $S\in \mathbf{X}$  is a conditional IV w.r.t.  $W\to Y$  if there exists a set of measured variables  $\mathbf{C}\subseteq \mathbf{X}$  such that (i)  $S\perp_{d}W\mid \mathbf{C}$ , (ii)  $S\perp_{d}Y\mid \mathbf{C}$  in  $\mathcal{G}_{\underline{W}}$ , and (iii)  $\forall C\in \mathbf{C}$ ,  $C$  is not a descendant of  $Y$ , where  $\mathcal{G}_{\underline{W}}$  is a manipulated DAG of  $\mathcal{G}$  by deleting the direct edge  $W\to Y$  from  $\mathcal{G}$ , and  $\perp_{d}$  and  $\perp_{d}$  are  $d$ -connection and  $d$ -separation (Pearl, 2009), respectively.
+
+In the above definition,  $\mathbf{C}$  is known as the set of variables that instrumentalise  $S$ , or the conditioning set of  $S$ . Note that a CIV is a more general IV than the standard IV and when  $\mathbf{C}$  is an empty set in Definition 1,  $S$  is a standard IV. More definitions are provided in Appendix A.
+
+# 3 THE PROPOSED CBRL.CIV METHOD
+
+# 3.1 PROBLEM SETTING & ASSUMPTIONS
+
+We assume that a binary CIV  $S \in \mathbf{X}$  is known and further assume that the rest variables in  $\mathbf{X}$ , denoted as  $\mathbf{C}$ , are pre-IV variables, i.e., they are measured before  $S$  is measured. Our problem is illustrated in Fig. 1 (d), a typical setting where a CIV is used, and in the figure,  $S$  is a CIV that is instrumentalised by  $\mathbf{C}$  (i.e.,  $\mathbf{C}$  is the conditioning set of  $S$ ).  $\mathbf{C}$  affects both  $S$  and  $W$ , and  $\mathbf{C}$  affects both  $W$  and  $Y$ .  $\mathbf{C}$  has two roles here: observed confounders  $\mathbf{C}$  between  $W$  and  $Y$  and conditioning set of CIV  $S$  w.r.t.  $W$  and  $Y$ . As discussed previously, when a CIV is used, in addition to the imbalanced distributions of  $\mathbf{C}$  between  $W$  and  $Y$  as discussed in the previous work (Wu et al., 2022),  $\mathbf{C}$  is imbalanced among the groups specified by the different values of  $S$ . That is, we face a double distribution imbalance problem, one between  $S$  and  $W$ , and one between  $W$  and  $Y$ . Therefore the existing solution addressing the distribution imbalance of  $\mathbf{C}$  between  $W$  and  $Y$  such as CBIV (Wu et al., 2022) cannot be applied because the confounding bias resulted from the imbalance of  $\mathbf{C}$  among the groups specified by the different values of  $S$  remain unresolved.
+
+To address the double imbalance issue caused in using a CIV approach, especially in non-linear scenarios, we have designed a novel three-stage method based on balanced representation learning techniques (Shalit et al., 2017; Wu et al., 2022).
+
+The following assumptions are essential when using CIV method for causal effect estimation in data.
+
+Assumption 1. (IV positivity) It is almost surely that  $0 < P(S = 1 \mid \mathbf{C}) < 1$ .
+
+The IV positivity assumption is necessary for non-parametric identification (Greenland, 2000).
+
+Assumption 2. (Independent compliance type (Cui & Tchetgen Tchetgen, 2021; Hernán & Robins, 2006)) It is almost surely that  $\phi (\mathbf{C})\equiv P(W = 1\mid S = 1,\mathbf{C}) - P(W = 1\mid S = 0,\mathbf{C}) = \tilde{\phi} (\mathbf{C},\mathbf{U})$  where  $\tilde{\phi} (\mathbf{C},\mathbf{U})\equiv P(W = 1\mid S = 1,\mathbf{C},\mathbf{U}) - P(W = 1\mid S = 0,\mathbf{C},\mathbf{U})$
+
+This assumption means that there is no additive interaction between  $S$  on  $\mathbf{U}$  in a system for the treatment probability conditioning on  $\mathbf{C}$  and  $\mathbf{U}$ . It also demonstrates that  $\mathbf{U}$  is independent of an individual's compliance type (Imbens & Rubin, 2015). In a binary  $S$ , there are four compliance types for an individual: Compliers, Always-takers, Never-takers and Defiers (Angrist & Imbens, 1995; Imbens & Rubin, 2015). The details of compliance types are provided in Appendix A.
+
+# 3.2 CBRL.CIV
+
+Our proposed CBRL.CIV method tackles the double distribution imbalance issue by learning a confounding balancing representation of  $\mathbf{C}$ , denoted as  $\mathbf{Z}$  using a representation network  $\psi_{\theta}(\cdot)$ , i.e.,  $\mathbf{Z} = \psi_{\theta}(\mathbf{C})$  to achieve confounding balance in counterfactual inference and remove the confounding bias from unobserved confounders without linearity assumption. CBRL.CIV consists of three components, CIV regression for balancing the distributions of  $\mathbf{C}$  between  $S$  and  $W$ , treatment regression for balancing the distribution between  $W$  and  $Y$ , and outcome regression using the learned balancing representation  $\mathbf{Z}$ .
+
+CIV Regression for Confounding Balance. To balance the distributions of  $\mathbf{C}$  between different values of the CIV  $S$ , we need to ensure that  $S$  and the representation  $\mathbf{Z}$  are independent conditioning on  $\mathbf{C}$ . It is challenging to directly learn a representation  $\mathbf{Z}$  such that  $S \perp \perp \mathbf{Z} \mid \mathbf{C}$  from the data. We propose to achieve this in two steps. In Step one, we employ a logistic regression network  $\varphi_{\mu}(\mathbf{c}_i)$  with  $\mu$  as a parameter of the network for an individual  $i$  to predict the conditional probability distribution of the CIV  $S$ ,  $P(S \mid \mathbf{C})$ , and we sample  $\hat{S}$  from  $P(S \mid \mathbf{C})$ , i.e.,  $\hat{S} \sim P(S \mid \mathbf{C})$ . For the CIV regression model, we have the following loss function:
+
+$$
+L _ {S} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(s _ {i} \log \left(\varphi_ {\mu} \left(\mathbf {c} _ {i}\right)\right) + \left(1 - s _ {i}\right) \left(1 - \log \left(\varphi_ {\mu} \left(\mathbf {c} _ {i}\right)\right)\right)\right) \tag {1}
+$$
+
+where  $n$  is the number of samples in the data. Step two learns  $\mathbf{Z}$  such that  $\hat{S} \perp \perp \mathbf{Z}$ . For achieving this goal, we train the representation network  $\mathbf{Z} = \psi_{\theta}(\mathbf{C})$  with the learnable parameter  $\theta$  and minimise the discrepancy of the distributions of  $\mathbf{Z}$  for different values of  $\hat{S}$  by using  $IPM(\cdot)$ , which is the
+
+integral probability metric for measuring the discrepancy of distributions. The above requirement for confounding balance is formally stated as the following objective:
+
+$$
+D (\hat {S}, \psi_ {\theta} (\mathbf {C})) = I P M \left(\left\{\psi_ {\theta} (\mathbf {C}) P \left(\hat {s} _ {i} = 0 \mid \mathbf {c} _ {i}\right) \right\} _ {i = 1} ^ {n}, \left\{\psi_ {\theta} (\mathbf {C}) P \left(\hat {s} _ {i} = 1 \mid \mathbf {c} _ {i}\right) \right\} _ {i = 1} ^ {n}\right) \tag {2}
+$$
+
+where  $\{\psi_{\theta}(\mathbf{C})P(\hat{s}_i = s\mid \mathbf{c}_i)\}_{i = 1}^n$ ,  $s\in \{0,1\}$  is the distribution of the representation  $\mathbf{Z}$  (i.e.,  $\psi_{\theta}(\mathbf{C}))$  in the sub-group  $\hat{S} = s$ . In this way, we obtain  $\hat{S}\perp \mathbf{Z}$ , which implies  $S\perp \mathbf{Z}\mid \mathbf{C}$  because  $\hat{S}$  is obtained by conditioning on  $\mathbf{C}$  and  $\mathbf{Z}$  is also obtained by conditioning on  $\mathbf{C}$ . We utilise the Wasserstein distance as the metric for evaluating the discrepancies for CFR, CBIV, and our method CBRL.CIV, enabling a fair comparison in our experiments because Wasserstein distance is used in the implementations of CFR and CBIV.
+
+Treatment Regression with Confounding Balance. To address the distribution imbalance of  $\mathbf{C}$  between  $W$  and  $Y$  and the effect of unobserved confounders  $U$ , we follow the previous non-linear IV-based methods, such as DeepIV (Hartford et al., 2017) and CBIV (Wu et al., 2022), to build a treatment regression network by using the estimated  $\hat{S}$  obtained in the first stage and  $\mathbf{C}$  to regress  $W$  for obtaining an estimated  $\hat{W}$  that is not affected by  $\mathbf{U}$ . Specifically, we use a logistic regression network  $\varphi_{\nu}(\hat{s}_i,\mathbf{c}_i)$  with the learnable parameter  $\nu$  to calculate the conditional probability distribution of the treatment  $W$ , i.e.,  $\hat{W}\sim P(W\mid \hat{S},\mathbf{C})$ , and optimise the following loss function:
+
+$$
+L _ {W} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(w _ {i} \log \left(\varphi_ {\nu} \left(\hat {s} _ {i}, \mathbf {c} _ {i}\right)\right) + \left(1 - w _ {i}\right) \left(1 - \log \left(\varphi_ {\nu} \left(\hat {s} _ {i}, \mathbf {c} _ {i}\right)\right)\right)\right) \tag {3}
+$$
+
+After obtaining  $\hat{W}$ , to achieve  $\hat{W} \perp \mathbf{Z}$  for alleviating the effect of the distribution imbalance of  $\mathbf{C}$  between  $W$  and  $Y$ , we minimise the discrepancy of the distributions  $\mathbf{Z}$  for different values of  $\hat{W}$ , and the objective function can be formed as:
+
+$$
+D (\hat {W}, \psi_ {\theta} (\mathbf {C})) = I P M \left(\left\{\psi_ {\theta} (\mathbf {C}) P (\hat {w} _ {i} = 0 \mid \hat {s} _ {i}, \mathbf {c} _ {i}) \right\} _ {i = 1} ^ {n}, \left\{\psi_ {\theta} (\mathbf {C}) P (\hat {w} _ {i} = 1 \mid \hat {s} _ {i}, \mathbf {c} _ {i}) \right\} _ {i = 1} ^ {n}\right) \tag {4}
+$$
+
+where  $\{\psi_{\theta}(\mathbf{C})P(\hat{w}_i = w\mid \hat{s}_i,\mathbf{c}_i)\}_{i = 1}^n$ ,  $w\in \{0,1\}$  represents the distribution of the representation  $\mathbf{Z} = \psi_{\theta}(\mathbf{C})$  in the sub-group  $\hat{W} = w$ . The confounding balance term is a constraint term that is included in the learning process to ensure that  $\psi_{\theta}(\mathbf{C})$  and  $\hat{W}$  are independent, i.e.,  $\hat{W}\perp \perp \mathbf{Z}$ .
+
+Outcome Regression. In the final stage, we propose to regress the outcome using  $\mathbf{Z}$  (i.e.,  $\psi_{\theta}(\mathbf{C})$ ) and the estimated  $\hat{W}$  obtained in the treatment regression stage. Furthermore, we use two separate regression networks for counterfactual inference, as done in Hartford et al. (2017). Both networks, denoted as  $f_{\gamma^1}(\psi_\theta (\mathbf{C}))$  and  $f_{\gamma^0}(\psi_\theta (\mathbf{C}))$ , have two learnable parameters  $\gamma^1$  and  $\gamma^0$ , respectively.
+
+The estimated treatment  $\hat{W} \sim P(W \mid \hat{S}, \mathbf{C})$  and the balancing representation  $\psi_{\theta}(\mathbf{C})$  are used to regress  $Y$  in the outcome regression network. The objective function for the two separate regression networks is defined as follows:
+
+$$
+L _ {Y} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(y _ {i} - \sum_ {w _ {i} \in \{0, 1 \}} f _ {\gamma^ {w _ {i}}} \left(\psi_ {\theta} (\mathbf {C})\right) P \left(w _ {i} \mid \hat {s} _ {i}, \mathbf {c} _ {i}\right)\right) ^ {2} \tag {5}
+$$
+
+where  $y_{i}$  is the value of observed outcome, and  $P(w_{i} \mid \hat{s}_{i}, \mathbf{c}_{i})$  is obtained from the treatment regression network  $\varphi_{\nu}(\hat{s}_i, \mathbf{c}_i)$  in the second stage.
+
+**Optimisation.** We employ a three-stage optimisation for our proposed CBRL.CIV method. Firstly, we optimise the CIV regression network  $\varphi_{\mu}(\mathbf{c}_i)$  for minimising the loss function  $L_{S}$  in Eq. 1. Then, we optimise the treatment regression network  $\varphi_{\nu}(\hat{s}_i,\mathbf{c}_i)$  for minimising the loss function  $L_{W}$  in Eq. 3. Finally, we simultaneously optimise the two confounding balance representation learning networks and the outcome regression network by adding the two confounding balance terms as a type of regularisation to the objective function of the outcome regression network, as follows:
+
+$$
+\min  _ {\theta , \gamma^ {0}, \gamma^ {1}} L _ {Y} + \alpha D (\hat {W}, \psi_ {\theta} (\mathbf {C})) + \beta D (\hat {S}, \psi_ {\theta} (\mathbf {C})) \tag {6}
+$$
+
+where  $\alpha$  and  $\beta$  are tuning parameters.
+
+After we obtain  $\psi_{\theta}(\mathbf{C})$  and parameters  $\gamma^0$  and  $\gamma^{1}$ , the average causal effect (ACE) of  $W$  on  $Y$  can be calculated by
+
+$$
+A \hat {C} E = \mathbb {E} \left[ f _ {\gamma^ {1}} \left(\psi_ {\theta} (\mathbf {C})\right) - f _ {\gamma^ {0}} \left(\psi_ {\theta} (\mathbf {C})\right) \right] \tag {7}
+$$
+
+The pseudo-code of the proposed CBRL.CIV method is provided in Appendix B.1.
+
+# 3.3 THEORETICAL ANALYSIS
+
+The deep learning based IV estimator, DeepIV (Hartford et al., 2017), provides the counterfactual prediction function:
+
+$$
+f (W, \mathbf {C}) \equiv g (W, \mathbf {C}) + \mathbb {E} [ \mathbf {U} \mid \mathbf {C} ] \tag {8}
+$$
+
+where  $\mathbf{U}$  is only conditioning on  $\mathbf{C}$ , but not  $W$ .  $\mathbb{E}[\mathbf{U} \mid \mathbf{C}]$  is non-zero, but stays constant with the change of  $W = 1$  to  $W = 0$ . With a valid standard IV  $S$ , the counterfactual prediction function  $f(W, \mathbf{C})$  can be solved over the observed confounders:
+
+$$
+\mathbb {E} [ Y \mid S, \mathbf {C} ] = \int f (W, \mathbf {C}) d P (W \mid S, \mathbf {C}) \tag {9}
+$$
+
+where  $dP(W\mid S,\mathbf{C})$  is the conditional treatment distribution. The relationship in Eq.(9) defines an inverse problem for the function  $f$  by using two observed functions  $\mathbb{E}[Y\mid S,\mathbf{C}]$  and  $P(W\mid S,\mathbf{C})$  (Hartford et al., 2017; Newey & Powell, 2003). Most IV methods (Chen & Pouzo, 2012; Singh et al., 2019) employ a two-stage process for solving this problem: estimating  $\hat{P}(W\mid S,\mathbf{C})$  for the true  $P(W\mid S,\mathbf{C})$ , and then approximating  $f(W,\mathbf{C})$  given the estimated  $P(W\mid S,\mathbf{C})$ .
+
+We have the following theorem to identify (solve) the counterfactual prediction in Eq.(9) by using a CIV and deep neural networks Bennett et al. (2019); LeCun et al. (2015) in our problem setting.
+
+Theorem 1. Given the observational data  $\mathcal{O} = (\mathbf{C}, S, W, Y)$  generated from a causal DAG  $\mathcal{G} = (\mathbf{C} \cup \mathbf{U} \cup \{S, W, Y\}, \mathbf{E})$  where  $\mathbf{E}$  is the set of directed edges as shown in Fig. 1 (d), suppose that  $S$  is a CIV and  $\mathbf{C}$  is a set of pre-IV variables. Under Assumptions 1 and 2, if the learned representation  $\mathbf{Z} = \psi_{\theta}(\mathbf{C})$  of  $\mathbf{C}$  is independent of estimated  $\hat{S}$  and  $\hat{W}$ , then  $f(W, \mathbf{C})$  can be identified by using the  $\hat{S}$  and the representation  $\mathbf{Z}$  as:  $\mathbb{E}[Y \mid S, \mathbf{C}] = \int f(W, \mathbf{Z}) dP(W \mid \hat{S}, \mathbf{C})$ , where  $P(W \mid \hat{S}, \mathbf{C})$  is the estimated conditional treatment distribution by using  $\hat{S}$  and  $\mathbf{C}$ .
+
+The proof is given in Appendix B.2. Theorem 1 ensures the soundness of the proposed CBRL.CIV method by solving an inverse problem for counterfactual outcomes prediction.
+
+# 4 EXPERIMENTS
+
+The goal of the experiments is to assess the performance of the proposed CBRL.CIV method in ACE estimation from data in the presence of unobserved confounders. Thus, we conduct extensive experiments on synthetic datasets since they allow the true ACE to be known. To show the potential of the proposed CBRL.CIV method in real-world applications, we also evaluate the performance of CBRL.CIV on two real-world datasets that have been widely used in the evaluation of IV-based estimators (Abadie, 2003; Card, 1993; Cheng et al., 2024). Due to the page limitation, we report the ablation study, and the effect of  $\alpha$  and  $\beta$  on CBRL.CIV in Appendix C.3 and C.4, respectively.
+
+# 4.1 EXPERIMENTAL SETUP
+
+Baselines. We assess the performance of CBRL.CIV by comparing it with eight state-of-the-art causal effect estimators for causal effect estimation from observational data. These estimators can be categorised into three groups: (I) most closely related work: CBIV, the confounding balancing IV regression Wu et al. (2022); (II) Other IV methods using machine learning/representation learning, including (1) the DeepIV (Hartford et al., 2017) (a deep neural network based IV estimator), (2) OrthIV Sjolander & Martinussen (2019) (an orthogonal regularisation machine learning based IV estimator), (3) DMLIV (Syrgkanis et al., 2019) (double machine learning based IV estimator), (4) DRIV (Syrgkanis et al., 2019) (doubly robust learning based IV estimator), (5) SDRIV (Chernozhukov et al., 2018) (sparse linear doubly DRIV), and (6) ForestDRIV (Syrgkanis et al., 2019) (random forest DRIV); (III) Other estimator, not IV based, but a popular representation learning method which considers confounding balance, i.e., CounterFactual Regression by using deep representation learning (CFR) (Shalit et al., 2017).
+
+Implementation. We have implemented CBRL.CIV using Python programming language. We used multi-layer perceptrons with ReLU activation function and BatchNorm to implement the logistic regression networks for CIV regression and treatment regression. For optimising the CIV
+
+Table 1: The table summarises the estimation error (mean±STD) over 30 synthetic datasets with different settings. The smallest estimation error in each setting is highlighted. Note that CBRL.CIV obtains the smallest estimation error in terms of mean and STD.  
+
+<table><tr><td colspan="4">Within-Sample</td><td colspan="3">Out-of-Sample</td></tr><tr><td>Methods</td><td>Syn-4-4</td><td>Syn-8-8</td><td>Syn-12-12</td><td>Syn-4-4</td><td>Syn-8-8</td><td>Syn-12-12</td></tr><tr><td>DeepIV</td><td>1.58±0.11</td><td>1.49±0.14</td><td>1.45±0.17</td><td>1.58±0.17</td><td>1.45±0.18</td><td>1.45±0.22</td></tr><tr><td>OrthIV</td><td>0.69±0.43</td><td>1.86±0.15</td><td>1.72±0.08</td><td>3.86±5.42</td><td>2.1±0.29</td><td>1.86±0.11</td></tr><tr><td>DMLIV</td><td>0.69±0.45</td><td>1.84±0.14</td><td>1.73±0.08</td><td>2.06±1.66</td><td>2.05±0.32</td><td>1.86±0.12</td></tr><tr><td>DRV</td><td>1.96±0.51</td><td>1.14±0.54</td><td>0.6±0.35</td><td>2.03±0.34</td><td>1.42±0.39</td><td>0.92±0.43</td></tr><tr><td>SDRIV</td><td>1.91±0.36</td><td>0.72±0.5</td><td>0.46±0.35</td><td>2.13±0.39</td><td>1.09±0.44</td><td>0.52±0.49</td></tr><tr><td>ForestDRIV</td><td>1.87±0.34</td><td>0.72±0.4</td><td>0.46±0.36</td><td>1.99±0.41</td><td>0.94±0.6</td><td>0.75±0.52</td></tr><tr><td>CFR</td><td>0.6±0.38</td><td>0.58±0.3</td><td>0.51±0.24</td><td>0.6±0.38</td><td>0.58±0.31</td><td>0.51±0.25</td></tr><tr><td>CBIV</td><td>0.48±0.27</td><td>0.4±0.19</td><td>0.41±0.14</td><td>0.48±0.27</td><td>0.4±0.19</td><td>0.41±0.14</td></tr><tr><td>CBRL.CIV</td><td>0.14±0.05</td><td>0.14±0.03</td><td>0.12±0.03</td><td>0.14±0.05</td><td>0.14±0.03</td><td>0.12±0.03</td></tr></table>
+
+and treatment regression networks, we used stochastic gradient descent (SGD) (Duchi et al., 2011). Furthermore, we used the ADAM method (Kingma & Ba, 2014) to optimise the outcome network jointing the confounding balance modules. To prevent overfitting, we employed an  $l_{2}$ -regularisation term to regularise the objective function in Eq. (6). The code of CBRL.CIV, network parameters, and parameter tuning are provided in the supplementary materials. The implementations and parameter settings of compared estimators are provided in Appendix C.1.
+
+Evaluation Metrics. For the experiments on synthetic and two real-world datasets, we use the estimated ACE error,  $\left|(A\hat{C}E - ACE)\right|$  to evaluate the performance of all estimators in ACE estimation from data with unobserved confounders.
+
+# 4.2 SIMULATION STUDY
+
+We follow the data generation process in (Hassanpour & Greiner, 2020; Wu et al., 2022) to generate synthetic datasets for the evaluation. Note that the underlying causal DAG in Fig. 1 (d) is used to generate all synthetic datasets for evaluation of the performance of CBRL.CIV is relative to the double imbalance issues in causal effect estimation from data with unobserved confounders. The details of the synthetic data generation process and more results are provided in Appendix C.2.
+
+We assess the performance of CBRL.CIV under different settings of the number of observed and unobserved confounders, and we denote the settings as 'Syn-p-q', i.e., a setting/dataset with  $p$  observed confounders and  $q$  unobserved confounders. In our experiments, we sample 10k individuals with three settings, Syn-4-4, Syn-8-8, and Syn-12-12. We report the mean and the standard deviation (STD) of the estimation error of all methods in Table 1. Note that 'Within-Sample' denotes the estimation error calculated over the training samples, and 'Out-of-Sample' implies the estimation error calculated over the testing samples.
+
+From the estimation errors of all estimators in Table 1, we have four conclusions: (1) The proposed CBRL.CIV method obtains the smallest estimation error on all synthetic datasets because of CBRL.CIV balances the distribution of  $\mathbf{C}$  between  $\hat{W}$  and  $Y$  in the outcome regression stage, as well as the distribution of  $\mathbf{C}$  between  $\hat{S}$  and  $W$  in the treatment regression. (2) CBIV has the second smallest estimation error but is worse than CBRL.CIV since it ignores the distribution imbalance of  $\mathbf{C}$  between  $\hat{S}$  and  $W$  in the treatment regression. (3) The estimators in the second group have a large estimation error since these methods ignore the confounding bias in both stages. (4) The confounding balance-based estimator, CFR has the third smallest estimation error, but CFR does not deal with the affect of  $\mathbf{U}$ .
+
+We also evaluate the impact of sample size on the performance of CBRL.CIV uses synthetic datasets with a range of sample sizes:  $2k, 4k, 6k, 8k, 10k,$  and  $20k$ . The estimation errors of all estimators on Out-of-Sample are visualised with boxplots in Fig. 3. From Fig. 3, we have the following observations: (1) CBRL.CIV has the lowest error and smallest variance of error in different sample sizes. (2) As the sample size increases, the variance of the estimation error of CBRL.CIV decreases slightly. (3) CBRL.CIV has the narrowest upper and lower bounds compared to all other methods.
+
+In summary, CBRL.CIV achieves the best performance among all estimators in ACE estimation from synthetic data in the presence of unobserved confounders.
+
+![](images/572ee354c8c65d73a3b1e73cf0b93bd89708aad5db08e7fe96c9d35e1936a5d9.jpg)  
+Figure 2: The experimental results of all estimators on synthetic datasets with unobserved confounders in non-linear scenarios. Our CBRL.CIV achieves the lowest error in ACE estimation. And as the sample increases, CBRL.CIV has a slight decrease in variance slightly.
+
+Table 2: The estimation errors (mean±STD) over the out of samples on two real-world datasets with different dimensions of  $p$  and  $q$ .  
+
+<table><tr><td>Methods</td><td>IHDP-2-2</td><td>IHDP-4-4</td><td>IHDP-6-6</td><td>Twins-2-2</td><td>Twins-4-4</td><td>Twins-6-6</td></tr><tr><td>DeepIV</td><td>2.642±0.407</td><td>2.892±0.262</td><td>2.471±0.419</td><td>0.088±0.045</td><td>0.081±0.070</td><td>0.081±0.050</td></tr><tr><td>OrthIV</td><td>14.678±64.451</td><td>3.702±3.843</td><td>11.571±17.026</td><td>0.465±0.379</td><td>0.533±0.336</td><td>0.675±0.882</td></tr><tr><td>DMLIV</td><td>7.472±10.153</td><td>8.945±9.029</td><td>4.805±2.629</td><td>3.167±8.197</td><td>1.695±1.839</td><td>0.412±0.269</td></tr><tr><td>DRV</td><td>3.865±1.501</td><td>7.163±9.231</td><td>4.875±3.889</td><td>0.588±0.534</td><td>0.602±0.853</td><td>0.589±0.421</td></tr><tr><td>SDRIV</td><td>7.373±17.763</td><td>4.009±6.502</td><td>6.373±7.868</td><td>0.799±0.585</td><td>0.591±0.403</td><td>1.773±4.778</td></tr><tr><td>ForestDRIV</td><td>16.252±29.596</td><td>12.197±35.402</td><td>5.283±5.484</td><td>0.763±0.647</td><td>0.357±0.307</td><td>0.673±0.544</td></tr><tr><td>CFR</td><td>2.570±1.064</td><td>1.754±0.501</td><td>2.917±0.419</td><td>0.037±0.008</td><td>0.039±0.021</td><td>0.040±0.026</td></tr><tr><td>CBIV</td><td>2.312±0.934</td><td>1.349±0.452</td><td>2.846±0.490</td><td>0.033±0.011</td><td>0.034±0.026</td><td>0.035±0.016</td></tr><tr><td>CBRL.CIV</td><td>0.251±0.082</td><td>0.454±0.068</td><td>0.108±0.063</td><td>0.030±0.019</td><td>0.027±0.018</td><td>0.030±0.024</td></tr></table>
+
+# 4.3 EXPERIMENTS ON TWO REAL-WORLD DATASETS
+
+We also conduct experiments on two real-world datasets, IHDP (Hill, 2011) and Twins (Shalit et al., 2017), to evaluate the performance of CBRL.CIV method and demonstrate its potential in real-world applications. The two datasets are widely used in the evaluation of data-driven causal effect estimators (Cheng et al., 2024; Guo et al., 2020).
+
+IHDP: The Infant Health and Development Program (IHDP) dataset consists of 747 infants (139 treated infants and 608 control infants). A total of  $p + q$  variables, serving as confounders, are selected from the original covariates that are ancestors of, or direct causes of  $Y$ , in which  $p$  variables are observed confounders  $\mathbf{C}$ , and  $q$  variables are unobserved confounders  $\mathbf{U}$  (where  $p$  and  $q$  take values of 2, 4, and 6 respectively). Then we have the CIV assignment probability:  $P(S \mid \mathbf{C}) = \frac{1}{1 + \exp(-\left(\sum_{i=1}^{p} C_i + \sum_{i=1}^{q} U_i\right))}$ ,  $S \sim \text{Bern}(P(S \mid \mathbf{C}))$ , and the treatment assignment probability:  $P(W \mid S, \mathbf{C}) = \frac{1}{1 + \exp(-\left(S * \sum_{i=1}^{p} C_i + \sum_{i=1}^{p} C_i + \sum_{i=1}^{q} U_i\right))}$ ,  $W \sim \text{Bern}(P(W \mid S, \mathbf{C}))$ , where  $\text{Bern}(\cdot)$  is the Bernoulli distribution.
+
+Twins: The dataset is collected from the recorded twin births and deaths in the USA between 1989-1991 (Almond et al., 2005). Similar to Wu et al. Wu et al. (2022), we use 5,271 records from same-sex twins with weights less than  $2,000\mathrm{g}$  from the original data. Following the same process with IHDP, we choose  $p + q$  variables from the original covariates as confounders and generate the CIV  $S$  and the treatment  $W$ .
+
+The experiments on the two real-world datasets are also conducted over 30 replications with the 63/27/10 splitting of train/validation/test samples. Following the work in (Wu et al., 2022), in each replication, we shuffle the data and split the data into train/validation/test samples randomly.
+
+Results. Table 2 shows the results where IHDP- $p-q$  or Twins- $p-q$ , e.g. IHDP-2-4 denotes the use of a dataset with 2 observed confounders and 4 unobserved confounders. From Table 2, we have
+
+the following observations: (1) CBRL.CIV achieves the lowest estimation errors in all experimental settings of both datasets. (2) CBIV achieves the second-best performance on both datasets. (3) CFR and DeepIV outperform the traditional IV methods.
+
+In a word, our CBRL.CIV method obtains the best performance on both synthetic and real-world datasets compared to the state-of-the-art IV methods and confounding balance based methods.
+
+# 5 RELATED WORKS
+
+The IV approach is a useful way to eliminate the confounding bias due to unobserved confounders in causal effect estimation (Anderson & Rubin, 1949; Cheng et al., 2023; Wang et al., 2022). Two-Stage Least Squares (TSLS) is one of the most commonly used IV-based methods for causal effect estimation from data in the presence of unobserved confounders through linear regression (Angrist & Imbens, 1995; Imbens, 2014). In non-linear setting, some methods use non-linear regression to regress the treatment conditioning on the IV and the other observed confounders in the first stage, and then regress the outcome variable by using the resampled treatment and the observed confounders (Newey & Powell, 2003). Recently, machine learning based IV methods have been developed for reducing estimation bias, such as Kernel IV (Singh et al., 2019), OrthIV (Sjolander & Martinussen, 2019), and Deep Features IV Regress (Xu et al., 2021). Furthermore, DeepIV Hartford et al. (2017) and OneSIV (Lin et al., 2019) fit the treatment assignment based on the IV and the observed confounders, and then resample the treatment based on the fitted treatment assignment mechanism. CBIV (Wu et al., 2022) combines confounding balancing with IV regression to remove the confounding bias between  $W$  and  $Y$  caused by the unobserved confounders and the observed confounders. Different from these methods, our CBLR.CIV addresses the confounding bias between  $S$  and  $W$ , as well as the confounding bias between  $W$  and  $Y$ . Therefore, CBLR.CIV has a wider application than a method based on a standard IV and it also deals with non-linear cases.
+
+Another line of related works is confounding balance with representation learning (Hassanpour & Greiner, 2020; Shalit et al., 2017). They consider confounding balancing as a domain adaption problem (Daume III, 2007) under the unconfoundedness assumption (Imbens & Rubin, 2015). In contrast, our CBLR.CIV does not rely on the unconfoundedness, and it reduces the confounding bias due to both unobserved and observed confounders by using representation learning techniques.
+
+# 6 CONCLUSION
+
+In this paper, we propose a novel method called Confounding Balancing Representation Learning for non-linear CIV regression (CBRL.CIV) for causal effect estimation from data with unobserved confounders in non-linear scenarios. CBRL.CIV addresses the imbalance problems of observed confounders and the confounding bias of unobserved confounders by using a balancing representation learning network to obtain a balancing representation of the observed confounders and removing the confounding bias caused by unobserved confounders simultaneously in a non-linear system. Extensive experiments conducted on synthetic and two real-world datasets have demonstrated the effectiveness of CBRL.CIV in average causal effect estimation from data with unobserved confounders. The proposed CBRL.CIV method makes significant contributions to causal effect estimation by considering the distribution imbalance of observed confounders between the CIV and the treatment, and the imbalance distributions of observed confounders between the treatment and the outcome simultaneously, which is the first work to address the challenging problem in causal inference. The proposed CBRL.CIV method has potential applications in various fields, including healthcare, social sciences, and economics, where causal effect estimation is essential for decision-making. The CBRL.CIV method relies on some specific assumptions, such as the existence of a valid CIV and the assumption on independent compliance type (Assumption 2). If these assumptions fail, the method may produce incorrect causal effects. It is vital to verify these conditions and conduct robustness checks and sensitivity analyses to ensure the validity of the causal estimates by CBRL.CIV when the domain experts are not sure whether the conditions are satisfied.
+
+# ACKNOWLEDGMENTS
+
+We wish to acknowledge the support from the Australian Research Council Discovery Project 230101122. Thuc Duy Le is supported by DECRA (DE200100200).
+
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+
+# A APPENDIX
+
+This is the appendix to the paper 'Conditional Instrumental Variable Regression with Representation Learning for Causal Inference'. We provide additional definitions of causal DAG, the pseudocode of CBRL.CIV, the proof of Theorem 1, the details of synthetic and semi-synthetic real-world datasets, and more experimental results.
+
+# A PRELIMINARY
+
+The definitions of Markov property and faithfulness are usually required in causal graphical modelling.
+
+Definition 2 (Markov property Pearl (2009)). Given a DAG  $\mathcal{G} = (\mathbf{X},\mathbf{E})$  and the joint probability distribution of  $\mathbf{X}$ $(P(\mathbf{X}))$ ,  $\mathcal{G}$  satisfies the Markov property if for  $\forall X_{i}\in \mathbf{X}$ ,  $X_{i}$  is probabilistically independent of all of its non-descendants, given the set of parent nodes of  $X_{i}$ .
+
+Definition 3 (Faithfulness Spirtes et al. (2000)). A DAG  $\mathcal{G} = (\mathbf{X},\mathbf{E})$  is faithful to a joint distribution  $P(\mathbf{X})$  over the set of variables  $\mathbf{X}$  if and only if every independence present in  $P(\mathbf{X})$  is entailed by  $\mathcal{G}$  and satisfies the Markov property. A joint distribution  $P(\mathbf{X})$  over the set of variables  $\mathbf{X}$  is faithful to the DAG  $\mathcal{G}$  if and only if the DAG  $\mathcal{G}$  is faithful to the joint distribution  $P(\mathbf{X})$ .
+
+When the faithfulness assumption is satisfied between a joint distribution  $P(\mathbf{X})$  and a DAG of a set of variables  $\mathbf{X}$ , the dependency/independency relations among the variables can be read from the DAG Pearl (2009); Spirtes et al. (2000). In a DAG, d-separation is a well-known graphical criterion used to read off the conditional independence between variables entailed in the DAG when the Markov property and faithfulness are satisfied Pearl (2009); Spirtes et al. (2000).
+
+Definition 4 (d-separation Pearl (2009)). A path  $\pi$  in a DAG  $\mathcal{G} = (\mathbf{X},\mathbf{E})$  is said to be  $d$ -separated (or blocked) by a set of nodes  $\mathbf{M}$  if and only if (i)  $\pi$  contains a chain  $X_{i}\to X_{k}\to X_{j}$  or a fork  $X_{i}\gets X_{k}\to X_{j}$  such that the middle node  $X_{k}$  is in  $\mathbf{M}$ , or (ii)  $\pi$  contains a collider  $X_{k}$  such that  $X_{k}$  is not in  $\mathbf{M}$  and no descendant of  $X_{k}$  is in  $\mathbf{M}$ . A set  $\mathbf{M}$  is said to  $d$ -separate  $X_{i}$  from  $X_{j}$  ( $X_{i}\perp X_{j}\mid \mathbf{M}$ ) if and only if  $\mathbf{M}$  blocks every path between  $X_{i}$  to  $X_{j}$ . Otherwise, they are said to be  $d$ -connected by  $\mathbf{M}$ , denoted as  $X_{i}\nsubseteq X_{j}\mid \mathbf{M}$ .
+
+The instrumental variable (IV) approach is a powerful tool for estimating causal effects from observational data in the presence of unobserved confounders and noncompliance with the assigned treatment  $W$  Angrist & Imbens (1995); Imbens (2014); Imbens & Rubin (2015). The main strata are based on the instrument assignment  $S$  to follow the treatment  $W$  (or not). Compliers always comply with  $W$ :  $W(S = 1) = 1$  and  $W(S = 0) = 0$ ; Always-takers always follow  $W$ :  $W(S = 1) = W(S = 0) = 1$ ; Never-takers never follow  $W$ :  $W(S = 1) = W(S = 0) = 0$ ; Defiers always against  $W$ :  $W(S = 1) = 0$  and  $W(S = 0) = 1$ .
+
+# B THE PROPOSED CBRL.CIV METHOD
+
+In this section, the pseudo-code of CBRL.CIV, the hyperparameters, the tuning parameters setting, and the proof of Theorem 1 are provided.
+
+# B.1 PSEUDO-CODE OF CBRL.CIV METHOD
+
+The proposed three networks can be optimized sequentially. The loss functions of the three networks are formulated as follows.
+
+$$
+\min  _ {\mu} L _ {S} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(s _ {i} \log \left(\varphi_ {\mu} \left(\mathbf {c} _ {i}\right)\right) + \left(1 - s _ {i}\right) \left(1 - \log \left(\varphi_ {\mu} \left(\mathbf {c} _ {i}\right)\right)\right)\right) \tag {10}
+$$
+
+$$
+\min  _ {\nu} L _ {W} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(w _ {i} \log \left(\varphi_ {\nu} \left(\hat {s} _ {i}, \mathbf {c} _ {i}\right)\right) + \left(1 - w _ {i}\right) \left(1 - \log \left(\varphi_ {\nu} \left(\hat {s} _ {i}, \mathbf {c} _ {i}\right)\right)\right)\right) \tag {11}
+$$
+
+$$
+\begin{array}{l} \min  _ {\theta , \gamma^ {0}, \gamma^ {1}} L _ {Y} + \alpha L _ {C B W} + \beta L _ {C B S} \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(y _ {i} - \sum_ {\hat {w} _ {i} \in \{0, 1 \}} f _ {\gamma \hat {w} _ {i}} \left(\psi_ {\theta} (\mathbf {C})\right) P \left(w _ {i} \mid \hat {s} _ {i}, \mathbf {c} _ {i}\right)\right) ^ {2} \tag {12} \\ + \alpha D (\hat {W}, \psi_ {\theta} (\mathbf {C})) + \beta D (\hat {S}, \psi_ {\theta} (\mathbf {C})) \\ \end{array}
+$$
+
+where  $\alpha$  and  $\beta$  are tuning parameters.  $L_{CBW}$  and  $L_{CBS}$  are the confounding balance constraints in Eq.s (2) and (4), respectively. The pseudo-code of CBRL.CIV is listed in Algorithm 1.
+
+Network Structures & Tuning Parameters. For the CIV and treatment regression networks, we use multi-layer perceptrons with ReLU activation functions for the logistic regression networks  $\varphi_{\mu}$  and  $\varphi_{\nu}$  respectively. We optimize the loss functions  $L_{S}$  and  $L_{W}$  using SGD with a learning rate of 0.05. For the outcome regression network, we use ADAM with a learning rate of 0.0005 to optimize the loss function  $L_{Y} + \alpha L_{CBW} + \beta L_{CBS}$ . For the two tuning parameters  $\alpha$  and  $\beta$ , we select them from the range  $\{0.0001, 0.001, 0.01, 0.1, 1\}$ .
+
+# B.2 THEORETICAL ANALYSIS
+
+Theorem 2. Given the observational data  $\mathcal{O} = (\mathbf{C}, S, W, Y)$  generated from a causal DAG  $\mathcal{G} = (\mathbf{C} \cup \mathbf{U} \cup \{S, W, Y\}, \mathbf{E})$  where  $\mathbf{E}$  is the set of directed edges as shown in Fig. 1 (d) in the main text, suppose that  $S$  is a CIV and  $\mathbf{C}$  is a set of pre-CIV variables. Under Assumptions 1 and 2, if the learned representation  $\mathbf{Z} = \psi_{\theta}(\mathbf{C})$  of  $\mathbf{C}$  is independent of the estimated  $\hat{S}$  and  $\hat{W}$ , then  $f(W, \mathbf{C})$  can be identified by using  $\hat{S}$  and  $\mathbf{Z}$  by solving the inverse problem using  $\hat{S}$  and  $\mathbf{Z}$  as:  $\mathbb{E}[Y \mid S, \mathbf{C}] = \int f(W, \mathbf{Z}) dP(W \mid \hat{S}, \mathbf{C})$ , where  $P(W \mid \hat{S}, \mathbf{C})$  is the estimated conditional treatment distribution by using  $\hat{S}$  and  $\mathbf{C}$ .
+
+Proof. We prove that the estimated  $\hat{S},\hat{W}$ , and the learned representation  $\mathbf{Z}$  by CBRL.CIV can be used to identify  $f(W,\mathbf{C})$  from the observational data  $\mathcal{O}$ . We first note that  $S$  and  $W$ , as well as  $S$  and  $Y$  are confounded by  $\mathbf{C}$  respectively in  $\mathcal{O}$  and the CBRL.CIV method uses the previously described three-stage strategy to adjust for the confounding bias due to  $\mathbf{C}$  and  $\mathbf{U}$ .
+
+In the CIV regression stage, we aim to obtain  $S \perp \mathbf{Z} \mid \mathbf{C}$  to eliminate the confounding bias between  $S$  and  $W$ . For this, we first perform logistic regression using deep neural networks for predicting the conditional probability distribution of the CIV  $S$  conditioning on  $\mathbf{C}$ , namely  $P(S \mid \mathbf{C})$ . Next,  $\hat{S}$  is sampled from  $P(S \mid \mathbf{C})$  assuming  $P(S \mid \mathbf{C}) = P(S \mid \mathbf{C}, \mathbf{Z})$  since then in the confounding balance phase, the learned representation  $\mathbf{Z}$  and the estimated  $\hat{S}$  are forced to be independent, i.e.,  $\hat{S} \perp \mathbf{Z}$  when the optimal solution of Eq. (6) in the main text is obtained. Therefore, the causal relationship between  $\hat{S}$  and  $W$  is not confounded by  $\mathbf{Z}$ .
+
+In the treatment regression stage, we aim to obtain  $W \perp \mathbf{Z} \mid \mathbf{C}$ . So we conduct logistic regression using deep neural networks for predicting the conditional probability distribution of the treatment  $W$
+
+Input: Observational dataset  $\mathcal{O} = (W,Y,\mathbf{X})$  with the treatment  $W$ , the outcome  $Y$ , the CIV  $S$  and the observed variables  $\mathbf{C}$ ; the maximum number of iterations  $t$ .
+
+Output:  $\hat{Y}_0 = f_{\gamma^0}(\psi_\theta (\mathbf{C}))$  and  $\hat{Y}_1 = f_{\gamma^1}(\psi_\theta (\mathbf{C}))$
+
+Algorithm 1 CBRL.CIV: CIV Regression with Confounding Balancing Representation Learning  
+CIV Regression:  
+for ite=1 to t do  $\{\mathbf{c}_i\}_{i = 1}^n\to \varphi_\mu (\mathbf{c}_i)\to P(s_i\mid \mathbf{c}_i)$ $L_{S} = \frac{1}{n}\sum_{i = 1}^{n}(s_{i}\log (\varphi_{\mu}(\mathbf{c}_{i})) + (1 - s_{i})(1 - \log (\varphi_{\mu}(\mathbf{c}_{i})))$  update  $\mu \leftarrow SGD(L_S)$    
+end for   
+Treatment Regression:   
+for ite=1 to t do  $\{\mathbf{c}_i\}_{i = 1}^n\to \varphi_\nu (\mathbf{c}_i)$ $\{\hat{s}_i,\mathbf{c}_i\}_{i = 1}^n\to \varphi_\nu (\hat{s}_i,\mathbf{c}_i)\to P(w_i\mid \hat{s}_i,\mathbf{c}_i)$ $L_{W} = \frac{1}{n}\sum_{i = 1}^{n}(w_{i}\log (\varphi_{\nu}(\hat{s}_{i},\mathbf{c}_{i})) + (1 - w_{i})(1 - \log (\varphi_{\nu}(\hat{s}_{i},\mathbf{c}_{i})))$  update  $\nu \gets SGD(L_W)$    
+end for   
+Outcome Regression:   
+for ite=1 to t do  $\{\mathbf{c}_i\}_{i = 1}^n\to \varphi_\mu (\mathbf{c}_i)\to P(s_i\mid \mathbf{c}_i)$ $\{\hat{s}_i,\mathbf{c}_i\}_{i = 1}^n\to \varphi_\nu (\hat{s}_i,\mathbf{c}_i)\to P(w_i\mid \hat{s}_i,\mathbf{c}_i)$ $\{\psi_{\theta}(\mathbf{c}_{i}),\hat{s}_{i}\}_{i = 1}^{n}\rightarrow D(\hat{S},\psi_{\theta}(\mathbf{C}))$ $\{\psi_{\theta}(\mathbf{c}_{i}),\hat{w}_{i}\}_{i = 1}^{n}\rightarrow D(\hat{W},\psi_{\theta}(\mathbf{C}))$ $L_{Y} + \alpha L_{CBW} + \beta L_{CBS} = \frac{1}{n}\sum_{i = 1}^{n}\Bigl (y_{i} - \sum_{\hat{w}_{i}\in \{0,1\}}f_{\gamma^{\hat{w}_{i}}}(\psi_{\theta}(\mathbf{C}))P(w_{i}\mid \hat{s}_{i},\mathbf{c}_{i})\Bigr)^{2}+$ $\alpha D(\hat{W},\psi_{\theta}(\mathbf{C})) + \beta D(\hat{S},\psi_{\theta}(\mathbf{C}))$  update  $\theta ,\gamma^0,\gamma^1\gets ADAM(L_Y + \alpha L_{CBW} + \beta L_{CBS})$    
+end for
+
+conditioning on  $\mathbf{C}$  and  $\hat{S}$ , namely  $P(W\mid \hat{S},\mathbf{C})$ . Next,  $\hat{W}$  is sampled from  $P(W\mid \hat{S},\mathbf{C})$  assuming  $P(W\mid \hat{S},\mathbf{C}) = P(W\mid \hat{S},\mathbf{C},\mathbf{Z})$  since then through the confounding balance phase, the learned representation  $\mathbf{Z}$  and the estimated  $\hat{W}$  are independent, i.e.,  $\hat{W} \perp \mathbf{Z}$  when the optimal solution of Eq. (6) in the main text is obtained. Furthermore,  $P(W\mid \hat{S},\mathbf{C}) \sim P(W\mid \hat{S},\mathbf{C}) \equiv P(W\mid \hat{S},\mathbf{C},\mathbf{U})$  as it is almost surely that  $\phi (\mathbf{C}) \equiv \tilde{\phi} (\mathbf{C},\mathbf{U})$  as stated in Assumption 2. Thus, the causal relationship between  $\hat{W}$  and  $Y$  is not confounded by  $\mathbf{Z}$  and  $\mathbf{U}$ .
+
+As the defined inverse problem of the standard IV in Eq.(9) in the main text, in our outcome regression stage,  $\hat{W} \sim P(W \mid \hat{S}, \mathbf{C})$  and  $\mathbf{Z} = \psi_{\theta}(\mathbf{C})$  are used to predict the counterfactual outcomes. Thus,  $\mathbb{E}[Y \mid S, \mathbf{C}] = \int f(W, \mathbf{Z}) dP(W \mid \hat{S}, \mathbf{C})$  is the inverse problem for  $f(W, \mathbf{Z})$  given  $\mathbb{E}[Y \mid S, \mathbf{C}]$  and  $P(W \mid \hat{S}, \mathbf{C})$ .
+
+# C EXPERIMENTS
+
+# C.1 EXPERIMENTAL SETUP
+
+Implementation sources for comparison methods. The implementations for DeepIV, OrthIV, DMLIV, DRIV, SDRIV, and ForestDRIV methods are sourced from EconML (Battocchi et al., 2019). The CBIV implementation is available on the authors' GitHub page at https://github.com/anpwu/CB-IV. Similarly, the CFR implementation can be found on the authors' GitHub repository at https://github.com/clinicalml/cfrnet.
+
+Parameters setting. To ensure a fair comparison, we use the common setting for the tuning parameters as done in Shalit et al. (2017); Wu et al. (2022), such as learning rate and number of
+
+![](images/b1e44dff2250f66251c1cb3dd10cefa983c0cb6c2da092ca8a13c1d94f43df7d.jpg)  
+Figure 3: The experimental results of all estimators on within-sample of synthetic datasets with unobserved confounders in non-linear scenarios. The CBRL.CIV obtained the lowest estimation error against all comparisons.
+
+epochs. Furthermore, we set the same parameters, such as the dimension of learned representation  $\mathbf{Z}$  for CFR, CBIV, and our CBRL.CIV.
+
+# C.2 SIMULATION STUDY
+
+We generate the synthetic datasets following the works Hassanpour & Greiner (2020); Wu et al. (2022) for the evaluation. Especially, the set of pre-IV variables  $\mathbf{C}$  and the set of unobserved variables  $\mathbf{U}$  drive from  $C_1, C_2, \ldots, C_p, U_1, U_2, \ldots, U_q \sim N(0, \Sigma p + q)$ , where  $p$  and  $q$  are the dimensions of  $\mathbf{C}$  and  $\mathbf{U}$ , respectively,  $\Sigma p + q = \mathbf{I}_{p+l} * 0.95 + \varrho_{p+q} * 0.05$  denotes that all values excluding those on the main diagonal are 0.05 in the covariance matrix, and  $\varrho_{p+q}$  is a  $p + q$  degree all-ones matrix. The CIV  $S$ , treatment variable  $W$ , and outcome variable  $Y$  are generated as follows:
+
+$$
+\begin{array}{l} P (S \mid \mathbf {C}) = \frac {1}{1 + e x p \left(- \left(\sum_ {i = 1} ^ {p} C _ {i} + \sum_ {i = 1} ^ {q} U _ {i}\right)\right)}, (13) \\ S \sim B e r n (P (S \mid \mathbf {C})) \\ P (W \mid S, \mathbf {C}) = \frac {1}{1 + e x p \left(- \left(S * \sum_ {i = 1} ^ {p} C _ {i} + \sum_ {i = 1} ^ {p} C _ {i} + \sum_ {i = 1} ^ {q} U _ {i}\right)\right)}, (14) \\ W \sim \operatorname {B e r n} (P (W \mid S, \mathbf {C})) \\ \end{array}
+$$
+
+$$
+f (W, \mathbf {U}, \mathbf {C}) = \frac {W}{p + q} \left(\sum_ {i = 1} ^ {p} C _ {i} ^ {2} + \sum_ {i = 1} ^ {q} U _ {i} ^ {2}\right) + \frac {1 - W}{p + q} \left(\sum_ {i = 1} ^ {p} C _ {i} + \sum_ {i = 1} ^ {q} U _ {i}\right) \tag {15}
+$$
+
+where  $\text{Bern}(P(S \mid \mathbf{C}))$  and  $\text{Bern}(P(W \mid S, \mathbf{C}))$  are the true CIV and treatment assignment, respectively. In order to avoid bias caused by the data generation process, we repeated the data generation process 30 times (i.e., 30 are generated datasets for each setting) for all experiments. We randomly split a dataset into training (70%) and testing (30%) samples for all our simulations. The generation process is widely used in the evaluation of data-driven causal effect estimation methods Hassanpour & Greiner (2020); Wu et al. (2022); Cheng et al. (2022).
+
+Results. The estimation errors of all estimators on within-sample are visualised with boxplots in Fig. 3. Based on Fig. 3, we have the same conclusion drawn in the main text.
+
+Table 3: The table summarises the estimation error of CBRL.CIV with the original  $S$  (mean  $\pm$  STD) over 30 synthetic datasets.  
+
+<table><tr><td>Syn-4-4</td><td>Syn-8-8</td><td>Syn-12-12</td></tr><tr><td>0.48 ± 0.27</td><td>0.40 ± 0.19</td><td>0.41 ± 0.14</td></tr></table>
+
+Table 4: The table summarises the estimation error (mean ± STD) over 30 synthetic datasets with different settings and sample sizes. The compared methods have CBRL.CIV, CBRL.CIV with Equation 2 unused (CBRL), and CBRL.CIV with both Equations 2 & 4 unused  $(\overline{CBRL})$  
+
+<table><tr><td></td><td></td><td>Syn-4-4</td><td>Syn-8-8</td><td>Syn-12-12</td></tr><tr><td rowspan="3">2k</td><td>CBRL.CIV</td><td>0.19 ± 1.93</td><td>0.13 ± 1.95</td><td>0.09 ± 1.97</td></tr><tr><td>CBRL</td><td>0.26 ± 1.77</td><td>0.45 ± 1.70</td><td>0.54 ± 1.72</td></tr><tr><td>CBRL</td><td>0.32 ± 1.63</td><td>0.61 ± 1.68</td><td>0.76 ± 1.68</td></tr><tr><td rowspan="3">4k</td><td>CBRL.CIV</td><td>0.18 ± 0.07</td><td>0.14 ± 0.03</td><td>0.13 ± 0.03</td></tr><tr><td>CBRL</td><td>0.43 ± 0.29</td><td>0.35 ± 0.20</td><td>0.39 ± 0.11</td></tr><tr><td>CBRL</td><td>0.56 ± 0.43</td><td>0.44 ± 0.32</td><td>0.61 ± 0.24</td></tr><tr><td rowspan="3">6k</td><td>CBRL.CIV</td><td>0.14 ± 0.05</td><td>0.14 ± 0.03</td><td>0.12 ± 0.03</td></tr><tr><td>CBRL</td><td>0.48 ± 0.27</td><td>0.40 ± 0.19</td><td>0.41 ± 0.14</td></tr><tr><td>CBRL</td><td>0.60 ± 0.38</td><td>0.58 ± 0.31</td><td>0.51 ± 0.25</td></tr><tr><td rowspan="3">8k</td><td>CBRL.CIV</td><td>0.16 ± 0.05</td><td>0.14 ± 0.03</td><td>0.11 ± 0.03</td></tr><tr><td>CBRL</td><td>0.56 ± 0.33</td><td>0.47 ± 0.13</td><td>0.38 ± 0.12</td></tr><tr><td>CBRL</td><td>0.68 ± 0.40</td><td>0.59 ± 0.25</td><td>0.47 ± 0.25</td></tr><tr><td rowspan="3">10k</td><td>CBRL.CIV</td><td>0.18 ± 0.05</td><td>0.14 ± 0.04</td><td>0.11 ± 0.04</td></tr><tr><td>CBRL</td><td>0.59 ± 0.25</td><td>0.40 ± 0.15</td><td>0.46 ± 0.14</td></tr><tr><td>CBRL</td><td>0.74 ± 0.34</td><td>0.53 ± 0.24</td><td>0.60 ± 0.20</td></tr><tr><td rowspan="3">20k</td><td>CBRL.CIV</td><td>0.15 ± 0.06</td><td>0.13 ± 0.04</td><td>0.11 ± 0.03</td></tr><tr><td>CBRL</td><td>0.57 ± 0.20</td><td>0.49 ± 0.14</td><td>0.48 ± 0.08</td></tr><tr><td>CBRL</td><td>0.60 ± 0.24</td><td>0.61 ± 0.28</td><td>0.58 ± 0.21</td></tr></table>
+
+# C.3 ABLATION STUDY
+
+We have tried using Eq. 2 (i.e., no longer using loss 1) on the synthetic datasets used in Section 4.2. The experimental results on datasets with out-of-samples are reported in Table 3.
+
+Results. From Table 3, we know that CBRL.CIV without loss 1 has a larger estimation error than CBRL.CIV with loss 1, i.e.,  $S \perp \mathbf{Z}$  is not enough for mitigating the confounding bias between  $S$  and  $W$ .
+
+To show the necessity of Eq. 2 (and Eq. 4), we conduct an ablation study and report the experimental results in Table 4. The synthetic dataset generation process is the same as that in Section 4.2.
+
+Results. From Table 3, CBRL.CIV has the smallest estimation error on all experimental settings than CBRL (CBRL.CIV without CIV regression, i.e. Eq. 2 is not used) and  $\overline{CBRL}$  (CBRL.CIV with both Eqs. 2 & 4 unused). The results indicate that the CIV regression for confounding balance is necessary (and that the treatment regression with confounding balance is also needed).
+
+# C.4 THE EFFECTS OF  $\alpha$  AND  $\beta$
+
+We have conducted the experiment to examine the effects of  $\alpha$  and  $\beta$  on the performance of CBRL.CIV. The results are summarised in Table 5. Note that we only adjust the values of  $\alpha$  and  $\beta$ , and Eqs. 1 and 3 were still worked in Eq. 5.
+
+Table 5: The table summarises the estimation error (mean ± STD) over 30 synthetic datasets with different values for  $\alpha, \beta$ . The sample size is 6k with the setting Syn-4-4.  
+
+<table><tr><td></td><td>Within-Sample</td><td>Out-of-Sample</td></tr><tr><td>α,β=0</td><td>0.1594 ± 0.0371</td><td>0.1597 ± 0.037</td></tr><tr><td>α,β=0.01</td><td>0.1553 ± 0.0418</td><td>0.1556 ± 0.0411</td></tr><tr><td>α,β=0.1</td><td>0.1406 ± 0.0533</td><td>0.1412 ± 0.0525</td></tr><tr><td>α,β=1</td><td>0.1877 ± 0.0691</td><td>0.1878 ± 0.0679</td></tr><tr><td>α,β=10</td><td>0.2198 ± 0.0511</td><td>0.2198 ± 0.0511</td></tr></table>
+
+Table 6: The estimation errors (mean±STD) over the within-sample on two real-world datasets with different dimensions of  $p$  and  $q$ .  
+
+<table><tr><td>Methods</td><td>IHDP-2-2</td><td>IHDP-4-4</td><td>IHDP-6-6</td><td>Twins-2-2</td><td>Twins-4-4</td><td>Twins-6-6</td></tr><tr><td>DeepIV</td><td>2.692±0.407</td><td>2.900±0.290</td><td>2.516±0.396</td><td>0.091±0.045</td><td>0.068±0.048</td><td>0.082±0.041</td></tr><tr><td>OrthIV</td><td>0.773±0.407</td><td>2.135±0.210</td><td>3.899±0.735</td><td>0.256±0.158</td><td>0.434±0.079</td><td>0.352±0.113</td></tr><tr><td>DMLIV</td><td>1.134±0.881</td><td>5.010±2.313</td><td>4.239±1.315</td><td>0.244±0.156</td><td>0.425±0.079</td><td>0.343±0.118</td></tr><tr><td>DRV</td><td>3.872±1.157</td><td>3.956±0.740</td><td>4.483±2.550</td><td>0.285±0.228</td><td>0.246±0.205</td><td>0.478±0.329</td></tr><tr><td>SDRIV</td><td>8.212±8.106</td><td>2.932±4.171</td><td>3.733±4.470</td><td>0.347±0.242</td><td>0.286±0.193</td><td>0.572±0.414</td></tr><tr><td>ForestDRIV</td><td>10.176±9.892</td><td>6.333±7.838</td><td>4.669±4.17071</td><td>0.377±0.283</td><td>0.255±0.228</td><td>0.490±0.355</td></tr><tr><td>CFR</td><td>2.568±1.083</td><td>1.757±0.459</td><td>2.917±0.428</td><td>0.037±0.008</td><td>0.039±0.021</td><td>0.040±0.027</td></tr><tr><td>CBIV</td><td>2.307±0.975</td><td>1.344±0.443</td><td>2.836±0.486</td><td>0.033±0.011</td><td>0.034±0.026</td><td>0.035±0.016</td></tr><tr><td>CBRL.CIV</td><td>0.251±0.083</td><td>0.455±0.068</td><td>0.120±0.078</td><td>0.030±0.019</td><td>0.027±0.018</td><td>0.031±0.024</td></tr></table>
+
+Results. From Table 5, we observe that when  $\alpha$  and  $\beta$  are set to 0.1, the minimum estimation error is achieved. Furthermore, even when  $\alpha$  and  $\beta$  are both set to 0, CBRL.CIV still exhibits minimal estimation error.
+
+# C.5 TWO REAL-WORLD DATASETS
+
+IHDP. IHDP is the Infant Health and Development Program (IHDP) dataset collected from a randomised controlled experiment that investigated the effect of home visits by health professionals on future cognitive test scores Hill (2011). There are 24 pretreatment variables and 747 infants, including 139 treated (having home visits by health professionals) and 608 controls.
+
+Twins. Twins is a benchmark dataset frequently used in causal inference literature. It focuses on twin births and deaths in the USA between 1989 and 1991 Almond et al. (2005). Each twin pair in the dataset includes 40 pretreatment variables that provide information about the parents, pregnancy, and birth of the twins. Both the treated group  $(\mathrm{W} = 1$  , representing the heavier twin) and the control group  $(\mathrm{W} = 0$  , representing the lighter twin) are observed. The true outcome of interest is the mortality rate after one year for each twin.
+
+Results. Table 2 shows the estimation errors over the within-sample on two real-world datasets with different dimensions of  $p$  and  $q$ , where  $p$  is and  $q$  is. From Table 6, we have the same conclusions as described in the main text.

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+# CONFORMAL INDUCTIVE GRAPH NEURAL NETWORKS
+
+Soroush H. Zargarbashi
+
+CISPA Helmholtz Center for Information Security
+
+zargarbashi@cs.uni-koeln.de
+
+Aleksandar Bojchevski
+
+University of Cologne
+
+bojchevski@cs.uni-koeln.de
+
+# ABSTRACT
+
+Conformal prediction (CP) transforms any model's output into prediction sets guaranteed to include (cover) the true label. CP requires exchangeability, a relaxation of the i.i.d. assumption, to obtain a valid distribution-free coverage guarantee. This makes it directly applicable to transductive node-classification. However, conventional CP cannot be applied in inductive settings due to the implicit shift in the (calibration) scores caused by message passing with the new nodes. We fix this issue for both cases of node and edge-exchangeable graphs, recovering the standard coverage guarantee without sacrificing statistical efficiency. We further prove that the guarantee holds independently of the prediction time, e.g. upon arrival of a new node/edge or at any subsequent moment.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) are used in many applications however without a reliable estimation of their output's uncertainty. We can not rely solely on the predicted distribution of labels  $\pi(y \mid x)$  (e.g. from the softmax) as it is often uncalibrated. Therefore, it is crucial to find confidence estimates aligned with the true  $p(y \mid x)$ . There is a rich literature on uncertainty quantification methods many of which require retraining or modifications to the model architecture. Among them, almost none come with a guarantee, and many rely on the i.i.d. assumption. Given the interdependency structure (adjacency) we cannot easily adopt these methods for node classification. As a result, uncertainty quantification methods for GNNs are very limited (Stadler et al., 2021).
+
+Conformal prediction (CP) is an alternative approach that uses the model as a black box and returns prediction sets guaranteed to cover the true label without any assumptions on the model's architecture or the data generating process. This guarantee is probabilistic and works for any user-specified probability  $1 - \alpha$ . To apply CP, we need a conformity score function  $s: \mathcal{X} \times \mathcal{Y} \mapsto \mathbb{R}$  that quantifies the agreement between the input and the candidate label. Additionally, we need a held-out calibration set. The only assumption for a valid coverage guarantee is exchangeability between the calibration set and the test set. This makes CP applicable to non-i.i.d settings like transductive node classification – Zargarbashi et al. (2023) and Huang et al. (2023) showed that with a permutation-equivariant model (like almost all GNNs) CP obtains a valid coverage guarantee.
+
+Given the dynamic nature of real-world graphs which evolve over time, inductive node classification is more reflective of actual scenarios than transductive. In the inductive setting, we are given an initial graph (used for training and calibration) which is progressively expanded by set of nodes and edges introduced to the graph over time. Importantly test nodes are not present at the training and calibration stage. For GNNs, updates in the graph cause an implicit shift in the embeddings of existing nodes and consequently the softmax outputs and conformity scores. Therefore, in the inductive setting, even assuming node/edge exchangeability (see § B for a formal definition), the coverage guarantee is no longer valid. Intuitively, calibration scores are no longer exchangeable with test scores, as soon as new nodes or edges are introduced (see Fig. 1 right).
+
+To address this issue, Clarkson (2023) adapt the beyond exchangeability approach applying weighted CP for inductive node-classification aiming to recover the guarantee up to a bounded (but unknown) error (Barber et al., 2023). Unfortunately, this approach has an extremely limited applicability when applied with a realistic (sparse) calibration set. In sparse networks and limited labels, the method fails to predict any prediction set for a large number of nodes; and for the rest, the statistical efficiency is significantly low (see § C for details).
+
+![](images/2abafa6d63c95fac898124eeef761f078531a9fb22a526ce22413d490b778dac.jpg)  
+Figure 1: [Left] Each vertical line on the heatmap shows sorted true test scores at each timestep. The dashed line shows the true (unknown)  $\alpha$ -quantile and the quantile from each approach is also shown alongside. NodeEx CP (ours) closely tracks the true quantile, while naive CP deviates over time. [Upper right] Distributions from selected timesteps marked by the same color on the heatmap. The distribution shift is observable over time with new nodes appearing. [Lower right] The earth mover distance (EMD) between naive CP calibration scores and shifted true scores, denoted as “Test”; and EMD between naive and NodeEx CP scores, denoted as “Cal”. Details in § D.5.
+
+![](images/467071384d3f7e5bf206fde966ab6b0d2e90e753d6344337f2719c5c29140220.jpg)
+
+We show that for a node or edge exchangeable graph sequence (without distribution shift), we can adapt CP w.r.t. the implicit shift in score space, recovering the desired coverage guarantee. Node-exchangeability assumes that the joint distribution of nodes is invariant to any permutation. In the inductive setting, this means that any node is equally likely to appear at any timestep. We show that under node-exchangeability, the effect of new coming nodes is symmetric to the calibration set and the existing nodes. Hence, computing the conformity scores conditional to the subgraph at any timestep recovers the guarantee. Previous works under the transductive setting (Zargarbashi et al., 2023) and (Huang et al., 2023) also assume node-exchangeability.
+
+While real-world graphs are sparse, all node-exchangeable generative processes produce either empty or dense graphs (see § 3). Therefore, we extend our study to the edge-exchangeable setting which can generate sparse graphs. In § 4.2 we show that w.r.t. scores, any edge-exchangeable sequence is equivalent to a weighted node-exchangeable sequence. Therefore, through weighted CP (weighted quantile lemma from Tibshirani et al. (2019)) we recover the  $1 - \alpha$  coverage. Importantly, the improvement of our method is orthogonal to whether there is a distribution shift over time. Therefore, to address shift, we can still apply the weighting scheme from Barber et al. (2023) on top.
+
+We focus on inductive node-classification. We define node-exchangeable (NodeEx) CP, and we show that (i) for node-exchangeable sequences, NodeEx CP obtains  $1 - \alpha$  guarantee by calibrating over shifted conformal scores conditional to subgraphs in each timestep (Fig. 1 left), (ii) the  $1 - \alpha$  guarantee is achievable via weighted CP on edge-exchangeable sequences, and (iii) The guarantee holds independent of the prediction time – the prediction set for a node at any timestep after its appearance benefits from the same coverage guarantee (e.g. prediction upon arrival or all nodes at once both result in the same coverage). We justify our approach with both theoretical and empirical results.
+
+# 2 BACKGROUND: CONFORMAL PREDICTION AND TRANSDUCTIVE GNNS
+
+Here we recall the general guarantee of CP (e.g. for image dataset). Through this paper, we assume a continuous score function without ties. This can apply to any function via adding noise. With weights  $w_{i} \in [0,1]$ , and the sorting permutation  $\tau^{\star}$ , we define the weighted quantile  $\mathrm{Quant}\left(\cdot ;\cdot ;\cdot\right)$  as
+
+$$
+\operatorname {Q u a n t} \left(\alpha ; \{s _ {i} \} _ {i = 1} ^ {N}; \{w _ {i} \} _ {i = 1} ^ {N}\right) = \inf  \left\{s _ {\tau^ {\star} (i)}: \frac {\sum_ {j = 1} ^ {i} w _ {\tau^ {\star} (j)}}{\sum_ {j = 1} ^ {N} w _ {j} + 1} \geq \alpha \right\} \tag {1}
+$$
+
+Theorem 1 (Vovk et al. (2005)). Let  $\{(\pmb{x}_i, y_i)\}_{i=1}^n$ , and  $(\pmb{x}_{n+1}, y_{n+1})$  be exchangeable. With any continuous function  $s: \mathcal{X} \times \mathcal{Y} \mapsto \mathbb{R}$  measuring the agreement between  $\pmb{x}$ , and  $\pmb{y}$ , and user-specified significance level  $\alpha \in (0,1)$ , with prediction sets defined as  $\mathcal{C}_{\alpha}(\pmb{x}_{n+1}) = \{y: s(\pmb{x}_{n+1}, y) \geq \hat{q}\}$  where  $\hat{q} := \operatorname{Quant}(\tilde{\alpha}; \{s(x_i, y_i)\}_{i=1}^n; 1)$ . We have  $\operatorname{Prob}[y_{n+1} \in \mathcal{C}(\pmb{x}_{n+1})] \geq 1 - \alpha$ .
+
+The marginal coverage probability is upper-bounded by  $1 - \alpha + 1 / (n + 1)$ . With infinite samples, the coverage is distributed as a  $\operatorname{Beta}(n + 1 - l, l)$  with  $l = \lfloor (n + 1)\alpha \rfloor$  (Vovk, 2012), while with a fixed population  $|\mathcal{D}| = n + m$ , and an exchangeable calibration set  $|\mathcal{D}_{\mathrm{cal}}| = n$ , the probability of coverage  $\operatorname{Cov}(\mathcal{D} \setminus \mathcal{D}_{\mathrm{cal}}) := (1/m)\mathbf{1}[y_i \in \mathcal{C}(\boldsymbol{x}_i)]_{i \in \mathcal{D} - \mathcal{D}_{\mathrm{cal}}}$  derives from a collection of hypergeometric distributions (Huang et al., 2023).
+
+Conformity scores. The guarantee from Theorem 1 holds regardless of how the conformity score function  $s: \mathcal{X} \times \mathcal{Y} \to R$  is defined. A wise choice of  $s(\cdot, \cdot)$  is reflected in other metrics like the prediction set size. We use adaptive prediction sets (APS) with a score function defined as  $s(\boldsymbol{x}, y) := -(\rho(\boldsymbol{x}, y) + u \cdot \pi(\boldsymbol{x})_y)$ . Here  $\rho(\boldsymbol{x}, y) := \sum_{c=1}^{K} \pi(\boldsymbol{x})_c 1[\pi(\boldsymbol{x})_c > \pi(\boldsymbol{x})_y]$  is the sum of all classes predicted as more likely than  $y$ , and  $u \in [0,1]$  is a uniform random value that breaks the ties between different scores to allow exact  $1 - \alpha$  coverage (Stutz et al., 2022). Our method is independent of the choice of score function and we show its applicability on other (network-agnostic and network-aware) scores in § D.3.
+
+CP beyond exchangeability. For cases where the calibration and test points are not exchangeable, there is a coverage gap  $\Delta_{\alpha}$ , i.e. Prob  $[y_{n + 1} \in \mathcal{C}_{\alpha}(\pmb{x}_{n + 1})] \geq 1 - \alpha - \Delta_{\alpha}$ . Barber et al. (2023) shows that  $\Delta_{\alpha}$  has an upper bound that depends on how far the data deviates from exchangeability. This upper bound can be further controlled via weighted conformal prediction. For related works see § F.
+
+# 2.1 CONFORMAL PREDICTION FOR TRANSDUCTIVE NODE-CLASSIFICATION
+
+Consider a graph  $\mathcal{G}(\mathcal{V},\mathcal{E},\mathbf{A},\mathbf{y})$  with  $\mathbf{X}\in \mathbb{R}^{n\times d}$  as node-features matrix,  $\mathbf{A}\in \mathbb{R}^{n\times n}$  as adjacency matrix, and  $\mathbf{y}\in \mathbb{R}^n$  as labels.  $\mathcal{V} = \mathcal{V}_{\mathrm{tr}}\cup \mathcal{V}_{\mathrm{cal}}\cup \mathcal{V}_u$  is the set of vertices (training, calibration, and test). Let  $f$  be a black-box permutation-equivariant model (e.g. a GNN) trained on labels of  $\mathcal{V}_{\mathrm{tr}}$  and  $s$  be a continuous score function with access to the outputs of  $f$ . The score function  $s$  may or may not use information about the graph structure (see § D.3). In the transductive setup the graph  $\mathcal{G}$  is fixed, and training and calibration is performed with all observed nodes (including features and structure). The labels of  $\mathcal{V}_{\mathrm{tr}}$  and  $\mathcal{V}_{\mathrm{cal}}$  are used respectively for training and calibration, while all other labels remain unseen. Here when  $\mathcal{V}_{\mathrm{cal}}$  and  $\mathcal{V}_u$  are exchangeable, standard CP can be applied (Zargarbashi et al., 2023; Huang et al., 2023). For a set with a fixed number of nodes  $\mathcal{V}'$ , we define the discrete coverage as  $\operatorname {Cov}(\mathcal{V}') = \frac{1}{|\mathcal{V}'|}\sum_{v_j\in \mathcal{V}'}\mathbf{1}\left[y_j\in \hat{\mathcal{C}}(v_j)\right]$ . The following theorem shows that under transductive calibration, CP yields valid prediction sets.
+
+Theorem 2 (Rephrasing of Theorem 3 by Huang et al. (2023)). For fixed graph  $\mathcal{G}$ , and a permutation-equivariant score function  $s(\cdot, \cdot)$ , with  $\mathcal{V}_{\mathrm{cal}} = \{(v_i, y_i)\}_{i=1}^N$  exchangeably sampled from  $\mathcal{V} \setminus \mathcal{V}_{\mathrm{tr}}$ , and prediction sets  $\hat{\mathcal{C}}(v) = \{y : s(v, y) \geq \operatorname{Quant}(\alpha; \{s(v_i, y_i)\}_{i=1}^N; 1)\}$  we have
+
+$$
+\operatorname {P r o b} \left[ \operatorname {C o v} \left(\mathcal {V} _ {u}\right) \leq t \right] = 1 - \Phi_ {H G} \left(i _ {\alpha} - 1; M + N, N, \lceil M t \rceil + i _ {\alpha}\right) \tag {2}
+$$
+
+where  $i_{\alpha} = \lceil (N + 1)(1 - \alpha) \rceil$  is the unweighted  $\alpha$ -quantile index of the calibration scores, and  $\Phi_{HG}(\cdot; N, n, K)$  is the c.d.f. of a hyper-geometric distribution with parameters  $N$  (population),  $n$  number of samples, and  $K$  number of successful samples among the population.
+
+The issue with inductive node-classification. The above approach does not directly translate to the inductive setting. As soon as the graph is updated, the implicit shift in the nodes embeddings results in a distribution shift w.r.t. the calibration scores that are computed before the update. This breaks the exchangeability as shown by Zargarbashi et al. (2023) and in Fig. 1. To address this issue, Clarkson (2023) adopts weighted CP (Barber et al., 2023) with neighborhood-dependent weights (NAPS), limiting the calibration nodes to those inside an immediate neighborhood of a test node. Applying NAPS on sparse graphs or with small (realistic) calibration sets, leaves a significant proportion of test nodes with an "empty" calibration set. Hence, its applicability is limited to very special cases (see § C for details). Moreover, NAPS does not quantify the coverage gap  $\Delta_{\alpha}$ . In contrast, when assuming either node or edge-exchangeability, the gap for our approach is zero regardless of the weights.
+
+# 3 INDUCTIVE GNNS UNDER NODE AND EDGE EXCHANGEABILITY
+
+In the inductive scenario, the graph changes after training and calibration. Therefore, the model  $f$  is trained, and CP is calibrated only on a subgraph before the changes. We track these updates in a sequence of graphs  $\mathcal{G}_1, \mathcal{G}_2, \ldots$  where  $\mathcal{G}_t = (\mathcal{V}_t, \mathcal{E}_t)$  is a graph with a finite set of vertices and edges. We focus on progressive updates meaning  $\forall t: \mathcal{V}_t \subseteq \mathcal{V}_{t+1}$  and  $\mathcal{E}_t \subseteq \mathcal{E}_{t+1}$ . At timestep  $t$  in a node-inductive sequence, the update adds a node  $v_t$  with all its connections to existing vertices. In an edge-inductive sequence, updates add an edge, which may or may not bring unseen nodes with it. W.l.o.g. update sets are singular. Node-inductive and edge-inductive sequences are formally defined in § B.
+
+We call a node-inductive sequence to be node-exchangeable if the generative process is invariant to the order of nodes, meaning that any permutation of nodes has the same probability. Analogously, a sequence is edge-exchangeable if all permutations of edges are equiprobable. For both cases, the problem is node-classification. Since split CP is independent of model training w.l.o.g. we assume that we train on an initial subgraph  $\mathcal{G}_0$ . We do not need to assume that  $\mathcal{G}_0$  is sampled exchangeably. Formal definitions of node-exchangeability and edge-exchangeability are provided in  $\S B$ .
+
+For transductive GNNs, both Zargarbashi et al. (2023) and Huang et al. (2023) assume that the calibration set is sampled node-exchangeably w.r.t. the test set, however the entire graph is fixed and given. In contract, we assume that the sequence of graphs  $\mathcal{G}_1, \mathcal{G}_2, \ldots$  itself, including calibration and test nodes, is either node- or edge-exchangeable. Transductive setting can be seen as a special case.
+
+Sparsity. Any node-exchangeable graph is equivalent to a mixture of graphons (Aldous, 1981; Hoover, 1979; Cai et al., 2016). A graphon (or graph limit) is a symmetric measurable function  $W:[0,1]^2\mapsto [0,1]$  and the graph is sampled by drawing  $u_{i}\sim \mathrm{Uniform}[0,1]$  for each vertex  $v_{i}$ , and an edge between  $v_{i}$ , and  $v_{j}$  with probability  $W(u_i,u_j)$ . We know that graphs sampled from a graphon are almost surely either empty or dense (Orbanz & Roy, 2014; Cai et al., 2016) – the number of edges grows quadratically w.r.t. the number of vertices. However, sparse graphs (with edges subquadratic in the number of vertices) are more representative to real-world networks. Therefore, we also consider edge-exchangeable graph sequences (Cai et al., 2016) that can achieve sparsity.
+
+# 4 CONFORMAL PREDICTION FOR EXCHANGEABLE GRAPH SEQUENCES
+
+The first  $N$  nodes of the sequence (and their labels) are taken as the calibration set, leaving the rest of the sequence to be evaluated. For easier analysis, we record the coverage of each node at each timestep. Let  $T_{0}$  be the time when the last calibration node arrived. Up to timestep  $T$ , let matrix  $C \in \{0,1\}^{\mathcal{V}_T \times T}$  with  $C[i,t]$  indicate whether the prediction set for test node  $v_{i}$  at timestep  $t$  covers the true label  $y_{i}$ . The time index in  $C$  is relative to  $T_{0}$  and w.l.o.g. we index nodes upon their appearance. Hence, in each column  $C[\cdot ,t]$ , the first  $|\mathcal{V}_t|$  elements are in  $\{0,1\}$  and the rest are N/A. Formally:
+
+$$
+\boldsymbol {C} [ i, t ] = \left\{ \begin{array}{l l} \mathbf {1} \left[ y _ {i} \in \left\{y: s \left(v _ {i}, y\right) \geq \operatorname {Q u a n t} \left(\alpha ; \left\{s \left(v _ {i}, y _ {i} \mid \mathcal {G} _ {t}\right) \right\} _ {v _ {i} \in \mathcal {V} _ {\mathrm {c a l}}} ; 1\right) \right\} \right] & i \geq t \\ \mathrm {N / A} & \text {o . w .} \end{array} \right. \tag {3}
+$$
+
+Here we assume that one can evaluate a node at any timestep after its appearance. We define  $\mathcal{V}_{\mathrm{eval}}^{(t)} \subseteq \mathcal{V}_t$  as the set of nodes evaluated at timestep  $t$ . For an edge-exchangeable sequence, a node  $v$  is considered as active (existing) upon the arrival of the first edge connected to it. After  $T$  timesteps, we call the (recorded) sub-partition  $\cup_{i=1}^{T} \mathcal{V}_{\mathrm{eval}}^{(i)}$  as an evaluation mask  $\mathbb{I}_T$ . Any node appears at most once in  $\mathbb{I}_T$  (we do not re-evaluate a node). We define the  $\mathrm{Cov}(\mathcal{V}_{\mathrm{eval}}^{(t)}) = (1 / |\mathcal{V}_{\mathrm{eval}}^{(t)}|) \sum_{v_i \in \mathcal{V}_{\mathrm{eval}}^{(t)}} C[i, t]$  as the empirical coverage over the mask  $\mathcal{V}_{\mathrm{eval}}^{(t)}$ . Similarly,  $\mathrm{Cov}(\mathbb{I}_T)$  is the empirical coverage for  $\mathbb{I}_T$  (average over all records). We visualize  $1 - C$  under node exchangeability on Fig. 2.
+
+![](images/65f8fbf171c5b4bcd75639a4b5c381c0cbdd74feedae4134485828d7fd4f1dd7.jpg)  
+Figure 2:  $1 - C$  for Cora. Details in  $\S B$
+
+# 4.1 NODE-EXCHANGEABLE SEQUENCES
+
+A node-inductive graph sequence is node-exchangeable if any permutation of node appearance is equally likely – at each step, any unseen node has the same probability of appearing (see § B).
+
+With the arrival of new nodes, the distribution of embeddings (and consequently scores) shifts (as shown in Fig. 1). This is why calibrating prior to the update fails to guarantee coverage for new nodes. However, with node-exchangeability, calibration and evaluation scores computed conditional to a specific subgraph are still exchangeable. We extend Theorem 2 to inductive GNNs on node-exchangeable graph sequences for any subset of evaluated vertices.
+
+Proposition 1. At time step  $t$ , with graph  $\mathcal{G}_t = (\mathcal{V}_t, \mathcal{E}_t)$  from a node-exchangeable sequence, and a calibration set  $\mathcal{V}_{\mathrm{cal}}$  consisting of the first  $n$  nodes in  $\mathcal{G}_t$ , for any  $v_j \in \mathcal{V}_{\mathrm{eval}}^{(t)} \subseteq \mathcal{V}_t \setminus \mathcal{V}_{\mathrm{cal}}$  define
+
+$$
+\mathcal {C} ^ {(t)} \left(v _ {j}\right) = \left\{y: s \left(v _ {j}, y \mid \mathcal {G} _ {t}\right) \geq \operatorname {Q u a n t} \left(\alpha ; \left\{s \left(v _ {i}, y _ {i} \mid \mathcal {G} _ {t}\right) \right\} _ {i \in \mathcal {V} _ {\mathrm {c a l}}}; 1\right) \right\} \tag {4}
+$$
+
+Then  $\operatorname{Prob}\left[y_j \in \mathcal{C}^{(t)}(v_j) \mid v_j \in \mathcal{V}_{\mathrm{eval}}^{(t)}\right] \geq 1 - \alpha$ . Moreover with  $m := |\mathcal{V}_{\mathrm{eval}}^{(t)}|$ ,
+
+$$
+\operatorname {P r o b} \left[ \operatorname {C o v} \left(\mathcal {V} _ {\text {e v a l}} ^ {(t)}\right) \leq \beta \right] = 1 - \Phi_ {\mathrm {H G}} \left(i _ {\alpha} - 1; m + n, n, i _ {\alpha} + \lceil \beta t \rceil\right) \tag {5}
+$$
+
+We defer proofs to § B. Here we provide an intuitive justification of the theorem. First, if  $\mathcal{V}_{\mathrm{eval}}^{(t)}$  includes all vertices the problem reduces to the transductive case, for which Theorem 2 applies with scores conditional on the current graph. Otherwise,  $\mathcal{V}' := \mathcal{V}_t \setminus \mathcal{V}_{\mathrm{eval}}^{(t)}$  is not empty. Here the effect of  $\mathcal{V}'$  is symmetric to  $\mathcal{V}_{\mathrm{cal}}$  and  $\mathcal{V}_{\mathrm{eval}}^{(t)}$ . For instance, consider a linear message passing  $Z = AXW$  with weight matrix  $W$  (our results hold in general). From standard CP we know that for row-exchangeable matrix  $X$ ,  $XW$  is also row-exchangeable (a linear layer is permutation equivariant). Hence, we just question the row-exchangeability of  $AX$ . Split  $X$  into  $[X_{\mathrm{cal}}^{\top} \mid X_{\mathrm{eval}}^{\top} \mid X_{\mathcal{V}'}^{\top}]^{\top}$ , and similarly split  $A$  into nine blocks based on the endpoints of each edge. We have that  $AX$  equals
+
+$$
+\left[ \begin{array}{c c c} \boldsymbol {A} _ {\mathrm {c a l} \cdot \mathrm {c a l}} & \boldsymbol {A} _ {\mathrm {c a l} \cdot \mathrm {e v a l}} & \boldsymbol {A} _ {\mathrm {c a l} \cdot \mathcal {V} ^ {\prime}} \\ \boldsymbol {A} _ {\mathrm {e v a l} \cdot \mathrm {c a l}} & \boldsymbol {A} _ {\mathrm {e v a l} \cdot \mathrm {e v a l}} & \boldsymbol {A} _ {\mathrm {e v a l} \cdot \mathcal {V} ^ {\prime}} \\ \boldsymbol {A} _ {\mathcal {V} ^ {\prime} \cdot \mathrm {c a l}} & \boldsymbol {A} _ {\mathcal {V} ^ {\prime} \cdot \mathrm {e v a l}} & \boldsymbol {A} _ {\mathcal {V} ^ {\prime} \cdot \mathcal {V} ^ {\prime}} \end{array} \right] \boldsymbol {X} = \underbrace {\left[ \begin{array}{c} \boldsymbol {A} _ {\mathrm {c a l} \cdot \mathrm {c a l}} \boldsymbol {X} _ {\mathrm {c a l}} + \boldsymbol {A} _ {\mathrm {c a l} \cdot \mathrm {e v a l}} \boldsymbol {X} _ {\mathrm {e v a l}} \\ \boldsymbol {A} _ {\mathrm {e v a l c a l}} \boldsymbol {X} _ {\mathrm {c a l}} + \boldsymbol {A} _ {\mathrm {e v a l} \cdot \mathrm {e v a l}} \boldsymbol {X} _ {\mathrm {e v a l}} \\ ... \end{array} \right]} _ {\operatorname {M a t} (1)} + \underbrace {\left[ \begin{array}{c} \boldsymbol {A} _ {\mathrm {c a l} \cdot \mathcal {V} ^ {\prime}} \boldsymbol {X} _ {\mathcal {V} ^ {\prime}} \\ \boldsymbol {A} _ {\mathrm {e v a l} \cdot \mathcal {V} ^ {\prime}} \boldsymbol {X} _ {\mathcal {V} ^ {\prime}} \\ ... \end{array} \right]} _ {\operatorname {M a t} (2)}
+$$
+
+For the guarantee to be valid, the first  $|\mathcal{V}_{\mathrm{cal}}| + |\mathcal{V}_{\mathrm{eval}}|$  rows should be exchangeable. For Mat(1), this already holds due to node-exchangeability and Theorem 2. With  $X_{\mathcal{V}'}$  being common in both blocks of Mat(2), we only need  $A_{\mathrm{cal}\cdot \mathcal{V}'}$  and  $A_{\mathrm{eval}\cdot \mathcal{V}'}$  to be row-exchangeable which again holds due to node exchangeability. Hence, the effect of  $\mathcal{V}'$  is symmetric to calibration and evaluation sets. In other words, the shifted embedding still preserves the guarantee conditional to the graph at timestep  $t$ . The only requirement is to compute the conformal scores for all nodes including the calibration set and dynamically update the quantile threshold – the threshold depends on  $t$ . Another way to understand the theorem is that for any column in  $C$ , the expectation of any subset of elements  $\geq 1 - \alpha$ . Next, we generalize this guarantee to any evaluation mask independent of time.
+
+Theorem 3. On a node-exchangeable graph sequence, with exchangeably sampled  $\mathcal{V}_{\mathrm{cal}}$  consisting of the first  $n$  nodes in  $\mathcal{G}_t$ , for any valid partition  $\mathbb{I}_T = \cup_{i=1}^T \mathcal{V}_{\mathrm{eval}}^{(i)}$  we have
+
+$$
+\operatorname {P r o b} \left[ y _ {i} \in \mathcal {C} ^ {(t)} \left(v _ {i}\right) \mid v _ {i} \in \mathcal {V} _ {\text {e v a l}} ^ {(t)} \right] \geq 1 - \alpha \tag {6}
+$$
+
+Moreover, it holds that  $\operatorname{Prob}[\operatorname{Cov}(\mathbb{I}_T) \leq \beta] = 1 - \Phi_{\mathrm{HG}}(i_\alpha; |\mathbb{I}_T| + n, n, i_\alpha + \lceil |\mathbb{I}_T| \cdot \beta \rceil)$ .
+
+Theorem 3 indicates that CP applies to inductive GNNs conditional to the subgraph in which each node is evaluated. We will leverage this insight in § 5. In conclusion, with a node-exchangeable graph sequence, for any node  $v_{i}$  appearing at timestep  $t$ , and evaluated at any timestep  $t' \geq t$ , it holds that  $\operatorname{Prob}\left[y_i \in \mathcal{C}(t')(v_i)\right] \geq 1 - \alpha$ . Two special cases of this result are evaluation upon appearance (the diagonal of  $C$ ) and evaluation all at once (the last column of  $C$ ) which follows from Theorem 2. The latter is also equivalent to the simultaneous-inductive setting in Zargarbashi et al. (2023). The user is flexible to delay the prediction to any time, still preserving the guarantee.
+
+# 4.2 EDGE-EXCHANGEABLE SEQUENCES
+
+In an edge-inductive sequence at each timestep an edge is added,  $\mathcal{G}_{t + 1} = (\mathcal{V}_t\cup V(e_{t + 1}),\mathcal{E}_t\cup e_{t + 1})$  .2 This edge may introduce new nodes or connect existing ones. When all permutations  $\mu$  of the edge-sequence are equally likely, Prob  $[(e_1,\dots e_m)] = \mathrm{Prob}\left[(\mu (e_1),\dots ,\mu (e_m))\right]$  , the sequence is edge-exchangeable (see  $\S$  B for a formal definition). We address this setting using a special case of weighted quantile lemma (Tibshirani et al., 2019) with weights defined by the frequency of elements.
+
+Lemma 1. Let  $\mathcal{X} = \{x_1, \dots, x_m\}$  be exchangeable random variables and  $f: 2^{\mathcal{X}} \mapsto \mathbb{R}$  be a mapping defined on subsets of  $\mathcal{X}$ . For any partitioning  $\cup_{i=1}^{n+1} \mathcal{X}_i = \mathcal{X}$  and  $z_i := f(\mathcal{X}_i)$  we have:
+
+$$
+\operatorname {P r o b} \left[ z _ {n + 1} \leq \operatorname {Q u a n t} \left(\beta ; \{z _ {i} \} _ {i = 1} ^ {n} \cup \{\infty \}; \left\{\frac {1}{| \mathcal {X} _ {i} |} \right\} _ {i = 1} ^ {n + 1}\right) \right] \geq \beta
+$$
+
+We introduce the edge-exchangeable (EdgeEx) CP and prove its guarantee by showing that at any timestep  $t$ , an edge exchangeable sequence, is decomposed into weighted node exchangeable subsequences with weights equal to  $1 / \deg(v)$ . Then, on each subsequence weighted CP maintains a valid guarantee. In this setup, there are no isolated nodes, any node can be evaluated upon appearance.
+
+Theorem 4. At each timestep  $t$ , given graph  $\mathcal{G}_t = (\mathcal{V}_t, \mathcal{E}_t)$  from an edge-exchangeable sequence and with a calibration set  $\mathcal{V}_{\mathrm{cal}}$ , define  $q = \mathrm{Quant}\left(\alpha; \{s_i\}_{v_i \in \mathcal{V}_{\mathrm{cal}}}; \left\{1 / \deg(v_i)\right\}_{v_i \in \mathcal{V}_{\mathrm{cal}}}\right)$ , and for any  $v_j \in \mathcal{V}_{\mathrm{eval}}^{(t)}$  define  $\mathcal{C}^{(t)}(v_j) := \{y : s(v_j, y) \geq q\}$ . We have  $\mathrm{Prob}\left[y_j \in \mathcal{C}^{(t)}(v_j)\right] \geq 1 - \alpha$ .
+
+Via Theorem 4, EdgeEx CP is guaranteed to provide  $1 - \alpha$  coverage. The result holds for each timestep  $t$ , conditional to that timestep.
+
+# 5 NODE EXCHANGEABLE AND EDGE EXCHANGEABLE CP
+
+Building upon the theory in § 4, for node-exchangeable graph sequences we define node-exchangeable (NodeEx) CP with coverage guarantee conditional to the subgraph at each timestep. Recall that the shift in scores upon changes in the graph is symmetric for calibration and evaluation nodes. Hence, NodeEx recomputes the calibration scores with respect to the additional context. In an edge-exchangeable sequence via Theorem 4, EdgeEx CP - weighted NodeEx CP with  $w_{i} = 1 / \deg(v_{i})$ , results in a similar valid guarantee. Any further details about NodeEx also generalizes to the EdgeEx via Theorem 4.
+
+With both settings at each timestep  $t$  we have seen a set of nodes  $\mathcal{V}_t$ . Let  $\mathcal{V}_{\mathrm{eval}}^{(t)} \subset \mathcal{V}_t$  be the set of nodes evaluated at timestep  $t$ . This set can contain any node from the past as long as they are not already evaluated. Specifically, we cannot use the prediction sets of a particular node multiple times to compare and pick a specific one (see § D.1 for a discussion). We always assume that the calibration set is the union of all nodes appeared up to timestep  $t = T_0$  and the test set contains the remaining nodes. For a sequence of timesteps and corresponding evaluation sets  $\{(t_i, \mathcal{V}_{\mathrm{eval}}^{t_i})\}_{i = T_0 + 1}^T$  where  $\cup_{i = T_0 + 1}^T \mathcal{V}_{\mathrm{eval}}^{t_i} = \mathcal{V}_T - \mathcal{V}_{\mathrm{cal}}$ , NodeEx CP (see § A for algorithm) returns valid prediction sets. We use EdgeEx CP for edge-exchangeable sequences.
+
+CP on node-conditional random subgraphs. Another application of Proposition 1 is to produce subgraph conditional prediction sets even for transductive node-classification. For each node  $v_{\mathrm{test}}$ , we define  $K$  subgraphs each including  $\{v_{\mathrm{test}}\}$ ,  $\mathcal{V}_{\mathrm{cal}}$  and randomly sampled  $\mathcal{V}' \subset \mathcal{V}_{\mathcal{G}}$ . Then, we average the scores and return prediction sets. In another approach, we can run  $K$  concurrent CPs, each resulting in a prediction set. Then we combine them with a randomized voting mechanism to a final prediction set. The resulting prediction sets include true labels with  $1 - \alpha$  probability. This approach works for any number of subgraphs. See § D.1 for further explanations.
+
+Relation to full conformal prediction. In our paper, we technically adopted the split conformal<sup>3</sup> approach where model training is separate from calibration due to its computational efficiency. An alternative is full conformal prediction, where to obtain a score for a (calibration) node  $v_{i} \in \mathcal{V}_{\mathrm{cal}}$  and any candidate label  $y'$  w.r.t. a test node  $v_{\mathrm{test}}$  we train a model from scratch on  $\mathcal{V}_{\mathrm{cal}} \cup \{(v_{\mathrm{test}}, y')\}$  and
+
+consider its prediction for  $v_{i}$ . This involves training  $|\mathcal{V}_{\mathrm{cal}}| + 1$  models for each  $y'$  and each test node, which is extremely expensive but can use all available labels. Transductive node classification can be seen as a middle ground between split and full conformal as explained by Huang et al. (2023). Our approach is similarly in between – instead of retraining the model from scratch we simply update the embeddings (and thus scores) by performing message-passing with the newly arrived nodes/edges.
+
+# 6 EXPERIMENTAL EVALUATION
+
+# 6.1 EXPERIMENTAL SETUP AND DISCUSSION OF METRICS
+
+We evaluate our approach for both node and edge exchangeable sequences where we have three options to make predictions: (i) upon node arrival, (ii) at a given fixed timestep shared for all nodes, (iii) at an arbitrary node-specific timestep (we choose at random). We compare our approach with naive CP where the coverage guarantee will not hold. Later we discuss why both over- and undercoverage are invalid. For completeness, we compare NodeEx CP with NAPS in § C in addition to a detailed overview of its drawbacks. However, as discussed in § 2.1, NAPS is not a suitable baseline. We use APS (Romano et al., 2020) as the base score function while other scores are tested in § D.
+
+Models and datasets. We consider 9 different datasets and 4 models: GCN Kipf & Welling (2017), GAT Velicković et al. (2018), and APPNPKlicpera et al. (2019) as structure-aware and MLP as a structure-independent model. We evaluate our NodeEx, and EdgeEx CP on the common citation graphs CoraML McCallum et al. (2004), CiteSeer Sen et al. (2008), PubMed Namata et al. (2012), Coauthor Physics and Coauthor CS Shchur et al. (2018), and co-purchase graphs Amazon Photos and Computers McAuley et al. (2015); Shchur et al. (2018) (details in § E). Results for Flickr (Young et al., 2014), and Reddit2 (Zeng et al., 2019) (with GCN model and their original split) datasets are in § D.5. Here, the model only affects the efficiency and not the validity.
+
+Evaluation procedure. For any of the mentioned datasets, we sample 20 nodes per class for training and 20 nodes for validation with stratified sampling. For the node-exchangeable setup, the calibration set has the same size as the training. For the edge-exchangeable sequence, the same number of edges are sampled. Therefore, each round of simulation has a potentially different number of calibration nodes for the edge-exchangeable setup. First, we take a random sample of nodes as train/val set and train the model on the resulting subgraph  $\mathcal{G}_0$ . Then, for the node-exchangeable sequence, we first sample a calibration set randomly from the remaining nodes. Then, at each timestep, we add a random unseen node to the graph (with all edges to the existing nodes) and predict the class probability given the updated subgraph. We use the updated conformal scores to create prediction sets for all existing test nodes until time  $t$  and record the empirical coverage results in column  $t$  of the coverage matrix  $C$ . Similarly, for the edge-exchangeable sampling, we sample calibration edges, and take both ends as calibration nodes. The remaining edges are sampled one at a time. The rest of the procedure is the same as node-exchangeable setting and results in an analogous  $C$ .
+
+Challenges with small calibration sets. With a limited number of samples in the calibration set, there will be an additional error in the coverage due to the discrete quantile function with a low sample rate. In practice, we take the  $\lfloor n / (n + 1) \cdot \alpha \rfloor$  index of a discrete array as the conformal threshold which is often not exactly equal to  $1 - \alpha$  quantile in the continuous domain. Therefore, we expect  $1 / (|\mathcal{V}_{\mathrm{cal}}| + 1)$  error around  $1 - \alpha$ . This error is in addition to the variance of the Beta or Hyper-geometric distribution and will converge to 0 with increasing calibration set size.
+
+Distance from target coverage. Our main evaluation criteria is the distance of empirical coverage w.r.t. the target  $1 - \alpha$ . For all datasets we set the desired coverage to  $90\%$ , and we report the (absolute) distance w.r.t. this value (see § E). Each reported result is an average of 10 different random node-exchangeable sequences and 15 different random edge-exchangeable sequences. Note that due to homophily, we observe a higher empirical coverage in the non-exchangeable case which can wrongly be interpreted as being better. To address any potential confusion, first the guarantee is invalid if it breaks either the lower or the upper-bound. In real-world deployment we do not have ground-truth labels to calculate empirical coverage, and if the guarantee is broken (in either direction) the output is unreliable - nullifying the main goal of CP. Second, a higher empirical coverage also results in larger set sizes making the prediction sets less useful (see also Fig. 4).
+
+![](images/d3c37507c7b41022bbe7955e967e3413809cfb8daa807be926893632407a26ad.jpg)
+
+![](images/879b23376711fd151b6f959e2f0201f35868203893feccd86c6be7ddcd89027c.jpg)
+
+![](images/0a46e4f3b1007c96d4610a19f43d87ac50a32257a01e782ff80a3897fb003286.jpg)  
+Figure 3: [Upper left] Coverage over time under node exchangeability when predicting upon node arrival (diagonals of  $C$ ). [Upper right] Coverage when we instead predict at a fixed time (columns of  $C$ ). [Lower right] Same as upper right but under edge-exchangeability. [Lower left] The empirical distribution of coverage when we predict at node-specific times (fixed entries of  $C$ ), compared to the theoretical distribution. Sample size results in a slight expected shift. The transparent lines show a particular sequence, and the thick solid lines show the average over 10 (15) sequences.
+
+![](images/1655b479e3cf469e1abe8e68402c350b7a432430f4bc5b1208356730f85d67f4.jpg)
+
+Efficiency and singletons. In addition to coverage, which is the main goal of our work, we also briefly study the efficiency (average set size) and the singleton hit - fraction of correct sets of size 1. As we will see in § 6.2, our NodeEx and EdgeEx CP improve these metrics as a byproduct. Some scoring functions such as RAPS (Angelopoulos et al., 2021) and DAPS (Zargarbashi et al., 2023) directly target these metrics and can be used on top of our approach. We explore DAPS in § D.3.
+
+# 6.2 EVALUATION OF EMPIRICAL COVERAGE, SET SIZE AND SINGLETON HIT
+
+Table 1 shows the deviation from the coverage guarantee for different datasets when the label of each node is predicted upon its arrival. As shown in the table, while naive CP shows a significant shift from the  $1 - \alpha = 0.9$  coverage, NodeEx CP maintains the empirical coverage close to the desired value. In Fig. 3 (upper left) we show the temporal evolution of coverage. We show the coverage for all nodes that arrived until time  $t$  and see that the guarantee is preserved for each timestep  $t$ . As mentioned before, the goal is to achieve near  $1 - \alpha$  empirical coverage, the over-coverage of standard CP might misleadingly appear as a better result. Not only does over-coverage come at the cost of less efficiency as shown in Fig. 4 (see § D.2 for details), but the empirical coverage of non-guaranteed prediction sets is not predictable apriori.
+
+Fig. 3(lower left) shows the same experiment for edge-exchangeability. While NodeEx CP guarantees coverage under node-exchangeability, the weights specified in our EdgeEx CP are necessary for the guarantee to hold on an edge-exchangeable sequence. Standard CP fails again.
+
+As explained in § 4 any node can be evaluated at any timestep after its appearance. Fig. 3 (upper right) shows the result when predicting at a fixed time (instead of upon arrival). Additionally, Fig. 3 (lower right) shows the empirical distribution of coverage for node-specific times (random subsets of nodes, each node predicted at a random time after appearance). For NodeEx CP, the empirical distribution matches the theoretical one, while the coverage of naive CP substantially diverges.
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Acc</td><td colspan="2">Node Exch. Sequence</td><td colspan="3">Edge Exch. Sequence</td></tr><tr><td>NodeEx CP</td><td>Naive CP</td><td>EdgeEx CP</td><td>NodeEx CP</td><td>Naive CP</td></tr><tr><td>Cora-ML</td><td>0.800</td><td>0.280</td><td>5.860</td><td>1.929</td><td>3.883</td><td>6.860</td></tr><tr><td>PubMed</td><td>0.777</td><td>1.254</td><td>4.649</td><td>1.241</td><td>3.405</td><td>5.315</td></tr><tr><td>CiteSeer</td><td>0.816</td><td>0.039</td><td>4.150</td><td>0.335</td><td>1.572</td><td>3.460</td></tr><tr><td>Coauth-CS</td><td>0.914</td><td>0.397</td><td>4.082</td><td>3.024</td><td>4.662</td><td>7.835</td></tr><tr><td>Coauth-Phy.</td><td>0.940</td><td>0.555</td><td>2.689</td><td>2.240</td><td>4.378</td><td>6.123</td></tr><tr><td>Amz-Comp.</td><td>0.788</td><td>0.263</td><td>6.373</td><td>2.687</td><td>5.727</td><td>7.036</td></tr><tr><td>Amz-Photo</td><td>0.868</td><td>0.127</td><td>3.483</td><td>2.546</td><td>4.130</td><td>6.613</td></tr></table>
+
+Table 1: Average absolute deviation from guarantee (in %, for GCN model and 1 train/val split).
+
+Unsurprisingly, under node exchangeability, in Fig. 4 we see that our NodeEx CP improves both additional metrics - has smaller sets and larger singleton hit ratio. The same holds for other settings.
+
+In all experiments, we consider a sparse (and thus realistic) calibration set size, e.g. for PubMed we sample 60 nodes for calibration, which is a significantly lower number compared to other tasks like image classification with 1000 datapoints for same purpose (Angelopoulos et al., 2021). The calibration size controls the concentration (variance) around  $1 - \alpha$  as reflected by the transparent lines on Fig. 3. Increasing the set size reduces the variance but the mean is always  $1 - \alpha$ .
+
+Limitations. We identified three main limitations. First, the guarantee is marginal. Second, real-world graphs may not satisfy node- or edge-exchagability. This can be partially mitigated by the beyond exchangeability framework. Finally, the guarantee does not hold for adversarially chosen evaluation sets  $\mathcal{V}_{\mathrm{eval}}$ . In other words, the choice of which nodes is evaluated at which timesteps must be prior to observing the prediction set. We provide a longer discussion on each of these limitations in § A. Our main focus is on the validity. However, we also reported other metrics such as the average set size, singleton hit ratio and other score functions in § D.
+
+# 7 CONCLUSION
+
+We adapt conformal prediction to inductive node-classification for both node and edge exchangeable graph sequences. We show that although introducing new nodes/edges causes a distribution shift in the conformity scores, this shift is symmetric. By recomputing the scores conditional to the evaluation subgraph, we recover the coverage guarantee. Under edge-exchangeability, we need to also account for the node degrees to maintain validity. Importantly, our approach affords flexibility – the guarantee holds regardless of prediction time which can be chosen differently for each node.
+
+![](images/f1ad646d04ba2fc18d2cc64764cf513cf84f8f3d6bc39ec6fef25381ac2661a6.jpg)  
+Figure 4: [Left] Average set size (lower is better) and [Right] singleton hits ratio (higher is better) of naive CP vs. NodeEx CP for CiteSeer and GCN. Our approach improves both metrics.
+
+![](images/30c23a6232e455b7bbeee91ab5f5d6c244084c4f2732b7af60b2c0b2e4518221.jpg)
+
+# REPRODUCIBILITY STATEMENT
+
+All datasets of our experiments are publicly available and referenced in § 6.1 in addition to details on experimental setups. The pseudocode of our approach is provided in § A. The hyperparameters of models are given in § E. The code of experiments is also uploaded as supplementary material.
+
+# REFERENCES
+
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+Anastasios Nikolas Angelopoulos and Stephen Bates. A gentle introduction to conformal prediction and distribution-free uncertainty quantification. ArXiv, abs/2107.07511, 2021b.  
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+Julian McAuley, Christopher Targett, Qinfeng Shi, and Anton van den Hengel. Image-based recommendations on styles and substitutes. Proceedings of the 38th International ACM SIGIR Conference on Research and Development in Information Retrieval, 2015.  
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+Peter Orbanz and Daniel M Roy. Bayesian models of graphs, arrays and other exchangeable random structures. IEEE transactions on pattern analysis and machine intelligence, 37, 2014.  
+Yaniv Romano, Matteo Sesia, and Emmanuel J. Candès. Classification with valid and adaptive coverage. In NeurIPS, 2020.  
+Maurizio Sadinle, Jing Lei, and Larry A. Wasserman. Least ambiguous set-valued classifiers with bounded error levels. Journal of the American Statistical Association, 2018.  
+Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI Magazine, Sep. 2008. doi: 10.1609/aimag.v29i3.2157.  
+Glenn Shafer and Vladimir Vovk. A tutorial on conformal prediction. Journal of Machine Learning Research, 2008.  
+Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Gunnemann. Pitfalls of graph neural network evaluation. ArXiv, abs/1811.05868, 2018.  
+Maximilian Stadler, Bertrand Charpentier, Simon Geisler, Daniel Zügner, and Stephan Gunnemann. Graph posterior network: Bayesian predictive uncertainty for node classification. Advances in Neural Information Processing Systems, 2021.  
+David Stutz, Krishnamurthy Dj Dvijotham, Ali Taylan Cemgil, and Arnaud Doucet. Learning optimal conformal classifiers. In International Conference on Learning Representations, 2022.  
+Ryan J Tibshirani, Rina Foygel Barber, Emmanuel Candes, and Aaditya Ramdas. Conformal prediction under covariate shift. Advances in neural information processing systems, 2019.  
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+Vladimir Vovk. Conditional validity of inductive conformal predictors. In Steven C. H. Hoi and Wray Buntine (eds.), Proceedings of the Asian Conference on Machine Learning, volume 25 of Proceedings of Machine Learning Research, pp. 475-490, Singapore Management University, Singapore, 04-06 Nov 2012. PMLR.  
+Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world, volume 29. Springer, 2005.  
+Pivithuru Wijegunawardana, Ralucca Gera, and Sucheta Soundarajan. Node classification with bounded error rates. 2020.  
+Peter Young, Alice Lai, Micah Hodosh, and Julia Hockenmaier. From image descriptions to visual denotations: New similarity metrics for semantic inference over event descriptions. Transactions of the Association for Computational Linguistics, 2, 2014.  
+Soroush H. Zargarbashi, Simone Antonelli, and Aleksandar Bojchevski. Conformal prediction sets for graph neural networks. In Proceedings of the 40th International Conference on Machine Learning, 2023.  
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+
+# A ALGORITHM FOR NODEX CP AND WEIGHTED EDGEEX CP
+
+At each timestep  $t$  with node-exchangeable graph  $\mathcal{G}_t$ , we first run the GNN model to extract conformity scores for all nodes in  $\mathcal{V}_t$ . We compute the  $\alpha$ -quantile of the conformity scores conditional to the current graph, and for each new test node  $v_{\mathrm{test}} \in \mathcal{V}_{\mathrm{eval}}^{(t)}$  we return any candidate label with a score larger than the conditional threshold. For an edge-exchangeable graph, the algorithm is still the same unless we compute the conditional weighted quantile with weights equal to  $1 / d(v)$  for all calibration vertices  $v$ . Algorithm 1 shows the pseudocode for NodeEx CP. The code is accessible at github.com/soroushzargar/conformal-node-classification.
+
+Algorithm 1: NodeEx and EdgeEx CP for inductive node classification  
+Data: Graph  $\mathcal{G}_t$  at timestep  $t$ . Calibration vertex set  $\nu_{\mathrm{cal}}$ , Permutation equivariant score function  $s$  (e.g. a function of GNN), Evaluation vertex set  $\nu_{\mathrm{eval}}^{(t)}$  Result:  $C^t (v_j):v_j\in \nu_{\mathrm{eval}}^{(t)}$  1  $S_{t}\gets s(\mathcal{G}_{t})$  // Get score functions for all vertices 2 if node-exchangeable sequence then 3 Set  $\tau_{t}\gets \mathrm{Quant}\left(\alpha ;\{\pmb {S}_{t}[i,y_{i}]\}_{v_{i}\in \nu_{\mathrm{cal}}};1\right)$  4 else if edge-exchangeable sequence then 5 Set  $\tau_{t}\gets \mathrm{Quant}\left(\alpha ;\{\pmb {S}_{t}[i,y_{i}]\}_{v_{i}\in \nu_{\mathrm{cal}}};1 / \deg (v_{i}\mid \mathcal{G}_{t})\right)$  6  $\forall v_{j}\in \mathcal{V}_{\mathrm{eval}}^{(t)}$ , set  $C^t (v_j) = \{y:\pmb {S}_t[j,y]\geq \tau^t\}$ ;
+
+Naive CP. In naive CP we compute calibration scores before the arrival of any new node, on the inductive subgraph over  $\mathcal{V}_{\mathrm{tr}} \cup \mathcal{V}_{\mathrm{cal}}$ . Prior to the evaluation of any node, we compute the calibration quantile  $q$  from calibration scores. For any test node, the conformity score of all its classes is compared to  $q$ . Clarkson (2023) shows that this approach fails to provide valid prediction sets (the guarantee is broken due to the shift caused by the arrival of new nodes).
+
+Computational complexity. Conformal prediction has two main computational routines in addition to model's inference: computing scores and quantiles. For the calibration set we only need the scores of the true class, but for test nodes, we should compute scores for all classes. Depending on the score function the complexity can be different. For standard CP we compute calibration scores once, but for NodeEx CP for each evaluation (timestep) we need to recompute the predictions and the conformity scores. Hence, Node(Edge)Ex CP has an overhead of  $O(n \times t_s \times t)$  for  $t$  timesteps,  $n$  calibration nodes, and  $t_s$  the time for computing a conformity score for one class and one node. At each step we should also evaluate the model once if the model's prediction changes across time. Computing the quantile threshold takes  $\mathcal{O}(n)$  steps. For standard CP we have to compute this value once, but in NodeEx CP this values is updated upon each evaluation. The total wall-clock overhead is less than a second. Additionally all mentioned complexities are for serial computation and some of CP procedures can run significantly faster via parallel computation.
+
+Limitations of NodeEx and EdgeEx CP. Our study mainly focuses on approaches that can provide a valid CP for changing subgraphs under certain assumptions of node-or edge-exchangeability. There are three main limitations for using NodeEx (or EdgeEx) CP (i) The guarantee provided by NodeEx (and EdgeEx) CP is marginal. However, it is shown that exact conditional coverage Prob  $[y\in \mathcal{C}(\pmb {x})\mid \pmb {x}]$  is not achievable. Still, there are methods that tend to get better approximations of conditional coverage. (ii) In real-life graph sequences, it is hard to determine whether the graph sequence is node- or edge-exchangeable or neither. Here a follow-up works is to use the beyond exchangeability approach (Barber et al., 2023) to achieve valid coverage guarantees without the need for node- or edge-exchangeability assumption. (iii) Our result maintains validity as long as the selection evaluation timestep for each node is done without any knowledge of the prediction set. An adversarial selection can cause a significant deviation from the guaranteed coverage. For instance, an adversary can observe prediction sets for a particular node during various timesteps and pick the timestep of the smallest prediction set similar to the two false examples in § D.1. This results can lead to a significant miscoverage rate compared to the guarantee.
+
+# B ADDITION TO THE THEORY
+
+Matric  $C$ . For timestep  $t$  and index  $i$ ,  $C[i, t]$  indicates that whether node  $v_i$  if evaluated at timestep  $t$  is going to be covered or not. Hence this value can be either 1, indicating CP covers node  $v_i$  at timestep  $t$ , 0 showing that CP does not cover this node, or  $N / A$  showing that this node has not appeared at this timestep. This matrix is constructed using ground-truth labels and is shown only for the purpose of sanity check and evaluation. The matrix  $C$  is a very dense matrix since  $1 - \alpha$  entities at each column should be 1. That is why in Fig. 2 we showed  $1 - C$  where each column is expected to have  $\alpha$ -percentage of ones. A valid CP approach should result in a similar percentage of ones in each column. For an invalid CP, if the trend is gradually moving toward over-coverage, the matrix  $1 - C$  becomes more sparse in later columns.
+
+Node- and edge-exchangeability. A sequence  $\mathcal{Z} = (z_1, \ldots, z_n)$  is called exchangeable if for any permutation  $\mu$ , the joint probability of the permuted sequence remains the same, i.e.  $\operatorname{Prob}\left[(z_1, \ldots, z_n)\right] = \operatorname{Prob}\left[(\mu(z_1), \ldots, \mu(z_n))\right]$ . Before considering inductive graph sequences, we first define node- and edge-exchangeability on a fixed given graph  $\mathcal{G}$ .
+
+Definition 1 (Node-exchangeability). Let  $\mathcal{G}(\mathcal{V},\mathcal{E},\boldsymbol {X},\boldsymbol {y})$  be a graph where  $i$ -th row of  $\pmb{X}$  is the feature vector for node  $v_{i}$ , and similarly,  $i$ -th element of  $\pmb{y}$  is the label of that node. Let  $\mu_V:\{1,\ldots ,n\} \times \{1,\ldots ,n\}$  with  $n = |\mathcal{V}|$  be a permutation. We define  $\mu_V(\mathcal{G})$  to be a relabeling of vertices  $\mathcal{V}$  with  $\mu_V(\mathcal{E}) = \{(\mu_V(v_i),\mu_V(v_j)): \forall (v_i,v_j)\in \mathcal{E}\}$ ,  $\mu_V(\pmb {X})[i,\cdot ] = \pmb {X}[\mu_V(i),\cdot ]$ , and  $\mu_V(\pmb {y})[i] = \pmb {y}[\mu_V(i)]$ . The permuted graph is  $\mu_V(\mathcal{G})(\mu_V(\mathcal{V}),\mu_V(\mathcal{G}),\mu_V(\pmb {X}),\mu_V(\pmb {y}))$ .  $\mathcal{G}$  is called node exchangeable if  $\mathrm{Prob}[G = \mathcal{G}] = \mathrm{Prob}[G = \mu_V(\mathcal{G})]$ , where  $G$  is sampled from a generative graph distribution.
+
+Definition 2 (Edge-exchangeability). Let  $\mathcal{G}(\mathcal{V},\mathcal{E},\boldsymbol {X},\boldsymbol {y})$  where  $\mathcal{E} = \{e_1,\dots e_m\}$  is the set of edges. Each edge  $e_k$  is defined as a pair  $e_{k} = ((v_{i},\pmb{x}_{i},y_{i}),(v_{j},\pmb{x}_{j},y_{j}))$ . For each node  $v_{i}\in \mathcal{V}$  let  $\pmb{X}$  be the feature matrix with rows  $\pmb{x}_i$ , and  $\pmb{y}$  be the vector of labels with entries  $y_{i}$ . Let  $\mu :\{1,\ldots ,m\} \times \{1,\ldots ,m\}$  with  $m = |\mathcal{E}|$  be a permutation. We define  $\mu (\mathcal{G})$  to be a relabeling of edges  $\mathcal{E}$  with  $\mu_E(\mathcal{E}) = \{e_{\mu_E(i)}:\forall e_i\in \mathcal{E}\}$ . The permuted graph is  $\mu_E(\mathcal{G})(\mathcal{V},\mu_E(\mathcal{E}),\boldsymbol {X},\boldsymbol {y})$ .  $\mathcal{G}$  is called edge exchangeable if  $\mathrm{Prob}[G = \mathcal{G}] = \mathrm{Prob}[G = \mu_E(\mathcal{G})]$ , where  $G$  is sampled from a generative graph distribution.
+
+By an inductive sequence, we refer to a progressive sequence of graphs meaning that for each timestep  $t$ , the graph  $\mathcal{G}_{t-1}$  is a subgraph of  $\mathcal{G}_t$ . Node- and edge-inductive sequences are defined as follows:
+
+Definition 3 (Node-inductive sequence). A node-inductive sequence  $\mathcal{G}_0, \mathcal{G}_1, \ldots$  is a sequence starting from an empty graph  $\mathcal{G}_0 = (\emptyset, \emptyset)$ . For each timestep  $t$ , the graph  $\mathcal{G}_t$  is defined by adding a vertex  $v_t$  with all its connections to the vertices in the previous timestep. The vertex set is then  $\mathcal{V}_t = \mathcal{V}_{t-1} \cup \{v_t\}$  and the edge set is the union of all previous edges  $\mathcal{E}_{t-1}$  and any edge between  $v_t$  and  $\mathcal{V}_{t-1}$ ;  $\mathcal{E}_t = \mathcal{E}_{t-1} \cup (\cap_{i=1}^{\infty} \{e = (v_t, v_i) : e \in \mathcal{E}_i, v_i \in \mathcal{V}_{i-1}\})$ .
+
+Similarly an edge-inductive sequence is defined as follows:
+
+Definition 4 (Edge-inductive sequence). An edge inductive sequence  $\mathcal{G}_0, \mathcal{G}_1, \ldots$  starts from an empty graph  $\mathcal{G}_0 = (\emptyset, \emptyset)$ . For each timestep  $t$ , the graph  $\mathcal{G}_t$  is defined by adding an edge  $e_t = (v_i, v_j)$  to the graph at the previous timestep. Hence  $\mathcal{E}_t = \mathcal{E}_{t-1} \cup \{e_t\}$ , and  $\mathcal{V}_t = \mathcal{V}_{t-1} \cup \{v_i, v_j\}$ .
+
+A node-inductive sequence is called node-exchangeable if for all graphs  $\mathcal{G}_t$  in the sequence,  $\mathcal{G}_t$  is node-exchangeable. Similarly, if all graphs in the sequence are edge-exchangeable, the sequence is also edge-exchangeable.
+
+Note, an inductive sequence could be equivalently defined with respect to a "final" graph  $\mathcal{G}_n$ , where  $n$  can be infinity, by starting from an empty graph and adding a random node/edge at each timestep. If the final graph is node-exchangeable any node-inductive subgraph of it  $\mathcal{G}' \subseteq \mathcal{G}_t$ ,  $\mathcal{G}'$  is also node exchangeable. A similar argument holds for edge-inductive subgraphs of a "final" edge-exchangeable graph.
+
+Permutation invariance and equivariance. A permutation invariant function  $f$  returns the same output for any permutation applied on its inputs,  $f(z_{1},\ldots ,z_{n}) = f(z_{\mu (1)},\ldots z_{\mu (n)})$  for any permutation  $\mu$ . A function  $f$  is permutation-equivariant if permuting the input results in the same permutation on the output, i.e. for any  $f(x_{1},\ldots ,x_{n}) = (y_{1},\ldots ,y_{n})$  we have  $f(x_{\mu (1)},\ldots x_{\mu (n)}) =$
+
+$(y_{\mu (1)},\ldots y_{\mu (n)})$  for any  $\mu$ . A permutation-equivariant GNN assigns the same predictions (and thus the same scores) to each node even when nodes/edges are relabeled.
+
+Proof for Proposition 1. First assume  $\mathcal{V}_{\mathrm{eval}}^{(t)} = \mathcal{V}_t$  meaning that any node appeared at timestep  $t$  is either in the calibration or the evaluation set. Due to node-exchangeability,  $\mathcal{V}_t$  and  $\mathcal{V}_{\mathrm{cal}}$  are exchangeable. Hence, we adapt the proof of Theorem 2. Now consider the general case where  $\mathcal{V}' := \mathcal{V}_t \setminus \mathcal{V}_{\mathrm{cal}} \neq \emptyset$ . Any subset  $\mathcal{A}'$  of an exchangeable set  $\mathcal{A}$  is still exchangeable. For any permutation of  $\mathcal{A}'$  there are  $k$  permutations in  $\mathcal{A}$  all having the same ordering over elements of  $\mathcal{A}'$ . Since  $k$  is a constant for all permutations and function of  $|\mathcal{A}|$  and  $|\mathcal{A}'|$ , any permutation in  $\mathcal{A}'$  has the same probability. Which implies the exchangeability of its elements. This implies that with non-empty set  $\mathcal{V}'$ ,  $\mathcal{V}_{\mathrm{cal}}$  and  $\mathcal{V}_t \setminus \mathcal{V}'$  are still exchangeable. As a result, again we adapt Theorem 2 with the effect of  $\mathcal{V}'$  being symmetric to the calibration and evaluation sets.
+
+![](images/7f2de57ff6eb5f5815897189fe60148c3c7952a5519491d0295a608f95295d44.jpg)
+
+Proof for Theorem 3. We break the proof to two parts.
+
+Part 1. For a fixed node  $v_{i}$  at timestep  $t_1$  let  $\alpha_{i} \coloneqq \mathbb{E}[c_{i,t_{1}}]$ . For any timestep  $t_2$  with which  $v_{i}$  is existing in, we have  $\mathbb{E}[c_{i,t_2}] = \alpha_i$ .
+
+Note that since  $v_{i}$  is fixed its expected coverage probability is not exactly  $1 - \alpha$ . As in definition,
+
+$$
+\alpha_ {i} = \operatorname {P r o b} \left[ \operatorname {Q u a n t} \left(\alpha ; \left\{s \left(v _ {j}, y _ {j} \mid \mathcal {G} _ {t _ {1}}\right) \right\} _ {v _ {j} \in \mathcal {V} _ {\mathrm {c a l}}}; 1\right) \leq s \left(v _ {i}, y _ {i} \mid \mathcal {G} _ {t _ {1}}\right) \right]
+$$
+
+Let  $\mathcal{V}_j' = \mathcal{V}_{t_j} \setminus (\mathcal{V}_{\mathrm{cal}} \cup \{v_i\})$ . Node-exchangeability implies that  $\mathcal{V}_1'$  and  $\mathcal{V}_2'$  have the same and also symmetric effect on all scores. Hence, with either of sets as context, the expected order of elements remains similar and  $\mathbb{E}[c_{i,t_2}] = \alpha_i$ . Moreover, for each node  $v_i$  and all  $t$ ,  $c_{i,t} \sim \operatorname{Bernoulli}(\alpha_i)$ .
+
+Part 2. The coverage of any sub-partitioning  $\mathbb{I}_T$  can be written as an average over a set of elements in  $C_T$ . In other words define  $\iota(v_i) = j \Leftrightarrow (v_i \in \mathcal{V}_{\mathrm{eval}}^j)$  we have
+
+$$
+\operatorname {C o v} \left(\mathbb {I} _ {T}\right) = \frac {1}{| \mathbb {I} _ {T} |} \sum_ {i = 1} ^ {| \mathbb {I} _ {T} |} \boldsymbol {C} [ i, \iota (v _ {i}) ] \stackrel {\text {P a r t}} {=} \sum_ {i = 1} ^ {| \mathbb {I} _ {T} |} \boldsymbol {C} [ i, T ]
+$$
+
+Following Proposition 1 we have
+
+$$
+\operatorname {P r o b} \left[ \operatorname {C o v} \left(\mathbb {I} _ {T}\right) \leq \beta \right] = \frac {1}{\left| \mathbb {I} _ {T} \right|} \sum_ {i = 1} ^ {\left| \mathbb {I} _ {T} \right|} \boldsymbol {C} [ i, \iota (v _ {i}) ] = 1 - \Phi_ {\mathrm {H G}} \left(i _ {\alpha}; \left| \mathbb {I} _ {T} \right| + n, n, i _ {\alpha} + \lceil | \mathbb {I} _ {T} | \cdot \beta \rceil\right)
+$$
+
+![](images/da85ff3b4c55bcbfe4663c98b07d48ab825ea0405e1df6e358468363d96495f8.jpg)
+
+Proof for Theorem 4. We divide the proof into two cases:
+
+Graph where all nodes have degree 1. In this case  $\mathcal{G}_t$  is a union of disjoint edges  $(e_1, \ldots, e_m)$ . We divide  $\mathcal{V}_t = \mathcal{V}_1 \cup \mathcal{V}_2$  including one and only one endpoint of each edge decided at random. The vertex set in this case has the same size as the edge set, and there is a one-to-one mapping between any permutation  $\mu$  over the edge set to vertices in  $\mathcal{V}_1$  or  $\mathcal{V}_2$  since only one endpoint is present in each set. Hence  $\operatorname{Prob}[\mathcal{E}] = \operatorname{Prob}[\mathcal{E}_{\mu}]$  implies  $\operatorname{Prob}[\mathcal{V}_i] = \operatorname{Prob}[\mathcal{V}_{i,\mu}]_{i \in \{1,2\}}$ . Similarly, selection  $\mathcal{V}_{\mathrm{cal}}$  decomposes to  $\mathcal{V}_{\mathrm{cal},1} \cup \mathcal{V}_{\mathrm{cal},2}$  and again  $\mathcal{V}_{\mathrm{cal},i}$ , and  $\mathcal{V}_i \setminus \mathcal{V}_{\mathrm{cal},i}$  are exchangeable due to node-exchangeability in  $\mathcal{V}_i$ . Any test node belongs to either of the subsequences which means is guaranteed via the CP applied to that subset.
+
+The vertex set is divided into two subsets, and the intersection of the calibration set to each subset is exchangeable. Hence calibrating on each subset result in  $1 - \alpha$  coverage for all the nodes in the subset. Both calibrations have the same expected quantile threshold  $q$ . This is because each calibration node has  $1/2$  probability of being included in each of the sets.
+
+General graphs. We follow the same approach. We divide the endpoints of edges into two multisets. For each edge we run this division and this decision is made for each vertex  $\deg(v_i)$  times. In each partition, each node  $v_i$  has the expected frequency  $\deg(v_i)/2$ . Via Lemma 1 we show that weighted CP is valid for each subsequence  $\mathcal{V}_i$  and hence for  $\mathcal{V}_t$ .
+
+![](images/c1364110fcbb7522ec8a23fee80c815784f0e26dd455861ba0c46f8f11d2307a.jpg)  
+Figure 5: [Left] The proportion of nodes without a prediction set due to empty calibration on CoraML dataset. The plot shows the inapplicability of  $k$ -hop NAPS with  $k \in \{1,2\}$ . [Right] The average size of each node's filtered calibration set. Note that this plot excludes non-applicable nodes.
+
+![](images/4193741a56bd40adb3fb9dfe534b45d8bc084f209a2f233310ecffa54d236ff3.jpg)
+
+![](images/e1acdbb75ff809eb0e2cb089e58059f569b268141f7535f2887ceeeb3d913e9a.jpg)  
+Figure 6: [Left] Comparison of NAPS and NodeEx CP in empirical coverage for CoraML dataset and GCN model across different permutations of nodes. [Right] The number of nodes without prediction set for different permutations on the same dataset/model.
+
+![](images/e85ccbb85f9a2ae3b1a8ad0e9af3c5b94a687e4c2157126f13748f12761ac02c.jpg)
+
+# C LIMITATIONS OF NAPS
+
+For the inductive scenario, NAPS (Clarkson, 2023) adapts the beyond exchangeability approach (Barber et al., 2023) without any computed bounds on the coverage gap (the distance between the coverage of the given non-exchangeable sequence and  $1 - \alpha$ ). For each test node, calibration weights are assigned equal to 1 in case they are in the immediate neighborhood of the test node and 0 otherwise. In other words, this approach filters out the non-neighbor calibration nodes for each test node. This weight assignment is generalized to any  $k$ -hop neighbor while it is originally suggested to use  $k = 1$  or 2. In a sparse graph, or with a limited calibration set, the probability of having calibration points in the immediate neighborhood is significantly low which leaves many of the test nodes with empty calibration sets. Fig. 5 shows that for our experimental setup, until the end of the graph sequence, a notable proportion of the nodes are left without a prediction set. For the same reason, even for nodes in the neighborhood of the calibration set, the statistical efficiency is very low. Since this inapplicability is reported on a node-exchangeable sequence, it is independent of the prediction timestep – a node disconnected from the calibration set will remain disconnected as the graph grows.
+
+Setting aside non-applicable nodes we compared NAPS with NodeEx CP in empirical coverage. Fig. 6 shows that NAPS is significantly far from the coverage guarantee compared to NodeEx CP. Note that this experiment is very biased in favor of NAPS as it excludes nodes without a prediction set while evaluating the same nodes for NodeEx CP.
+
+# D ADDITIONAL EXPERIMENTS
+
+# D.1 CP WITH SUBGRAPH SAMPLING
+
+Although the theory in §4 concerns inductive graph sequences, we can leverage its results to transductive node-classification via subgraph sampling. For each test node  $v_{\mathrm{test}}$ , we sample  $K$  inductive subgraphs all including  $\{v_{\mathrm{test}}\} \cup \mathcal{V}_{\mathrm{cal}}$ . Considering Theorem 3, in each subgraph, calibration scores and test scores are exchangeable. This leaves  $K$  prediction sets all with a coverage guarantee of  $1 - \alpha$ . We consider each prediction set as a vote for all its elements. Each label  $y'$  will be selected in the final prediction set with a Bernoulli experiment with parameter  $p = \sum_{i=1}^{K} \mathbf{1}[y_j \in C_i(v_{\mathrm{test}})] / K$ . Again the resulting prediction set contains the true label with  $1 - \alpha$  probability. It is important to note that Theorem 3 does not imply any result about the probability of a node being covered in all of  $K$  prediction sets. Specifically, the probability of a node being covered in all prediction sets simultaneously is less than
+
+(or equal to) the probability of the same event in each single CP. As shown in Fig. 7 the subgraph Bernoulli prediction sets result in the same coverage as standard CP for transductive node-classification. Additionally, the union and intersection of CPs respectively result in over-coverage and under-coverage.
+
+![](images/2d525256526ac87bce77deff9643c19febdaa79f267c3e390101ad4b15e15a12.jpg)  
+Figure 7: Comparison between standard CP, Bernoulli prediction sets from NodeEx CP on random subgraphs, the union, and intersection of CPs.
+
+# D.2 SET SIZE AND SINGLETON HIT RATIO
+
+In § 6 we focused on empirical coverage as the main evaluation criteria since our goal is to recover the conformal guarantee. When the guarantee holds, this metric is often reported as a sanity check. In that case, the average prediction set size (efficiency) is considered as a direct indicator of how efficient (and therefore useful) prediction sets are. Another important metric called singleton hit ratio is the frequency of singleton sets covering the true label. In Fig. 8 we show the coverage matrix alongside the prediction set size and singleton hits indicator for each node at each timestep. As shown in the figure, since the standard CP breaks the guarantee toward over-coverage, it results in larger prediction sets on average and hence, a lower number of singleton hits. The same result is also shown in Fig. 4 over different timesteps.
+
+# D.3 DIFFERENT SCORE FUNCTIONS
+
+The threshold prediction sets (TPS) approach (Sadinle et al., 2018) directly takes the softmax output as a conformity score  $s(\pmb{x},y) = \pi(\pmb{x})_y$ , where  $\pi(\pmb{x})_y$  is the predicted probability for class  $y$ . Although TPS produces small prediction sets, its coverage is biased toward easier examples leaving the hard examples under-covered (Angelopoulos & Bates, 2021b). Romano et al. (2020) define adaptive prediction sets (APS) with a score function defined as  $s(\pmb{x},y) := -(\rho(\pmb{x},y) + u \cdot \pi(\pmb{x})_y)$ . Here  $\rho(\pmb{x},y) := \sum_{c=1}^{K} \pi(\pmb{x})_c 1[\pi(\pmb{x})_c > \pi(\pmb{x})_y]$  is the sum of all classes predicted as more likely than  $y$ , and  $u \in [0,1]$  is a uniform random value that breaks the ties between different scores to allow exact  $1 - \alpha$  coverage (Stutz et al., 2022). APS returns the smallest prediction sets satisfying conditional coverage if the model returns the ground truth conditional probability  $p(y \mid x)$ , otherwise Barber et al. (2019b) show that achieving conditional coverage is impossible without strong unrealistic assumptions. Here we use APS as baseline scoring function, but our method is orthogonal to this choice and works with any score.
+
+![](images/0c91f37eec53745454f0d2d37e8c2a648f54cac97f950e248ae649b95ab87bc4.jpg)  
+Figure 8: [Left column] The coverage matrix  $1 - C$  (colored points show miscoverage). [Middle column] The prediction set size for each point at each timestep. [Right column] The matrix of singleton hits. A cell is colored if the node at the timestep is predicted with a singleton set covering the true label. [Upper row] The NodeEx CP approach. [Lower row] The standard CP. The result is shown for CiteSeer dataset and GCN model.
+
+Although we used APS as the base score function for evaluation, our results are general to any other scoring function. Here we evaluate our approach with two other score functions called threshold prediction sets (TPS) (Sadinle et al., 2018), and diffused adaptive prediction sets DAPS Zargarbashi et al. (2023). TPS (Fig. 9 - left) simply applies the softmax function on model's results (logits) and uses it as the conformity score. Although it produces small prediction sets its coverage is biased toward easier examples (Angelopoulos & Bates, 2021b). DAPS (Fig. 9 - right) works as a structure-aware extension scoring over APS. It diffuses conformity scores over the network structure
+
+![](images/0d251eaae7517c7538f81a76a3051b19763b7903b02248b07ac54a15584da69a.jpg)  
+Figure 9: [Left] Comparison of naive CP and NodeEx CP with TPS score function and [Right] DAPS score function. Results are for CiteSeer dataset and GCN model.
+
+to leverage the uncertainty information in the neighborhood and produce a more efficient prediction set. Since DAPS also incorporates the structure in the scores' space (in addition to the implicit effect via message passing) it is even more influenced by the changes in the graph structure while using standard CP. However, as shown in the Fig. 9, we can recover the coverage guarantee via NodeEx CP regardless of the utilized score function.
+
+# D.4 DIFFERENT MODELS AND INITIAL Splits
+
+The coverage guarantee in CP holds regardless of the model structure. CP uses the model as a black-box and just performs the quantile calibration over the output of the model which is the input conformity score function. However, better model structure, training procedure, etc are reflected in other metrics like set size and singleton hits. As shown in Fig. 10, NodeEx CP results in similar coverage values for all the models while standard CP results in different coverage for each model (note the significant distance between structure-aware models and MLP). As MLP does not take the adjacency structure into account, its empirical coverage is still guaranteed even with the standard CP. Note that different models result in different efficiencies in the prediction sets. Fig. 11 shows the same experiment conducted on edge-exchangeable sampling. Again similar results are observed for edge-exchangeable sequences.
+
+As pointed out in (Shchur et al., 2018), GNNs are sensitive to the initial train/Validation sampling. This however does not impact the coverage of NodeEx (and EdgeEx) CP as it is guaranteed agnostic to the model and the initial split. However similar to different model architectures, different initial splits also affect the model's accuracy which is reflected in the efficiency of the prediction set.
+
+![](images/88248459700589bbaf3697fcf4e2ded7e3aa070efefcf8d9ff211d82766ccb7b.jpg)
+
+![](images/5a08d263d81c76af23501355a3dad55508c9d5a5fcc1866cdf9de0f6502df676.jpg)
+
+![](images/4979b39d80d940b94aae9b5d5d872b8e58be20416d54e009dbae2674a14cf70a.jpg)
+
+![](images/9ddd6f94178ea85ae1b725623f3164ebb888e110ec0d08a49b8702694a8e3cb3.jpg)
+
+![](images/724d3aef46e9de0b3f6bab165cb920ba38e030485f18ba303ec7cf70515115c6.jpg)
+
+![](images/7b92f67853b522377d2cec3e78313f8d9af9b2d7788527e2d55937b233ff6c67.jpg)
+
+![](images/87e2b0fa20f0be611ef2b231f739f90a84b33dbd7d2313afc79b118cb4479b6c.jpg)  
+No. Nodes
+
+![](images/09f2c0b7f11eeafce96c24aac0a06e713bdeba3e9d1b0cd44f4972e34b7e5c7f.jpg)  
+No. Nodes  
+Figure 10: [Left] The empirical coverage for different models. [Right] The average set size. The results are shown for the CoraML dataset with node-exchangeable sampling.
+
+![](images/9ca66959b7268735e41923aa7d2c052a180e52abaf7bce5ee3b5582362994859.jpg)
+
+![](images/9d5b3a02f1f30fcd8ce1b1ea034c91bf5afe09796dd1958b8f6fce7c757996d4.jpg)
+
+![](images/7f18bc0ff11b3db7b3fed4e3b24cf41f7c1bfc013f6c95e6e785f15c357eca30.jpg)
+
+![](images/9250fee79b7b4e49647d9a22df65abe889c2b35a2d0bafc29f430d27e9aae5bb.jpg)
+
+![](images/b7b66aa057314e59aa0bdf54bc9480c548e793b9a912fd63a2ec38ef69332a98.jpg)
+
+![](images/0d7d5f8dcf4c73c96ae063995115c9b6e41722405e124aa90e72cfb00eeaca4b.jpg)
+
+![](images/2e23d1b6d229ce387a0be175d067f96c52c6a6c62b19739c577b0f5c1620ca80.jpg)  
+Figure 11: [Left] The empirical coverage for different models. [Right] the average set size. The results are shown for the CoraML dataset with edge-exchangeable sampling.
+
+![](images/fa7d742b8b60c90e26db8b7beb3837dd3c58aa5e97309d7b59787a839be107db.jpg)
+
+Fig. 12 verifies this fir the node-exchangeable sampling. We also evaluated our method compared to standard CP for edge-exchangeable sequences over different initial sampling. The result is in Fig. 13.
+
+# D.5 OTHER EXPERIMENTS
+
+Experiment corresponding to Fig. 1. We performed a node-exchangeable sampling on CiteSeer dataset. At each timestep, we ran the model (GCN trained on the train/val nodes) on the existing subgraph extracting conformal scores for all existing test nodes. The heatmap in Fig. 1 (left) shows the distribution of test scores (sorted) at each timestep. An oracle CP (with access to test nodes and their label) will choose the line labeled as "ground truth" as the conformal threshold since it is the exact  $\alpha$  quantile of existing test scores. This is shown as the ideal reference to evaluate each CP approach with respect to it. We draw the threshold computed by standard CP and NodeEx CP. It is shown that the NodeEx CP picks thresholds close to the ideal line while the naive CP shifts from it due to the message passing with the new nodes.
+
+We drew the distribution of conformity score for calibration nodes at some selected times (sampled with equal distance across from all timesteps) and compared it with calibration scores used by standard CP in Fig. 1(upper right). A distribution shift is clearly observable. We showed this shift in Fig. 1(upper right) by plotting the distributions across various timesteps. The corresponding timestep for each distribution is marked in the left subplot with the same color. Here we computed the EMD (earth mover distance) between the scores of standard CP and conformity scores (of true labels) for calibration and test points. It is shown that the distribution shift is increasing by the number of new nodes introduced to the graph. This shift is almost similar in calibration and test. One source of
+
+![](images/82d838b5e29a35b863a6c4d4443b53bddec51a82c8bac04a68c380652dce5987.jpg)  
+Figure 12: [Left] The empirical coverage of standard CP and NodeEx CP with different initial train/val splits. [Right] The average set size. The result is shown for CoraML dataset and GCN model for node-exchangeable samplings.
+
+![](images/60fe848e2db3259515eed560e44be90f61ea8762ee2c6f4bc362be2d278f8d70.jpg)
+
+![](images/21ca2cccb90d3648e0d1294e82979a7fbd349af1a7fabfb56a794b90a303e701.jpg)  
+Figure 13: [Left] The empirical coverage of standard CP and EdgeEx CP with different initial train/val splits. [Right] The average set size. The result is shown for CoraML dataset and GCN model for edge-exchangeable samplings.
+
+![](images/f0559d8bb0a785f3928ecb90867a74a71cdf6702aa67eaa2aec6b607896572a6.jpg)
+
+the noise in thresholds and EMD is the uniform random value in APS scoring function. EMD is smoothed over 10 steps.
+
+Large Datasets. We ran NodeEx CP and compared it with the baseline for two large datasets: Flickr (Young et al., 2014), Reddit2 (Zeng et al., 2019). Our results for Flicker and Reddit2 datasets under node exchangeable sampling are shown in Fig. 14. Both NodeEx CP and naive CP show similar results. In the edge-exchangeable sampling Fig. 15 shows a significant deviation from the guarantee for NodeEx CP and naive CP, while EdgeEx CP maintains a valid coverage.
+
+Class-conditional coverage. Conformal prediction comes with a marginal guarantee which means that it does not guarantee conditional to group or class. In Fig. 16 we compare this metric. In most of the classes, we observe the standard CP to be closer to the guaranteed line.
+
+Effect of calibration set size. In non-graph data as mentioned in § 2 the coverage probability follows a Beta distribution. In graph data with a fixed number of test nodes (see Theorem 2) the distribution of coverage over test nodes follows a collection of hyper-geometric distributions. In both scenarios, there is a possible variance around predefined  $1 - \alpha$  probability which is a function
+
+![](images/54c2bc5367e1e83819b594c4c927f8fdc844683b2828ea02a6e90e1c08f3e61a.jpg)  
+Figure 14: Coverage Results for [Left] Flickr, and [Right] Reddit2 dataset under node-exchangeable sampling. The results are shown for GCN model.
+
+![](images/a7b9da44daa27c161599d6722cb516eb087717925ab2f398fdd1a80f3a54715e.jpg)
+
+![](images/59df3be3d7700b8c6d5253be1caf3c5eb2ecb3c7cc210e53bb567d258a82a3c8.jpg)  
+Figure 15: Coverage Results for [Left] Flickr, and [Right] Reddit2 dataset under edge-exchangeable sampling. The results are shown for GCN model.
+
+![](images/52b4277543a8707eeb8ada641d453d62109215fe38ae269c325cce3f597fac36.jpg)
+
+of calibration set size. As the number of calibration nodes increases, the distribution of coverage probability concentrates around  $1 - \alpha$ . Since in our experiments we followed a realistic setup (not allowing calibration set to be larger than training set) there is a variance observed around the guarantee line – each line converges to a value close to but not exactly equal to  $1 - \alpha$ . Fig. 17 shows that as we increase the calibration set size the result concentrates around the guarantee.
+
+# E SUPPLEMENTARY DETAILS OF EXPERIMENTS
+
+Datasets. Table 2 provides specifications of datasets used in our experimental evaluations, including the number of nodes, edges, and homophily. The details of label sampling are provided in § 6. For each experiment, we ran each CP approach on 10 different sequences in node-exchangeable and 15 different sequences on edge-exchangeable setup. All transparent lines in plots show the result for one sequence. In those experiments, the solid line shows the average of sequences at each timestep.
+
+Models. For all architectures, we built one hidden layer of 64 units and one output layer. We applied dropout on the hidden layer with probability 0.6 for GCN, and GAT, 0.5 for APPNPNet, and 0.8 for
+
+![](images/73085ce7f8c4ba7ebb698c47517d93728caba261b0d133e7eea3f1a9f850589f.jpg)
+
+![](images/41d94988c4425f0a7c445a8b9fe3ffcc960ea132d68dc84400239571009fe1c5.jpg)
+
+![](images/da1cd97aa036682fd2c2d8f3621e4ffb5793b930c88fb395d30726802d555ada.jpg)  
+Figure 16: NodeEx CP and standard CP in class-conditional coverage. The result is for CiteSeer dataset and GCN model. Plots show classes 1 and 2 [Upper row left to right], 3 and 4 [Lower row].
+
+![](images/79199dd9a45626cca705f6bdfb78f88fc6fb3b5da79d6e14eb337781e80e6de7.jpg)
+
+![](images/64d608cbb8b05f4391d536b2ed43dc21bab38747715fd971f41d1ea67ac4ebbd.jpg)  
+Figure 17: Effect of different calibration set sizes on the concentration of coverage probability. The results are on Cora-ML dataset and for 200 [Left] and 1000 [Right] calibration nodes.
+
+![](images/799d8ac4b6f6cf28ba28a5a8093ba3f6c50622450288e2adc51f723e0c903aa4.jpg)
+
+MLP. For GAT we used 8 heads. We trained all models with categorical cross-entropy loss, and Adam optimizer with  $L_{2}$  regularization.
+
+Adaptive and constant coverage. Despite many studies on CP, Zargarbashi et al. (2023) chooses an adaptive value for  $1 - \alpha$  which is conditional to the model accuracy. This is a suitable choice when the main comparison is based on set size and other efficiency-related metrics. The main supporting idea is that efficiency should be compared with a guarantee that is not trivially achievable by the normal model prediction. However, our main concern is to recover the guarantee meaning that the empirical coverage is the main comparison criteria. Hence we choose 0.9 as the required coverage for all datasets and models regardless of the accuracy. Furthermore, NodeEx CP works for any user-specified coverage guarantee including model-conditional values.
+
+Table 2: Statistics of the datasets.  
+
+<table><tr><td>Dataset Name</td><td>Vertices</td><td>Attributes</td><td>Edges</td><td>Classes</td><td>Homophily</td></tr><tr><td>CoraML</td><td>2995</td><td>2879</td><td>16316</td><td>7</td><td>78.85%</td></tr><tr><td>PubMed</td><td>19717</td><td>500</td><td>88648</td><td>3</td><td>80.23%</td></tr><tr><td>CiteSeer</td><td>4230</td><td>602</td><td>10674</td><td>6</td><td>94.94%</td></tr><tr><td>Coaut. CS</td><td>18333</td><td>6805</td><td>163788</td><td>15</td><td>80.80%</td></tr><tr><td>Coauth. Physics</td><td>34493</td><td>8415</td><td>495924</td><td>5</td><td>93.14%</td></tr><tr><td>Amz. Comp.</td><td>13752</td><td>767</td><td>491722</td><td>10</td><td>77.72%</td></tr><tr><td>Amz. Photo</td><td>7650</td><td>745</td><td>238162</td><td>8</td><td>82.72%</td></tr></table>
+
+# F RELATED WORKS
+
+Standard conformal prediction. CP is introduced by Vovk et al. (2005) and further developed in Lei & Wasserman (2014); Shafer & Vovk (2008); Barber (2020). Different variants of CP are yet defined ranging in the trade-off between statistical and computational efficiency. While full conformal prediction requires multiple training rounds for each single test point, Jackknife+ Barber et al. (2019a), and split conformal prediction sacrifice statistical efficiency, defining faster and hence more scalable CP algorithms. A comprehensive survey of CP can be found in Angelopoulos & Bates (2021a). In this study, we focus on split conformal prediction.
+
+Further contributions in this area include generalization of the guarantee from including the true label to any risk function Angelopoulos et al. (2024), improving the efficiency (reducing the average set size) by simulating calibration during training Stutz et al. (2022), limiting the false positive rate Fisch et al. (2022), etc.
+
+CP without exchangeability. Standard CP requires exchangeability for datapoints and assumes the model to treat datapoints symmetrically. The latter assumption ensures the exchangeability of datapoints even after observing the fitted model. Tibshirani et al. (2019) extend CP to cases where exchangeability breaks via different  $p(X)$  for calibration set and the test data, given  $p(Y \mid X)$  is still the same. In this case, CP is adapted via reweighing the calibration set using the likelihood ratio to compare the training and test covariate distributions. This requires the high-dimensional likelihood to be known or well-approximated. Barber et al. (2023) does neither rely on known likelihood between the original and shifted distribution nor the symmetry in the learning algorithm and proposes a coverage gap based on the total variation distance between calibration points and the test point. The main upperbound on the coverage requires a predefined fixed weight function. To adapt this bound to data-dependent weights, the  $\mathrm{d}_{\mathrm{TV}}$ -distance must be computed conditional to the assigned weights.
+
+CP for graphs. Recently adapting CP to graphs got increasing attention. Wijegunawardana et al. (2020) adapted conformal prediction for node classification to achieve bounded error. Clarkson (2023) assumed exchangeability violated for node classification hence adapting weighted exchangeability without any lowerbound on the coverage, however, Zargarbashi et al. (2023) and Huang et al. (2023) proved the applicability of standard CP for transductive setting in GNNs. Moreover, Huang et al. (2023) proposed a secondary GNN trained to increase the efficiency of prediction sets, and Zargarbashi et al. (2023) stated that in homophily networks, diffusion of conformal score can increase the efficiency significantly.
+
+For the inductive GNNs, the only work we are aware of is the neighborhood APS (NAPS) approach which is an adaptation of weighted CP Clarkson (2023), however, it is shown that NAPS can not be applied on a significant fraction of nodes if the network is sparse or the quantity of calibration nodes are limited Zargarbashi et al. (2023) (this number increases to over  $70\%$  in some benchmark datasets). As our approach is adaptable to sparse networks and works for any calibration set size, we do not compare our approach with NAPS.

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+# CONFORMAL PREDICTION VIA REGRESSION-AS-CLASSIFICATION
+
+# Etash Guha
+
+RIKEN Center for AI Project, SambaNova Systems {etash.guha}@sambanovasystems.com
+
+# Shlok Natarajan
+
+Salesforce {shloknatarajan}@salesforce.com
+
+# Thomas Møllenhoff
+
+RIKEN Center for AI Project {thomas.moellenhoff}@riken.jp
+
+# Mohammad Emtiyaz Khan
+
+RIKEN Center for AI Project {emtiyaz.khan}@riken.jp
+
+# Eugene Ndiaye
+
+Apple {e_ndiaye}  $@$  apple.com
+
+# ABSTRACT
+
+Conformal prediction (CP) for regression can be challenging, especially when the output distribution is heteroscedastic, multimodal, or skewed. Some of the issues can be addressed by estimating a distribution over the output, but in reality, such approaches can be sensitive to estimation error and yield unstable intervals. Here, we circumvent the challenges by converting regression to a classification problem and then use CP for classification to obtain CP sets for regression. To preserve the ordering of the continuous-output space, we design a new loss function and make necessary modifications to the CP classification techniques. Empirical results on many benchmarks show that this simple approach gives surprisingly good results on many practical problems.
+
+# 1 INTRODUCTION
+
+Quantifying and estimating the uncertainty of machine-learning models is an important task for many problems, especially mission-critical applications where reliable predictions are required. Conformal Prediction (CP) (Vovk et al., 2005) has recently gained popularity and has been used successfully in applications such as breast cancer detection (Lambrou et al., 2009), stroke risk prediction (Lambrou et al., 2010), and drug discovery (Cortés-Ciriano & Bender, 2020). Under mild conditions, CP techniques aim to construct a prediction set that, for given test inputs, is guaranteed to contain the true (unknown) output with high probability. The set is built using a conformity score, which, roughly speaking, indicates the similarity between a new test example and the training examples. The conformal set merely gathers examples that have large conformity scores. Despite its popularity, CP for regression can be challenging, especially when the output distribution is heteroscedastic, multimodal, or skewed (Lei & Wasserman, 2014). The main challenge lies in the design of the conformity score. It is common to use a simple choice for score functions such as distance to mean regressor, but such choices may ignore the subtle features of the shape of the output distribution. For instance, this could lead to symmetric intervals or ignoring the heteroscedasticity. In theory, it is better to estimate the (conditional) distribution over the output, for example, by using kernel density estimation and directly using it to build a confidence interval. However, such estimation approaches are also challenging, and estimates can be sensitive to the choice of kernel and hyperparameters, which can yield unstable results.
+
+We circumvent the challenges by exploiting the existing CP techniques for classification. We proceed by first converting regression to a classification problem and then using CP techniques for classification to obtain a conformal set. Regression-as-classification approaches are popular for various applications in computer vision and have led to more accurate training than only-regression training (Stewart et al., 2023; Zhang et al., 2016; Fu et al., 2018; Rothe et al., 2015; Van Den Oord
+
+![](images/d0bce1e14c951493c1b13d47e0380a2028efeffb4a3a546e562aa1102c131aa4.jpg)  
+(a) Heteroskedastic
+
+![](images/5d183140cfa8d5ee26e0108a709344989abe9258274f3065a00ee6106824af5b.jpg)  
+(b) Bimodal  
+Figure 1: We show two examples where the output distribution is heteroskedastic (left) and bimodal (right). In both cases, our method is able to change the interval (shaded gray region) adaptively as the input values  $x$  are increased. Examples outside the gray regions (white dots) are deemed different from those inside it (black dots).
+
+et al., 2016; Diaz & Marathe, 2019). We leverage them to construct a distribution-based conformal set that can flexibly capture the shape of the output distribution while preserving the simplicity and efficiency of CP for classification. First, we discretize the output space into bins, treating each bin as a distinct class. Second, to preserve the ordering of the continuous output space, we design an alternative loss function that penalizes the density on bins far from the true output value but also facilitates variability by using an entropy regularization. The loss design is similar in spirit to Weigend & Srivastava (1995); Diaz & Marathe (2019). The resulting method can adapt to heteroscedasticity, bimodality, or both in the label distribution. We verify this on synthetic and real datasets where we achieve the shortest intervals compared to other CP baselines. See examples in Figure 1.
+
+# 2 BACKGROUND ON CONFORMAL PREDICTION
+
+Given a new input  $x_{\mathrm{new}}$ , CP techniques aim to construct a set that contains the true but unknown output  $y_{\mathrm{new}}$  with high probability. Assuming that a pair of input-output variables  $(x, y)$  has a joint density  $p(x, y)$  and a conditional density  $p(y \mid x)$ , oracle prediction sets (with joint and conditional coverage) for the output  $y$  can be constructed as
+
+$$
+\{z \in \mathbb {R}: p (x, z) \geq \tau_ {\alpha} \} \text {o r} \{z \in \mathbb {R}: p (z \mid x) \geq \tau_ {\alpha , x} \}, \tag {1}
+$$
+
+where the thresholds  $\tau_{\alpha}$  and  $\tau_{\alpha, x}$  are selected to ensure that the corresponding sets have a probability mass that meets or exceeds prescribed confidence level  $1 - \alpha \in (0, 1)$ . As the ground-truth distribution is unknown, we rely solely on estimating these uncertainty sets using the density estimators  $\hat{p}(x, y)$  and  $\hat{p}(y \mid x)$ . The latter can be inaccurate due to numerous sources of errors such as model misspecification, small sample size, high optimization errors during training, and overfitting. Without a stronger distribution assumption, the finite-sample guarantee is typically not upheld.
+
+Conformal Prediction has arisen as a method for yielding sets that do hold finite-sample guarantees. Given a partially observed instance  $(x_{\mathrm{new}},y_{\mathrm{new}})$  where  $y_{\mathrm{new}}$  is unknown, Conformal Prediction (CP) (Vovk et al., 2005) constructs a set of values that contains  $y_{\mathrm{new}}$  with high probability without knowing the underlying data distribution. Under conformal prediction, this property is guaranteed under the mild assumption that the data satisfies exchangeability. The set is called the conformal set and is built using a conformity score, denoted by  $\sigma (x,y)$ , which measures how appropriate an output value is for a given input example. There are many ways to build the conformity score, but they all involve splitting the data into a training set  $\mathcal{D}_{\mathrm{tr}}$  and a calibration set  $\mathcal{D}_{\mathrm{cal}}$ . Often, a prediction model  $\mu_{\mathrm{tr}}(x)$  is built using the training set, and then a conformity score is obtained using this model along with the calibration set (we will shortly give an example). The conformal set merely gathers the points with larger conformity scores:
+
+$$
+\{z \in \mathbb {R}: \sigma (x _ {\mathrm {n e w}}, z) \geq Q _ {1 - \alpha} (\mathcal {D} _ {\mathrm {c a l}}) \},
+$$
+
+where  $Q_{1 - \alpha}(\mathcal{D}_{\mathrm{cal}})$  is the  $(1 - \alpha)$ -quantile of the conformity scores on the calibration data. This set provably contains  $y_{\mathrm{new}}$  with probability larger than  $1 - \alpha$  for any finite sample size and without assumption on the ground-truth distribution.
+
+There are many design choices for this conformity score. For example, one can choose a prediction model  $\mu_{\mathrm{tr}}(x)$  as an estimate of the conditional expectation and measure conformity as the absolute value of the residual, i.e.,  $\sigma (x,y) = -|y - \mu_{\mathrm{tr}}(x)|$ . The corresponding conformal set is a single interval centered around the prediction  $\mu_{\mathrm{tr}}(x)$  and of constant length  $Q_{1 - \alpha}(\mathcal{D}_{\mathrm{cal}})$  for any example  $x_{\mathrm{new}}$ , without taking into account its variability. However, in situations where the underlying data distribution demonstrates skewness or heteroscedasticity, we may desire a more flexible conformity score. We wish to generate an approach to approximate this density function that is versatile in the distributions it can represent and avoids the difficulties seen in prior methods. We explore established density estimation in Classification Conformal Prediction that are already performing effectively.
+
+Related Works Since their introduction, a lot of work has been done to improve the set of conformal predictions. As simple score function, distance to conditional mean is  $\sigma(x,y) = |y - \mu(x)|$  where  $\mu(x)$  is an estimate of  $\mathbb{E}(y \mid x)$  was prominently used (Papadopoulos et al., 2002; Lei et al., 2018; Guha et al., 2023). Instead, (Romano et al., 2019) suggests estimating a conditional quantile instead and a conformity score function based on the distance from a trained quantile regressor, i.e.  $\sigma(x,y) = \max(\mu_{\alpha/2}(x) - y, y - \mu_{1 - \alpha/2}(x))$  where  $\mu_{\alpha}$  are the  $\alpha$ -th quantile regressors. Within the literature, our strategy is more closely related to the distribution-based methods (Lei et al., 2013). Following a similar line of work (Chernozhukov et al., 2021) argued that the conformal quantile regression score function might be less adaptive since the distance of the quantile behaves similarly to the distance of the mean estimate. Instead, they suggested estimating the cumulative (conditional) distribution function and directly outputting  $\{y : F(x,y) \in [\alpha/2,1 - \alpha/2]\}$ . However, their method cannot account for bimodality since it can only output a single interval by design. An equally interesting approach for distribution-based prediction sets is based on learning a Bayesian estimator, which, however, may require a well-specified prior and can be computationally expensive (Fong & Holmes, 2021). Variants based on estimating the conditional density of the response using histogram regressors (Sesia & Romano, 2021) could detect a possible asymmetry of the ground-truth distribution. However, our densities are estimated through classification techniques, whereas their densities are learned through a histogram of many regressors. Smoothness over the distribution is encoded in our loss function, whereas they use a post hoc subinterval finding algorithm through linear programming to prevent many disjoint intervals. Our techniques have some overlap with ordinal regression and ordinal classification. Xu et al. (2023) discusses various risk categories (similar to coverage) for ordinal classification, while our work considers different score functions where coverage serves as the loss function. Lu et al. (2022) discuss the adaptation of APS style score functions to accommodate the ordinal structure of classes, which was subsequently utilised in (Sesia & Romano, 2021).
+
+# 3 CP VIA REGRESSION-AS-CLASSIFICATION
+
+# 3.1 CLASSIFICATION CONFORMAL PREDICTION
+
+We aim to compute a conformity function that accurately predicts the appropriateness of a label for a specific data point. Given that the distribution function across labels can adopt diverse forms, such as being bimodal, heavy-tailed, or heteroscedastic, our approach must effectively factor in such shapes while upholding its coverage precision. A frequently employed technique involves using the conditional label density as a conformity function, leading to reliable results in the classification context. Typically, practitioners perform conformal prediction for classification with probability estimates from a Softmax neural network that covers  $K$  output logits using cross-entropy loss. Let us denote the parametrized density
+
+$$
+q _ {\theta} (\cdot \mid x) = \operatorname {s o f t m a x} (f _ {\theta} (x)), \text {w h e r e} \operatorname {s o f t m a x} (v) _ {j} = \frac {\exp (v _ {j})}{\sum_ {k = 1} ^ {K} \exp (v _ {k})},
+$$
+
+as the outputted discrete probability distribution over the labels of the input  $x$ . Traditionally, we fit our neural network by minimizing the cross-entropy loss on the training set:
+
+$$
+\hat {\theta} \in \underset {\theta} {\arg \min} \sum_ {i = 1} ^ {n} \mathrm {K L} (\delta_ {y _ {i}} | | q _ {\theta} (\cdot | x _ {i})).
+$$
+
+Here,  $\delta_{y_i}$  is the Dirac Distribution with all of the probabilistic mass on  $y_i$ . Let us assume that we have trained and acquired a  $\hat{\theta}$  that has minimized the traditional cross-entropy loss function on the training dataset. A natural conformity score is simply the probability of a label according to the learned conditional distribution, i.e.,  $\sigma(x,y) = q_{\hat{\theta}}(y \mid x)$ . This approach is both straightforward and efficient. The neural network's flexibility allows it to learn numerous label distributions with adaptivity across examples with less explicit design of specific prior structure.
+
+# 3.2 REGRESSION TO CLASSIFICATION APPROACH
+
+Naturally, we strive to embody such practicality and effectiveness in regression settings. However, the distribution of labels in the regression scenario is continuous, and learning a continuous distribution directly using a neural network is challenging (Rothfuss et al., 2019). Often, Bayesian or kernel density estimators are employed to estimate this distribution. Other techniques acquire knowledge of this distribution by training numerous regressors and categorizing the regressors, for example, Conformal Histogram Regression (Sesia & Romano, 2021).
+
+However, these methods look different from the conformal prediction approaches for classification. It would be desirable to be able to use similar methods for both classification and regression conformal prediction. One method of unifying classification and regression problems outside the conformal prediction literature is known as Regression-as-Classification. We simply turn a regression problem into a classification problem by binning the range space. Specifically, we generate  $K$  bins with  $K$  equally spaced numbers covering the interval  $\mathcal{V} = [y_{\mathrm{min}}, y_{\mathrm{max}}]$ , where  $y_{\mathrm{min}}$  (or  $y_{\mathrm{max}}$ ) is the minimum (or maximum) value of the labels observed in the training set. More explicitly, we define our discretization of the label space as
+
+$$
+\hat {\mathcal {Y}} = \left\{\hat {y} _ {1}, \dots , \hat {y} _ {K} \right\} \text {w h e r e} \hat {y} _ {k + 1} = \hat {y} _ {k} + \frac {\hat {y} _ {K} - \hat {y} _ {1}}{K - 1} \text {w i t h} \hat {y} _ {1} = y _ {\min } \text {a n d} \hat {y} _ {K} = y _ {\max }.
+$$
+
+These values  $\hat{y} \in \hat{\mathcal{V}}$  form the midpoints for each bin of our discretization. Naturally, the  $k$ th bin is all the labels in the range space  $\mathcal{V}$  closest to  $\hat{y}_k$ . Intuitively, we can treat each bin as a class. Thus, we have turned a regression problem into a classification problem. This method is simple but has yielded surprising results. Some work has suggested that this form of binning results in more stable training (Stewart et al., 2023) and gives significantly better results for learning conditional expectations. To unify classification and regression conformal prediction, a simple solution is to employ the Classification Conformal Prediction model with discrete labels  $\tilde{y}_i = \arg \min_{\hat{y} \in \hat{\mathcal{V}}} |y_i - \hat{y}|$ . This will aid in training the neural network with modified labels through cross-entropy loss, resulting in a discrete distribution of  $q_\theta(\cdot \mid x)$ , as outlined in the previous section.
+
+To compute conformity scores for all labels, we employ linear interpolation from the discrete probability function  $q_{\theta}(\cdot \mid x)$  to generate the continuous distribution  $\bar{q}_{\theta}(\cdot \mid x)$ . Nevertheless, this approach encounters a critical issue when employed for regression.
+
+# 3.3 DATA FITTING
+
+A critical problem with employing CrossEntropy loss in the classification conformal prediction context is that any structural relationships between classes are disregarded. This is not surprising given that in the classification context, no structure exists between classes, and each class is independent. Therefore, the CrossEntropy loss does not need to differentiate between whether  $q_{\theta}$  allocates probable mass far away or close to the actual label. Instead, it only incentivizes the allocation of probable mass on the correct label. However, within the regression setting, despite a multitude of labels, the labels adhere to an ordinal structure. Thus, to enhance the accuracy of labeling, it is imperative to devise a loss function that incentivizes the allocation of probabilistic mass not only to the correct bin but also to the neighboring bins. Formally, given an input and output pair  $(x,y)$ , our goal is to determine a density estimate  $q_{\theta}(\cdot \mid x)$  that assigns low (resp. high) probability to points that are far (resp. close) to the true label  $y$ , i.e.,
+
+$$
+q _ {\theta} (\hat {y} \mid x _ {i}) \text {h i g h (r e s p . s m a l l) w h e n t h e l o s s} \ell (\hat {y}, y _ {i}) \text {i s s m a l l (r e s p . h i g h)}.
+$$
+
+Hence, a natural desideratum for learning the probability density function  $q_{\theta}$  is that their product  $\ell(y, \hat{y})q(\hat{y} \mid x)$  is small in expectation. We propose to find a distribution  $q_{\theta}$  minimizing the loss
+
+$$
+\mathbb {E} _ {\hat {y} \sim q (\cdot | x)} [ \ell (y, \hat {y}) ] = \sum_ {k = 1} ^ {K} \ell (y, \hat {y} _ {k}) q (\hat {y} _ {k} \mid x)
+$$
+
+Minimizing this loss in the space of all possible distributions  $q(\cdot \mid x)$  is equivalent to minimizing the original loss  $\ell(y, \cdot)$ , where the minimizing distribution is a Dirac delta  $\delta_{\hat{y}_{\star}}$  at the minimizer  $\hat{y}_{\star}$  of  $\ell(y, \cdot)$ . Therefore, we expect an unregularized version of this loss to share similarities with the typical empirical risk minimization on  $\ell(y, \cdot)$ . However, a key difference is that when minimizing in a restricted family of distributions (for example, those representable by a neural network with a fixed architecture), the distributional output can represent multi-modal or heavy-tailed label distributions. Minimizing the original loss  $\ell(y, \cdot)$ , one would always be confined to a point estimate.
+
+# 3.4 ENTROPY REGULARIZATION
+
+Although the proposed loss function better encodes the connection between bins, it tends towards outputting Dirac distributions, as outlined in the previous paragraph. Overconfidence in our neural network is a commonly reported problem in the literature (Wei et al., 2022). Nevertheless, smoothness has been a traditional requirement in density learning. As such, we rely on a classical entropy regularization technique for learning density estimators (Wainwright & Jordan, 2008). Given the set of density estimators that match the training label distribution well and put high probability mass on the best bins, we prefer the probability distribution that maximizes the entropy since this intuitively takes fewer assumptions on the data distribution structure. Our choice is based on selecting density estimators that effectively match the distribution of the training labels and assign a higher probability to the best bins. Formally, we can calculate the entropy of our probability distribution by using the Shannon entropy  $\mathcal{H}$  of the produced probability distribution  $q(\cdot |x)$  as a penalty term as follows:
+
+$$
+\mathcal {H} \left(q _ {\theta} (\cdot | x)\right) = \sum_ {k = 1} ^ {K} q _ {\theta} \left(\hat {y} _ {k} | x\right) \log q _ {\theta} \left(\hat {y} _ {k} | x\right).
+$$
+
+Summary We learn a distribution by minimizing the following expected loss over a given training data  $\mathcal{D}_{\mathrm{tr}} = \{(x_1,y_1),\ldots ,(x_{n_{\mathrm{tr}}},y_{n_{\mathrm{tr}}})\}$  of size  $n_{\mathrm{tr}}$
+
+$$
+\mathcal {L} (\theta) = \sum_ {i = 1} ^ {n _ {\mathrm {t r}}} \sum_ {k = 1} ^ {K} \ell \left(y _ {i}, \hat {y} _ {k}\right) q _ {\theta} \left(\hat {y} _ {k} \mid x _ {i}\right) - \tau \mathcal {H} \left(q _ {\theta} (\cdot \mid x _ {i})\right), \tag {2}
+$$
+
+In particular, we choose  $\ell(y_i, \hat{y}_k)$  as  $\ell(y_i, \hat{y}_k) = |y_i - \hat{y}_k|^p$  where  $p > 0$  is a hyperparameter. This selection functions as a natural distance metric that meets all required objectives. As a result, we can employ a technique similar to the aforementioned Classification Conformal Prediction methods. Initially, we train a Softmax neural network  $f_\theta$  with  $K$  logits, grounded on the loss function Equation (2) on the training dataset. To calculate conformity scores for the calibration set, we utilize the linearly interpolated  $\sigma(x,y) = \bar{q}_\theta(y|x)$  for a specific data point  $(x,y)$ . Namely, for any  $y$  between  $\hat{y}_k$  and  $\hat{y}_{k+1}$ ,  $\bar{q}_\theta(y|x)$  is defined as
+
+$$
+\bar {q} _ {\theta} (y \mid x) = \gamma_ {k} q _ {\theta} (\hat {y} _ {k} \mid x) + (1 - \gamma_ {k}) q _ {\theta} (\hat {y} _ {k + 1} \mid x)
+$$
+
+$$
+\text {w h e r e} \gamma_ {k} = \frac {\hat {y} _ {k + 1} - y}{\hat {y} _ {k + 1} - \hat {y} _ {k}}.
+$$
+
+Complete details of this procedure are outlined in Algorithm 1.
+
+# 4 EXPERIMENTS
+
+All code to run our method can be installed via
+
+$$
+p i p \quad \text {i n s t a l l} \quad r 2 c c p.
+$$
+
+We investigate the empirical behavior of our R2CCP (Regression-to-Classification Conformal Prediction) method, which we have explained in detail in Algorithm 1. We have three sets of experiments. The first one is described in Section 4.1 and presents empirical evidence of the algorithm's ability to produce narrower intervals by utilizing label density characteristics, including
+
+Algorithm 1 Regression to Classification Conformal Prediction (R2CCP).
+
+1: Input:
+
+- Dataset  $\mathcal{D}_n = \{(x_1, y_1), \ldots, (x_n, y_n)\}$  and new input  $x_{n+1}$  
+- Desired confidence level  $\alpha \in (0,1)$
+
+2: Hyperparameters: temperature  $\tau > 0$ ,  $p > 0$ , number of bins  $K > 1$  
+3: Discretize the output space  $[y_{\min}, y_{\max}]$  into  $K$  equidistant bins with midpoints  $\{\hat{y}_1, \dots, \hat{y}_K\}$  
+4: Randomly split the dataset  $\mathcal{D}_n$  in training  $\mathcal{D}_{\mathrm{tr}}$  and calibration  $\mathcal{D}_{\mathrm{cal}}$  
+5: Find a distribution  $q_{\hat{\theta}}(\cdot \mid x)$  by (approximately) optimizing on the training set  $\mathcal{D}_{\mathrm{tr}}$
+
+$$
+\hat {\theta} \in \arg \min  _ {\theta \in \mathbb {R} ^ {d}} \sum_ {i = 1} ^ {n _ {\mathrm {t r}}} \sum_ {k = 1} ^ {K} | y _ {i} - \hat {y} _ {k} | ^ {p} q _ {\theta} (\hat {y} _ {k} \mid x _ {i}) - \tau \mathcal {H} (q _ {\theta} (\cdot \mid x _ {i}))
+$$
+
+where  $q_{\theta}(\hat{y}_k|x) = \mathrm{softmax}(f_{\theta}(x))_k$  for a model (e.g., neural net)  $f_{\theta}:\mathbb{R}^{d}\to \mathbb{R}^{K}$ .
+
+6:  $S \gets \left\{\bar{q}_{\hat{\theta}}(y \mid x) \text{ for } (x, y) \in \mathcal{D}_{\mathrm{cal}}\right\} \# \bar{q}_{\hat{\theta}}(\cdot \mid x)$  is linear interpolation of softmax probabilities.
+
+7:  $Q_{\alpha}(\mathcal{D}_{\mathrm{cal}})\gets$  quantile  $(S,\alpha)$  
+8: return Conformal Set  $\Gamma^{(\alpha)}(x_{n + 1}) = \{z\in \mathbb{R}\mid \bar{q}_{\hat{\theta}}(z\mid x_{n + 1})\geq Q_{\alpha}(\mathcal{D}_{\mathrm{cal}})\}$
+
+heteroscedasticity, bimodality, or a combination. Section 4.2 demonstrates the effectiveness of our algorithm on synthetic and real data by comparing it with various benchmarks from the Conformal Prediction literature in terms of length and coverage. Section 4.3 evaluates the effect of different loss functions on the final learned densities and their impact on the intervals produced. All experiments were run over 5 different seeds at a coverage level of  $90\%$ , and the standard error over the experiments is reported in the subscript. We do not tune the hyperparameters and keep values of  $K = 50$ ,  $p = 0.5$ , and  $\tau = 0.2$  constant across all experiments. For all experiments, we report length, meaning the length of all the sets predicted, and coverage, the percent of instances where the true label is contained in the predicted intervals.
+
+# 4.1 SPECIFIC CHARACTERISTICS OF LABEL DISTRIBUTION
+
+Heteroscedasticity We generate a toy dataset where the input is one-dimensional. It contains samples from the following distribution:  $y \sim \mathcal{N}(0, |x|)$ . The label distribution is heteroscedastic, meaning the variance of the labels changes as the input changes. In traditional Conformal Prediction literature, many existing algorithms fail to capture heteroscedasticity, resulting in wide intervals for inputs  $x$  where the label distribution of  $y$  has low variance. However, our learned algorithm can directly learn this relation and adjust the outputted probability distribution accordingly. Thus, we see that the lengths of the intervals will increase as the variance of the label distribution increases, which is desirable. We see this relation in Figure 1a. Moreover, we also use the dataset generated from (Lei & Wasserman, 2014) as discussed in Appendix D. This dataset exhibits heteroskedasticity and bimodality as  $X$  passes  $-0.5$ . We see that our learned method can adjust intervals accordingly to maintain coverage and length for all  $X$ . We plot this in Figure 1b. We plot how the intervals (grey) change as the data distribution (black) changes. As the variance of the labels increases as  $x$  increases, the produced intervals adaptively get wider, taking advantage of the heteroscedasticity.
+
+Bimodality We showcase our algorithm's capability to address labels with a bimodal distribution. Our bimodal dataset is generated by repeatedly (1) sampling two sets of random features that are close geometrically and (2) giving one set of features a label of 1 plus some Gaussian noise and giving the other a label of  $-1$  plus some Gaussian noise. Therefore, our dataset is comprised of many similar data points with bimodally distributed labels. This bimodal distribution is particularly hard for many existing CP algorithms to solve since it requires outputting two disjoint conformal sets to achieve low length. CQR, for instance, cannot deal with this circumstance and will generate a conformal set that covers the entire range space. However, our method is flexible enough to assign a low probability to labels between  $-1$  and  $1$ , and our resulting conformal set will not include these intermediate labels. We see this in Figure 6d. In Figure 6d, our outputted probability distribution has two modes around labels  $-1$  and  $1$  and assigns a low probability value to the valley between the modes.
+
+# 4.2 COMPARISON TO OTHER CONFORMAL PREDICTION ALGORITHMS
+
+The crucial criteria for assessing a Conformal Prediction algorithm consist of (1) coverage, representing the percentage of generated conformal sets that incorporate accurate labels, and (2) the length of the generated conformal sets. Our baseline techniques consist of the Kernel Density Estimator (KDE) as proposed by Lei & Wasserman (2014), alongside a conformity score shown by the estimated probability density. Furthermore, we take into consideration Fong & Holmes (2021) (CB), which employs the likelihood of a posterior distribution in their conformity function, and Sesia & Romano (2021) (CHR), which uses quantile regressors on every bin of a histogram density estimator. We have included the Conformal Quantile Regression as described by Romano et al. (2019) (CQR), which employs conformity based on the labels' distance from quantile regressors. Moreover, we had the Distributional Conformal Prediction which uses a Cumulative Distribution Function to form its intervals from Chernozhukov et al. (2021) (DCP). Additionally, we have used the Lasso Conformal Predictor with a distance to mean regressors, which is the most straightforward option (LASSO). We use synthetic data exhibiting bimodally distributed and log-normally distributed to illustrate particular weaknesses of existing methods. We use real datasets also in Romano et al. (2019). Specifically, these are several datasets from the UCI Machine Learning repository (Bio, Blog, Concrete, Community, Energy, Forest, Stock, Cancer, Solar, Parkinsons, Pendulum) (Nottingham et al., 2023) and the medical expenditure panel survey number 19-21 (MEPS-19-21) (Cohen et al., 2009). These regression datasets are commonly used to benchmark regression models. Our approach yields tighter intervals on real datasets than some of the strongest baselines.
+
+Results We report lengths and coverages results in Table 1 respectively. We added figures depicting example probability distributions on these datasets in Figure 6 in Appendix H. The intervals produced by our method are the shortest over 10 of the 16 datasets. From Figure 6, we see that our method can learn many different shaped distributions well, which accounts for this significant improvement in intervals. Overall, the Kernel Density Estimation and our method can accurately predict the best intervals on datasets where the connection between the data and feature is simple, such as Bimodal or Log-Normal. While the Kernel Density Estimator can fail when the labels are complexly related to inputs, no such connection exists in these datasets since the labels are independent of the features. Thus, the Kernel Density Method's simplicity allows it to learn the label density quickly. Our method also seems to handle the case where there is no connection between the data and the labels, as seen in Figure 6d in Appendix H. Moreover, on datasets where the label density is smooth and close to the Laplace prior, such as on Figure 6c and Figure 6i, Conformal Bayes and our method can accurately learn this distribution since both methods can output smooth distributions. Moreover, on very sharp datasets such as on Concrete in Figure 6j and Energy in Figure 6h, both our method and CQR can capture the sharp and unnoisy distribution needed to achieve strong length. Moreover, on complex distributions such as in Pendulum in Figure 6e and Bio in Figure 6f, we see that both CHR and our method have the flexibility to portray complex distributions resulting in the most accurate intervals. Thus, the flexibility of our algorithm to smoothly learn sharp, wide, complex, and simple conditional label densities results in our method achieving the best length most consistently over the entire dataset.
+
+# 4.3 ABLATION STUDIES
+
+Our loss function consists of an error and entropy terms. The error term penalizes distributions that put weight far away from the true label, whereas the entropy term acts as a regularizing term in the probability distribution space. We will do ablations on both the error and entropy terms to illustrate the importance of each part. For the error term, there are several notable alternatives that a practitioner may use. An alternative is the log maximum likelihood or cross-entropy formulations of the error term, which we denote as
+
+$$
+\mathcal {L} _ {\mathrm {M L E}} (\theta) = - \sum_ {i = 1} ^ {N} \log \left(q _ {\theta} \left(\tilde {y} _ {i} \mid x _ {i}\right)\right),
+$$
+
+where we remind the reader that  $\tilde{y}_i = \arg \min_{\hat{y} \in \hat{Y}} |\hat{y} - y_i|$ . We note that both MLE and CE are equivalent formulations in this setting. These two are standard error terms often used in practice. We will train our models with this loss function over all datasets to see how the change of error term affects the intervals' length and the learned density. We will show that our chosen error term is better
+
+<table><tr><td>DATASET</td><td>BIMODAL</td><td>LOGNORM</td><td>CONCRETE</td><td>MEPS-19</td><td>MEPS-20</td><td>MEPS-21</td><td>Bio</td><td>COMMUNITY</td></tr><tr><td>CQR</td><td>2.14(0.01)</td><td>1.58(0.03)</td><td>0.39(0.01)</td><td>1.87(0.05)</td><td>2.00(0.07)</td><td>1.99(0.03)</td><td>1.34(0.01)</td><td>1.44(0.03)</td></tr><tr><td>KDE</td><td>0.35(0.01)</td><td>1.40(0.04)</td><td>1.54(0.03)</td><td>2.16(0.02)</td><td>2.51(0.05)</td><td>2.39(0.03)</td><td>2.27(0.00)</td><td>2.23(0.09)</td></tr><tr><td>LASSO</td><td>2.14(0.00)</td><td>3.30(0.06)</td><td>2.74(0.03)</td><td>4.64(0.06)</td><td>4.79(0.04)</td><td>4.92(0.04)</td><td>3.89(0.00)</td><td>3.26(0.05)</td></tr><tr><td>CB</td><td>2.16(0.00)</td><td>1.45(0.01)</td><td>0.96(0.01)</td><td>4.47(0.02)</td><td>4.50(0.02)</td><td>4.51(0.01)</td><td>2.09(0.00)</td><td>1.80(0.00)</td></tr><tr><td>CHR</td><td>2.14(0.00)</td><td>1.52(0.05)</td><td>0.47(0.02)</td><td>2.60(0.08)</td><td>2.53(0.03)</td><td>2.75(0.03)</td><td>1.59(0.01)</td><td>1.49(0.05)</td></tr><tr><td>DCP</td><td>2.14(0.01)</td><td>1.74(0.07)</td><td>0.47(0.01)</td><td>68.64(0.15)</td><td>66.71(0.40)</td><td>67.56(0.33)</td><td>1.74(0.01)</td><td>1.59(0.03)</td></tr><tr><td>R2CCP (OURS)</td><td>0.46(0.01)</td><td>1.96(0.03)</td><td>0.38(0.01)</td><td>1.60(0.01)</td><td>1.70(0.03)</td><td>1.72(0.03)</td><td>1.11(0.01)</td><td>1.47(0.03)</td></tr></table>
+
+<table><tr><td>DATASET</td><td>DIABETES</td><td>SOLAR</td><td>PARKINSONS</td><td>STOCK</td><td>CANCER</td><td>PENDULUM</td><td>ENERGY</td><td>FOREST</td></tr><tr><td>CQR</td><td>1.30(0.05)</td><td>1.98(0.22)</td><td>0.42(0.01)</td><td>1.85(0.23)</td><td>3.09(0.13)</td><td>2.25(0.31)</td><td>0.19(0.01)</td><td>3.18(0.19)</td></tr><tr><td>KDE</td><td>1.34(0.05)</td><td>0.50(0.01)</td><td>3.79(0.02)</td><td>4.72(0.29)</td><td>3.82(0.09)</td><td>3.96(0.09)</td><td>2.72(0.07)</td><td>2.90(0.17)</td></tr><tr><td>LASSO</td><td>3.01(0.04)</td><td>3.54(0.12)</td><td>3.46(0.03)</td><td>1.39(0.04)</td><td>3.55(0.14)</td><td>3.99(0.07)</td><td>1.29(0.03)</td><td>3.97(0.29)</td></tr><tr><td>CB</td><td>1.19(0.01)</td><td>3.78(0.02)</td><td>3.42(0.01)</td><td>1.32(0.01)</td><td>3.14(0.04)</td><td>3.71(0.03)</td><td>1.26(0.01)</td><td>3.75(0.03)</td></tr><tr><td>CHR</td><td>1.40(0.02)</td><td>1.49(0.23)</td><td>0.68(0.02)</td><td>1.59(0.07)</td><td>3.42(0.12)</td><td>1.69(0.11)</td><td>0.23(0.01)</td><td>3.03(0.15)</td></tr><tr><td>DCP</td><td>1.29(0.02)</td><td>15.69(0.06)</td><td>0.83(0.04)</td><td>1.69(0.10)</td><td>3.57(0.07)</td><td>1.76(0.10)</td><td>0.23(0.01)</td><td>6.00(0.02)</td></tr><tr><td>R2CCP (OURS)</td><td>1.34(0.02)</td><td>3.80(2.61)</td><td>0.50(0.00)</td><td>0.92(0.02)</td><td>3.21(0.08)</td><td>1.60(0.07)</td><td>0.20(0.02)</td><td>3.80(0.26)</td></tr></table>
+
+Table 1: This is the length results over all datasets. We see that our method achieves the best length on 10 of the 16 datasets. Meanwhile, CQR is best at 5, CHR is best at 3, CB is best at 1, and KDE is the best at 3. Our method achieves the shortest intervals across these datasets.
+
+![](images/43521dfb048538867fd9311555e14db37d8b0a1fd1f4310b26f0d793cd56cdad.jpg)  
+(a) Original
+
+![](images/e50152fec68f06ae74e39245dc73b72c49141571c73a2b6d196f7169d779a0d1.jpg)  
+(b) No Entropy
+
+![](images/e3863ad504eb2e0657f4ef7de4eaed12f2e0e379c874c13b23aa4bf7d41af9fe.jpg)  
+(c) MLE
+
+![](images/3e42da70a0b703e0662f5f5b9a847e6907c7d78c0a0f96a28fa13b5b0f8ae7d6.jpg)  
+(d) MLE with Entropy  
+Figure 2: The resulting density estimates with different loss functions. We see that removing entropy from our loss function or using MLE as error terms causes sharp density estimates. Moreover, adding in the entropy regularization with MLE does not smooth the density estimate but instead raises the entire distribution uniformly; which does not provide valuable information for CP.
+
+for producing optimal intervals empirically. Moreover, we will also demonstrate the importance of our entropy term. We do this by retraining our models with the entropy part omitted. We also test by combining the MLE error term with entropy regularization. Therefore, we will perform 4 different
+
+<table><tr><td>DATASET</td><td>BIMODAL</td><td>LOG-NORMAL</td><td>CONCRETE</td><td>MEPS-19</td><td>MEPS-20</td><td>MEPS-21</td><td>Bio</td><td>COMMUNITY</td></tr><tr><td>LNE</td><td>0.430.001</td><td>3.250.058</td><td>1.890.037</td><td>3.730.057</td><td>3.790.061</td><td>8.780.163</td><td>2.010.006</td><td>3.640.054</td></tr><tr><td>LMLE</td><td>0.350.001</td><td>1.760.020</td><td>0.740.016</td><td>1.580.008</td><td>1.590.009</td><td>1.710.034</td><td>2.710.001</td><td>2.320.047</td></tr><tr><td>LMLE+E</td><td>0.360.001</td><td>3.520.008</td><td>1.910.017</td><td>1.600.008</td><td>1.630.027</td><td>1.700.008</td><td>2.320.003</td><td>3.770.013</td></tr><tr><td>L</td><td>0.440.002</td><td>1.820.034</td><td>0.370.004</td><td>1.600.009</td><td>1.590.011</td><td>1.690.013</td><td>1.100.004</td><td>1.500.025</td></tr><tr><td>DATASET</td><td>DIABETES</td><td>SOLAR</td><td>PARKINSONS</td><td>STOCK</td><td>CANCER</td><td>PENDULUM</td><td>ENERGY</td><td>FOREST</td></tr><tr><td>LNE</td><td>1.920.052</td><td>2.200.233</td><td>4.760.033</td><td>10.170.165</td><td>3.260.207</td><td>12.820.479</td><td>3.300.093</td><td>3.330.254</td></tr><tr><td>LMLE</td><td>1.560.031</td><td>0.140.005</td><td>0.340.006</td><td>4.480.214</td><td>3.260.117</td><td>3.000.203</td><td>0.250.011</td><td>3.040.110</td></tr><tr><td>LMLE+E</td><td>1.960.012</td><td>0.150.005</td><td>0.230.001</td><td>9.730.074</td><td>3.950.022</td><td>13.400.221</td><td>0.250.009</td><td>5.240.055</td></tr><tr><td>L</td><td>1.370.037</td><td>0.660.092</td><td>0.460.036</td><td>1.950.039</td><td>3.040.142</td><td>1.620.042</td><td>0.210.025</td><td>2.920.127</td></tr></table>
+
+Table 2: We present the length results over all of the variant loss functions. We find that our loss function delivers the best over 12 datasets, demonstrating that our chosen loss function often generates the best intervals. For datasets where our method does not deliver the best results, it is likely that tuning the weight on the entropy  $\tau$  and the smoothing term  $p$  would likely have improved the results, but we do not do this for the sake of evaluation.
+
+ablation experiments on the loss function by retraining on different variations of the loss functions and reporting the final length and coverage generated. Overall, the four loss functions we will use are the original loss function  $\mathcal{L}$ , the original loss function without entropy  $\mathcal{L}_{\mathrm{NE}}$ , the MLE loss function  $\mathcal{L}_{\mathrm{MLE}}$ , and the MLE loss function with entropy added  $\mathcal{L}_{\mathrm{MLE} + \mathrm{E}}$ . We will demonstrate that our chosen loss function delivers the best results across all the loss functions. We report our results in Table 2.
+
+Results We see that our chosen loss function achieves the best length across most loss functions in Table 2. The only datasets where our chosen loss function does not achieve the best length are Energy, Solar, Bimodal, and Log-Normal. We now discuss the individual differences between our loss function and each variant loss function. When comparing our loss  $\mathcal{L}$  with the variant without entropy  $\mathcal{L}_{\mathrm{NE}}$ , we see the lengths constantly increase except for the bimodal distribution. Without an entropy regularizing term, the outputted probability distributions are not smooth, placing much mass on a single data point. This overconfidence results in the  $\alpha$ th quantile of the low probability of the true label as in Figure 2.
+
+When looking at the differences between our loss function when changing the error term to Maximum Log Likelihood, we see similar overconfidence. While MLE loss works on several datasets, such as Energy, Solar, Bimodal, and Log-Normal, it also performs poorly on other datasets. MLE works well for datasets with less simple label distributions but fails otherwise due to similar overconfidence. Even when adding entropy as a regularizer to the MLE loss as in  $\mathcal{L}_{\mathrm{MLE} + \mathrm{E}}$ , we see that the addition of entropy does not improve the length. Since MLE loss does not treat nearby bins differently than far away bins, regularization decreases the overconfidence but does so uniformly across all bins. The result is a roughly uniform distribution with a single spike. This does not utilize the structure of regression where bins near the best bin are preferable. Thus, prioritizing nearby bins is crucial for achieving strong length. Moreover, when adding entropy regularization to our error term, we do not see a uniform increase in probability across all bins but instead a smoothening of a direct. This connection between entropy regularization and our distance-based loss function appears to be a powerful synergy per the intervals produced. Thus, the error term utilizing the regression structure by prioritizing nearby bins and the regularizing entropy term preventing overconfidence seem crucial for producing good intervals.
+
+Another important insight is that the datasets where other loss functions excel are the same ones where other CP methods perform better than ours. In particular, on the Bimodal, Log-Normal, Parkinson's, and Solar datasets, our CP method is not superior, and other loss functions outperform it. This indicates that the MLE's sharp distributions result in greater accuracy on these specific datasets. Hence, adjusting the weighting terms for entropy  $\tau$  and smoothing  $p$  could enhance the smoothness of the learned distribution and, thereby, the performance of our algorithm across these datasets, leading to the best outcomes. Nevertheless, to maintain fairness, we avoid this approach.
+
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+
+# A AUTHOR CONTRIBUTION STATEMENT
+
+Etash Guha, Shlok Natarajan, Thomas Möllenhoff, Eugene Ndiaye, and Mohammad Emtiyaz Khan were all responsible for the development of the main idea and writing of the paper. Etash Guha and Shlok Natarjan were responsible for the experiments and implementation of the algorithm. Etash Guha, Thomas Möllenhoff, and Eugene Ndiaye were responsible for the design of the loss function, binning, and entropic regularization. Etash Guha and Eugene Ndiaye were responsible for writing the related works.
+
+# B LIMITATIONS
+
+There are several limitations to our algorithm. The loss function we choose to optimize needs larger representational power due to increased complexity of the output. Therefore, the neural networks we used in practice to minimize this loss function are larger than the ones used for CQR. Moreover, to achieve strong training, we had to slightly vary the hyperparameters for a dataset depending on its size. For example, larger datasets needed models to be trained with smaller initial learning rate to avoid divergence. This was not needed for Quantie Regressors. If the network does not train well or achieve good training or validation loss, the intervals produced will be suboptimal, so finding a training setup that minimizes the loss effectively is crucial for our algorithm to produce meaningful intervals. Moreover, we do not directly learn the label distribution and our produced probability distributions do not directly mirror that of the ground truth distribution. One reason for this is the use of entropy regularization. While using entropy regularization introduces bias, we found it to be necessary to prevent the neural network from being overconfident on a simple bin in practice.
+
+# C MORE DETAILS ON CONFORMAL PREDICTION AND VARIANTS
+
+Given an input observation  $x$  without its label  $y$ , Conformal Prediction method Vovk et al. (2005) allows to select set of values that its unobserved response  $y$  are likely to take. For an arbitrary value  $z$ , the rational is to score how the example  $(x_{n + 1},z)$  is a typical/conformal observation given a sequence of iid data  $\mathcal{D}_n = \{(x_i,y_i)\}_{i\in [n]}$ . Intuitively, the scoring function, that we denote  $\sigma$ , measures how different is the new example compared to examples that we have already observed. We compute the score function  $\sigma (x,y)$  for every example in the augmented dataset  $\mathcal{D}_{n + 1}(z) = \mathcal{D}_n\cup (x,z)$  and report the fraction of time previous example score better than the new one.
+
+This define the so called conformity function  $\pi (\cdot)\in [0,1]$  as
+
+$$
+\pi (z) = 1 - \frac {\operatorname {R a n k} (z)}{n + 1} \text {w h e r e} \operatorname {R a n k} (z) = \sum_ {(x, y) \in D _ {n + 1} (z)} \mathbb {I} \left\{\sigma (x, y) \leq \sigma \left(x _ {n + 1}, z\right) \right\}. \tag {3}
+$$
+
+If  $\pi (z)$  is small (resp. large) then  $z$  is non conformal (resp. conformal). Therefore, for a threshold level  $\alpha \in (0,1)$ , a simple way to find our conformal set  $\Gamma^{(\alpha)}(x_{n + 1})$  for the response  $y_{n + 1}$  of the input  $x_{n + 1}$  is to select all the values  $z$  whose conformity exceeds the threshold i.e.
+
+$$
+\Gamma^ {(\alpha)} (x _ {n + 1}) = \{z \in \mathbb {R}: \pi (z) \geq \alpha \}.
+$$
+
+The validity of the method relies on the fact that if we assume that order in which the observations arrive is irrelevant and that the score function  $\sigma$  is invariant wrt permutation of the data, then the sequence of scores  $\{\sigma(x_1, y_1), \ldots, \sigma(x_n, y_n), \sigma(x_{n+1}, y_{n+1})\}$  are also equally likely and thus the  $\operatorname{Rank}(y_{n+1})$  is uniformly distributed in  $\{1, \ldots, n+1\}$ . As such, for any confidence level  $\alpha \in (0, 1)$ ,
+
+$$
+\mathbb {P} (y _ {n + 1} \in \Gamma^ {(\alpha)} (x _ {n + 1})) \geq \mathbb {P} (\pi (y _ {n + 1}) \geq \alpha) = \mathbb {P} (\operatorname {R a n k} (y _ {n + 1}) \leq (n + 1) (1 - \alpha)) \geq 1 - \alpha .
+$$
+
+Conditional validity The previous coverage guarantee may not always be ideal because it is not conditional on the particular input  $x_{n + 1}$ . Unfortunately, it is not possible to have such conditional validity when the distribution of the inputs is continuous. Nevertheless, it can be approximated using some localisation strategies based on a weighted conformity function; see (Lin et al., 2021; Ghosh et al., 2023). This also allows to further reduce the size of the conformal prediction sets.
+
+Validity can be conditional on a discrete attribute of the input,  $x_{n + 1}$ , through the use of Mondrian Conformal Prediction. This method involves a slight modification of the conformity function. Let
+
+a taxonomy  $\kappa$  be a function that partitions the data into a finite number of categories  $\operatorname{eg} \kappa(x,.)$  is a classification of the example  $x$ . This allows to modify the pi-value function in order to achieve validity conditional on the category of the test example:
+
+$$
+\pi (z) = \sum_ {(x, y) \in \mathcal {D} _ {n + 1} (z)} \frac {\mathbb {I} \left\{\sigma (x , y) \leq \sigma \left(x _ {n + 1} , z\right) , \kappa (x , y) = \kappa \left(x _ {n + 1} , z\right) \right\}}{\sum_ {(x , y) \in \mathcal {D} _ {n + 1} (z)} \mathbb {I} \left\{\kappa (x , y) = \kappa \left(x _ {n + 1} , z\right) \right\}}
+$$
+
+Computational issues As mentioned in the introduction, the selection of the score function affects the size and shape of the prediction sets. There are also computational issues that arise when the full data set is used to fit the function. For example, consider using the estimated distance to the conditional mean  $\sigma (x,y) = |y - \hat{E} [y|x]|$ . The estimator  $\hat{E} [\cdot |x]$  is learned on the extended dataset  $\mathcal{D}_{n + 1}(z)$  for all possible values  $z$ , which is computationally infeasible for most methods without stronger assumptions; see (Ndiaye & Takeuchi, 2023), (Ndiaye, 2022). As such, most of the popular strategies are based on the fitting of the prediction model to an left-out independent set of data, as we did in this paper.
+
+# D DETAILS ON DATASET FROM LEI & WASSERMAN (2014)
+
+For the dataset referenced in Figure 1b, we generate the dataset by sampling many  $(X,Y)$  pairs from the following distribution.
+
+$$
+X \sim \mathrm {U n i f} [ - 1. 5, 1. 5 ]
+$$
+
+$$
+Y \mid X \sim \frac {1}{2} \mathcal {N} (f (X) - g (X), \sigma^ {2} (X)) + \frac {1}{2} \mathcal {N} (f (X) + g (X), \sigma^ {2} (X))
+$$
+
+$$
+f (X) = (X - 1) ^ {2} (X + 1), g (X) = 4 \sqrt {(X + 1 / 2) \mathbb {I} (X \geq - 1 / 2)}, \sigma^ {2} (X) = 1 / 4 + | X |
+$$
+
+This dataset demonstrates bimodality after  $x$  passes the threshold of -0.5. This dataset was similarly used by the Lei & Wasserman (2014).
+
+# E COVERAGE DATA
+
+We have presented the coverage data for the Conformal Prediction comparison as well as the ablation experiments in Table 3 and Table 4, respectively. Since all methods obey the classical Conformal Prediction framework, we expect coverage at the guaranteed level of  $1 - \alpha$ . This guarantee is indeed what we see across all experiments. This confirms that our coverage guarantee indeed holds in practice.
+
+<table><tr><td>DATASET</td><td>BIMODAL</td><td>LOGNORM</td><td>CONCRETE</td><td>MEPS-19</td><td>MEPS-20</td><td>MEPS-21</td><td>BIO</td><td>COMMUNITY</td></tr><tr><td>CQR</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>KDE</td><td>0.93(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.92(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>LASSO</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>CB</td><td>0.90(0.00)</td><td>0.91(0.01)</td><td>0.88(0.01)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.89(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>CHR</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>DCP</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.91(0.00)</td><td>1.00(0.00)</td><td>1.00(0.00)</td><td>1.00(0.00)</td><td>1.00(0.00)</td><td>0.90(0.00)</td></tr><tr><td>R2CCP (OURS)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>DATASET</td><td>DIABETES</td><td>SOLAR</td><td>PARKINSONS</td><td>STOCK</td><td>CANCER</td><td>PENDULUM</td><td>ENERGY</td><td>FOREST</td></tr><tr><td>CQR</td><td>0.91(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.92(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td></tr><tr><td>KDE</td><td>0.91(0.00)</td><td>0.71(0.15)</td><td>0.89(0.00)</td><td>0.90(0.00)</td><td>0.92(0.01)</td><td>0.90(0.01)</td><td>0.91(0.00)</td><td>0.81(0.08)</td></tr><tr><td>LASSO</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.92(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.90(0.00)</td></tr><tr><td>CB</td><td>0.89(0.00)</td><td>0.92(0.01)</td><td>0.90(0.00)</td><td>0.88(0.01)</td><td>0.87(0.02)</td><td>0.88(0.01)</td><td>0.89(0.01)</td><td>0.90(0.01)</td></tr><tr><td>CHR</td><td>0.91(0.00)</td><td>0.91(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.92(0.00)</td><td>0.91(0.00)</td><td>0.91(0.00)</td><td>0.91(0.00)</td></tr><tr><td>DCP</td><td>0.91(0.00)</td><td>1.00(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>0.92(0.00)</td><td>0.90(0.00)</td><td>0.91(0.00)</td><td>1.00(0.00)</td></tr><tr><td>R2CCP (OURS)</td><td>0.90(0.00)</td><td>0.92(0.02)</td><td>0.96(0.00)</td><td>0.92(0.02)</td><td>0.90(0.00)</td><td>0.90(0.00)</td><td>0.91(0.01)</td><td>0.89(0.00)</td></tr></table>
+
+Table 3: This is the coverage data over all datasets. We see that all methods achieve the roughly  $1 - \alpha$  coverage guaranteed by conformal prediction.
+
+<table><tr><td>DATASET</td><td>BIMODAL</td><td>LOGNORM</td><td>CONCRETE</td><td>MEPS-19</td><td>MEPS-20</td><td>MEPS-21</td><td>BIO</td><td>COMMUNITY</td></tr><tr><td>L</td><td>0.90(0.02)</td><td>0.90(0.02)</td><td>0.90(0.02)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.90(0.00)</td><td>0.90(0.01)</td></tr><tr><td>LNE</td><td>0.90(0.02)</td><td>0.81(0.03)</td><td>0.84(0.03)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.68(0.01)</td><td>0.88(0.02)</td></tr><tr><td>LMLE</td><td>0.90(0.02)</td><td>0.90(0.02)</td><td>0.90(0.02)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.90(0.00)</td><td>0.90(0.01)</td></tr><tr><td>LMLE+E</td><td>0.90(0.02)</td><td>0.90(0.02)</td><td>0.90(0.02)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.90(0.01)</td><td>0.92(0.00)</td><td>0.90(0.01)</td></tr><tr><td>DATASET</td><td>DIABETES</td><td>SOLAR</td><td>PARKINSONS</td><td>STOCK</td><td>CANCER</td><td>PENDULUM</td><td>ENERGY</td><td>FOREST</td></tr><tr><td>L</td><td>0.90(0.03)</td><td>0.90(0.02)</td><td>0.95(0.01)</td><td>0.90(0.03)</td><td>0.90(0.05)</td><td>0.90(0.03)</td><td>0.90(0.03)</td><td>0.89(0.03)</td></tr><tr><td>LNE</td><td>0.94(0.02)</td><td>0.90(0.02)</td><td>0.73(0.01)</td><td>0.89(0.03)</td><td>0.90(0.05)</td><td>0.90(0.03)</td><td>0.90(0.02)</td><td>0.77(0.04)</td></tr><tr><td>LMLE</td><td>0.90(0.03)</td><td>0.90(0.02)</td><td>0.90(0.01)</td><td>0.91(0.03)</td><td>0.90(0.05)</td><td>0.90(0.03)</td><td>0.90(0.02)</td><td>0.89(0.03)</td></tr><tr><td>LMLE+E</td><td>0.90(0.03)</td><td>0.90(0.02)</td><td>0.90(0.01)</td><td>0.90(0.03)</td><td>0.90(0.05)</td><td>0.90(0.03)</td><td>0.90(0.03)</td><td>0.90(0.03)</td></tr></table>
+
+Table 4: This is the coverage data for different loss functions from the ablation.
+
+# F MORE EXPERIMENTAL DETAILS
+
+In order to maintain fairness across all baselines, we use the same size neural network for our method across all experiments. Specifically, we discretize the range space into  $K = 50$  points, weight the entropy term by  $\tau = 0.2$ , use a 1000 hidden dimension, use 4 layers, use weight decay of  $1e - 4$ , use  $p = .5$ , and use AdamW as an optimizer. For most of the experiments, we use learning rate  $1e - 4$  and batch size 32. However, for certain datasets, namely the MEPS datasets, we used a larger batch size of 256 to improve training time and used a smaller learning rate to prevent training divergence. We did change any other parameter between all of our runs. For the baselines of CQR and CHR, we use the neural network configurations mentioned in the paper. We found that the parameterizations mentioned by the authors in the papers achieved the best performance and changing the parameterizations weakened their results.
+
+# G ADDITIONAL EXPERIMENTS
+
+We add several additional experiments to help contextualize our results. First, we add experiments tracking how our method performs over different coverage levels relative to CHR and CQR across several datasets in Appendix G.2. Second, in Appendix G.3, we analyze what percentage of the time the predicted intervals are singeltons, meaning there is only one continuous interval. Third, we analyze the correlation between the residual prediction error and the predicted interval's length with our method in Appendix G.4.
+
+# G.1 PERFORMANCE WITH DIFFERENT NUMBER OF BINS
+
+We set the number of bins at a constant  $K = 50$ . A natural question is how the number of bins affects the performance of our conformal prediction methodology. To test this, we retrain several models with varying number of bins on the Concrete dataset. We report the size of the predicted intervals for each number of bins in Figure 3. We observe that as the number of bins increases, the average length of the predicted intervals decreases and stabilizes. This suggests that after a sufficient number of bins, the length of the intervals becomes relatively constant. In practice, classification problems with 1000 bins or more have been routinely solved since 2012 (Krizhevsky et al., 2012). This suggests that the increasing number of bins does not make the learning task more difficult, explaining how performance remains relatively unaffected as the number of bins is increased.
+
+# G.2 DIFFERENT COVERAGE LEVELS
+
+While the coverage level of  $\alpha = .1$  is relatively standard across the Conformal Prediction literature, we want to see how our method compares to other methods across different coverage levels. We take the trained models from the original experiments and compute the intervals across different coverage levels for our method, CHR, and CQR. As expected, for all methods, the length of the predicted intervals increases as the required coverage level increases. We report our results in Figure 4.
+
+![](images/2888577743e44cd8712f7da50e10094ac97e9c4de94d8e7ee6ae08152c258adc.jpg)  
+Figure 3: We present an ablation of how the number of bins affects the average length of the generated intervals.
+
+# G.3 NUMBER OF SINGLETONS
+
+We also provide an analysis of the percentage of times that our predicted intervals are singleton, meaning only one interval is predicted. We present our results in Table 5. We see that for a majority of the datasets, our model tends to predict singleton intervals. However, for both the Bimodal dataset and the Bio dataset, we see that our method predicts far less singletons. This observation is intuitive for the Bimodal dataset as two intervals are expected to account for the different modes.
+
+# G.4 CORRELATION BETWEEN PREDICTION ERROR AND LENGTH OF PREDICTED INTERVALS
+
+We want to see how the prediction error correlates with the predicted interval lengths. We define prediction error as the absolute error residual between predicted label and the true label. Given our model is taylored to generated intervals, there are many ways to predict a single label for a given set of features. For example, we can select the bin with the largest probability predicted by our model
+
+$$
+\hat {y} = \hat {y} _ {l} \text {w h e r e} k = \underset {k \in [ K ]} {\arg \max } q _ {\theta} (\hat {y} _ {k} | x).
+$$
+
+We can also use the expected label according to our distribution as the predicted label
+
+$$
+\hat {y} = \sum_ {k = 1} ^ {K} \hat {y} _ {k} q _ {\theta} (\hat {y} _ {k} | x).
+$$
+
+We choose the second method for the sake of these experiments. We compute the absolute residual error as  $\| \hat{y} - y \|$ . We plot our results in Figure 5. We see that the prediction error is not strongly correlated with the length of the predicted interval.
+
+# H OUTPUT DISTRIBUTION
+
+Here, we plot the distribution of lengths of all Conformal Prediction methods over all datasets. We also plot example density functions learned by out method, CHR, and KDE on all datasets. We note that KDE's learned densities seem to be relatively less informative. Moreover, the learned densities from CHR are noisy and not smooth. Moreover, we plot several examples of only our learned probability distribution. We use this when referencing the experiments to demonstrate the different shapes of label distributions from the data.
+
+![](images/f31d349670ad3b39705d5551fd28ae186f57c52b5deb6805a01c7c5ba5a16e69.jpg)  
+(a) MEPS-20
+
+![](images/835b18e8a99086b065fd5f30e25a02f1fa730f9e6e2b2877fa3d9a8096c7eecf.jpg)  
+(b) MEPS-21
+
+![](images/108ef3dc615e8f724a4d54e1715540106bf22e5ae5a74dbc0a442056f838fe42.jpg)  
+(c) DIABETES
+
+![](images/eb2d6c97cd9165d3fff4ac1206cebf386059342c8d90b262c77c806fd62a1c4f.jpg)  
+(d) BIMODAL
+
+![](images/5c43542ee5c63949a3cd9e0237143b963de6d041177e6a3bc155475fd0df0773.jpg)  
+(e) PENDULUM
+
+![](images/8a47a551fead73c28d38dcb74902b8efc8a64c92fee06413f4b8acd409ef5263.jpg)  
+(f) BIO
+
+![](images/85c5f92ed66ea71b1f785a6b0e266f847056e485d671506e1c96cf6aab65de12.jpg)  
+(g) LOG NORMAL
+
+![](images/82f8e2f9b485d5e6d90b9a3e92bb1a59db4dd88f1bf12695b2a28ba9ae4b3efc.jpg)  
+(h) ENERGY
+
+![](images/9a5070a66885649753d8f00f81db3e4b07d318bd10e4b78282a382a0cf8e9245.jpg)  
+(i) BREAST CANCER
+
+![](images/4fcb374a09d3968a39783fd83d0500f28e6c374a47f30a5ac2197897b0aff403.jpg)  
+(j) CONCRETE
+
+![](images/e30b2797a18d8ade8012be0298e5e103c2e1a4b4c994b23ed309218ba6e0de6c.jpg)  
+(k) STOCK
+
+![](images/81ffd5dbeae6122ad180d9cd14ec8999eea9bfdf2dc10f87c4df3c209e1104d7.jpg)  
+(1) COMMUNITY
+
+![](images/7addef4e4b2712b566209f8774a1f84bc7c4913c52758672729a83f87056c2f6.jpg)  
+(m) FOREST
+
+![](images/c54782d7f9dc7cf1c022c94641820ed65a88a6a919ba02bb0b807da856cd67c6.jpg)  
+(n) PARKINSON'S
+
+![](images/e8cdef07c270509532454037eb12cd6cb4d5711b7979d1bc71ad8370659e3692.jpg)  
+(o) MEPS-21  
+Figure 4: We plot the width of the predicted intervals for our method, CHR, and CQR across different coverage levels.
+
+<table><tr><td>DATASET</td><td>BIMODAL</td><td>LOG-NORMAL</td><td>CONCRETE</td><td>MEPS-19</td><td>MEPS-20</td><td>MEPS-21</td><td>BIO</td><td>COMMUNITY</td></tr><tr><td>NUMBER OF SINGLETONS</td><td>5</td><td>197</td><td>194</td><td>3157</td><td>3509</td><td>3132</td><td>6725</td><td>382</td></tr><tr><td>NUMBER OF EXAMPLES</td><td>400</td><td>200</td><td>206</td><td>3157</td><td>3509</td><td>3132</td><td>9146</td><td>399</td></tr><tr><td>DATASET</td><td>DIABETES</td><td>SOLAR</td><td colspan="2">PARKINSONSSTOCK</td><td>CANCER</td><td>PENDULUM</td><td>ENERGY</td><td>FOREST</td></tr><tr><td>NUMBER OF SINGLETONS</td><td>81</td><td>174</td><td>1103</td><td>105</td><td>37</td><td>115</td><td>153</td><td>96</td></tr><tr><td>NUMBER OF EXAMPLES</td><td>89</td><td>214</td><td>1175</td><td>108</td><td>39</td><td>126</td><td>154</td><td>104</td></tr></table>
+
+Table 5: We present an analysis of the number of singletons predicted by our method over all datasets. For most datasets, we see that our method tends to predict singletons.
+
+![](images/e5cb142d4cd26e5b982a68030d68bb108fe47217484390002ac3de0c8935d46c.jpg)  
+(a) MEPS-20
+
+![](images/f6d20a8929a9773faa330822c0ba6297cd806309886a610167fc3ee2e16e60c4.jpg)  
+(b) MEPS-21
+
+![](images/ab90db12ca29169b192bfe38b66f2691a2a98b2de3771bc635451ab426fb06a3.jpg)  
+(c) DIABETES
+
+![](images/26e2c63614c5e2a01ca424092578bcfa47dd76e99123aa011a44928dff829e8c.jpg)  
+(d) BIMODAL
+
+![](images/963d5d7712f826891bafc8544034b85e6314841b45be15382509c244bc09f4d1.jpg)  
+(e) PENDULUM
+
+![](images/a0b2f60fbdd607b1d45a293a7969b09e60d30d0937ac461d2bbd0c48210482d2.jpg)  
+(f) BIO
+
+![](images/c35cf5e27e2b263f5d673383b4f2593a03b79e6b8647361e159654fbb5067ac6.jpg)  
+(g) LOG NORMAL
+
+![](images/6c944541ea2960cf3baabc680ae09c812588188ed49d301465ff4fe4e9f8628a.jpg)  
+(h) ENERGY
+
+![](images/6956510258ed8f7f5283ce895970022081acd631d54e96f09977d564ee6645d6.jpg)  
+(i) BREAST CANCER
+
+![](images/a09258455ac0f23c52bde6c0274ce91ef25e2a8b62e90b089c7a281d99ee1ba1.jpg)  
+(j) CONCRETE
+
+![](images/8842d90a765fba060888e20d4d35156c134593082bf3bf46d94fc9ff458ded03.jpg)  
+(k) STOCK
+
+![](images/20c45ec11df7f8fc567a8f40680ec81c1f920af5b1c4705dcbc086b807f703d8.jpg)  
+(1) COMMUNITY
+
+![](images/6a45e027c17bed02d4b3fff9e958115075b3cae92313d32e1b00add682ec9e60.jpg)  
+(m) FOREST
+
+![](images/a0041827b72b176b2d83909dfd6ade292290b4ff22129fe1a918ba99e502dcb3.jpg)  
+(n) PARKINSON'S
+
+![](images/8af492ffd8eb80780326133c1d1215ffb7df2bdf583bd83579eb2217d98172a9.jpg)  
+(o) SOLAR  
+Figure 5: We plot the correlation between absolute residual error and the length of our predicted intervals over several datasets. We see that there is not a strong correlation.
+
+![](images/2f19b8716806b4dfc2434eb8a4fb09132510836838b52d94bf8e263329ceaab2.jpg)  
+(a) MEPS-20
+
+![](images/5a079eacf1dd972fcc84033fab48d7dc64995b452675acc723f235ccab33d905.jpg)  
+(b) MEPS-21
+
+![](images/9a235cc5465921270953cf535f2cd510994923f3256eae425ef67d9b74397a8a.jpg)  
+(c) DIABETES
+
+![](images/c7f61f334716135d58081a92cf636d7bfbebfd605988318a25ff28ccfad018a0.jpg)  
+(d) BIMODAL
+
+![](images/484fee9386fda10fc3b610230e77aa933dfe3c7916e56f09c25fe193a0e9293d.jpg)  
+(e) PENDULUM
+
+![](images/c77864ea58df83d3d58479f741525c766cfadc337b8070a395bc2af19635da5c.jpg)  
+(f) BIO
+
+![](images/453eecd1e902d5f8ccb5bbc047b43e70f806340e8d13b6e441bb410a6847a0c1.jpg)  
+(g) LOG NORMAL
+
+![](images/dc13d3da3d685531a18b9ef51b11e886c03837f9eef34a502a649d54932be228.jpg)  
+(h) ENERGY
+
+![](images/a5f385b82d43bb9c6f636354bce597826181f0909669538ee02acc67530146ca.jpg)  
+(i) BREAST CANCER
+
+![](images/cb560ab7cde9e60bad3881e88b92d245776b1979b9ff1d8e5985e3f9fa8bc587.jpg)  
+(j) CONCRETE
+
+![](images/2042d6f7669cdb4d89fbf4fd30fc98333a99e209a86431392ce0e8e8115a4fcd.jpg)  
+(k) STOCK
+
+![](images/9af994d64c4a9e8f531a686abca85662ea8e7e77f816d9cc63eb618a70b5d0da.jpg)  
+(1) COMMUNITY
+
+![](images/8291a1b65df35535174879a17d2cb1aec02c86defb82f1f64b70de4b08506c28.jpg)  
+(m) FOREST
+
+![](images/b20d77fb1c8c12b70c35e2f5ecf811b915bd4723fb5cb84ffe50c26dbb9ee8ae.jpg)  
+(n) PARKINSON'S
+
+![](images/62023e46dcf471da956cececfb5e16981973ba2c13952e1c4298a062146f0879.jpg)  
+(o) SOLAR  
+Figure 6: Example Probability Distributions outputted by our learned methodology on different datapoints from different datasets. We can see there are a variety of different shapes of distributions learned.
+
+![](images/c69157443bada77f8128a2451e4c73a562657b7922e998a90c4d24c43d78ea8f.jpg)  
+(a) Bimodal Length Distribution
+
+![](images/fcee5a32f3cf49c18d2575f962e8d548ff4762cae46e86e264c9727e71d57759.jpg)  
+(b) Bio Length Distribution
+
+![](images/b3d284599d346fb410441aba24a0ca83dfd2050573aa52487fbd958fdfecbe87.jpg)  
+(a) Breast Cancer Length Distribution
+
+![](images/d106d2b7d9e355c4ff2a305c88fb89da80f85aaa5c7f43f4c1d58fd5ab199e7a.jpg)  
+(b) Community Length Distribution
+
+![](images/d6858f4a449d859de910d84e8d0309ee96edda201d9fc63f9b7df6c72d03a555.jpg)  
+(a) Concrete Length Distribution
+
+![](images/4c7fad729b1291dba3a2b151cae3b13e0c25c8514604ef36ec63550339af2601.jpg)  
+(b) Diabetes Length Distribution
+
+![](images/5831294756c7d4386931030f4027b0579dc9c01e546683a4f2ba6d523087b58b.jpg)  
+(a) Energy Length Distribution
+
+![](images/c5a3331783c5d28e7a10eca314a28af64e6f2ecd62a81ddc30e828e3b6b2dad0.jpg)  
+(b) Forest Length Distribution
+
+![](images/79e24417b60a04bd2e25c9e1892290230411cba5ba8bc9b943d4ea58d5867e8d.jpg)  
+(a) Log Normal Length Distribution
+
+![](images/fe95bdd2a49e5166820952b3a624c1aa2f6a2ecd643a95d011b37bb845f89a02.jpg)  
+(b) MEPS 19 Length Distribution
+
+![](images/c94d80b965114a728a37bcc1bbf2a3dc633fd0aa494ef52a437487731ef23e63.jpg)  
+(a) MEPS 20 Length Distribution
+
+![](images/cefdfe22fb962f1eeafba41e6e340ae9a13e0c91698f2a190da817e6d0ea0695.jpg)  
+(b) MEPS 21 Length Distribution
+
+![](images/1693abe8f0a6588db2e4a8acd6a7e1be85fa4f3417e6287b03365841c2a40987.jpg)  
+(a) Parkinsons Length Distribution
+
+![](images/10b266277871954161b028aee352a36fc6cc033c06451ab7310df405648fb05e.jpg)  
+(b) Pendulum Length Distribution
+
+![](images/7801aa6b1c329a663eec8c71645a8d93198ce5d8b029b9282060f4f08df45de3.jpg)  
+(a) Solar Length Distribution
+
+![](images/2de747e4e7b644e458fb786fe0d1eaed95bc72b3ca1bb82bdbf479c0b64d4f06.jpg)  
+(b) Stock Length Distribution
+
+![](images/b818a0e497129946d4153d9901b89779be163f095a83ee58f4524d7bbd633954.jpg)  
+(a) BIO
+
+![](images/0912b4efac518291e48c5069fec53a7417ea5fc5d82a1662649554d8a16700d4.jpg)  
+(b) BREAST CANCER  
+Figure 7: Plot of outputted density functions for ours, KDE, and CHR on for BIO and BREAST CANCER
+
+![](images/6f9ac6b403679f9fea10cd8352503cc1d4e3ce8df289dc1999e81106ab050fa0.jpg)  
+(a) BIMODAL
+
+![](images/bf79f4a895bcbbb74adba880254b5d5a9a2dcface36b1901d3b5a196d1ff7696.jpg)  
+(b) COMMUNITY  
+Figure 8: Plot of outputted density functions for ours, KDE, and CHR on for BIMODAL and COMMUNITY
+
+![](images/674d482764268c1310722bfa6b358234479b557fd98a0babad7cfc7fd20167bc.jpg)  
+(a) CONCRETE
+
+![](images/7770a70479e7c818e8364d402b5c143c576d6e6b7f4bfa82c57889ce630e7af6.jpg)  
+(b) DIABETES
+
+![](images/480c0ecd89c04c4f795e84814f5bfb657749c27ddf5795bf39f4cbbf7e8479b7.jpg)  
+Figure 9: Plot of outputted density functions for ours, KDE, and CHR on for CONCRETE and DIABETES  
+(a) ENERGY  
+Figure 10: Plot of outputted density functions for ours, KDE, and CHR on for ENERGY and FOR-EST
+
+![](images/187f88779da7189d20bc325d082539e39b417c3dc80d50f19db8ad6783cd913f.jpg)  
+(b) FOREST
+
+![](images/3459272ed802c4b327b450067094f1ebc4ec4fcd414dd181ddc39092c1315a97.jpg)  
+(a) LOG NORMAL
+
+![](images/171ab19f549314c2c532723505ac523a3fd62cf0ec45baed4c81735056386151.jpg)  
+(b) MEPS-21  
+Figure 11: Plot of outputted density functions for ours, KDE, and CHR on for LOG NORMAL and MEPS-19
+
+![](images/75296075cbd1381000be7e94ab5dfa0f6679ae7705aa50c082e01ba1e02a43e2.jpg)  
+(a) PARKINSON'S
+
+![](images/b5d0f490b0d4f17f861885233846ec44d65dad678ee7b1d80d043ac129984dc0.jpg)  
+(b) PENDULUM
+
+![](images/7ccbba47c2e58c23f8690efdf41dd7b4c2275d1ada4a80e2c1c6804098839581.jpg)  
+Figure 12: Plot of outputted density functions for ours, KDE, and CHR on for PARKINSON'S and PENDULUM  
+(a) SOLAR
+
+![](images/8e11331097941faf72e16065088ab9101ea81a872e4d7dfb3d89af6a7548a6d3.jpg)  
+(b) STOCK  
+Figure 13: Plot of outputted density functions for ours, KDE, and CHR on for SOLAR and STOCK

2024/Conformal Prediction via Regression-as-Classification/images.zip

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2024/ConjNorm_ Tractable Density Estimation for Out-of-Distribution Detection/a7b7903b-a2f9-4bb3-ae1a-734885d929e5_origin.pdf

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2024/ConjNorm_ Tractable Density Estimation for Out-of-Distribution Detection/full.md

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+# CONJNORM: TRACTABLE DENSITY ESTIMATION FOR OUT-OF-DISTRIBUTION DETECTION
+
+Bo Peng $^{1*}$ ; Yadan Luo $^{2*}$ ; Yonggang Zhang $^{3}$ , Yixuan Li $^{4}$ , Zhen Fang $^{1\dagger}$
+
+University of Technology Sydney, Australia
+
+The University of Queensland, Australia
+
+Hong Kong Baptist University, Hong Kong<sup>3</sup>
+
+University of Wisconsin-Madison, USA
+
+bo.peng-7@student.uts.edu.au, y.luo@uq.edu.au
+
+czygzhang@comp.hkbu.edu.hk, sharonli@cs.wisc.edu
+
+zhen.fang@uts.edu.au
+
+# ABSTRACT
+
+Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a CONJNORM method, reframing density function design as a search for the optimal norm coefficient  $p$  against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed CONJNORM has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to  $13.25\%$  and  $28.19\%$  (FPR95) on CIFAR-100 and ImageNet-1K, respectively.
+
+# 1 INTRODUCTION
+
+Despite the significant progress in machine learning that has facilitated a broad spectrum of classification tasks (Gaikwad et al., 2010; Huang et al., 2014; Zhao et al., 2019; Shantaiya et al., 2013; Krizhevsky et al., 2012; Masana et al., 2022), models often operate under a closed-world scenario, where test data stems from the same distribution as the training data. However, real-world applications often entail scenarios in which deployed models may encounter unseen classes of samples during training, giving rise to what is known as out-of-distribution (OOD) data. These OOD instances have the potential to undermine a model's stability and, in certain cases, inflict severe damage upon its performance. To identify and safely remove these OOD data in decision-critical tasks (Chen et al., 2022; Zimmerer et al., 2022), OOD detection techniques have been proposed. To facilitate easy separation of in-distribution (ID) and OOD data, mainstream OOD approaches either leverage post-hoc analysis or model re-training (Ming et al., 2022a; Wei et al., 2022; Chen et al., 2021; Huang & Li, 2021; Du et al., 2022; Katz-Samuels et al., 2022; Wang et al., 2023; Lee et al., 2017) by using density-based (Morteza & Li, 2022), output-based (Liu et al., 2020), distance-based (Lee et al., 2018) and reconstruction-based strategies (Zhou, 2022).
+
+Following previous works (Liu et al., 2020; Liang et al., 2017; Hendrycks & Gimpel, 2016; Sun et al., 2021; Ahn et al., 2023; Djurisic et al., 2022; Lee et al., 2018), we focus on the post-hoc OOD detection strategy, which offers more practical advantages than learning-based OOD approaches
+
+without requiring resource-intensive re-training processes. Our key research question of this approach centers on how to derive a proper scoring functions to indicate the ID-ness of the input for effectively discerning OOD samples during testing. By definition, OOD data inherently diverges from ID data by means of their data density distributions, rendering estimated density an ideal metric for discrimination. Nevertheless, it is non-trivial to parameterize the unknown ID data distribution for density estimation since the computation of normalization constants tends to be costly and even intractable (Gutmann & Hyvarinen, 2012a). While recent attempts have been made by modeling ID data as some specific prior distributions, i.e., the Gibbs-Boltzmann distribution in (Liu et al., 2020) and the mixture Gaussian distribution in (Morteza & Li, 2022), to factitiously make normalization constants sample-independent or known, this practice imposes strong distributional assumptions on the underlying feature space. Furthermore, it offers no theoretical guarantee that those pre-defined distributions necessarily hold in practice.
+
+In this paper, we introduce an innovative Bregman divergence-based (Banerjee et al., 2005) theoretical framework aimed at providing a unified perspective for designing density functions within an expansive exponential family of distributions (Amari, 2016). This framework not only bridges the gap between existing post-hoc OOD approaches (Liu et al., 2020; Morteza & Li, 2022) but also highlights a valuable conjugation constraint for tailoring density functions to given datasets. Without loss of generality, we focus on the conjugate pair of  $l_{p}$  and  $l_{q}$  norms and propose the CONJNORM method. This approach reframes the density function design as a search for the optimal norm coefficient within a narrow range. To facilitate tractable estimation of the partition function for normalization, we compare two existing estimation baselines and put forward a Monte Carlo-based importance sampling technique, which yields an unbiased and analytically tractable estimator.
+
+# 2 PRELIMINARIES
+
+Let  $\mathcal{X}$  and  $\mathcal{Y} = \{1, \dots, K\}$  represent the input space and ID label space, respectively. The joint ID distribution, represented as  $P_{X_1Y_1}$ , is a joint distribution defined over  $\mathcal{X} \times \mathcal{Y}$ . During testing time, there are some unknown OOD joint distributions  $D_{X_0Y_0}$  defined over  $\mathcal{X} \times \mathcal{Y}^c$ , where  $\mathcal{Y}^c$  is the complementary set of  $\mathcal{Y}$ . We also denote  $p_1(\mathbf{x})$  as the density of the ID marginal distribution  $P_{X_1}$ . According to (Fang et al., 2022), OOD detection can be formally defined as follows:
+
+Problem 1 (OOD Detection). Given labelled  $ID$  data  $\mathcal{D}_{\mathrm{in}} = \{(\mathbf{x}_1,\mathbf{y}_1),\dots,(\mathbf{x}_N,\mathbf{y}_N)\}$ , which is drawn from  $P_{X_1Y_1}$  independent and identically distributed, the aim of OOD detection is to learn a predictor  $g$  by using  $\mathcal{D}_{\mathrm{in}}$  such that for any test data  $\mathbf{x}$ : 1) if  $\mathbf{x}$  is drawn from  $D_{X_1}$ , then  $g$  can classify  $\mathbf{x}$  into correct  $ID$  classes, and 2) if  $\mathbf{x}$  is drawn from  $D_{X_0}$ , then  $g$  can detect  $\mathbf{x}$  as OOD data.
+
+Post-hoc Detection Strategy. Many representative OOD detection methods (Liu et al., 2020; Liang et al., 2017; Hendrycks & Gimpel, 2016; Sun et al., 2021; Ahn et al., 2023; Djurisic et al., 2022; Lee et al., 2018) follow a post-hoc strategy, i.e., given a well-trained model  $\mathbf{f}_{\theta}$  using  $\mathcal{D}_{\mathrm{in}}$ , and a scoring function  $S$ , then  $\mathbf{x}$  is detected as ID data if and only if  $S(\mathbf{x};\mathbf{f}_{\theta})\geq \lambda$ , for some given threshold  $\lambda$ :
+
+$$
+g (\mathbf {x}) = \mathrm {I D}, \text {i f} S (\mathbf {x}; \mathbf {f} _ {\boldsymbol {\theta}}) \geq \lambda ; \text {o t h e r w i s e}, g (\mathbf {x}) = \mathrm {O O D}. \tag {1}
+$$
+
+Following the representative work (Morteza & Li, 2022), a natural view for the motivation of the post-hoc strategy is to use a level set for ID density  $p_{\mathrm{I}}(\mathbf{x})$  to discern ID and OOD data. Its main objective is to construct an efficient scoring function  $S$ , that can effectively replicate the behavior of the ID density function,  $p_{\mathrm{I}}(\mathbf{x})$ , i.e.,  $S(\mathbf{x}; \mathbf{f}_{\theta}) \propto p_{\mathrm{I}}(\mathbf{x})$ . Therefore, using the density-based framework, (Morteza & Li, 2022) rewrites the post-hoc strategy as follows: given ID data density function  $\hat{p}_{\theta}(\cdot)$  estimated by well-trained model  $\mathbf{f}_{\theta}$  and a pre-defined threshold  $\lambda$ , then for any data  $\mathbf{x} \in \mathcal{X}$ ,
+
+$$
+g (\mathbf {x}) = \mathrm {I D}, \text {i f} \hat {p} _ {\boldsymbol {\theta}} (\mathbf {x}) \geq \lambda ; \text {o t h e r w i s e}, g (\mathbf {x}) = \mathrm {O O D}. \tag {2}
+$$
+
+In this work, we mainly utilize the density-based framework to design our theory and algorithm.
+
+Density Estimation Modeling. The performance of density-based OOD detection heavily relies on the alignment between the estimated data density  $\hat{p}_{\theta}(\mathbf{x})$  and the true density  $p_{\mathrm{I}}(\mathbf{x})$ . Considering a commonly used assumption in OOD detection, i.e., the uniform class prior on ID classes (Jiang et al., 2023),  $\hat{p}_{\theta}(\mathbf{x})$  can be expressed as the aggregate of the ID class-conditioned distributions  $\hat{p}_{\theta}\left(\mathbf{x}|k\right)$ :
+
+$$
+\hat {p} _ {\boldsymbol {\theta}} (\mathbf {x}) = \sum_ {k = 1} ^ {K} \hat {p} _ {\boldsymbol {\theta}} (\mathbf {x} | k) \cdot \hat {p} _ {\boldsymbol {\theta}} (k) \propto \sum_ {k = 1} ^ {K} \hat {p} _ {\boldsymbol {\theta}} (\mathbf {x} | k). \tag {3}
+$$
+
+Based on Eq. 3, our main objective is to estimate the class-conditional distribution of ID data, in order to effectively construct the data density  $\hat{p}_{\theta}$  for discriminating between ID and OOD data.
+
+Without loss of generality, we employ latent features  $\mathbf{z}$  extracted from deep models as a surrogate for the original high-dimensional raw data  $\mathbf{x}$ . This is because  $\mathbf{z}$  is deterministic within the post-hoc framework. Consistent with probabilistic theory, we express  $\hat{p}_{\theta}(\mathbf{z}|k)$  in the following general form:
+
+$$
+\hat {p} _ {\boldsymbol {\theta}} (\mathbf {z} | k) = \frac {g _ {\boldsymbol {\theta}} (\mathbf {z} , k)}{\Phi (k)}, \tag {4}
+$$
+
+where  $g_{\theta}(\mathbf{z},k)$  represents a non-negative density function, and  $\Phi (k) = \int g_{\theta}(\mathbf{z},k)d\mathbf{z}$  denotes the partition function for normalization. According to prior works, the design principle for  $g_{\theta}(\mathbf{z},k)$  can be divided into 3 categories: logit-based, distance-based and density-based methods.
+
+Logit-based OOD methods (Liu et al., 2020; Hendrycks et al., 2019) resort to derive  $g_{\theta}(\mathbf{z},k)$  from logit outputs. As a representative work, energy-based method (Liu et al., 2020) explicitly acknowledges  $g_{\theta}(\mathbf{z},k)$  by fitting to the Gibbs-Boltzmann distribution, i.e.,  $g_{\theta}(\mathbf{z},k) = \exp (f_{\theta}^{k} / T)$  where  $f_{\theta}^{k}$  is the kth coordinate of  $\mathbf{f}_{\theta}$  and  $T$  is a temperature parameter. This directly results in an energy-based scoring function  $E(\mathbf{z}) = -T\log \sum_{k = 1}^{K}g_{\theta}(\mathbf{z},k)$ . However, it can be easily checked from Eq. 4 that  $E(\mathbf{z})\propto -\log p(\mathbf{z})$  holds if and only if  $\Phi (k) = \mathrm{constant},\forall k\in \mathcal{V}$ . While the energy-based method has demonstrated empirical effectiveness, it is essential to recognize that this condition, i.e.,  $\Phi (k) = \mathrm{constant},\forall k\in \mathcal{V}$ , may not always hold in practical scenarios. Differently, Hendrycks & Gimpel (2016) proposes the maximum softmax score (MSP) to estimate OOD uncertainty:
+
+$$
+\operatorname {M S P} (\mathbf {z}) = \max  _ {k = 1, \dots , K} \hat {p} _ {\boldsymbol {\theta}} (k | \mathbf {z}) = \max  _ {k = 1, \dots , K} \frac {g _ {\boldsymbol {\theta}} (\mathbf {z} , k)}{\sum_ {k ^ {\prime} = 1} ^ {K} g _ {\boldsymbol {\theta}} (\mathbf {z} , k ^ {\prime})} \not \prec \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}). \tag {5}
+$$
+
+where  $g_{\theta}(\mathbf{z},k) = \exp (f_{\theta}^{k})$ . However, as shown in Eq. 5, there exists a misalignment between MSP and the true data density, making ultimately MSP a suboptimal solution to OOD detection.
+
+Distance-based OOD methods (Lee et al., 2017) target on deriving  $g_{\theta}(\mathbf{z},k)$  by assessing the proximity of the input to the  $k$  -th prototype  $\mu_{k}$  . The selection of appropriate similarity metrics is crucial in capturing the intrinsic geometric data relationships. One of the most representative metrics used is the maximum Mahalanobis distance (Lee et al., 2017), which is formally defined as,
+
+![](images/798b4a9fb88df69ab9bf2b1a930fdd69e7c425dd5653ea96bbe2d552af020f0d.jpg)  
+Figure 1: Illustration of the alignment of GEM score and true density of Gaussian (Left) and Gamma (Right) distributions.
+
+![](images/aebc05443539e3df9cb557f5a0b53593bb1d941df8120147e80f36efd992f17f.jpg)
+
+$$
+\begin{array}{l} \operatorname {M a h a} (\mathbf {z}) = \max  _ {k = 1, \dots , K} - \left(\mathbf {z} - \boldsymbol {\mu} _ {k}\right) ^ {\top} \Sigma^ {- 1} \left(\mathbf {z} - \boldsymbol {\mu} _ {k}\right) \\ = \max  _ {k = 1, \dots , K} \log g _ {\boldsymbol {\theta}} (\mathbf {z}, k) \not \propto \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}). \\ \end{array}
+$$
+
+The distance metric can be considered as the density function  $g_{\theta}(\mathbf{z},k)$  in Eq. 4. This interpretation allows us to bypass the estimation of the partition function and leads to a significant observation: the distance measures are not directly proportional to the true data density.
+
+Density-based OOD methods have rarely been studied compared to the previous two groups, primarily because of the complexities involved in estimating  $\Phi(k)$ . Recently, a method called GEM (Morteza & Li, 2022) has been proposed, with the assumption that the class-conditional density conforms to a Gaussian distribution: let  $g_{\theta}(\mathbf{z}, k) = \exp \left(-\frac{1}{2} (\mathbf{z} - \boldsymbol{\mu}_k)^{\top} \Sigma^{-1} (\mathbf{z} - \boldsymbol{\mu}_k)\right)$ ,
+
+$$
+\operatorname {G E M} (\mathbf {z}) = \sum_ {k = 1} ^ {K} \frac {\exp \left(- \frac {1}{2} \left(\mathbf {z} - \boldsymbol {\mu} _ {k}\right) ^ {\top} \Sigma^ {- 1} \left(\mathbf {z} - \boldsymbol {\mu} _ {k}\right)\right)}{\sqrt {(2 \pi) ^ {d} | \Sigma |}} = \sum_ {k = 1} ^ {K} \frac {g _ {\boldsymbol {\theta}} (\mathbf {z} , k)}{\Phi (k)} \propto \frac {1}{K} \sum_ {k = 1} ^ {K} \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z} | k) = \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}),
+$$
+
+where  $\Sigma \in \mathbb{R}^{d\times d}$  is the covariance matrix. Note that  $\Phi (k) = \sqrt{(2\pi)^{d}|\Sigma|}$  in this case. This Gaussian assumption, while simplifying the estimation of  $\Phi (k)$ , enables the direct utilization of the Mahalanobis distance as  $g_{\theta}(\mathbf{z},k)$ . However, it is crucial to acknowledge that this methodology may impose constraints on its ability to generalize effectively across a wide range of testing scenarios due to the strict Gaussian assumption it relies upon.
+
+Discussion. Empirical examination of a toy dataset presented below reveals a case where the GEM's Gaussian assumption may prove inadequate, as demonstrated in Fig 1. For the purpose of visualization, we begin by considering a simple scenario in which the input distribution is a mixture of two-dimensional Gaussians with means  $\mu_{2} = 2\mu_{1} = 8$  and variances  $\sigma_{1} = \sigma_{2} = 1$  respectively. While the GEM can well align with the true data density function, the alignment of GEM scores with the true data density is noticeably compromised when the  $p(\mathbf{x})$  is changed to a mixture of Gamma and Gaussian distributions (as shown on the right). In order to ensure an accurate estimation of the ID class-conditional density, two fundamental questions arise:
+
+Can we develop a unified framework that guides the design of  $g_{\theta}(\mathbf{z},k)$ ?  
+$\spadesuit$  Within this framework, how can we obtain a tractable estimate for  $\Phi (k)$  without presuming any particular prior distribution of  $\hat{p}_{\theta}(\mathbf{z}|k)$ ?
+
+In the following section, we propose a novel theoretical framework to answer the above questions.
+
+# 3 METHODOLOGY
+
+In this section, we first present the main Bregman Divergence-based theoretical framework of OOD detection in Sec. 3.1. This framework unifies density function formulation and connects with prior OOD techniques, leveraging the expansive exponential distribution family. Motivated by theory, we introduce a novel approach called CONJNORM to determine the desired  $g_{\theta}$  through an exhaustive search for the best norm coefficient  $p$ . To enable tractable density estimation, we explore two partition function estimation baselines and propose our importance sampling in Sec. 3.2.
+
+# 3.1 BREGMAN DIVERGENCE-GUIDED DESIGN OF  $g_{\theta}(\mathbf{z},k)$
+
+In formulating our theoretical framework, it is imperative to adopt a universal distribution family to model the ID class-conditioned distributions  $\hat{p}_{\theta}(\mathbf{x}|k)$  without constraining ourselves to any particular choice. In this work, we consider the broad Exponential Family of Distributions (Brown, 1986). The family encompasses a wide range of probability distributions frequently employed in prior OOD investigations, such as Gaussian, Gibbs-Boltzmann, and gamma distributions. To be precise, the exponential family of distribution can be formally defined as follows:
+
+Definition 1 (Exponential Family of Distribution (Brown, 1986)). A regular exponential family  $\hat{p}_{\theta}(\mathbf{z}|k)$  is a family of probability distributions with density function with the parameters  $\boldsymbol{\eta}_k$ :
+
+$$
+\hat {p} _ {\boldsymbol {\theta}} (\mathbf {z} | k) = \exp \left\{\mathbf {z} ^ {\top} \boldsymbol {\eta} _ {k} - \psi \left(\boldsymbol {\eta} _ {k}\right) - g _ {\psi} (\mathbf {z}) \right\}, \tag {6}
+$$
+
+where  $\psi (\cdot)$  is the so-called cumulant function and is a convex function of Legendre type.
+
+By employing different cumulant functions  $\psi(\cdot)$  and parameters  $\eta_k$ , one can create diverse class-conditioned distributions  $\hat{p}_{\theta_k}(\mathbf{z}|k)$ . Nevertheless, it has been argued by Azoury & Warmuth (2001); Chowdhury et al. (2023) that directly learning  $\eta_k$  to fit the ID data is computationally costly and even intractable. To mitigate this challenge, a corresponding dual theorem (referred to as Theorem 1) has been developed. This theorem asserts that any regular exponential family distribution can be presented through a uniquely determined Bregman divergence (Bregman, 1967), as defined below:
+
+Definition 2 (Bregman Divergence (Bregman, 1967)). Let  $\varphi(\cdot)$  be a differentiable, strictly convex function of the Legendre type, the Bregman divergence is defined as:
+
+$$
+d _ {\varphi} (\mathbf {z}, \mathbf {z} ^ {\prime}) = \varphi (\mathbf {z}) - \varphi \left(\mathbf {z} ^ {\prime}\right) - \left(\mathbf {z} - \mathbf {z} ^ {\prime}\right) ^ {\top} \nabla \varphi \left(\mathbf {z} ^ {\prime}\right), \tag {7}
+$$
+
+where  $\nabla \varphi (\mathbf{z}^{\prime})$  represents the gradient vector of  $\varphi (\cdot)$  evaluated at  $\mathbf{z}'$ .
+
+The choices of the convex function  $\varphi$  in Bregman divergence can result in diverse distance metrics. For instance, 1) When  $\varphi(\mathbf{z}) = \| \mathbf{z} \|^2$ , the resulting  $d_{\varphi}$  corresponds to the squared Euclidean distance; 2) When  $\varphi(\mathbf{z})$  is the negative entropy function,  $d_{\varphi}$  represents the KL divergence; and 3) When  $\varphi(\mathbf{z})$  be expressed as a quadratic form,  $d_{\varphi}$  represents the Mahalanobis distance. Next, Theorem 1 bridges the Bregman divergence and the exponential family of distributions.
+
+Theorem 1 (Forster & Warmuth (2002)). Suppose that  $\psi(\cdot)$  and  $\varphi(\cdot)$  are conjugate Legendre functions. Let  $\hat{p}_{\theta}(\mathbf{z}|k)$  be a member of the exponential family conditioned on the  $k$ -th ID class with cumulant function  $\varphi$  and parameters  $\eta_{k}$  ( $k = 1, \dots, K$ ),  $d_{\varphi}$  be the Bregman divergence, then  $\hat{p}_{\theta}(\mathbf{z}|k)$
+
+can be represented as follows:  $\hat{p}_{\boldsymbol{\theta}}(\mathbf{z}|k) = \exp(-d_{\boldsymbol{\varphi}}(\mathbf{z}, \boldsymbol{\mu}(\boldsymbol{\eta}_k)) - g_{\boldsymbol{\varphi}}(\mathbf{z}))$ , where  $\boldsymbol{\mu}(\boldsymbol{\eta}_k)$  is the expectation parameter corresponding to  $\boldsymbol{\eta}_k$  (Banerjee et al., 2005),  $g_{\boldsymbol{\varphi}}(\cdot)$  is a function uniquely determined by  $\varphi(\cdot)$ , and agnostic to  $\boldsymbol{\mu}(\boldsymbol{\eta}_k)$ .
+
+Remark. As a direct implication of Theorem 1, a unified theoretical principle emerges for the design of  $g_{\theta}(\mathbf{z},k)$  for OOD detection, owing to the conjugate relationship between  $\psi$  and  $\varphi$ . In essence, when seeking an appropriate  $\psi$  for a given dataset, the optimal design of  $g_{\theta}(\mathbf{z},k)$  should inherently adhere to the requirements of the corresponding Bregman divergence: let  $\varphi (\cdot) = \psi^{*}(\cdot)$ , then
+
+$$
+g _ {\boldsymbol {\theta}} (\mathbf {z}, k) = \exp \left(- d _ {\varphi} (\mathbf {z}, \boldsymbol {\mu} (\boldsymbol {\eta} _ {k}))\right). \tag {8}
+$$
+
+Given that  $g_{\varphi}(\mathbf{z})$  is agnostic to the choice of  $z$ , we exclude this term from consideration by treating it as a constant in our analysis, which gives us a systematic approach to answering the question.
+
+CONJNORM. Given the expansive function space for the selection of the convex function  $\psi$ , our focus is on simplifying the search process by utilizing the  $l_p$  norm as  $\psi$ , denoted as CONJNORM, where  $\psi(\pmb{\eta}_k) = \frac{1}{2} \|\pmb{\eta}_k\|_p^2$ . Therefore, the task of selecting an appropriate  $\psi$  is equivalent to identifying a suitable  $p$  from the range of  $(1, +\infty)$  for the given dataset. The  $l_p$  norm offers several advantageous properties, including convexity and simplicity in its conjugate pair. Firstly, the  $l_p$  norm is convex for all  $p \geq 1$ , ensuring the presence of a global minimum during optimization. Secondly, the  $l_p$  norm has a well-defined and simple conjugate pair, namely the  $l_q$  norm, where  $q$  represents the conjugate exponent of  $p$  such that  $1/p + 1/q = 1$ . This simplicity in the conjugate pair enhances computational tractability and facilitates the determination of  $\varphi = \psi^*$ :
+
+$$
+\varphi (\mathbf {z}) = \psi^ {*} (\mathbf {z}) = \frac {1}{2} \| \mathbf {z} \| _ {q} ^ {2}, \text {w h e r e} q = \frac {p}{p - 1}. \tag {9}
+$$
+
+To this end, the desired Bregman divergence  $d_{\varphi}$  can be determined as
+
+$$
+d _ {\varphi} (\mathbf {z}, \boldsymbol {\mu} (\boldsymbol {\eta} _ {k})) = \frac {1}{2} \| \mathbf {z} \| _ {q} ^ {2} + \frac {1}{2} \| \boldsymbol {\mu} (\boldsymbol {\eta} _ {k}) \| _ {q} ^ {2} - \langle \mathbf {z}, \nabla \frac {1}{2} \| \boldsymbol {\mu} (\boldsymbol {\eta} _ {k}) \| _ {q} ^ {2} \rangle . \tag {10}
+$$
+
+Hence, our final ID density can be estimated by combining Eq. 6 and Theorem 1
+
+$$
+\hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) = \frac {1}{K} \sum_ {k = 1} ^ {K} \frac {g _ {\boldsymbol {\theta}} (\mathbf {z} , k)}{\Phi (k)} = \frac {1}{K} \sum_ {k = 1} ^ {K} \frac {\exp \left(- d _ {\varphi} (\mathbf {z} , \boldsymbol {\mu} (\boldsymbol {\eta} _ {k}))\right)}{\int \exp \left(- d _ {\varphi} \left(\mathbf {z} ^ {\prime} , \boldsymbol {\mu} (\boldsymbol {\eta} _ {k})\right)\right) \mathrm {d} \mathbf {z} ^ {\prime}}. \tag {11}
+$$
+
+In the context of our CONJNORM framework, where we treat  $p$  as a hyperparameter, the process of searching for the optimal  $p^{opt}$  and identifying the most suitable density function  $d_{\varphi}$  for a given dataset becomes straightforward. We present experimental results that explore the effects of varying  $p$  as illustrated in Fig. 4.
+
+# 3.2 ESTIMATION OF PARTITION FUNCTION  $\Phi (\cdot)$
+
+To an estimate of  $\hat{p}_{\theta}$  in Eq. 11, it is imperative to accurately approximate the partition function  $\Phi(k)$ . The most straightforward approach to address this challenge involves fitting  $k$  distinct kernel density functions, each corresponding to a different class. By employing this method, the density function  $g_{\theta}$  can be effectively normalized:
+
+Baselines 1: Self-Normalization (SN). Following Gutmann & Hyvarinen (2012b); Wu et al. (2018); Mnih & Kavukcuoglu (2013), we assume the pre-trained neural network is perfectly expressively such that the unnormalized density function  $d_{\varphi}$  is self-normalized, i.e.,  $\Phi(k) = \text{constant}, \forall k \in \mathcal{V}$ . In this way, there is no need to explicitly compute the partition function  $\Phi(k)$ , which is given by
+
+$$
+\hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) \propto \sum_ {k = 1} ^ {K} \exp \left(- d _ {\varphi} (\mathbf {z}, \boldsymbol {\mu} (\boldsymbol {\eta} _ {k}))\right). \tag {12}
+$$
+
+Baselines 2: Normalization via Kernel Density Estimation. Kernel density estimation (KDE) (Chen, 2017; Kim & Scott, 2012) is a statistical method that is commonly used for probability density estimation. This approach is inherently non-parametric, providing flexibility in the choice of kernel functions (e.g., linear, Gaussian, exponential). Mathematically, the KDE-based estimation of partition function  $\Phi(k)$  can be formulated as follows: let  $\mathcal{D}_{\mathrm{in}}^k$  be the ID training data with label  $k$ ,
+
+$$
+\Phi_ {\mathrm {K D E}} (k) = \frac {1}{h \left| \mathcal {D} _ {\mathrm {i n}} ^ {k} \right| g _ {\boldsymbol {\theta}} (\mathbf {z} , k)} \sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {\mathrm {i n}} ^ {k}} \mathcal {K} \left(\frac {g _ {\boldsymbol {\theta}} (\mathbf {z} , k) - g _ {\boldsymbol {\theta}} \left(\mathbf {z} ^ {\prime} , k\right)}{h}\right), \tag {13}
+$$
+
+![](images/7a8d0f8f327a82702834b6bf741c76e0e47bfcc01b4aa5b4ad1c8ad10cfaff00.jpg)  
+Figure 2: Evaluations of different partition function estimation baselines on ImageNet: Left: MobileNetV2 and Right: ResNet50.
+
+![](images/9df13ac7b1b531062b2c6830a0d018eb374999d215cd9280362bc8a2af13ed9d.jpg)
+
+with  $h > 0$  as the bandwidth that determines the smoothing of the resulting density function.
+
+Experimental comparisons w.r.t. different baselines on the OOD detection benchmarks are summarized in Fig 2. As we demonstrate later, we propose to leverage the means of importance sampling for theoretically unbiased estimation instead, which provides stronger flexibility and generality.
+
+Ours: Importance Sampling-Based Approximation. To enhance the flexibility of density-based OOD detection, we consider a Monte Carlo method and construct a simple and analytically tractable estimator to theoretically unbiasedly approximate them by means of importance sampling (IS) (Liu et al., 2015; Tokdar & Kass, 2010; Ben Alaya et al., 2023). Specifically, let  $\hat{p}_o(\mathbf{z})$  be a tractable distribution that has been properly normalized such that  $\int \hat{p}_o(\mathbf{z})\mathrm{d}\mathbf{z} = 1$ , we draw data  $S = \left(\left(\mathbf{z}_o^1,\mathbf{y}_o^1\right),\dots,\left(\mathbf{z}_o^n,\mathbf{y}_o^n\right)\right)$  from the ID training data following the distribution  $\hat{p}_o(\mathbf{z})$ , and estimate  $\Phi (k)$  by
+
+$$
+\Phi_ {\mathrm {I S}} (k; S) = \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {g _ {\boldsymbol {\theta}} \left(\mathbf {z} _ {o} ^ {i} , k\right)}{\hat {p} _ {o} \left(\mathbf {z} _ {o} ^ {i}\right)}. \tag {14}
+$$
+
+For simplicity, we set  $\hat{p}_o$  as a uniform distribution over the training ID data  $\mathcal{D}_{\mathrm{in}}$  and  $n = N\times \alpha$  with  $\alpha$  as the sampling ratio. In practice, we find that  $\alpha = 10\%$  is sufficient to get decent performance. Besides, a desirable property of importance sampling is that the estimator  $\Phi_{\mathrm{IS}}(k;S)$  is theoretically unbiased (Liu et al., 2015), i.e.,  $\mathbb{E}_{S\sim \hat{p}_o}[\Phi_{\mathrm{IS}}(k;S)] = \Phi (k)$ . IS provides us with a simple but effective approach to answering the question  $\spadesuit$ .
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTS SETUP
+
+Baseline Methods. We compare our method with representative methods, including MSP (Hendrycks & Gimpel, 2016), ODIN (Liang et al., 2017), Energy (Liu et al., 2020), ASH (Djurisic et al., 2022), DICE (Sun & Li, 2022), ReAct (Sun et al., 2021), Mahalanobis (Maha) (Lee et al., 2018), GEM (Morteza & Li, 2022), KNN (Sun et al., 2022) and SHE (Zhang et al., 2022). It is worth noting that we have adopted the recommended configurations proposed by prior works, while concurrently standardizing the backbone architecture to ensure equitable comparisons.
+
+Evaluation Metrics. The detection performance is evaluated via two threshold-independent metrics: the false positive rate of OOD data is measured when the true positive rate of ID data reaches  $95\%$  (FPR95); and the
+
+Table 1: OOD detection on CIFAR benchmarks. We average the results across 6 OOD datasets.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>48.73</td><td>92.46</td><td>80.13</td><td>74.36</td></tr><tr><td>ODIN</td><td>24.57</td><td>93.71</td><td>58.14</td><td>84.49</td></tr><tr><td>Energy</td><td>26.55</td><td>94.57</td><td>68.45</td><td>81.19</td></tr><tr><td>DICE</td><td>20.83</td><td>95.24</td><td>49.72</td><td>87.23</td></tr><tr><td>ReAct</td><td>26.45</td><td>94.67</td><td>62.27</td><td>84.47</td></tr><tr><td>ASH</td><td>15.05</td><td>96.61</td><td>41.40</td><td>90.02</td></tr><tr><td>Maha</td><td>31.42</td><td>89.15</td><td>55.37</td><td>82.73</td></tr><tr><td>GEM</td><td>29.56</td><td>92.14</td><td>49.31</td><td>86.45</td></tr><tr><td>KNN</td><td>17.43</td><td>96.74</td><td>41.52</td><td>88.74</td></tr><tr><td>SHE</td><td>23.26</td><td>94.40</td><td>54.66</td><td>82.60</td></tr><tr><td>Ours</td><td>13.92</td><td>97.15</td><td>28.27</td><td>92.50</td></tr><tr><td>Ours+ASH</td><td>12.14</td><td>97.64</td><td>25.66</td><td>93.21</td></tr></table>
+
+area under the receiver operating characteristic curve (AUROC) is computed to quantify the probability of the ID case receiving a higher score compared to the OOD case. Reported performance
+
+Table 2: OOD detection results on the ImageNet benchmark with MobileNet-V2.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">iNaturalist</td><td colspan="2">SUN</td><td colspan="2">Places</td><td colspan="2">Textures</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>64.29</td><td>85.32</td><td>77.02</td><td>77.10</td><td>79.23</td><td>76.27</td><td>73.51</td><td>77.30</td><td>73.51</td><td>79.00</td></tr><tr><td>ODIN</td><td>55.39</td><td>87.62</td><td>54.07</td><td>85.88</td><td>57.36</td><td>84.71</td><td>49.96</td><td>85.03</td><td>54.20</td><td>85.81</td></tr><tr><td>Energy</td><td>59.50</td><td>88.91</td><td>62.65</td><td>84.50</td><td>69.37</td><td>81.19</td><td>58.05</td><td>85.03</td><td>62.39</td><td>84.91</td></tr><tr><td>ReAct</td><td>42.40</td><td>91.53</td><td>47.69</td><td>88.16</td><td>51.56</td><td>86.64</td><td>38.42</td><td>91.53</td><td>45.02</td><td>89.60</td></tr><tr><td>DICE</td><td>43.09</td><td>90.83</td><td>38.69</td><td>90.46</td><td>53.11</td><td>85.81</td><td>32.80</td><td>91.30</td><td>41.92</td><td>89.60</td></tr><tr><td>ASH</td><td>39.10</td><td>91.94</td><td>43.62</td><td>90.02</td><td>58.84</td><td>84.73</td><td>13.12</td><td>97.10</td><td>38.67</td><td>90.95</td></tr><tr><td>Maha</td><td>62.11</td><td>81.00</td><td>47.82</td><td>86.33</td><td>52.09</td><td>83.63</td><td>92.38</td><td>33.06</td><td>63.60</td><td>71.01</td></tr><tr><td>GEM</td><td>65.77</td><td>79.82</td><td>45.53</td><td>87.45</td><td>82.85</td><td>68.31</td><td>43.49</td><td>86.22</td><td>59.39</td><td>80.25</td></tr><tr><td>KNN</td><td>46.78</td><td>85.96</td><td>40.18</td><td>86.28</td><td>62.46</td><td>82.96</td><td>31.79</td><td>90.82</td><td>45.30</td><td>86.51</td></tr><tr><td>SHE</td><td>47.61</td><td>89.24</td><td>42.38</td><td>89.22</td><td>56.62</td><td>83.79</td><td>29.33</td><td>92.98</td><td>43.98</td><td>88.81</td></tr><tr><td>Ours</td><td>29.06</td><td>93.89</td><td>46.74</td><td>87.10</td><td>62.07</td><td>81.41</td><td>10.30</td><td>97.53</td><td>37.04</td><td>89.98</td></tr><tr><td>Ours+ASH</td><td>24.08</td><td>94.36</td><td>30.19</td><td>92.63</td><td>46.26</td><td>87.57</td><td>12.70</td><td>97.20</td><td>28.31</td><td>92.94</td></tr></table>
+
+results for our method are averaged over 5 independent runs for robustness. Due to the space limit, we provide the implementation details in the Appendix.
+
+# 4.2 MAIN RESULTS
+
+Evaluation on CIFAR Benchmarks. Following the setup in Sun & Li (2022), we consider CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009) as ID data and train DenseNet-101 (Huang et al., 2017) on them respectively using the cross-entropy loss. The feature dimension of the penultimate layer is 342. For both CIFAR-10 and CIFAR-100, the model is trained for 100 epochs, with batch size 64, weight decay 1e-4, and Nesterov momentum 0.9. The start learning rate is 0.1 and decays by a factor of 10 at 50th, 75th, and 90th epochs. There are six datasets for OOD detection with regard to CIFAR benchmarks: SVHN (Netzer et al., 2011), LSUN-Crop (Yu et al., 2015), LSUN-Resize (Yu et al., 2015), iSUN (Xu et al., 2015), Places (Zhou et al., 2017), and Textures (Cimpoi et al., 2014). At test time, all images are of size  $32 \times 32$ . Table 1 presented the performance of our approach and existing competitive baselines, where the proposed approach significantly outperforms existing methods. Specifically, comparing with the standard post-hoc methods, our method reveals  $3.51\%$  and  $0.41\%$  average improvements w.r.t. FPR95 and AUROC on the CIFAR-10 dataset, and  $13.25\%$  and  $3.76\%$  of the average improvements on the CIFAR-100 dataset. For advanced works that consider post-hoc enhancement, e.g., ASH and DICE, our method still significantly performs better on both datasets
+
+Evaluation on ImageNet Benchmark. We conduct experiments on the ImageNet benchmark, demonstrating the scalability of our method. Specifically, we inherit the exact setup from (Djurisic et al., 2022), where the ID dataset is ImageNet-1k (Krizhevsky et al., 2012), and OOD datasets include iNaturalist (Xiao et al., 2010b), SUN (Xiao et al., 2010a), Places365 (Zhou et al., 2017), and Textures (Cimpoi et al., 2014). We use the pre-trained MobileNetV2 (Sandler et al., 2018) models for ImageNet-1k provided by Pytorch (Paszke et al., 2019). At test time, all images are resized to  $224 \times 224$ . In Table 2, we reported the performances of four OOD test datasets respectively. It can be seen that our method reaches state-of-the-art with  $21.51\%$  FPR95 and  $95.48\%$  AUROC on average across four OOD datasets. Besides, we notice that ASH can further considerably improve our method by  $9.27\%$  and  $2.06\%$  w.r.t. FPR95 and AUROC respectively. We suspect that removing a large portion of activations at a late layer helps to improve the representative ability of features.
+
+# 4.3 ABLATION STUDY
+
+Extracted Features z. This paper follows the convention in feature-based OOD detectors (Sun et al., 2022; Zhang et al., 2022; Djurisic et al., 2022), where features from the penultimate layer are utilized to estimate uncertain scores for OOD detection. Fig 3 provides an experimental evaluation on the choice of working placement. It can be seen that feature from the deeper layer contributes to better OOD detection performance than shallower ones. This is likely due to the penultimate layer preserves more information than shallower layers.
+
+Sampling Ratio  $\alpha$ . In Fig 4, we analyze the effect of the sampling ratio  $\alpha$  on CIFAR-100 and ImageNet-1k datasets. We vary the random sampling ratio  $\alpha$  within  $\{1\%, 5\%, 10\%, 50\%, 100\}$ . We note several interesting observations: (1) The optimal OOD detection (measured by FPR95) remains similar under different random sampling ratios  $\alpha$  especially when  $\alpha \geq 10\%$ , which demon
+
+![](images/a6dec65beb3f8c718c864a811ffa40b5193ba47f722c1eeea89bcb26957a262c.jpg)  
+(a)
+
+![](images/ca90fc7ca257fdb740c154f8ffb951cf441ca2417c63bac1ccc725a67fd46aec.jpg)  
+(b)
+
+![](images/233308fccd02372de76c5118f892b76fb2a3c2afff22cd59dbf9de7a812ea609.jpg)  
+(c)
+
+![](images/03d1f61cdadced8ee86b13e90cb21a422f117493515f6349d4b2dd2617ac548b.jpg)  
+Figure 3: Ablation study using feature extractions from (a) the first, (b) the second, and (c) the last dense block of the DenseNet on the CIFAR-10.  
+(a) CIFAR-100  
+Figure 4: Ablation study w.r.t varying sampling ratio  $\alpha$  in red; and the norm coefficient  $p$  in blue.
+
+![](images/9b7bd73b1040a51823566504e656b82ed2791b8db6d05df2ae564a63836259c1.jpg)  
+(b) ImageNet-1K
+
+![](images/b94e2a1ac92a6e8e498ea6881226499c4bebef50ee39bf88ff024dcc3ff6f6ea.jpg)  
+(c) CIFAR-10
+
+![](images/40db03c6a52309f1a96dc6277fc7f0aa0925550aa2b9395aa3b0498c0cba009d.jpg)  
+(d) CIFAR-100
+
+strates the robustness of our method to the sampling ratio. (2) our method still achieves competitive performance on benchmarks even when sampling  $1\%$  total number of ID training data.
+
+Parameter Sensitivity of  $l_{p}$ . We conduct a comparative assessment of OOD performance while varying the  $l_{p}$  norm coefficient  $p$ , which directly governs the density function  $g_{\theta}$  on CIFAR benchmarks. The results, as depicted in Fig 4, reveal a consistent trend across both datasets. Notably, the FPR95 scores exhibit a clear minimum within the range of (2,3), suggesting that our proposed CONJNORM approach can efficiently identify the optimal normalization without the computational overhead. It is worth highlighting that when  $p = 2$ , signifying the Bregman divergence in Eq.(10) degenerates into the squared Euclidean distance (corresponding to Gaussian densities), the OOD performance does not attain its peak. This observation underscores the limitation of Gaussian assumptions and underscores the generality and effectiveness of our CONJNORM.
+
+Sensitivity of  $q$  with fixed  $l_{p}$  norm. We also conduct a comparative assessment of OOD performance by varying the value of  $q$  while fixing the  $l_{p}$  norm. The experimental results on CIFAR-100 under two cases where  $p = 2.5$  and  $p = 3.0$ . Note that the performance of our method tends to be more appealing when  $q$  satisfies the conjugate condition that  $q = p / (p - 1)$ , exceeding the case where  $q = 2.0$  by nearly  $10\%$  on FPR95. This empirically echoes Theorem 1.
+
+# 4.4 EXTENSION TO MORE PROTOCOLS
+
+In this section, we assess the versatility of the proposed CONJNORM approach in (1) Hard OOD detection and (2) Long-tailed OOD settings. For more extensions, please refer to the Appendix.
+
+# 4.4.1 HARD OOD DETECTION
+
+We consider hard OOD scenarios (Tack et al., 2020), of which the OOD data are semantically similar to that of the ID cases. With the CIFAR-100 as the ID dataset for training ResNet-50, we evaluate our
+
+Table 3: Evaluation on hard OOD detection tasks.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">LSUN-Fix</td><td colspan="2">ImageNet-Fix</td><td colspan="2">ImageNet-Resize</td><td colspan="2">CIFAR-10</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>90.43</td><td>63.97</td><td>88.46</td><td>67.32</td><td>86.38</td><td>71.24</td><td>89.67</td><td>66.47</td><td>88.73</td><td>67.25</td></tr><tr><td>ODIN</td><td>91.28</td><td>66.53</td><td>82.98</td><td>72.89</td><td>72.71</td><td>82.19</td><td>88.27</td><td>71.30</td><td>83.81</td><td>73.23</td></tr><tr><td>Energy</td><td>91.35</td><td>66.52</td><td>83.02</td><td>72.88</td><td>72.45</td><td>82.22</td><td>88.17</td><td>71.29</td><td>83.75</td><td>73.23</td></tr><tr><td>ReAct</td><td>93.70</td><td>64.52</td><td>83.36</td><td>73.47</td><td>62.85</td><td>85.79</td><td>89.09</td><td>69.87</td><td>82.25</td><td>73.41</td></tr><tr><td>Maha</td><td>90.54</td><td>56.43</td><td>83.24</td><td>61.84</td><td>75.83</td><td>64.71</td><td>90.27</td><td>54.36</td><td>84.97</td><td>59.33</td></tr><tr><td>KNN</td><td>91.70</td><td>69.70</td><td>80.58</td><td>76.46</td><td>68.90</td><td>85.98</td><td>83.28</td><td>75.57</td><td>81.12</td><td>76.93</td></tr><tr><td>SHE</td><td>93.52</td><td>63.56</td><td>85.62</td><td>70.75</td><td>81.54</td><td>76.97</td><td>89.32</td><td>71.52</td><td>87.50</td><td>70.70</td></tr><tr><td>Ours</td><td>85.80</td><td>72.48</td><td>76.14</td><td>78.77</td><td>65.38</td><td>86.29</td><td>84.87</td><td>75.88</td><td>78.05</td><td>78.35</td></tr></table>
+
+![](images/e297b1e113cbdedeafc7ce43c55695e5ba358655ed36ffad7570c0b4e23fbfe2.jpg)  
+Figure 5: Comparisons of varying  $q$  when  $p$  is fixed at 2.5 (Left) and 3.0 (Right) on CIFAR-100.
+
+![](images/c038dbe614121ec2f027e11a154050721035919231b3a4346d763c06256e4fbf.jpg)
+
+Table 4: Evaluation on long-tailed OOD detection tasks.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">LSUN</td><td colspan="2">iSUN</td><td colspan="2">Texture</td><td colspan="2">Places365</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>97.82</td><td>56.45</td><td>82.48</td><td>73.54</td><td>97.61</td><td>54.95</td><td>95.51</td><td>54.53</td><td>92.49</td><td>60.08</td><td>93.18</td><td>59.91</td></tr><tr><td>ODIN</td><td>98.70</td><td>48.32</td><td>64.80</td><td>83.70</td><td>97.47</td><td>52.41</td><td>95.99</td><td>49.27</td><td>91.56</td><td>58.49</td><td>89.70</td><td>58.44</td></tr><tr><td>Energy</td><td>98.81</td><td>43.10</td><td>47.03</td><td>89.41</td><td>97.37</td><td>50.77</td><td>95.82</td><td>46.25</td><td>91.73</td><td>57.09</td><td>86.15</td><td>57.32</td></tr><tr><td>Maha</td><td>74.01</td><td>83.67</td><td>77.88</td><td>72.63</td><td>49.42</td><td>85.52</td><td>63.10</td><td>81.28</td><td>94.37</td><td>53.00</td><td>71.76</td><td>75.22</td></tr><tr><td>MSP+RP</td><td>97.76</td><td>56.45</td><td>82.33</td><td>73.54</td><td>97.51</td><td>54.76</td><td>95.73</td><td>54.13</td><td>92.65</td><td>59.73</td><td>93.20</td><td>59.72</td></tr><tr><td>ODIN+RW</td><td>98.82</td><td>52.94</td><td>83.79</td><td>69.28</td><td>96.10</td><td>50.64</td><td>96.95</td><td>45.14</td><td>93.36</td><td>51.5</td><td>93.80</td><td>53.90</td></tr><tr><td>Energy+RW</td><td>98.86</td><td>49.07</td><td>77.30</td><td>77.32</td><td>96.25</td><td>50.91</td><td>96.86</td><td>45.27</td><td>93.03</td><td>54.02</td><td>92.46</td><td>55.32</td></tr><tr><td>KNN</td><td>64.39</td><td>86.16</td><td>56.13</td><td>84.24</td><td>45.36</td><td>88.39</td><td>34.36</td><td>89.86</td><td>90.31</td><td>60.09</td><td>58.11</td><td>81.75</td></tr><tr><td>Ours</td><td>40.16</td><td>91.00</td><td>45.72</td><td>87.64</td><td>41.89</td><td>90.42</td><td>40.50</td><td>86.80</td><td>91.74</td><td>58.44</td><td>52.00</td><td>82.86</td></tr></table>
+
+method on 4 hard OOD datasets, namely, LSUN-Fix (Yu et al., 2015), ImageNet-Fix (Krizhevsky et al., 2012), ImageNet-Resize (Krizhevsky et al., 2012), and CIFAR-10. The model is trained for 200 epochs, with batch size 128, weight decay 5e-4 and Nesterov momentum 0.9. The start learning rate is 0.1 and decays by a factor of 5 at 60th, 12th, 160th epochs. We select a set of strong baselines that are competent in hard OOD detection, and the experiments are summarized in Table 3. It can be seen that our method can beat the state-of-the-art across the considered datasets, even for the challenging CIFAR-100 versus CIFAR-10 setting. The reason is that our  $l_{p}$  norm-induced density function can better capture the ID data distribution.
+
+# 4.4.2 LONG-TAILED OOD DETECTION
+
+We consider long-tailed OOD scenarios (Wang et al., 2022; Bai et al., 2023), of which the ID training data exhibits an imbalanced class distribution. We use the long-tailed versions of CIFAR datasets with the setting in Cao et al. (2019); Zhong et al. (2021). It is by controlling the degrees of data imbalance with an imbalanced factor  $\beta = N_{\mathrm{max}} / N_{\mathrm{min}}$ , where  $N_{\mathrm{max}}$  and  $N_{\mathrm{min}}$  are the numbers of training samples belonging to the most and the least frequent classes. Following (Zhong et al., 2021; Zhou et al., 2020), we pre-train the ResNet-32 (He et al., 2016) network with  $\beta = 50$  on CIFAR-100 for 200 epochs with batch size 128, weight decay 2e-4 and Nesterov momentum 0.9. The start learning rate is 0.1 and decays by a factor of 5 at the 160-th, 180-th epochs. The performance of our methods and baselines are shown in Table 4, where we introduce the strategies of Replacing (RP) and Reweighting (RW) in (Jiang et al., 2023) to modify previous OOD scoring functions. The performance gain in Table 4 empirically demonstrates that using a uniform ID class distribution does not make our method incompatible with the model that is pre-trained with class-imbalanced data.
+
+# 5 CONCLUSION
+
+In this paper, we present a theoretical framework for studying density-based OOD detection. By establishing connections between the exponential family of distributions and Bregman divergence, we provide a unified principle for designing scoring functions. Given the expansive function space for selecting Bregman divergence, we propose a pair of conjugate functions to simplify the search process. To address the challenging problem of the partition function, we introduce a computationally tractable and theoretically unbiased estimator through importance sampling. Empirically, our method outperforms numerous prior methods by a significant margin on several standard benchmark datasets using various protocols. Since we only consider a pair of conjugate functions in finding Bregman divergence and evaluate our method on Convolutional neural networks. In the future, it is interesting to delve deeper into the design of Bregman divergence and incorporate large-scale pre-trained Vision-Language Models (VLMs).
+
+# 6 ACKNOWLEDGEMENT
+
+This work is partially supported by the Australian Research Council (CE200100025 and DE240100105). Li gratefully acknowledges the support of the AFOSR Young Investigator Program under award number FA9550-23-1-0184, National Science Foundation (NSF) Award No. IIS-2237037 & IIS-2331669, and Office of Naval Research under grant number N00014-23-1-2643.
+
+# 7 ETHIC STATEMENT
+
+This paper does not raise any ethical concerns. This study does not involve any human subjects practices to data set releases, potentially harmful insights, methodologies and applications, potential conflicts of interest and sponsorship, discrimination/bias/fairness concerns, privacy and security issues. legal compliance, and research integrity issues.
+
+# 8 REPRODUCIBILITY STATEMENT
+
+To make all experiments reproducible, we have listed all detailed hyper-parameters. We upload source codes and instructions in the supplementary materials.
+
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+
+Table 5: OOD detection on CIFAR100 benchmarks. We average the results across 6 OOD datasets.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td>Metrics</td><td>MSP</td><td>ODIN</td><td>Energy</td><td>ReAct</td><td>DICE</td><td>Maha</td><td>GEM</td><td>KNN</td><td>ASH</td><td>Ours</td></tr><tr><td>FPR95↓</td><td>79.29</td><td>77.47</td><td>76.66</td><td>69.95</td><td>73.06</td><td>71.75</td><td>65.21</td><td>62.22</td><td>65.29</td><td>54.47</td></tr><tr><td>AUROC↑</td><td>79.9</td><td>74.36</td><td>80.14</td><td>81.99</td><td>79.86</td><td>63.14</td><td>77.81</td><td>84.22</td><td>81.36</td><td>87.26</td></tr></table>
+
+# A APPENDIX
+
+# A.1 LIMITATIONS
+
+The limitation of this work lies in manually searching a good value of  $p$  to determine Bregman divergence. We do not test our method on large-scale models.
+
+# A.2 OOD DATASET
+
+For experiments where CIFAR benchmarks are the ID data, we adopt SVHN (Netzer et al., 2011), LSUN-Crop (Yu et al., 2015), LSUN-Resize (Yu et al., 2015), iSUN (Xu et al., 2015), Places (Zhou et al., 2017), and Textures (Cimpoi et al., 2014) as the OOD datasets. For experiments where ImageNet-1K is the ID data, we adopt iNaturalist (Xiao et al., 2010b), SUN (Xiao et al., 2010a), Places365 (Zhou et al., 2017), and Textures (Cimpoi et al., 2014) and the OOD dataset.
+
+# A.3 IMPLEMENTATION DETAILS.
+
+Similar to DICE (Sun & Li, 2022), we adopt Tiny-ImageNet-200 (Le & Yang, 2015) as the auxiliary OOD data with the searching space of  $p$  as (1,3]. We remove those data whose labels coincide with ID cases. We set  $p = 2.2$  for experiments in CIFAR-10,  $p = 2.5$  for experiments in CIFAR-100,  $p = 1.5$  and  $p = 1.8$  for experiments in ImageNet-1k on ResNet50 and MobileNetv2 respectively.
+
+# A.4 RESULTS WITH DIFFERENT BACKBONES
+
+In the main paper, we have shown that our method is competitive on DenseNet and MobileNet. In this section, we show in Table 5 and Table 6 that the strong performance of our method holds on ResNet50 (He et al., 2016). All the numbers reported are averaged over OOD test datasets described in Section 4.2. For ImageNet-1k, We use the pre-trained models provided by Pytorch. At test time, all images are resized to  $224 \times 224$ . For CIFAR-100, the model is trained for 200 epochs, with batch size 128, weight decay 5e-4 and Nesterov momentum 0.9. The start learning rate is 0.1 and decays by a factor of 5 at 60th, 120th and 160th epochs. At test time, all images are of size  $32 \times 32$ .
+
+# A.5 OOD DETECTION WITH CONTRASTIVE REPRESENTATIONS
+
+We explore the compatibility of our method with contrastive representations. Closely following the training protocol in Ming et al. (2022b); Khosla et al. (2020), we pre-train ResNet-34 (He et al., 2016) on CIFAR-100 with the SupCon (Khosla et al., 2020) and CIDER (Ming et al., 2022b) losses respectively. We train the model using stochastic gradient descent for 500 epochs with batch size 512, Nesterov momentum 0.9, and weight decay 1e-4. The initial learning rate is 0.5 with cosine scheduling. Table 7 demonstrates, under both SupCon and CIDER settings, ours consistently outperforms the Maha, SHE and KNN scores by a large margin, highlighting our method's effectiveness.
+
+# A.6 MORE RESULTS ON LONG-TAILED OOD DETECTION
+
+Continuing from Section 4.4.2, we further test the compatibility of our method to the model that is pre-trained with class-imbalanced data. In this section, we use the long-tailed versions of CIFAR-10 datasets with the setting in Cao et al. (2019); Zhong et al. (2021) with an imbalanced factor  $\beta = 50$ . Following (Zhong et al., 2021; Zhou et al., 2020), we pre-train the ResNet-32 (He et al., 2016) network with  $\beta = 50$  on CIFAR-10 for 200 epochs with batch size 128, weight decay 2e-4 and Nesterov momentum 0.9. The start learning rate is 0.1 and decays by a factor of 5 at the 160-th, 180-th epochs. The performance of our methods and baselines are shown in Table 8.
+
+Table 6: OOD detection results on the ImageNet benchmark with ResNet-50.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">iNaturalist</td><td colspan="2">SUN</td><td colspan="2">Places</td><td colspan="2">Textures</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>54.99</td><td>87.74</td><td>70.83</td><td>80.86</td><td>73.99</td><td>79.76</td><td>68.00</td><td>79.61</td><td>66.95</td><td>81.99</td></tr><tr><td>ODIN</td><td>47.66</td><td>89.66</td><td>60.15</td><td>84.59</td><td>67.89</td><td>81.78</td><td>50.23</td><td>85.62</td><td>56.48</td><td>85.41</td></tr><tr><td>Energy</td><td>55.72</td><td>89.95</td><td>59.26</td><td>85.89</td><td>64.92</td><td>82.86</td><td>53.72</td><td>85.99</td><td>58.41</td><td>86.17</td></tr><tr><td>ReAct</td><td>20.38</td><td>96.22</td><td>24.20</td><td>94.20</td><td>33.85</td><td>91.58</td><td>47.30</td><td>89.80</td><td>31.43</td><td>92.95</td></tr><tr><td>DICE</td><td>25.63</td><td>94.49</td><td>35.15</td><td>90.83</td><td>46.49</td><td>87.48</td><td>31.72</td><td>90.30</td><td>34.75</td><td>90.77</td></tr><tr><td>Maha</td><td>97.00</td><td>52.65</td><td>98.50</td><td>42.41</td><td>98.40</td><td>41.79</td><td>55.80</td><td>85.01</td><td>87.43</td><td>55.47</td></tr><tr><td>GEM</td><td>51.67</td><td>81.66</td><td>68.87</td><td>73.78</td><td>79.52</td><td>67.34</td><td>35.73</td><td>86.54</td><td>58.95</td><td>77.33</td></tr><tr><td>KNN</td><td>59.77</td><td>85.89</td><td>68.88</td><td>80.08</td><td>78.15</td><td>74.10</td><td>10.90</td><td>97.42</td><td>54.68</td><td>84.37</td></tr><tr><td>SHE</td><td>45.35</td><td>90.15</td><td>45.09</td><td>87.93</td><td>54.19</td><td>84.69</td><td>34.22</td><td>90.18</td><td>44.71</td><td>88.24</td></tr><tr><td>Ours</td><td>9.62</td><td>97.97</td><td>37.75</td><td>90.13</td><td>48.99</td><td>86.60</td><td>9.61</td><td>97.74</td><td>26.49</td><td>93.11</td></tr></table>
+
+Table 7: OOD detection results on CIFAR-100 under contrastive learning.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">LSUN</td><td colspan="2">iSUN</td><td colspan="2">Texture</td><td colspan="2">Places365</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>SupCon+Maha</td><td>14.47</td><td>97.31</td><td>92.81</td><td>67.81</td><td>79.33</td><td>80.71</td><td>50.35</td><td>79.79</td><td>95.93</td><td>54.17</td><td>66.58</td><td>75.96</td></tr><tr><td>SupCon+SHE</td><td>28.35</td><td>94.34</td><td>72.53</td><td>78.81</td><td>99.18</td><td>19.31</td><td>72.16</td><td>68.07</td><td>83.17</td><td>75.41</td><td>71.08</td><td>67.19</td></tr><tr><td>SupCon+KNN</td><td>38.68</td><td>92.61</td><td>43.16</td><td>91.43</td><td>67.89</td><td>85.04</td><td>58.46</td><td>86.65</td><td>75.18</td><td>77.22</td><td>56.67</td><td>86.59</td></tr><tr><td>SupCon+Ours</td><td>28.74</td><td>94.58</td><td>15.43</td><td>97.28</td><td>56.97</td><td>88.54</td><td>46.03</td><td>90.11</td><td>74.97</td><td>77.65</td><td>44.43</td><td>89.63</td></tr><tr><td>CIDER+Maha</td><td>18.52</td><td>96.36</td><td>88.86</td><td>71.93</td><td>79.55</td><td>79.76</td><td>54.31</td><td>77.89</td><td>95.47</td><td>52.93</td><td>67.34</td><td>75.77</td></tr><tr><td>CIDER+SHE</td><td>28.16</td><td>94.09</td><td>69.06</td><td>82.60</td><td>99.69</td><td>15.51</td><td>75.57</td><td>62.84</td><td>87.21</td><td>72.41</td><td>71.94</td><td>65.51</td></tr><tr><td>CIDER+KNN</td><td>22.93</td><td>95.17</td><td>16.17</td><td>96.33</td><td>71.62</td><td>80.85</td><td>45.35</td><td>90.08</td><td>74.12</td><td>67.25</td><td>46.23</td><td>87.31</td></tr><tr><td>CIDER+Ours</td><td>15.87</td><td>96.55</td><td>6.04</td><td>98.76</td><td>52.45</td><td>88.32</td><td>31.97</td><td>93.31</td><td>72.31</td><td>75.94</td><td>35.73</td><td>90.58</td></tr></table>
+
+# A.7 DETAILED CIFAR RESULTS
+
+Table 10 and Table 11 supplement Table 1 in the main text, as they display the full results on each of the 6 OOD datasets for DenseNet trained on CIFAR-10 and CIFAR-100 respectively. Table 12 supplement Table 5, as it displays the full results on each of the 6 OOD datasets for ResNet50 trained on CIFAR-100 respectively
+
+# A.8 DISCUSSION ON DEEP GENERATIVE MODELS FOR DENSITY ESTIMATION
+
+Since density estimation plays a key role in our method, our work is related to deep generative models that achieve empirically promising results based on neural networks. Generally, there are two families of DGMs for density estimation: 1) autoregressive models (Germain et al., 2015; Uria et al., 2016; Papamakarios et al., 2017) that decompose the density into the product of conditional densities based on probability chain rule where Each conditional probability is modeled by a parametric density (e.g., Gaussian or mixture of Gaussian) whose parameters are learned by neural networks, and 2) normalizing flows (Rezende & Mohamed, 2015; Ballé et al., 2015; Dinh et al., 2016; Grover et al., 2018; Albergo & Vanden-Eijnden, 2022) that represent input as an invertible transformation of a latent variable with known density with the invertible transformation as a composition of a series of simple functions. While using DGMs for density estimation seems to be a valid and intuitive option for density-based OOD detection, this requires training a DGM from scratch and therefore violates the principle of post-hoc OOD detection, i.e., only pre-trained models at hand are expected to be used to detect OOD data from streaming data at the interference stage. Besides, Zhang et al. (2021) finds that DGMS tend to assign higher probabilities or densities to OOD images than images from the training distribution. We also explore the possibility of integrating pre-trained Diffusion models (Peebles & Xie, 2023; Rombach et al., 2022) into zero-shot class-conditioned density estimation based on Eq.(1) in Li et al. (2023). Unfortunately, the computation is intractable due to the integral. Although authors in Li et al. (2023) use a simplified ELBO for approximation, there is no theoretical guarantee that the ELBO can well align with the data density not to mention the computational-inefficient inference of diffusion models. We will leave this challenge as our future work.
+
+Table 8: Evaluation on long-tailed OOD detection tasks on CIFAR-10.  $\uparrow$  indicates larger values are better and vice versa. The best result in each column is shown in bold.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">LSUN</td><td colspan="2">iSUN</td><td colspan="2">Texture</td><td colspan="2">Places365</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>90.12</td><td>78.00</td><td>83.62</td><td>86.98</td><td>53.29</td><td>90.00</td><td>77.23</td><td>77.24</td><td>87.42</td><td>77.88</td><td>78.34</td><td>82.02</td></tr><tr><td>ODIN</td><td>99.43</td><td>48.10</td><td>78.52</td><td>79.10</td><td>42.64</td><td>88.54</td><td>77.22</td><td>74.59</td><td>84.40</td><td>63.45</td><td>76.44</td><td>70.76</td></tr><tr><td>Energy</td><td>99.74</td><td>35.80</td><td>75.83</td><td>77.12</td><td>42.15</td><td>88.29</td><td>79.13</td><td>71.71</td><td>83.78</td><td>62.56</td><td>76.13</td><td>67.10</td></tr><tr><td>Maha</td><td>80.80</td><td>81.86</td><td>98.07</td><td>67.41</td><td>75.34</td><td>85.17</td><td>74.66</td><td>80.83</td><td>92.51</td><td>68.22</td><td>84.28</td><td>76.70</td></tr><tr><td>MSP+RP</td><td>82.64</td><td>78.00</td><td>67.21</td><td>86.98</td><td>50.30</td><td>90.00</td><td>75.80</td><td>77.24</td><td>80.71</td><td>77.88</td><td>71.33</td><td>82.02</td></tr><tr><td>ODIN+RW</td><td>88.64</td><td>70.74</td><td>52.37</td><td>84.93</td><td>63.97</td><td>84.37</td><td>82.62</td><td>68.93</td><td>74.70</td><td>79.47</td><td>72.46</td><td>77.69</td></tr><tr><td>Energy+RW</td><td>94.10</td><td>67.02</td><td>54.60</td><td>87.21</td><td>52.19</td><td>89.10</td><td>79.65</td><td>71.87</td><td>75.90</td><td>80.42</td><td>71.29</td><td>79.12</td></tr><tr><td>KNN</td><td>77.29</td><td>86.80</td><td>77.90</td><td>78.91</td><td>50.98</td><td>93.54</td><td>76.56</td><td>88.73</td><td>57.70</td><td>77.31</td><td>68.09</td><td>85.06</td></tr><tr><td>Ours</td><td>28.89</td><td>93.65</td><td>55.36</td><td>85.55</td><td>48.08</td><td>88.78</td><td>54.45</td><td>85.29</td><td>78.77</td><td>79.05</td><td>53.11</td><td>86.46</td></tr></table>
+
+# A.9 INTRACTABLE LEARNING OF THE EXPONENTIAL FAMILY NATURAL PARAMETER
+
+Given the fact that  $\int \hat{p}_{\boldsymbol{\theta}}(\mathbf{z}|k)\mathrm{d}\mathbf{z} = 1$  we then have:
+
+$$
+\int \exp \left\{\mathbf {z} ^ {\top} \boldsymbol {\eta} _ {k} - \psi (\boldsymbol {\eta} _ {k}) - g _ {\psi} (\mathbf {z}) \right\} d \mathbf {z} = 1 \tag {15}
+$$
+
+Eq. 15 means that, for any known  $\psi(\cdot)$  and  $g_{-}\psi(\cdot)$ , one can learn the natural parameter  $\pmb{\eta}_{k}$  by solving the following:
+
+$$
+\exp \psi (\boldsymbol {\eta} _ {k}) = \int \exp \left\{\mathbf {z} ^ {\top} \boldsymbol {\eta} _ {k} - g _ {\psi} (\mathbf {z}) \right\} d \mathbf {z} \tag {16}
+$$
+
+Since the right side of Eq. 16 includes the integral over latent feature space that is high-dimensional, learning the natural parameter of an Exp. Family is said to be intractable.
+
+# A.10 THEORETICAL JUSTIFICATION
+
+Let  $\mathcal{B}$  denotes the Borel  $\sigma$ -algebra on  $\mathcal{Z}$  and  $\mathcal{P}(\mathcal{Z})$  denotes the set of all probability measures on  $(\mathcal{Z},\mathcal{B})$ . We recall the following definitions:
+
+Definition 3 (Total Variation). Let  $\mathcal{P}_1, \mathcal{P}_2 \in \mathcal{P}(\mathcal{Z})$ . The total variation(TV) is defined by:
+
+$$
+\delta \left(\mathcal {P} _ {I}, \mathcal {P} _ {2}\right) = \sup  _ {A \in \mathcal {B}} | \mathcal {P} _ {I} (A) - \mathcal {P} _ {2} (A) | \tag {17}
+$$
+
+We use the following characterization of TV (See Müller (1997) Theorem 5.4):
+
+Lemma 1. Let  $\mathcal{P}_1, \mathcal{P}_2 \in \mathcal{P}(\mathcal{Z})$  and let  $\mathcal{F}$  denotes the unit ball in  $L^{\infty}(\mathcal{Z})$ , i.e.,
+
+$$
+\mathcal {F} := \{f \in L ^ {\infty} (\mathcal {Z}) | \| f \| _ {\infty} \leq 1 \} \tag {18}
+$$
+
+then we have the following characterization for the TV distance,
+
+$$
+\delta \left(\mathcal {P} _ {I}, \mathcal {P} _ {2}\right) = \sup  _ {f \in \mathcal {F}} \left| \mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {I}} f (\mathbf {z}) - \mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {2}} f (\mathbf {z}) \right| \tag {19}
+$$
+
+Next, let us recall the definition of Kullback-Leibler(KL) divergence,
+
+Definition 4 (KL Divergence). Let  $\mathcal{P}_1, \mathcal{P}_2 \in \mathcal{P}(\mathcal{Z})$  be two probability measures with density functions  $p_1$  and  $p_2$  respectively. The KL divergence is defined by
+
+$$
+K L \left(\mathcal {P} _ {I} \mid \mid \mathcal {P} _ {2}\right) := \int_ {\mathbf {z} \in \mathcal {Z}} p _ {1} (\mathbf {z}) \ln \frac {p _ {1} (\mathbf {z})}{p _ {2} (\mathbf {z})} d \mathbf {z} \tag {20}
+$$
+
+whenever the above integral is defined.
+
+Next, recall the following standard lemma that computes KL divergence between exponential family distributions.
+
+Lemma 2 (Relation between Bregman Divergences and KL Divergence (Banerjee et al., 2005)). Let  $\mathcal{P}_1, \mathcal{P}_2 \in \mathcal{P}(\mathcal{Z})$  conform to exponential family distributions with the corresponding density functions  $p_1$  and  $p_2$  are parameterized by  $\eta_1$  and  $\eta_2$  respectively, then we have the following:
+
+$$
+K L \left(\mathcal {P} _ {I} \mid \mid \mathcal {P} _ {2}\right) = d _ {\varphi} \left(\boldsymbol {\mu} \left(\boldsymbol {\eta} _ {1}\right), \boldsymbol {\mu} \left(\boldsymbol {\eta} _ {2}\right)\right) \tag {21}
+$$
+
+where  $d_{\varphi}(\cdot ,\cdot)$  is the so-called Bregman Divergence.
+
+Next, we recall the following inequality that bounds the TV by KL divergence. (see Tsybakov (2008), Lemma 2.5 and Lemma 2.6])
+
+Lemma 3 (Pinsker inequality). Let  $\mathcal{P}_1, \mathcal{P}_2 \in \mathcal{P}(\mathcal{Z})$  then we have the following:
+
+$$
+\delta \left(\mathcal {P} _ {I}, \mathcal {P} _ {2}\right) \leq \sqrt {\frac {1}{2} K L \left(\mathcal {P} _ {I} \mid \mid \mathcal {P} _ {2}\right)} \tag {22}
+$$
+
+Theorem 2. We have the following bound:
+
+$$
+\mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {l n}} \left[ \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) \right] - \mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {o u t}} \left[ \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) \right] \leq \alpha \tag {23}
+$$
+
+where  $\alpha := \frac{1}{K} \sum_{k=1}^{K} \sqrt{\frac{1}{2} d_{\varphi}(\pmb{\mu}(\pmb{\eta}_k), \pmb{\mu}(\pmb{\eta}_{out}))}$
+
+Theorem 2 bounds the measure  $D$  in terms of Bregman divergence between  $\mu(\eta_k)$  and  $\mu(\eta_{\mathrm{out}})$ . It can be observed that  $D$  will converge to 0 as  $\alpha \to 0$ . This indicates that the performance of our method can be guaranteed by a sufficiently discriminative feature space where the averaged Bergman divergence between ID-class means and OOD data mean is sufficiently large. This theory is empirically justified by our results in Section A.5 where CIDER are more beneficial to our method than SupCon with the former learning more powerful feature representations than the latter.
+
+Proof. First, notice that  $\hat{p}_{\boldsymbol{\theta}}(\mathbf{z}) \in [0,1], \forall \mathbf{z} \in \mathcal{Z}$ . Therefore, by Lemma 1, we have,
+
+$$
+\mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {\mathrm {i n}}} [ \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) ] - \mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {\mathrm {o u t}}} [ \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) ] \leq \delta \left(\mathcal {P} _ {\mathrm {i n}}, \mathcal {P} _ {\mathrm {o u t}}\right) \tag {24}
+$$
+
+Next, recall that  $p_{\mathrm{in}}(\mathbf{z}) = \frac{1}{K}\sum_{k=1}^{K}p_{\mathrm{in}}(\mathbf{z}|k)$ , let  $\mathcal{P}_{\mathrm{in}}^k$  denotes the probability distribution corresponding to  $p_{\mathrm{in}}(\cdot|k)$  and by triangle inequality and the definition of total variation we obtain
+
+$$
+\begin{array}{l} \delta \left(\mathcal {P} _ {\text {i n}}, \mathcal {P} _ {\text {o u t}}\right) = \delta \left(\frac {1}{K} \sum_ {k = 1} ^ {K} \mathcal {P} _ {\text {i n}} ^ {k}, \mathcal {P} _ {\text {o u t}}\right) = \sup  _ {A \in \mathcal {B}} \left| \frac {1}{K} \sum_ {k = 1} ^ {K} \mathcal {P} _ {\text {i n}} ^ {k} (A) - \mathcal {P} _ {\text {o u t}} (A) \right| (25) \\ \leq \frac {1}{K} \sum_ {k = 1} ^ {K} \sup  _ {A \in \mathcal {B}} \left| \mathcal {P} _ {\text {i n}} ^ {k} (A) - \mathcal {P} _ {\text {o u t}} (A) \right| (26) \\ = \frac {1}{K} \sum_ {k = 1} ^ {K} \delta \left(\mathcal {P} _ {\text {i n}} ^ {k} - \mathcal {P} _ {\text {o u t}}\right) (27) \\ \end{array}
+$$
+
+Finally, by Lemma 2 and Lemma 3, we have:
+
+$$
+\delta \left(\mathcal {P} _ {\text {i n}} ^ {k} - \mathcal {P} _ {\text {o u t}}\right) \leq \sqrt {\frac {1}{2} K L \left(\mathcal {P} _ {1} \mid \mid \mathcal {P} _ {2}\right)} = \sqrt {\frac {1}{2} d _ {\varphi} \left(\boldsymbol {\mu} \left(\boldsymbol {\eta} _ {k}\right) , \boldsymbol {\mu} \left(\boldsymbol {\eta} _ {\text {o u t}}\right)\right)} \tag {28}
+$$
+
+Putting all together, we obtain
+
+$$
+\mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {\text {i n}}} [ \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) ] - \mathbb {E} _ {\mathbf {z} \in \mathcal {P} _ {\text {o u t}}} [ \hat {p} _ {\boldsymbol {\theta}} (\mathbf {z}) ] \leq \frac {1}{K} \sum_ {k = 1} ^ {K} \sqrt {\frac {1}{2} d _ {\varphi} (\boldsymbol {\mu} (\boldsymbol {\eta} _ {k}) , \boldsymbol {\mu} (\boldsymbol {\eta} _ {\text {o u t}}))} \tag {29}
+$$
+
+and the proof is complete.
+
+# A.11 CONTRIBUTION SUMMARY
+
+The contributions of our method are summarised as follows:
+
+- It is always non-trivial to generalize from a specific distribution/distance to a broader distribution/distance family since this will trigger an important question to the optimal design of the underlying distribution  $(\clubsuit)$ . To answer this question, we explore the conjugate relationship as a guideline for the design. Compared with other hand-crafted choices, our proposed  $l_{p}$  norm is general and well-defined, offering simplicity in determining its conjugate pair. By searching the optimal value of  $p$  for each dataset, we can flexibly model ID data in a data-driven manner instead of blindly adopting a narrow Gaussian distributional assumption in prior work, i.e., GEM (Morteza & Li, 2022) and Maha (Lee et al., 2018).
+
+Table 9: Additional results of long-tailed OOD detection on Cifar-100, where we consider two baselines: (a) the ID training data is with class imbalance and (b) the ID training data is with class balance.  $\uparrow$  indicates larger values are better and vice versa.  
+
+<table><tr><td rowspan="2">Baseline</td><td colspan="2">Maha</td><td colspan="2">GEM</td><td colspan="2">KNN</td><td colspan="2">Ours</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>(a)</td><td>71.76</td><td>75.22</td><td>66.82</td><td>76.97</td><td>58.11</td><td>81.75</td><td>52.00</td><td>82.86</td></tr><tr><td>(b)</td><td>67.39</td><td>77.16</td><td>56.67</td><td>82.93</td><td>60.11</td><td>79.22</td><td>48.46</td><td>84.02</td></tr></table>
+
+- Our proposed framework reveals the core components in density estimation for OOD detection, which was overlooked by most heuristic-based OOD papers. In this way, The framework not only inherits prior work including GEM (Morteza & Li, 2022) and Maha (Lee et al., 2018) but also motivates further work to explore more effective designing principles of density functions for OOD detection.  
+- We demonstrate the superior performance of our method on several OOD detection benchmarks (CIFAR10/100 and ImageNet-1K), different model architectures (DenseNet, ResNet, and MobileNet), and different pre-training protocols (standard classification, long-tailed classification and contrastive learning).
+
+# A.12 LIST OF ASSUMPTIONS
+
+The assumptions made in our method are given as follows:
+
+1. The ID class prior is uniform, i.e.,  $\hat{p}_{\theta}(k) = \frac{1}{K}$ .  
+2.  $g_{\varphi}(\cdot) = const$  and  $\psi (\cdot) = \frac{1}{2}\| \big\| \cdot \| \| _p^2$
+
+We note that 1) Assumption 1 is made in many post-hoc OOD detection methods either explicitly or implicitly (Jiang et al., 2023). Experiments in Section 4.4.2 show that our method still outperforms in long-tailed scenarios with Assumption 1, and 2) Assumption 2 helps to reduce the complexity of the exponential family distribution. While it is possible to parameterize the exponential family distribution in a more complicated manner, our proposed simple version suffices to perform well.
+
+# A.13 A CLOSER LOOK AT EXPERIMENTS ON LONG-TAILED OOD DETECTION
+
+As shown in Table 9, all methods that involve the use of ID training data suffer from a decrease in their averaged OOD detection performance when the ID training data is with class imbalance. Note that we keep using the network pre-trained on the long-tailed version of CIFAR-100 for fair comparison. Even so, our method consistently outperforms in both scenarios, which implies the robustness of our method. We suspect the reason is that the flexibility of the norm coefficient provides us with the chance to find a compromised distribution from the exponential family.
+
+Table 10: Detailed results on six common OOD benchmark datasets: Textures, SVHN, Places365, LSUN-Crop, LSUN-Resize, and iSUN. For each ID dataset, we use the same DenseNet pretrained on CIFAR-100.  $\uparrow$  indicates larger values are better and vice versa.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">LSUN-C</td><td colspan="2">LSUN-R</td><td colspan="2">iSUN</td><td colspan="2">Texture</td><td colspan="2">Places365</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>81.70</td><td>75.40</td><td>60.49</td><td>85.60</td><td>85.24</td><td>69.18</td><td>85.99</td><td>70.17</td><td>84.79</td><td>71.48</td><td>82.55</td><td>74.31</td><td>80.13</td><td>74.36</td></tr><tr><td>ODIN</td><td>41.35</td><td>92.65</td><td>10.54</td><td>97.93</td><td>65.22</td><td>84.22</td><td>67.05</td><td>83.84</td><td>82.34</td><td>71.48</td><td>82.32</td><td>76.84</td><td>58.14</td><td>84.49</td></tr><tr><td>Energy</td><td>87.46</td><td>81.85</td><td>14.72</td><td>97.43</td><td>70.65</td><td>80.14</td><td>74.54</td><td>78.95</td><td>84.15</td><td>71.03</td><td>79.20</td><td>77.72</td><td>68.45</td><td>81.19</td></tr><tr><td>React</td><td>83.81</td><td>81.41</td><td>25.55</td><td>94.92</td><td>60.08</td><td>87.88</td><td>65.27</td><td>86.55</td><td>77.78</td><td>78.95</td><td>82.65</td><td>74.04</td><td>62.27</td><td>84.47</td></tr><tr><td>DICE</td><td>54.65</td><td>88.84</td><td>0.93</td><td>99.74</td><td>49.40</td><td>91.04</td><td>48.72</td><td>90.08</td><td>65.04</td><td>76.42</td><td>79.58</td><td>77.26</td><td>49.72</td><td>87.23</td></tr><tr><td>ASH</td><td>25.02</td><td>95.76</td><td>5.52</td><td>98.94</td><td>51.33</td><td>90.12</td><td>46.67</td><td>91.30</td><td>34.02</td><td>92.35</td><td>85.86</td><td>71.62</td><td>41.40</td><td>90.02</td></tr><tr><td>Maha</td><td>22.44</td><td>95.67</td><td>68.90</td><td>86.30</td><td>23.07</td><td>94.20</td><td>31.38</td><td>93.21</td><td>62.39</td><td>79.39</td><td>92.66</td><td>61.39</td><td>55.37</td><td>82.73</td></tr><tr><td>SHE</td><td>41.98</td><td>91.02</td><td>1.01</td><td>99.66</td><td>77.63</td><td>74.14</td><td>72.36</td><td>76.42</td><td>60.05</td><td>76.49</td><td>74.94</td><td>77.90</td><td>54.66</td><td>82.60</td></tr><tr><td>KNN</td><td>17.67</td><td>96.40</td><td>31.62</td><td>92.85</td><td>46.95</td><td>90.65</td><td>39.41</td><td>39.41</td><td>24.73</td><td>93.44</td><td>88.72</td><td>67.19</td><td>41.52</td><td>88.75</td></tr><tr><td>Ours</td><td>5.37</td><td>97.05</td><td>15.94</td><td>96.97</td><td>20.34</td><td>96.11</td><td>18.77</td><td>96.36</td><td>20.41</td><td>94.77</td><td>80.29</td><td>73.99</td><td>28.27</td><td>92.50</td></tr><tr><td>Ours+ASH</td><td>10.53</td><td>97.83</td><td>12.15</td><td>97.85</td><td>21.34</td><td>96.93</td><td>18.27</td><td>97.41</td><td>19.40</td><td>94.73</td><td>72.27</td><td>74.51</td><td>25.66</td><td>93.21</td></tr></table>
+
+Table 11: Detailed results on six common OOD benchmark datasets: Textures, SVHN, Places365, LSUN-Crop, LSUN-Resize, and iSUN. For each ID dataset, we use the same DenseNet pretrained on CIFAR-10.  $\uparrow$  indicates larger values are better and vice versa.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">LSUN-C</td><td colspan="2">LSUN-R</td><td colspan="2">iSUN</td><td colspan="2">Texture</td><td colspan="2">Places365</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>47.24</td><td>93.48</td><td>33.57</td><td>95.54</td><td>42.10</td><td>94.51</td><td>42.31</td><td>94.52</td><td>64.15</td><td>88.15</td><td>63.02</td><td>88.57</td><td>48.73</td><td>92.46</td></tr><tr><td>ODIN</td><td>25.29</td><td>94.57</td><td>4.70</td><td>98.86</td><td>3.09</td><td>99.02</td><td>3.98</td><td>98.90</td><td>57.50</td><td>82.38</td><td>52.85</td><td>88.55</td><td>24.57</td><td>93.71</td></tr><tr><td>Energy</td><td>40.61</td><td>93.99</td><td>3.81</td><td>99.15</td><td>9.28</td><td>98.12</td><td>10.07</td><td>98.07</td><td>56.12</td><td>86.43</td><td>39.40</td><td>91.64</td><td>26.55</td><td>94.57</td></tr><tr><td>DICE</td><td>25.99</td><td>95.90</td><td>0.26</td><td>99.92</td><td>3.91</td><td>99.20</td><td>4.36</td><td>99.14</td><td>41.90</td><td>88.18</td><td>48.59</td><td>89.13</td><td>20.83</td><td>95.24</td></tr><tr><td>React</td><td>41.64</td><td>93.87</td><td>5.96</td><td>98.84</td><td>11.46</td><td>97.87</td><td>12.72</td><td>97.72</td><td>43.58</td><td>92.47</td><td>43.31</td><td>91.03</td><td>26.45</td><td>94.67</td></tr><tr><td>ASH</td><td>6.51</td><td>98.65</td><td>0.90</td><td>99.73</td><td>4.96</td><td>98.92</td><td>5.17</td><td>98.90</td><td>24.34</td><td>95.09</td><td>48.45</td><td>88.34</td><td>15.05</td><td>96.61</td></tr><tr><td>Maha</td><td>6.42</td><td>98.31</td><td>56.55</td><td>86.96</td><td>9.14</td><td>97.09</td><td>9.78</td><td>97.25</td><td>21.51</td><td>92.15</td><td>85.14</td><td>63.15</td><td>31.42</td><td>89.15</td></tr><tr><td>SHE</td><td>30.02</td><td>94.48</td><td>0.58</td><td>99.85</td><td>9.03</td><td>98.23</td><td>10.49</td><td>98.05</td><td>51.10</td><td>83.52</td><td>38.32</td><td>92.39</td><td>23.26</td><td>94.40</td></tr><tr><td>KNN</td><td>7.64</td><td>98.52</td><td>0.81</td><td>99.76</td><td>15.24</td><td>97.35</td><td>13.77</td><td>97.65</td><td>27.09</td><td>95.09</td><td>40.00</td><td>92.06</td><td>17.43</td><td>96.74</td></tr><tr><td>Ours</td><td>5.37</td><td>98.95</td><td>7.13</td><td>98.63</td><td>5.18</td><td>98.92</td><td>6.42</td><td>98.65</td><td>17.06</td><td>96.72</td><td>42.37</td><td>91.82</td><td>13.92</td><td>97.15</td></tr><tr><td>Ours+ASH</td><td>6.61</td><td>98.70</td><td>1.87</td><td>99.56</td><td>1.65</td><td>99.58</td><td>2.77</td><td>99.39</td><td>17.22</td><td>96.83</td><td>47.72</td><td>91.83</td><td>12.14</td><td>97.65</td></tr></table>
+
+Table 12: Detailed results on six common OOD benchmark datasets: Textures, SVHN, Places365, LSUN-Crop, LSUN-Resize, and iSUN. For each ID dataset, we use the same DenseNet pretrained on CIFAR-100.  $\uparrow$  indicates larger values are better and vice versa.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">LSUN-C</td><td colspan="2">LSUN-R</td><td colspan="2">iSUN</td><td colspan="2">Texture</td><td colspan="2">Places365</td><td colspan="2">Average</td></tr><tr><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td><td>FPR95↓</td><td>AUROC↑</td></tr><tr><td>MSP</td><td>63.14</td><td>58.98</td><td>91.36</td><td>58.71</td><td>75.09</td><td>81.99</td><td>78.36</td><td>79.93</td><td>87.50</td><td>73.10</td><td>87.01</td><td>69.27</td><td>79.29</td><td>79.29</td></tr><tr><td>ODIN</td><td>88.46</td><td>77.68</td><td>77.68</td><td>90.50</td><td>72.78</td><td>84.61</td><td>76.01</td><td>83.04</td><td>87.61</td><td>74.81</td><td>87.30</td><td>69.78</td><td>77.47</td><td>79.90</td></tr><tr><td>Energy</td><td>40.61</td><td>93.99</td><td>3.81</td><td>99.15</td><td>9.28</td><td>98.12</td><td>10.07</td><td>98.07</td><td>56.12</td><td>86.43</td><td>39.40</td><td>91.64</td><td>26.55</td><td>94.57</td></tr><tr><td>DICE</td><td>86.75</td><td>78.47</td><td>18.07</td><td>95.95</td><td>79.00</td><td>81.75</td><td>81.76</td><td>81.16</td><td>81.13</td><td>75.84</td><td>91.65</td><td>65.97</td><td>73.06</td><td>79.86</td></tr><tr><td>React</td><td>91.73</td><td>78.45</td><td>40.08</td><td>90.98</td><td>67.93</td><td>84.65</td><td>66.36</td><td>85.30</td><td>68.60</td><td>81.60</td><td>84.99</td><td>70.96</td><td>69.95</td><td>81.99</td></tr><tr><td>ASH</td><td>59.22</td><td>88.20</td><td>26.21</td><td>95.37</td><td>77.01</td><td>77.03</td><td>79.35</td><td>76.27</td><td>61.13</td><td>84.47</td><td>84.47</td><td>66.83</td><td>65.29</td><td>81.36</td></tr><tr><td>Maha</td><td>57.16</td><td>58.98</td><td>91.36</td><td>58.71</td><td>74.26</td><td>65.38</td><td>71.69</td><td>65.45</td><td>53.40</td><td>69.04</td><td>82.65</td><td>61.31</td><td>71.75</td><td>63.14</td></tr><tr><td>KNN</td><td>46.63</td><td>89.84</td><td>60.87</td><td>84.50</td><td>65.13</td><td>87.76</td><td>65.49</td><td>86.56</td><td>54.15</td><td>85.65</td><td>81.04</td><td>70.98</td><td>62.22</td><td>84.22</td></tr><tr><td>Ours</td><td>31.35</td><td>93.26</td><td>40.53</td><td>91.10</td><td>68.25</td><td>87.54</td><td>67.51</td><td>86.97</td><td>42.41</td><td>88.94</td><td>76.77</td><td>75.77</td><td>54.47</td><td>87.26</td></tr></table>

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