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+# Causal Discovery with Fewer Conditional Independence Tests
+
+Kirankumar Shiragur *1 Jiaqi Zhang *12 Caroline Uhler 2
+
+# Abstract
+
+Many questions in science center around the fundamental problem of understanding causal relationships. However, most constraint-based causal discovery algorithms, including the well-celebrated PC algorithm, often incur an exponential number of conditional independence (CI) tests, posing limitations in various applications. Addressing this, our work focuses on characterizing what can be learned about the underlying causal graph with a reduced number of CI tests. We show that it is possible to learn a coarser representation of the hidden causal graph with a polynomial number of tests. This coarser representation, named Causal Consistent Partition Graph (CCPG), comprises of a partition of the vertices and a directed graph defined over its components. CCPG satisfies consistency of orientations and additional constraints which favor finer partitions. Furthermore, it reduces to the underlying causal graph when the causal graph is identifiable. As a consequence, our results offer the first efficient algorithm for recovering the true causal graph with a polynomial number of tests, in special cases where the causal graph is fully identifiable through observational data and potentially additional interventions.
+
+# 1. Introduction
+
+Causal discovery is a fundamental task in various scientific disciplines including biology, economics, and sociology (King et al., 2004; Cho et al., 2016; Tian, 2016; Sverchkov & Craven, 2017; Rotmensch et al., 2017; Pingault et al., 2018; de Campos et al., 2019; Reichenbach, 1956; Woodward, 2005; Eberhardt & Scheines, 2007; Hoover, 1990;
+
+*Equal contribution; alphabetic order<sup>1</sup> Eric and Wendy Schmidt Center, Broad Institute<sup>2</sup> Laboratory for Information & Decision Systems, Massachusetts Institute of Technology. Correspondence to: Kirankumar Shiragur <kshiragu@broadinstitute.org>, Jiaqi Zhang <viczhang@mit.edu>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+Friedman et al., 2000; Robins et al., 2000; Spirtes et al., 2000; Pearl, 2003). Directed acyclic graphs (DAGs) stand out as a popular choice for representing causal relations, with edge directions signifying the flow of information between variables. The core objective of causal discovery is to identify both the edges and their orientations based on available data. While certain structures can be recovered from observational data (Verma & Pearl, 1990), orienting the full graph often requires additional experiments or interventions.
+
+Research on causal structure learning from observational data dates back to the 1990s (Verma & Pearl, 1990; Spirtes et al., 1989). As a pioneering work in this direction, the PC algorithm (Spirtes et al., 2000), named after the authors Peter Spirtes and Clark Glymour, still remains one of most popular and widely used algorithms. It recovers the structure using observational data through conditional independence (CI) tests, with the number of tests being exponential in the degree of the graph. Following this, many causal discovery algorithms emerged (Kalisch & Buhlman, 2007; Brenner & Sontag, 2013; Alonso-Barba et al., 2013; Schulte et al., 2010), accommodating diverse and more general settings, including the presence of latent variables (Spirtes et al., 1999; 2013) and interventional data (Eberhardt et al., 2005; 2006). However, a common challenge shared by these algorithms is their reliance, to different extents, on an exponential number of CI tests in certain graph parameters. This inherent dependence on an exponential number of tests poses practical challenges, making them unsuitable for many real-world scenarios. Moreover, it suggests that achieving exact causal structure learning can be highly challenging.
+
+As performing exponential number of tests is limited in many applications, it motivates us to study the following question:
+
+What useful information about the underlying causal graph can be inferred with fewer conditional independence tests?
+
+Aligned with this motivation, our work also explores the role of interventions in the structure learning process.
+
+In our work, we study these questions under standard Markov, faithfulness and causal sufficiency assumptions (Lauritzen, 1996; Spirtes et al., 2000). The primary contribution of our work is an efficient algorithm that uses a polynomial number of CI tests and recovers a representation
+
+of the underlying causal graph with observational and optionally interventional datasets. This representation consists of a partition of the vertices and a DAG defined over its components which is consistent with the underlying causal graph. In addition, our representation is designed to avoid dummy partitions that group all the vertices into a single component. The definition of our representation ensures that the components in the partition satisfy several additional properties, guaranteeing that each component either contains a single vertex or comprises an edge that could only be oriented after an intervention is performed on one of its endpoints. We refer to this representation as the Causally Consistent Partition Graph (CCPG) representation.
+
+An important implication of our results is that if the underlying causal graph is fully identifiable using only observational data, our algorithm yields a CCPG with a partition containing components, each of which is of size one. The size-one property of each component means that our algorithm recovers the true causal graph using only a polynomial number of conditional independence tests. We extend this result in the presence of interventions and provide an algorithm that recovers the true causal graph, when the set of interventions provided is sufficient to identify the underlying causal graph. To the best of our knowledge, our algorithms present the first to guarantee recovering the true causal graph using a polynomial number of tests when the graph is either entirely identifiable from observational data or with an additional set of interventions.
+
+# 1.1. Related Works
+
+Efficient algorithms for causal structure learning (Spirtes et al., 2000; Claassen et al., 2013) exist for constant bounded degree graphs, recovering the causal graph with a polynomial number of CI tests. For general causal graphs, current methods often entail an exponential number of CI tests, where (Xie & Geng, 2008; Zhang et al., 2024) aimed to reduce such complexity. For Bayesian network learning, finding a minimal Bayesian network is NP-hard, even with a constant-time CI oracle and nodes with at most  $k \geq 3$  parents. Chickering et al. (2004) demonstrated this hardness through a polynomial reduction from the NP-complete problem, Degree-Bounded Feedback Arc Set. These findings highlight the contrast between causal structure and minimal Bayesian network learning, suggesting that causal structure learning is notably more straightforward. Our results further reinforce this notion by identifying a special class of causal graphs that can be recovered with a polynomial number of conditional independence tests. For other hardness results on Bayesian network learning, we refer readers to Bouckaert (1994); Chickering et al. (2004) and references therein.
+
+Learning causal relationships from observational data (Verma & Pearl, 1990; Spirtes et al., 1989; 2000; Chickering,
+
+2002; Geiger & Heckerman, 2002; Nandy et al., 2018) and interventional data (Eberhardt, 2010; Hu et al., 2014; Shanmugam et al., 2015; Greenewald et al., 2019; Squires et al., 2020; Choo et al., 2022; Choo & Shiragur, 2023; Shiragur et al., 2024) is a well studied problem with a rich literature. We encourage interested readers to explore Glymour et al. (2019); Squires & Uhler (2022) and references therein for a more comprehensive understanding and further details.
+
+# 1.2. Organization
+
+The rest of our paper is organized as follows. Section 2 is our preliminary section. In Section 3, we provide all the main results of the paper. In Section 4 and Section 5 combined, we provide our CCPG recovery algorithm when just observational data is available. In Section 6, we extend our results to the case of interventions. We provide numerical results in Section 7. Finally in Section 8, we conclude with a short discussion and few open directions.
+
+# 2. Preliminaries
+
+# 2.1. Graph Definitions
+
+Let  $\mathcal{G}$  be a directed acyclic graph (DAG) on  $n$  vertices in  $V$ . For a vertex  $v \in V$ , let  $\operatorname{Pa}(v), \operatorname{Anc}(v)$ , and  $\operatorname{Des}(v)$  denote the parents, ancestors, and descendants of  $v$  respectively. Let  $\operatorname{Anc}[v] = \operatorname{Anc}(v) \cup \{v\}$  and  $\operatorname{Des}[v] = \operatorname{Des}(v) \cup \{v\}$ .
+
+For a set of vertices  $S\subset V$  , denote  $\mathbb{P}\mathsf{a}(S) = \cup_{v\in S}\mathbb{P}\mathsf{a}(s)$  Similarly define  $\operatorname {Anc}(S),\operatorname {Des}(S),\operatorname {Anc}[S]$  , and  $\mathrm{Des}[S]$  We write  $\mathrm{src}(S)$  as the set of source nodes within  $S$  , that is,
+
+$$
+\operatorname {s r c} (S) = \left\{v \in S \mid \operatorname {A n c} (v) \cap S = \varnothing \right\}.
+$$
+
+Denote  $\bar{S} = V\backslash S$ . Let  $\mathcal{G}[S]$  be a subgraph of  $\mathcal{G}$  by removing all vertices in  $\bar{S}$ .
+
+A  $\nu$ -structure refers to three distinct vertices  $u, v, w$  such that  $u \to v \gets w$  and  $u, w$  are not adjacent. An edge  $u \to v$  is a covered edge (Chickering, 1995) if  $\mathbb{P}\mathsf{a}[u] = \mathbb{P}\mathsf{a}(v)$ . A path in  $\mathcal{G}$  is a list of distinct vertices, where consecutive vertices are adjacent. We can associate a topological ordering  $\pi : V \to [n]$  to any DAG  $\mathcal{G}$  such that any  $u \to v$  in  $\mathcal{G}$  satisfy  $\pi(u) < \pi(v)$ . Note that such topological ordering is not necessarily unique.
+
+# 2.2. D-Separation and Conditional Independence
+
+DAGs are commonly used in causality (Pearl, 2009), where vertices represent random variables and their joint distribution  $P$  factorizes according to the DAG:  $P(v_{1},\ldots ,v_{n}) = \prod_{i = 1}^{n}P(v_{i}\mid \mathsf{Pa}(v_{i}))$ . This factorization entails a set of conditional independencies (CIs) in the observational distribution  $P$ . These CI relations are fully characterized by  $d$ -separation (Geiger & Pearl, 1990). Formally, for disjoint vertex sets  $A,B,C\subset V$ , sets  $A,B$  are  $d$ -separated by  $C$
+
+if and only if any path connecting  $A$  and  $B$  in  $\mathcal{G}$  is inactive given  $C$ . A path is inactive given  $C$  when it has a collider<sup>1</sup>  $d \notin \operatorname{Anc}[C]$  or a non-collider  $c \in C$ ; otherwise the path is active given  $C$ . Figure 1 illustrates these concepts.
+
+![](images/44e5641723b54789446ad7f854fcb149d7cb7f8efa35dfb72fcb56809557f91a.jpg)  
+Figure 1. (Left).  $\{1\}$  and  $\{4\}$  are  $d$ -separated by  $\{2\}$ , as all paths are inactive given  $\{2\}$ . (Right).  $\{1\}$  and  $\{4\}$  are not  $d$ -separated by  $\{2,3\}$ , as path  $1 \to 3 \gets 4$  is active given  $\{2,3\}$  by collider 3.
+
+![](images/e639a3b644f283e00096ac271f0030b8a1c01bc0f0a556acc8f22e9be8ab76b6.jpg)
+
+We write  $A \perp B \mid C^2$  when  $A, B$  are conditionally independent given  $C$  in the observational distribution  $P$ . If any set among  $A, B, C$  contains only one node, e.g.,  $A = \{a\}$ , we write  $a \perp B \mid C$  for simplicity. When  $C$  d-separates  $A, B$ , then it holds that  $A \perp B \mid C$  (known as the global Markov property (Geiger & Pearl, 1990)). Under the faithfulness assumption, the reverse also holds, i.e., all CI relations in  $P$  are implied by d-separation in  $\mathcal{G}$ .
+
+Setup. In this work, we assume that the causal DAG  $\mathcal{G}$  is unknown. But we assume causal sufficiency (i.e., no latent confounders), faithfulness and access to enough samples from  $P$  to determine if  $A \perp B \mid C$  for any  $A, B, C \subset V$ . As all CIs are implied by d-separations, we may infer information about  $\mathcal{G}$  using these tests.
+
+# 2.3. Interventions
+
+An intervention  $I \subset V$  is an experiment where the conditional distributions  $P(v \mid \mathbb{P}\mathsf{a}(v))$  for  $v \in I$  are changed into  $P^I (v)$ . Such interventions eliminate the dependency between  $v$  and  $\mathbb{P}\mathsf{a}(v)$ . Let  $\mathcal{G}^I$  denote the modified version of  $\mathcal{G}$ , where all incoming edges to  $v \in I$  are removed. Let  $P^I$  denote the interventional distribution, i.e.,  $P^I (v_1,\ldots ,v_n) = \prod_{v\in I}P^I (v)\prod_{v\notin I}P(v\mid \mathbb{P}\mathsf{a}(v))$ . Then  $P^I$  factorizes with respect to  $\mathcal{G}^I$ . We denote  $A \perp_I B \mid C$  for CI tests in the interventional distribution  $P^I$ .
+
+Setup with Interventions. Similar to the observational setting, we assume faithfulness of  $P^I$  to  $\mathcal{G}^I$  and access to enough samples from  $P^I$  to determine if  $A\perp_{I}B\mid C$
+
+# 2.4. Verifying Intervention Sets and Covered Edges
+
+When it is possible to perform any number of CI tests: with observational data, a DAG  $\mathcal{G}$  is in general only identifiable
+
+up to its skeleton, v-structures (Andersson et al., 1997), and possibly additional edges given by the Meek rules (Meek, 1995). Identifiability can be improved with interventional data (Hauser & Buhlmann, 2012), where  $I$  allows us to infer the edge orientation of any edge cut by  $I$  and  $V \setminus I$ .
+
+A verifying intervention set  $\mathcal{I}$  for a DAG  $\mathcal{G}$  (Choo et al., 2022) is a set of interventions that fully orients  $\mathcal{G}$ , possibly with repeated applications of the Meek rules. We will make use of the following result in our work.
+
+Proposition 2.1 (Theorem 9 in (Choo et al., 2022)). Set  $\mathcal{I}$  is a verifying intervention set if and only if for every covered edge  $u\to v$  in  $\mathcal{G}$ , there is  $|I\cap \{u,v\}| = 1$  for some  $I\in \mathcal{I}$ .
+
+The verification number  $\nu(\mathcal{G})$  is defined as the minimum size of any verifying intervention set of  $\mathcal{G}$ . This proposition tells us that  $\nu(\mathcal{G})$  equals to the minimum size of any vertex cover of the covered edges in  $\mathcal{G}$ .
+
+# 3. Main Results
+
+Here we present our main findings. As highlighted in the introduction, the key contribution of our work lies in recovering a representation of the underlying causal graph that satisfies various desirable properties with very few CI tests. We now provide a formal definition of this representation.
+
+Definition 3.1 (CCPG & I-CCPG). A Causally Consistent Partition Graph (CCPG) representation of a DAG  $\mathcal{G}$  on  $V$  consists of a partition of  $V$  into components  $V_{1},\ldots ,V_{k}$  and a DAG  $\mathcal{D}$  between the components such that,
+
+(intra-component property): for each  $i \in [k]$ , it holds that  $|\mathrm{src}(V_i)| = 1$ . Furthermore, if  $|V_i| > 1$ , then  $\mathcal{G}[V_i]$  has at least one covered edge.
+
+(inter-component property):  $\mathcal{D}$  is topologically ordered, i.e.,  $i\rightarrow j$  in  $\mathcal{D}$  only if  $V_{i} < V_{j}$ . It is also consistent with  $\mathcal{G}$ : (1) if there is no directed edge  $i\rightarrow j$  in  $\mathcal{D}$ , then there are no edges between  $V_{i}$  and  $V_{j}$  in  $\mathcal{G}$ ; (2) if there is a directed edge  $i\rightarrow j$  in  $\mathcal{D}$ , then there is  $u\in V_i$  such that  $u\in \operatorname {Pa}(\operatorname {src}(V_j))$
+
+We further define an Interventional Causally Consistent Partition Graph  $(\mathcal{I}$  -CCPG) representation of  $\mathcal{G}$  with respect to an intervention set  $\mathcal{I}$ : an  $\mathcal{I}$  -CCPG is a CCPG representation of  $\mathcal{G}$  that additionally satisfies the following strong intracomponent condition: for each  $i\in [k]$ , if  $|V_{i}| > 1$  then  $\mathcal{G}[V_i]$  has at least one unintervened $^4$  covered edge.
+
+Figure 2 illustrates these concepts. Note that when  $\mathcal{I} = \varnothing$ ,  $\mathcal{I}$ -CCPG reduces to CCPG.
+
+In Definition 3.1, the first property prefers finer partitions, while the second property ensures consistency. Formally, we can show the following properties of these representations, which establish the significance of CCPGs. Proofs for all lemmas in this section can be found in Appendix A.
+
+![](images/07be124445d9c900122ad1b46229993c85aa643c3a0195f45b24c6e557de13c1.jpg)  
+Figure 2. Example of CCPG &  $\mathcal{I}$ -CCPG. (Left). Ground-truth  $\mathcal{G}$ . (Right). A CCPG representation of  $\mathcal{G}$ , where  $V_{1}, V_{2}, V_{3}$  are indicated by green boxes and  $\mathcal{D}$  is illustrated in chalk strokes. Vertices 3, 4 can be in one component as  $3 \rightarrow 4$  is a covered edge. For  $\mathcal{I} = \{4\}$ , the only  $\mathcal{I}$ -CCPG is  $\mathcal{G}$  itself (due to strong intra-component condition in Definition 3.1).
+
+![](images/dab6341187bef32726d0bc2ff8f9cf775421d33a4d9b5108575d10797af4cf52.jpg)
+
+Lemma 3.2 (Properties of CCPG). For any intervention set  $\mathcal{I}$  (including  $\varnothing$ ), the following arguments hold:
+
+-  $\mathcal{D} = \mathcal{G}$  (i.e., partitioning  $V$  into individual vertices) is a valid  $\mathcal{I}$ -CCPG of  $\mathcal{G}$ .  
+- If the verification number of  $\mathcal{G}$  is zero, i.e.,  $\nu(\mathcal{G}) = 0$ , then  $\mathcal{D} = \mathcal{G}$  is the unique valid  $\mathcal{I}$ -CCPG of  $\mathcal{G}$ .  
+- If  $\mathcal{I}$  is a verifying intervention set of  $\mathcal{G}$ , then  $\mathcal{D} = \mathcal{G}$  is the unique valid  $\mathcal{I}$ -CCPG of  $\mathcal{G}$ .
+
+The key algorithmic contribution of our work, proven in Section 5, lies in an efficient algorithm that learns a valid CCPG with only polynomial number of CI tests.
+
+Theorem 3.3 (Learning CCPG). Given observational data, there exists an efficient algorithm that performs at most  $\mathcal{O}(n^5)$  CI tests, and outputs a CCPG representation.
+
+This result extends to the interventional setting as follows; the proof is given in Section 6.
+
+Theorem 3.4 ( $\mathcal{I}$ -Learning CCPG). Given observational data and interventional data from interventions in  $\mathcal{I}$ , there exists an efficient algorithm that performs at most  $\mathcal{O}(n^5) + |\mathcal{I}| \cdot \mathcal{O}(n^3)$  CI tests, and outputs an  $\mathcal{I}$ -CCPG representation.
+
+Combining these results with the properties of CCPG in Lemma 3.2, this provides an efficient algorithm for learning the causal graph with polynomial number of conditional independence tests under certain cases, detailed below.
+
+Corollary 3.5 (Causal Discovery with Polynomial CI Tests). For a DAG  $\mathcal{G}$  and its verifying intervention set  $\mathcal{I}$  (can be  $\varnothing$ ), our algorithm recovers the full causal graph with at most  $\mathcal{O}(n^5) + |\mathcal{I}| \cdot \mathcal{O}(n^3)$  CI tests.
+
+We remark here that Eberhardt et al. (2012) provides a construction of verifying intervention set of size  $\log_2(n)$  that is independent of the underlying DAG  $\mathcal{G}$ . Together with Corollary 3.5, this implies that our algorithm can learn the full causal graph with at most  $\mathcal{O}(n^5)$  CI tests.
+
+To the best of our knowledge, our results present the first formal characterization of the information recoverable about general causal graphs using polynomial number of CI tests. Prior works showed that it is possible to learn sparse causal graphs with  $n^{\mathcal{O}(k)}$  CI tests (Claassen et al., 2013; Spirtes et al., 2000). Here  $k$  is an upper bound on the vertex degrees. Note that, the number of CI tests our algorithm requires,  $\mathcal{O}(n^{5})$ , is a polynomial of  $n$  that is independent of any graph parameters.
+
+# 3.1. Proxy V-structure and Meek Rule Statements
+
+In our derivations, we will make use of the following results, which we believe is of separate interest well beyond the scope of this work.
+
+Lemma 3.6 (Proxy V-Structure). Let  $S \subseteq V$  and  $u, v, z \in V \setminus S$ . If  $u \perp v|S$  and  $u \not\perp v|S \cup \{z\}$ , then  $u, v \notin \mathsf{De}s[z]$ .<sup>5</sup>
+
+Definition 3.7 (Prefix Vertex Set). We call  $S \subseteq V$  a prefix vertex set if it satisfies: for all  $w \in S$ ,  $\operatorname{Anc}[w] \cap \bar{S} = \emptyset$  (vertices in  $S$  appear first in the topological order).
+
+Lemma 3.8 (Proxy Meek Rule 1). Let  $S$  be a prefix subset. If  $u, v$  and  $w$  are such that  $u \in S$ ,  $v, w \notin S$  and  $u \not\perp v|S$  and  $u \perp w|S \cup \{v\}$  then  $v \notin \mathsf{Des}[w]$ .
+
+The results stated above serve as proxy statements of v-structure and Meek Rule 1. Given the CI tests in the preceding lemmas, if we additionally have confirmed adjacencies between specific pairs of vertices, stronger statements could be made; e.g., in Lemma 3.6, we could conclude that  $z$  is a child (or descendant) of both  $u$  and  $v$ , and in Lemma 3.8 that  $w$  is a child of  $v$ . However, since our lemma statements do not assume any knowledge of adjacency, we can only ascertain weaker statements, in the first case that  $u$  and  $v$  are not descendants of  $z$ , and in the second case that  $v$  is not a descendant of  $w$ .
+
+While our proxy results reveal weaker relationships among variables, they achieve this using a constant number of CI tests. Uncovering stronger relationships requires adjacency information, which may entail an exponential number of CI tests. Our main contribution lies in leveraging these weaker relationships, along with other favorable properties embedded in our algorithm, to implement the prefix vertex procedure and reconstruct the CCPG representation of the underlying causal graph using few CI tests.
+
+# 4. Prefix Vertex Set
+
+The core component of our CCPG algorithm involves a procedure that produces a series of prefix vertex sets. We now show how such prefix vertex sets can be learned by performing few CI tests.
+
+This will be helpful to obtain a CCPG representation of  $\mathcal{G}$  since the components  $V_{1},\ldots ,V_{k}$  satisfy that  $V_{1}\cup \dots \cup V_{i}$  is a prefix vertex set for all  $i\in [k]$ .
+
+# 4.1. Algorithm for Learning
+
+We begin by presenting our algorithm for learning a prefix vertex set. This algorithm takes as input a prefix vertex set  $S \subsetneq V$  (which can be  $\varnothing$ ) and produces a larger prefix vertex set  $S'$ .
+
+The analysis of Algorithm 1 will be provided in the next section. For an input prefix vertex set  $S \subsetneq V$ , it makes use of three types of CI tests, which we formalize below.
+
+Definition 4.1 (Type-I Set  $D_S$ ). For all  $w \in \bar{S}$ , let  $w \in D_S$  if and only if  $u \perp v \mid S$  and  $u \not\perp v \mid S \cup \{w\}$  for some  $u \in V$  and  $v \in \bar{S}$ .
+
+By the proxy v-structure in Lemma 3.6, these two CI tests indicate that  $v \notin \operatorname{Des}[w]$ . Therefore  $w$  can potentially be a descendant of  $v$ . Thus  $D_S$  contains vertices that are potential descendants of some other vertex in  $\tilde{S}$ . We will rule out this set when searching for prefix  $S' \supseteq S$ . Similarly, we can define a type-II set  $E_S$ .
+
+Definition 4.2 (Type-II Set  $E_S$ ). For all  $w \in \bar{S} \setminus D_S$ , let  $w \in E_S$  if and only if  $u \perp v' \mid S \cup \{v\}$  and  $u \not\perp v' \mid S \cup \{v, w\}$  for some  $u \in S$  and  $v, v' \in \bar{S} \setminus D_S$ .
+
+We also exclude any type-III set  $F_{S}$ , defined as follows.
+
+Definition 4.3 (Type-III Set  $F_S$ ). For all  $w \in \bar{S} \setminus D_S$ , let  $w \in F_S$  if and only if  $u \not\perp v \mid S$ ,  $u \perp w \mid S \cup \{v\}$ , and  $v \not\perp w \mid S$  for some  $u \in S$  and  $v \in \bar{S} \setminus D_S$ .
+
+By the proxy Meek Rule 1 in Lemma 3.8, the first two CI tests  $u \not\perp v \mid S$  and  $u \perp w \mid S \cup \{v\}$  guarantee that  $v \notin \mathsf{Des}[w]$ . The remaining CI test  $v \not\perp w \mid S$  is to ensure that we do not exclude too many vertices in  $F_S$ , in particular  $\mathrm{src}(\bar{S})$ , as we show in the next section.
+
+# Algorithm 1 Learning a Prefix Vertex Set
+
+1: Input: A prefix vertex set  $S \subsetneq V$ . CI queries from  $\mathcal{G}$ .  
+2: Output: A prefix vertex set  $S'$  such that  $S' \supseteq S$ .  
+3: Compute type-I set  $D_S$ .  
+4: Compute type-II and III sets  $E_S, F_S$ .  
+5: Let  $U = \bar{S} \setminus (D_S \cup E_S \cup F_S)$ .  
+6: return  $S' = S \cup U$ .
+
+# 4.2. Correctness and Guarantees
+
+Algorithm 1 satisfies the following guarantees. All omitted proofs can be found in Appendix B.
+
+![](images/eb89890c9eb1b545118d428fde2ecec300c7438e2480d97ec27df42384764430.jpg)  
+Figure 3.  $D_{S}, E_{S}, F_{S}$  satisfy that (1) they contain all downstream vertices of any vertex in them; (2) they do not intersect with  $\mathrm{src}(\bar{S})$ .
+
+Theorem 4.4. Algorithm 1 outputs a prefix vertex set in  $\mathcal{O}(n^4)$  number of CI tests. In addition, this prefix vertex set contains all the remaining source nodes, i.e.,  $\mathrm{src}(\bar{S})\subset S'$ .
+
+For its proof, we will make use of the following properties of the type-I, II, and III sets (illustrated in Figure 3).
+
+Lemma 4.5. Let  $S$  be a prefix vertex set. If  $w \in D_S$ , then  $\text{Des}[w] \subset D_S$ . Furthermore,  $D_S \cap \text{src}(\bar{S}) = \emptyset$ . The same properties hold for  $E_S$ .
+
+Lemma 4.6. Let  $S$  be a prefix vertex set. If  $w \in F_S$ , then  $\mathsf{Des}[w] \subset E_S \cup F_S$ . Furthermore,  $F_S \cap \operatorname{src}(\bar{S}) = \varnothing$ .
+
+Proof of Theorem 4.4. We first show that  $S'$  returned by Algorithm 1 satisfies  $\operatorname{src}(\bar{S}) \subset S'$ . By Lemmas 4.5 and 4.6, we have  $(D_S \cup E_S \cup F_S) \cap \operatorname{src}(\bar{S}) = \emptyset$ . As  $\bar{S}' = D_S \cup E_S \cup F_S$ , it must hold that  $\operatorname{src}(\bar{S}) \subset S'$ .
+
+Next we show that  $S'$  is a prefix vertex set. For this, we only need to show that  $\forall w \in \bar{S}'$  and  $y \in \mathsf{Des}(w)$ , it holds that  $y \in \bar{S}'$ . Since  $\bar{S}' = D_S \cup E_S \cup F_S$ , one of the following three scenarios must hold: (1) if  $w \in D_S$ , then  $y \in D_S$  by Lemma 4.5; (2) if  $w \in E_S$ , then  $y \in E_S$  by Lemma 4.5; or (3) if  $w \in F_S$ , then  $y \in E_S \cup F_S$  by Lemma 4.6.
+
+Therefore  $S'$  must be a prefix vertex set. We now bound the number of CI tests performed by Algorithm 1: computing  $D_S$  takes  $\mathcal{O}(n^3)$  CI tests; computing  $E_S$  takes  $\mathcal{O}(n^4)$  CI tests; computing  $F_S$  takes  $\mathcal{O}(n^3)$  CI tests, which completes the proof.
+
+# 4.3. Relation to Covered Edges
+
+In Theorem 4.4, we showed that  $\mathrm{src}(\bar{S})\subset S^{\prime}$ . In fact, when there are no covered edges coming from  $\mathrm{src}(\bar{S})$ , one can show that  $S^{\prime} = \mathrm{src}(\bar{S})\cup S$  via the following Lemma 4.7.
+
+Lemma 4.7. Let  $S$  be a prefix vertex set. For  $w \in \bar{S} \setminus D_S$ , if  $w \notin \operatorname{src}(\bar{S})$  and there is no covered edge from  $\operatorname{Anc}[w] \cap \operatorname{src}(\bar{S})$  to  $\operatorname{Anc}[w]$ , then  $w \in E_S \cup F_S$ .
+
+Corollary 4.8. Let  $S$  be a prefix vertex set. If there is no covered edge in  $\bar{S}$ , then  $S' = \operatorname{src}(\bar{S}) \cup S$ .
+
+This result will be useful when deriving CCPG representations using Algorithm 1. To see this, consider the simple case where there is no covered edge in  $\mathcal{G}$ . Then running Algorithm 1 with  $S = \varnothing$  we can learn  $\operatorname{src}(V)$  by Corollary 4.8. Then running Algorithm 1 with  $S = \operatorname{src}(V)$ , we can learn the source vertices of  $V \setminus \operatorname{src}(V)$ . Applying this iteratively, we can obtain the ground-truth topological order of  $\mathcal{G}$ . As a consequence, one can easily learn  $\mathcal{G}$ , which is the sole CCPG representation as there is no covered edge.
+
+# 5. Learning Causally Consistent Partition Graph Representations
+
+We now present our algorithm for learning causally consistent partition graph representations. All omitted proofs can be found in Appendix C.
+
+Algorithm 2 contains three parts, indicated by colored boxes below. In the first part, we use Algorithm 1 iteratively to learn prefix vertex sets of the form  $\varnothing \subsetneq \ldots S \subsetneq S' \subsetneq \dots \subsetneq V$ . In the second part, we break each  $S' \setminus S$  into smaller components that are pairwise independent given  $S$ . As we will see, these components satisfy the property of CCPG in Definition 3.1. In the third part, we build the acyclic graph between the CCPG components.
+
+Algorithm 2 Learning a CCPG Representation
+
+1: Input: CI queries from  $\mathcal{G}$  
+2: Output: A CCPG representation of  $\mathcal{G}$ .  
+3: Set  $S = \varnothing$  and  $S$  as empty ordered list.  
+4: while  $S \neq V$  do  
+5: Run Algorithm 1 on  $S$  to obtain  $S'$ .  
+6: Add  $S^{\prime}\setminus S$  to the end of  $\mathcal{S}$  
+7: Update  $S = S^{\prime}$  
+8: end while  
+9: Initialize  $l_{1} = 1$  
+10: for  $S_{i}$  in  $\mathcal{S} = [S_1, \ldots, S_m]$  do  
+11: Create empty graph  $\mathcal{T}$  on vertices in  $S_{i}$ .  
+12: Add  $v - w$  to  $\mathcal{T}$  iff  $v \not\perp w \mid S_1 \cup \dots \cup S_{i-1}$ .  
+13: Split  $S_{i}$  into components  $V_{l_i}, V_{l_{i+1}}, \ldots, V_{l_{i+1}}$  based on connected components in  $\mathcal{T}$ .  
+14: end for  
+15: Create empty graph  $\mathcal{D}$  on vertices in  $[l_{m + 1}]$ .  
+16: for  $V_{i}, V_{j}$  with  $i < j$  do  
+17: Add  $i \to j$  to  $\mathcal{D}$  iff  $V_{i} \not\subset V_{j} \mid V_{1} \cup \dots \cup V_{j-1}$ .  
+18: end for  
+19: return  $V_{1}, \ldots, V_{l_{m+1}}$  and  $\mathcal{D}$
+
+To show that the components we obtain in the second part
+
+![](images/d4d82e83c83375fabc3699c8b5765864e3cacbe66c09675d2a0390213909d6a3.jpg)  
+Figure 4. Illustration of  $\bar{S} \setminus D_S$  (inside the dashed box). It can be split into connected subgraphs based on vertices in  $\mathrm{src}(\bar{S})$  (indicated by the fill color of each vertex in  $\bar{S} \setminus D_S$ ).
+
+satisfy the property of CCPG, we will use the following result. Essentially, we can show that the subgraph  $\mathcal{G}[S' \setminus S]$  can be split into connected subgraphs based on individual vertices in  $\operatorname{src}(S' \setminus S) (= \operatorname{src}(\tilde{S}))$  by Theorem 4.4). This is illustrated in Figure 4.
+
+Lemma 5.1. Let  $S$  be a prefix subset. For any  $w \in \bar{S} \backslash D_S$ , there is  $|\text{Anc}[w] \cap \text{src}(\bar{S})| = 1$ . Furthermore, for any other  $w' \in \bar{S} \setminus D_S$ , there is  $w \not\subset w' \mid S$  if and only if  $\text{Anc}[w] \cap \text{src}(\bar{S}) = \text{Anc}[w'] \cap \text{src}(\bar{S})$ .
+
+Based on this result, we can establish that Algorithm 2 outputs a CCPG representation by proving Theorem 3.3.
+
+Proof of Theorem 3.3. We show that Algorithm 2 outputs a CCPG representation of  $\mathcal{G}$ . Since it runs in polynomial time and uses  $\mathcal{O}(n^5)$  CI tests, this will prove Theorem 3.3.
+
+Intra-component Property. Consider  $S_i$ . Denote  $S_1 \cup \dots \cup S_{i-1} = S$ . We will show that  $S_i$  is split into  $\mathsf{Des}[s] \cap S_i$  for each  $s \in \mathrm{src}(\bar{S})$ . Since  $S_i \subset \bar{S}$ , we have  $S_i = \cup_{s \in \mathrm{src}(\bar{S})} (\mathsf{Des}[s] \cap S_i)$ . By Theorem 4.4, we know that  $S$  is a prefix vertex set. Since  $S_i \subset \bar{S} \setminus D_S$ , by Lemma 5.1,  $\mathsf{Des}[s] \cap S_i$  for different  $s \in \mathrm{src}(\bar{S})$  are disjoint. Furthermore, by Theorem 4.4, these disjoint sets are non-empty because  $s \in \mathsf{Des}[s] \cap S_i$ . Therefore  $\mathrm{src}(\mathsf{Des}[s] \cap S_i) = \{s\}$  and we have  $|\mathrm{src}(\mathsf{Des}[s] \cap S_i)| = 1$ . Thus by Lemma 5.1, each of  $V_{l_i}, V_{l_{i+1}}, \ldots, V_{l_{i+1}}$  corresponds to  $\mathsf{Des}[s] \cap S_i$  for some  $s \in \mathrm{src}(\bar{S})$ .
+
+If  $|\mathsf{Des}[s] \cap S_i| > 1$ , then it contains a vertex that is not  $s$ . Suppose  $w \in \mathsf{Des}(s) \cap S_i$ . By Lemmas 4.7 and 5.1, there is a covered edge from  $s$  to some vertex  $x \in \mathsf{Anc}[w]$ . Since  $S$  and  $S \cup S_i$  are prefix vertex sets,  $s, w \in S_i$  implies  $x \in S_i$ . Thus  $x \in \mathsf{Des}[s] \cap S_i$  and there is a covered edge  $s \to x$  in
+
+$\mathsf{Des}[s]\cap S_i$  . Thus, we have proven that each component in  $V_{1},\ldots ,V_{l_{m + 1}}$  satisfies the first property of CCPG.
+
+Inter-component Property. For each  $V_i$ , denote the subset that it came from as  $S_{h_i}$ , i.e.,  $V_i \subset S_{h_i}$ . By the construction of  $\mathcal{D}$  in Algorithm 2, no edge  $i \rightarrow j$  in  $\mathcal{D}$  means either  $i > j$  or  $V_i \perp V_j \mid V_1 \cup \dots \cup V_{j-1}$ . If  $i > j$  and  $h_i \neq h_j$ , then by the fact that  $S_1 \cup \dots \cup S_{h_i}$  is a prefix vertex set, there is no edge from  $V_i$  to  $V_j$  in  $\mathcal{G}$ . If  $h_i = h_j = h$ , then we know that there exists  $s_i \neq s_j$  such that  $V_i = \operatorname{Des}[s_i] \cap S_h$  and  $V_j = \operatorname{Des}[s_j] \cap S_h$ . If there is an edge from  $V_i$  to  $V_j$  in  $\mathcal{G}$ , denoted as  $v \rightarrow w$ , then  $w \in \operatorname{Des}[v] \subset \operatorname{Des}[s_i]$  which means  $V_i \cap V_j \neq \emptyset$ , a contradiction. Therefore there is no edge from  $V_i$  to  $V_j$  in  $\mathcal{G}$ . If  $V_i \perp V_j \mid V_1 \cup \dots \cup V_{j-1}$ , then clearly there is no edge from  $V_i$  to  $V_j$  in  $\mathcal{G}$ . Thus when there is no  $i \rightarrow j$  in  $\mathcal{D}$ , there is no edge from  $V_i$  to  $V_j$  in  $\mathcal{G}$ .
+
+If there is an edge  $i\to j$  in  $\mathcal{D}$ , then we have  $i < j$  and  $V_{i}\not\perp V_{j}\mid V_{1}\cup \dots \cup V_{j - 1}$ . Note that since  $S_{1}\cup \dots \cup S_{h_{j}}$  is prefixed, it holds that  $\mathrm{Pa}(V_j)\subset S_1\cup \dots \cup S_{h_j}$  and  $\mathrm{Des}[V_j]\cap (S_1\cup \ldots S_{h_j - 1}) = \emptyset$ . As shown above, there is no edge between any other  $V_{j^{\prime}}\subset S_{h_j}$  and  $V_{j}$ . Thus we have  $\mathrm{Pa}(V_j)\subset V_1\cup \dots \cup V_j$  and  $\mathrm{Des}[V_j]\cap (V_1\cup \dots \cup V_{j - 1}) = \emptyset$ . Similarly  $\mathrm{Pa}(V_i)\subset V_1\cup \dots \cup V_i$ . Thus by the local Markov property and Bayes rule, $^8$ $V_{i}\not\perp V_{j}\mid V_{1}\cup \dots \cup V_{j - 1}$  only if there is a direct edge  $u\rightarrow v$  from  $u\in V_i$  to  $v\in V_j$ . We now show that there is also an edge  $u\rightarrow s$  where  $\{s\} = \mathrm{src}(V_j)$ . Assume on the contrary that there is no edge  $u\rightarrow s$ . Since  $s$  is the source node of  $V_{j}$  and thus a source node of  $S_{h_j}$ , we have from  $\mathrm{Pa}(V_j)\subset S_1\cup \dots \cup S_{h_j}$  that  $\mathrm{Pa}(s)\subset S_1\cup \dots \cup S_{h_j - 1}$ . In addition,  $\mathrm{Des}[s]\cap (S_1\cup \dots \cup S_{h_j - 1}) = \emptyset$ . Thus by the local Markov property,  $u\perp s\mid S_1\cup \dots \cup S_{h_j - 1}$ . However as  $u\rightarrow v$  and  $v\in$ $\mathrm{Des}[s]$ , we have  $u\not\perp s\mid S_1\cup \dots \cup S_{h_j - 1}\cup \{v\}$ . As a consequence,  $v\in D_{S_1\cup \dots \cup S_{h_j - 1}}$ . By Algorithm 1, it is impossible that  $v\in S_{h_j}$ , a contradiction. Therefore we must have  $u\in \mathrm{Pa}(s)$ . This proves that  $\mathcal{D}$  satisfies the second property of CCPG.
+
+# 6. Interventions
+
+In Sections 4 and 5, we showed how to learn a CCPG representation using observational data. Here we generalize these methods to interventions. This results in a more refined  $\mathcal{I}$ -CCPG representation of  $\mathcal{G}$ . All omitted proofs can be found in Appendix D.
+
+# 6.1. Learning Refined Prefix Vertex Sets
+
+In Section 4.1, we obtained a prefix vertex set by excluding three types of sets  $D_S, E_S$  and  $F_S$ . With interventions, we can exclude an additional set. To define it, we first characterize what can be learned using interventions.
+
+![](images/a3ac630ebf1ec10ef8863180fb8439c63811781b82c66daa41dae360dfaed6ec.jpg)  
+Figure 5. Illustration of  $J_{S}^{I}$ , where  $I$  is indicated by the purple circle.  $J_{S}^{I}$  satisfies similar properties as  $D_{S}, E_{S}$  and  $F_{S}$ .
+
+Lemma 6.1. Let  $S$  be a prefixed vertex set. Given an intervention on vertices  $I \subseteq V$ . For any  $u \notin I$ , we have  $u \not\in I$ . If for some  $v \in I \setminus S$  if and only if  $u \in \operatorname{Des}(I \setminus S) \setminus (I \setminus S)$ ,
+
+Note that when the number of CI tests is not restricted, one can learn all edges cut by  $I$  as well as their orientations. Lemma 6.1 shows that it is possible to learn the joint set of descendants using  $\mathcal{O}(n^2)$  CI tests.
+
+With this set and a few more CI tests, it is possible to learn additional directional information.
+
+Lemma 6.2. Let  $S$  be a prefix subset. Given an intervention  $I$  and  $v \in I \setminus S$ , denote  $H_S^I(v) = \{ u \notin S \cup \text{Des}[I \setminus S] : u \not\subset v \mid V \setminus \text{Des}[I \setminus S] \}$ . Then  $Pa(v) \setminus (S \cup \text{Des}[I \setminus S]) \subseteq H_S^I(v) \subseteq \text{Anc}(v) \setminus (S \cup \text{Des}[I \setminus S])$ .
+
+Such sets can be useful when deciding if a target of  $I \setminus S$  should be excluded when learning a prefix vertex subset  $S' \supset S$ . Formally, we define the type-IV set as follows.
+
+Definition 6.3 (Type-IV Set  $J_S^I$ ). Let  $S$  be a prefix vertex set. For an intervention  $I$ , let  $J_S^I = \operatorname{Des}(I \setminus S) \cup \{v \in I \setminus S : H_S^I(v) \cap \bar{S} \neq \emptyset\}$ .
+
+Analogous to Lemma 4.5, we can show that  $J_S^I$  satisfies similar properties (illustrated in Figure 5).
+
+Lemma 6.4. Let  $S$  be a prefix vertex set. Then  $\forall w \in J_S^I$ , we have  $\mathsf{D} \ominus \mathsf{s}(w) \subseteq J_S^I$ . Furthermore,  $J_S^I \cap \mathrm{src}(\bar{S}) = \emptyset$ .
+
+Therefore we can exclude  $J_S^I$  as well, which results in the following modification of Algorithm 1 and for which we can show similar guarantees using Lemma 6.4.
+
+# Algorithm 3 Learning a Prefix Vertex Set (w. Interventions)
+
+1: Input: A prefix vertex set  $S \subsetneq V$ . CI queries from  $\mathcal{G}$  and  $\mathcal{G}^I$  for each intervention  $I \in \mathcal{I}$ .  
+2: Output: A prefix vertex set  $S'$  such that  $S' \supseteq S$ .  
+3: for  $I \in \mathcal{I}$  do  
+4: Compute type-IV set  $J_S^I$ .  
+5: end for  
+6: Compute type-I set  $D_S$ .  
+7: Compute type-II and III sets  $E_S, F_S$ .  
+8: Let  $U' = \tilde{S} \setminus \left( \cup_{I \in \mathcal{I}} J_S^I \cup D_S \cup E_S \cup F_S \right)$ .  
+9: return  $S' = S \cup U'$ .
+
+Theorem 6.5. Algorithm 3 outputs a prefix vertex set in  $\mathcal{O}(n^4) + |\mathcal{I}|\mathcal{O}(n^2)$  CI tests. In addition, this prefix vertex set contains all the remaining source nodes, i.e.,  $\mathrm{src}(\bar{S})\subset S'$ .
+
+Proof. Similar to the proof of Theorem 3.3, we can use the additional Lemma 6.4 to show that: (1)  $S'$  is a prefix vertex set, (2) it contains  $\operatorname{src}(\bar{S})$ . We now bound the number of CI tests performed by Algorithm 3: note that it bears the additional need to compute  $J_S^I$  compared to Algorithm 1. By Lemmas 6.1 and 6.2, computing  $J_S^I$  for each  $I \in \mathcal{I}$  takes  $\mathcal{O}(n^2)$ . Thus the total complexity is  $\mathcal{O}(n^4) + |\mathcal{I}| \mathcal{O}(n^2)$ .
+
+In addition, we can show the following property for covered edges coming from  $\mathrm{src}(\bar{S})$ .
+
+Lemma 6.6. Let  $S$  be a prefix vertex set,  $v \in \operatorname{src}(\bar{S})$  and  $v \to w$  be a covered edge. Given any intervention  $I$ , if  $|I \cap \{v, w\}| = 1$ , then  $w \in J_S^I$ .
+
+Proof. If  $I \cap \{v, w\} = \{v\}$ , then  $w \in \mathsf{Des}(I \setminus S)$ . If  $I \cap \{v, w\} = \{w\}$ , then  $w \in I \setminus S$ . We will show that  $H_S^I(w) \cap \bar{S} \neq \emptyset$ . This in turn proves that  $w \in J_S^I$ . Note that since  $v \in \operatorname{src}(\bar{S})$ , we have  $v \notin S \cup \mathsf{Des}[I \setminus S]$ . Furthermore, since  $v \to w$  is an edge, we have  $v \in \mathsf{Pa}(w)$ . Thus by Lemma 6.2, we have  $v \in \mathsf{Pa}(w) \setminus (S \cup \mathsf{Des}[I \setminus S]) \subseteq H_S^I(w)$ .
+
+# 6.2. Learning  $\mathcal{L}$ -CCPG Representations
+
+We obtain the final algorithm by plugging Algorithm 3 into Algorithm 2, i.e., replacing Algorithm 1 in line 5 by Algorithm 3.
+
+Proof of Theorem 3.4. By Theorem 6.5, for any  $i$ , set  $S = S_{1} \cup \dots \cup S_{i-1}$  is a prefix vertex set. Therefore using the proof in Theorem 3.3 and the fact that  $\mathrm{src}(\bar{S}) \subset S \cup S_{i}$  (Theorem 6.5), we immediately have  $|\mathrm{src}(V_{i})| = 1$  and  $\mathcal{D}$  satisfies the second property of  $\mathcal{I}$ -CCPG in Definition 3.1.
+
+We now show that when  $|V_i| > 1$ , subgraph  $\mathcal{G}[V_i]$  has at least one unintervened covered edge. Suppose on the contrary that  $\mathcal{G}[V_i]$  has no unintervened covered edge. Denote
+
+$\{s\} = \operatorname{src}(V_i)$ . Let  $w \in V_i$  such that  $w \neq s$ . Similar to Theorem 3.3, we can show that  $w \in \mathsf{Des}(s)$ . By Lemma 4.7, there is a covered edge from  $s$  to some vertex  $x \in \mathsf{Anc}[w]$  and  $x \in V_i$ . Since  $\mathcal{G}[V_i]$  has no unintervened covered edge,  $s \to x$  is intervened by some  $I \in \mathcal{I}$ . Then by Lemma 6.6, we have  $x \in J_S^I$ . This contradicts  $x \in V_i$ .
+
+# 7. Experiments
+
+In this section, we test our proposed method and compare it to existing causal discovery methods on synthetic data and a toy real-world example. Source code for these results can be found at https://github.com/uhlerlab/CCPG.
+
+# 7.1. Synthetic Data
+
+In these experiments, we consider the observational setting and generate samples from linear causal models with additive Gaussian noise, governed by identifiable causal graphs, in particular, in-star-shaped DAGs with varying number of nodes. In such settings, with enough samples, our algorithm is guaranteed to return the ground-truth DAG (Corollary 3.5).
+
+We compare our Algorithm 2, termed CCPG, to other constraint-based and hybrid methods that rely on conditional independence tests. The constraint-based methods we include are: PC (Spirtes et al., 2000), FCI (Spirtes et al., 1999), and RFCI (Colombo et al., 2012), where we use the order-independent variants from Colombo et al. (2014) (i.e., "stable" versions). The hybrid methods we included are: GSP with depth  $= 4$  and depth  $= \infty$  (Solus et al., 2021). Implementation details can be found in Appendix E.
+
+![](images/f3d87091cfab31db763fe83e9913f4537ddbcf37c797ced99c14a02dd6c557c8.jpg)  
+Figure 6. Runtime comparison across graphs of different sizes. CCPG is as fast as hybrid methods (GSP) and significantly more efficient than constraint-based methods (PC, FCI, and RFCI). The programming language behind each implementation is indicated by dashed or solid lines (see Appendix E for details).
+
+![](images/0e04e051e4d48eb94e1874e9db8d1a6d25ddce627973e734a0208d068c5f2ec6.jpg)  
+Figure 7. Sample complexity comparison. CCPG requires the least number of samples to recover the true graph. Methods that do not assume causal sufficiency are indicated by shading (see Appendix E for details).
+
+Runtime Analysis. To test the computational efficiency of our method, we generated 100k samples across graphs of different sizes and reported the runtime averaged across five runs. Figure 6 shows that CCPG is as fast as hybrid methods (GSP) and significantly more efficient than constraint-based methods (PC, FCI, and RFCI), which can only scale to 20 nodes with reasonable runtime.
+
+Sample Complexity Analysis. To test the sample efficiency of CCPG, we consider a 10-node in-star-shaped DAG. For each method, we increase the number of samples until it returns the ground-truth DAG and repeat this procedure for five runs. Figure 7 shows that CCPG requires the least number of samples to recover the true graph.
+
+In general, constraint-based methods (PC, FCI, and RFCI) have better sample efficiency than hybrid methods, while being significantly more runtime inefficient. In comparison, CCPG, with its polynomial-number of CI test guarantee, enjoys low sample complexity and low runtime complexity.
+
+# 7.2. A Real-world Example
+
+To illustrate the utility of the coarser representation learned by CCPG in real-world settings, we include a simple 6-variable Airfoil example (Asuncion & Newman, 2007; Lam et al., 2022); see Lam et al. (2022) for a detailed description of this example. Although there is no known ground-truth DAG in this setting, a few causal relations are known: (1) velocity, chord, and attack should be source nodes; (2) pressure is downstream of all other nodes.
+
+The coarser representation learned by CCPG is shown in Figure 8. Compared to PC, which returns the partially directed
+
+![](images/a673da5d09f28483cc65bc9fd2d4b64e6702bf4fec1728ee94724c12b2d92abe.jpg)  
+Figure 8. Learned coarser representation by CCPG in the Airfoil example.
+
+![](images/f2660c59d6133d58256b662351345f422cf622619a8558e1e6095851d054dba2.jpg)  
+Figure 9. Learned partially directed graph by PC in the Airfoil example.
+
+graph in Figure 9, CCPG seems to be more consistent with the known causal relations while it contains less information due to a coarser representation.
+
+# 8. Discussion
+
+In our work, we studied causal structure learning under the constraint of fewer CI tests. Since exact structure learning may demand an exponential number of CI tests, we defined a representation (CCPG) that captures partial but crucial information about the underlying causal graph. Moreover, we provided an efficient algorithm that recovers a CCPG representation in a polynomial number of CI tests. This result enabled us to design efficient algorithms for the full recovery of causal graphs in two specific settings, utilizing only a polynomial number of CI tests.
+
+We hope that our work will motivate further exploration of the causal discovery problem under the constraint of fewer CI tests, extending to various settings, including those involving latent variables. Furthermore, our research establishes a foundation for addressing the search problem<sup>10</sup> with reduced tests, suggesting the potential existence of search algorithms capable of recovering the causal graph with a polynomial number of independence tests while performing an approximately optimal number of interventions.
+
+# Acknowledgements
+
+We thank the anonymous reviewers for helpful feedback. J.Z. was partially supported by an Apple AI/ML PhD Fellowship. K.S. was supported by a fellowship from the Eric and Wendy Schmidt Center at the Broad Institute. C.U. was partially supported by NCCIH/NIH (1DP2AT012345), ONR (N00014-22-1-2116), DOE-ASCR (DE-SC0023187), the MIT-IBM Watson AI Lab, and a Simons Investigator Award.
+
+# Impact Statement
+
+This paper presents theoretical work whose goal is to advance the field of causal inference. There are many potential societal applications of our work, none of which we feel must be specifically highlighted here.
+
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+
+# A. Useful Lemmas
+
+# A.1. Proof of Lemma 3.2
+
+Lemma 3.2 (Properties of CCPG). For any intervention set  $\mathcal{I}$  (including  $\varnothing$ ), the following arguments hold:
+
+-  $\mathcal{D} = \mathcal{G}$  (i.e., partitioning  $V$  into individual vertices) is a valid  $\mathcal{L}$ -CCPG of  $\mathcal{G}$ .  
+- If the verification number of  $\mathcal{G}$  is zero, i.e.,  $\nu(\mathcal{G}) = 0$ , then  $\mathcal{D} = \mathcal{G}$  is the unique valid  $\mathcal{I}$ -CCPG of  $\mathcal{G}$ .  
+- If  $\mathcal{I}$  is a verifying intervention set of  $\mathcal{G}$ , then  $\mathcal{D} = \mathcal{G}$  is the unique valid  $\mathcal{I}$ -CCPG of  $\mathcal{G}$ .
+
+Proof. Note that if  $|V_i| = 1$ , then CCPG is actually  $\mathcal{G}$ . If  $|V_i| > 1$ , then it means that there is a covered edge in this subset that is not intervened. By Proposition 2.1, this is impossible when  $\mathcal{I}$  is a verifying intervention set.
+
+# A.2. Proof of Lemma 3.6
+
+Lemma 3.6 (Proxy V-Structure). Let  $S \subseteq V$  and  $u, v, z \in V \backslash S$ . If  $u \perp v|S$  and  $u \not\perp v|S \cup \{z\}$ , then  $u, v \notin \mathsf{Des}[z]$ .<sup>11</sup>
+
+Proof. Let  $P$  be an active path (that carries dependency) between  $u$  and  $v$  when conditioned on  $S \cup \{z\}$ . Since  $u \perp v|S$  and  $u \not\perp v|S \cup \{z\}$ , we get that there exist a set of vertices  $w_1 \ldots w_k$  that are colliders on  $P$  and satisfy:  $\mathsf{Des}[w_i] \cap S = \emptyset$  and  $z \in \mathsf{Des}[w_i]$ .
+
+Consider the path  $P$  and note that it takes the form  $P = u - \dots \to w_{1} \gets \dots \to w_{i} \gets \dots \to w_{k} \gets \dots - v$ . Define  $P_{1} = u - \dots \to w_{1}$  and  $P_{2} = w_{k} \gets \dots - v$ . Since all the colliders on paths  $P_{1}, P_{2}$  are in or have their descendants in  $S$  and all non-colliders do not belong to  $S$ , we have that  $P_{1}, P_{2}$  are active given  $S$ . We prove our lemma using the proof by contradiction strategy. For contradiction, let us assume that one of the vertices in  $\{u, v\}$  belong to the set  $\mathsf{Des}[z]$  and without loss of generality let that vertex be  $u$ . Then since  $z \in \mathsf{Des}[w_{i}]$  for all  $i$ , there must be  $u \in \mathsf{Des}[w_{i}]$  for all  $i$ .
+
+Since  $u \in \mathsf{D}\varepsilon \mathsf{s}[w_k]$ , let  $Q$  be the directed path in the graph that connects  $w_{k}$  to  $u$ . Note that all the vertices in path  $Q$  are all descendants of  $w_{k}$  and they do not belong to the set  $S$  (because  $\mathsf{D}\varepsilon \mathsf{s}[w_k] \cap S = \emptyset$ ). Now consider the path  $(P_2, Q)$  that connects vertices  $v$  and  $u$  and note that the vertex  $w$  is a non-collider on the new path  $(P_2, Q)$ . Since  $P_2$  and  $Q$  are active paths given  $S$  and since  $w \notin S$ , we immediately get that the path  $(P_2, Q)$  is an active path given  $S$ , which further implies that  $u \not\perp v|S$ ; a contradiction and we conclude the proof.
+
+# A.3. Proof of Lemma 3.8
+
+Lemma 3.8 (Proxy Meek Rule 1). Let  $S$  be a prefix subset. If  $u, v$  and  $w$  are such that  $u \in S$ ,  $v, w \notin S$  and  $u \not\perp v|S$  and  $u \perp w|S \cup \{v\}$  then  $v \notin Des[w]$ .
+
+Proof. Suppose on the contrary that there is  $u, v, w$  such that  $u \in S$ ,  $v, w \notin S$  and  $u \not\perp v|S$  and  $u \perp w|S \cup \{v\}$  and  $v \in \operatorname{Des}[w]$ .
+
+Since  $u \not\perp v \mid S$ , let  $P$  be the active path connecting  $u$  and  $v$  given  $S$ . As  $u \in S$  and  $v \notin S$ , there is an edge  $u' - v'$  on  $P$  such that  $u' \in S$  but  $v' \notin S$ .
+
+Now denote the vertex on  $P$  that is immediate next to  $v$  as  $x$ . If  $x \gets v$ , then there must be a collider on  $P$  between  $u'$  and  $v$  as  $S$  is a prefix subset where  $u' \in S$  and  $v \notin S$ . Let  $y$  be the collider on  $P$  between  $u'$  and  $v$  that is closest to  $u'$ , then  $u' \to v' \cdots \to y$ . Since  $P$  is active given  $S$ , the collider  $y \in \operatorname{Anc}[S]$ . However  $v' \in \operatorname{Anc}[y] \subseteq \operatorname{Anc}[S]$  and  $v' \notin S$ , a contradiction to  $S$  being prefix. Thus there must be  $x \to v$  on  $P$ .
+
+Since we assumed on the contrary that  $v \in \mathsf{Des}[w]$ , we can consider the path  $Q$  joined by  $P$  and the directed path from  $w$  to  $v$ . Compared to  $P$ , the path  $P$  has one additional collider  $v$ , and has a few additional colliders that lie between  $v$  and  $w$  which are not in  $S$  (as they are all descendants of  $w$  and  $w \notin S$ ). Therefore  $Q$  is active given  $S \cup \{v\}$ . This means  $u \not\perp w \mid S \cup \{v\}$ , a contradiction.
+
+# A.4. Additional Lemma
+
+In addition, we will make use of the following lemma.
+
+Lemma A.1. For disjoint sets  $A, B, C \subset V$ , if  $\mathbb{P}a(A) \subset C \cup A$ ,  $\mathbb{P}a(B) \subset C \cup B$  and  $\mathsf{Des}(B) \cap C = \emptyset$ , then  $A \perp B \mid C$ .
+
+Proof. Assume without loss of generality that the vertices in  $B$  have the following topological order  $b_{1},\ldots ,b_{m}$ . Then for any  $i\in [m]$ , by the local Markov property (Spirtes et al., 1989), we have  $A\perp b_{i}\mid C\cup \{b_{1},\ldots ,b_{i - 1}\}$ . Therefore using Bayes rule, we have
+
+$$
+\begin{array}{l} P (B \mid C, A) = P \left(b _ {1} \mid C, A\right) P \left(b _ {2} \mid C, A, b _ {1}\right) \dots P \left(b _ {m} \mid C, A, b _ {1}, \dots , b _ {m - 1}\right) \\ = P \left(b _ {1} \mid C, A\right) P \left(b _ {2} \mid C, b _ {1}\right) \dots P \left(b _ {m} \mid C, b _ {1}, \dots , b _ {m - 1}\right) \\ = P (B \mid C), \\ \end{array}
+$$
+
+and thus  $A\perp B\mid C$  , which completes the proof.
+
+# B. Missing Proofs of Prefix Vertex Set
+
+# B.1. Proof of Lemma 4.5
+
+We restate the lemma below.
+
+Lemma 4.5. Let  $S$  be a prefix vertex set. If  $w \in D_S$ , then  $\mathsf{D} \in \mathsf{S}[w] \subset D_S$ . Furthermore,  $D_S \cap \mathrm{src}(\bar{S}) = \emptyset$ . The same properties hold for  $E_S$ .
+
+# B.1.1. TYPE-I SET  $D_{S}$
+
+Proof of Lemma 4.5 for  $D_S$ . We first show that if  $w \in D_S$ , then it must hold that  $\mathsf{Des}[w] \subset D_S$ : since  $w \in D_S$ , there exists a vertex  $u \in V$  and  $v \in \bar{S}$  such that  $u \perp v \mid S$  and  $u \not\perp v \mid S \cup \{w\}$ . We now show that for any  $x \in \mathsf{Des}[w]$ , we have  $u \not\perp v \mid S \cup \{x\}$ .
+
+Since  $u \not\perp v \mid S \cup \{w\}$ , there is a path  $P$  from  $u$  to  $v$  that is active given  $S \cup \{w\}$ . Therefore, all non-colliders on  $P$  are not in  $S \cup \{w\}$  and all colliders on  $P$  are in  $\mathsf{Anc}[S \cup \{w\}]$ . Since  $x \in \mathsf{Des}[w]$ , all colliders on  $P$  are in  $\mathsf{Anc}[S \cup \{x\}]$ . If all non-colliders are not in  $S \cup \{x\}$ , then  $P$  is an active path from  $u$  to  $v$  given  $S \cup \{x\}$ , and thus  $u \not\perp v \mid S \cup \{x\}$ . Otherwise there is a non-collider on  $P$  that is  $x$ .
+
+Since  $u \perp v \mid S$ , the path  $P$  is inactive given  $S$ . From above we know that all non-colliders on  $P$  are not in  $S$ . Therefore there exists a collider on  $P$  that is not in  $\operatorname{Anc}[S]$ . Suppose the leftmost and rightmost such colliders are  $k, k'$  (it is possible that  $k = k'$ ), then  $k, k'$  must be in  $\operatorname{Anc}[S \cup \{w\}] \setminus \operatorname{Anc}[S] \subseteq \operatorname{Anc}[w] \subset \operatorname{Anc}[x]$ . Consider the path  $Q$  in the graph by cutting out the parts between  $k, x$  (and  $k', x$ ) on  $P$  and replacing them with directed edges from  $k$  to  $x$  (and from  $k'$  to  $x$ ). Compared to  $P$ , the additional non-colliders on  $Q$  are all on the directed path from  $k$  to  $x$  (or  $k'$  to  $x$ ). They are not in  $S$  since  $k, k' \notin \operatorname{Anc}[S]$ , and thus  $Q$  has no non-colliders in  $S$ .
+
+Compared to  $P$ , there is no collider on  $P$  that is not in  $\mathsf{Anc}[S]$  and is still on  $Q$  by the fact that  $k, k'$  are leftmost and rightmost colliders on  $P$  that are not in  $\mathsf{Anc}[S]$ . Therefore,  $x$  must be a collider on  $Q$ , or else  $Q$  is active given  $S$  and  $u \not\perp v \mid S$ . Therefore all non-colliders on  $Q$  are not in  $S \cup \{x\}$ . Every collider on  $Q$  is either  $x$  or a collider of  $P$ , which is in  $\mathsf{Anc}[S \cup \{x\}]$ . Thus  $Q$  is active given  $S \cup \{x\}$ . Therefore  $u \not\perp v \mid S \cup \{x\}$ .
+
+Next we show that  $D_S \cap \operatorname{src}(\bar{S}) = \emptyset$ : for contradiction assume that there exists a vertex  $a \in \operatorname{src}(\bar{S})$  such that  $a \notin \bar{S} \backslash D_S$ , that is  $a \in \operatorname{src}(\bar{S})$  and for some vertex  $u \in V, v \in \bar{S}, v \perp u|S$  and  $v \not\perp u|S \cup \{a\}$ .
+
+Since  $v \perp u|S$  and  $v \not\subset u|S \cup \{a\}$ , there exists a path  $P$  between  $v$  and  $u$  which is inactive when conditioned on  $S$  but is active upon conditioning on  $S \cup \{a\}$ . Moreover, this path contains a vertex  $b$  that is a collider on  $P$  and satisfies:  $a \in \mathsf{Des}[b]$  and  $\mathsf{Des}[b] \cap S = \emptyset$ . Since  $\mathsf{Des}[b] \cap S = \emptyset$ , we have that  $b \in \bar{S}$ . Furthermore, since  $b \in \bar{S}, a \in \mathrm{src}(\bar{S})$  and  $a \in \mathsf{Des}[b]$ , this implies that  $b = a$ . Therefore, the path  $P$  takes the form:  $P = v \dots \to a \gets \dots u$ . All the colliders on the path  $P$  either belong to or have descendant in the set  $S \cup \{a\}$ .
+
+Now consider the path  $v \cdots \to a$  and note that it is active given  $S$ . Let  $k$  be the number of vertices between  $v$  and  $a$  on this path  $v - v_1 \ldots v_k \to a$ . It is immediate that  $v_k \in S$  since  $a \in \operatorname{src}(\bar{S})$ . However, since  $v_k \in S$ , and since we condition on
+
+the set  $S$ , this should be a collider for the path  $Q$  to be active, which is not possible. Thus we get a contradiction, which completes the proof.
+
+# B.1.2. TYPE-II SET  $E_{S}$
+
+Proof of Lemma 4.5 for  $E_S$ . We first show that if  $w \in E_S$ , it must hold that  $y \in E_S$  for any  $y \in \mathsf{D} \in \mathsf{s}(w)$ : since  $w \in E_S$ , there is  $u \in S$  and  $v, v' \in \bar{S} \setminus D_S$ , such that  $u \perp v' \mid S \cup \{v\}$  and  $u \not\perp v' \mid S \cup \{v, w\}$ . We will show  $u \not\perp v' \mid S \cup \{v, y\}$ .
+
+Since  $u \perp v' \mid S \cup \{v\}$  and  $u \not\perp v' \mid S \cup \{v, w\}$ , by Lemma 3.6 (note that the set "S" in the exposition of Lemma 3.6 can be an arbitrary subset), we know that  $v' \notin \mathsf{Des}[w]$ . Assume on the contrary that  $u \perp v' \mid S \cup \{v, y\}$ . Since  $u \perp v' \mid S \cup \{v\}$  and  $u \not\perp v' \mid S \cup \{v, w\}$ , there is a path  $P$  between  $u, v'$  that is active given  $S \cup \{v, w\}$  but inactive given  $S \cup \{v\}$  or  $S \cup \{v, y\}$ . This means that (1)  $y$  is a non-collider on  $P$ , (2) all colliders on  $P$  are in  $\mathsf{Anc}[S \cup \{v, w\}]$ , (3)  $P$  has a collider that is in  $\mathsf{Anc}[w] \setminus \mathsf{Anc}[S \cup \{v\}]$ .
+
+Note that  $v' \notin \operatorname{Des}[y]$  since  $y \in \operatorname{Des}[w]$  but  $v' \notin \operatorname{Des}[w]$ . In addition, we also have  $u \notin \operatorname{Des}[w]$  since  $u$  is in the prefix vertex set  $S$  but  $w \in \bar{S}$ . Therefore  $y$  being a non-collider on  $P$  means there is a collider  $x$  on  $P$  such that  $y \in \operatorname{Anc}(x)$ . Note that this collider has to be in  $\operatorname{Anc}[S \cup \{v, w\}]$ . However, since  $y \notin \operatorname{Anc}[S \cup \{w\}]$  and  $y \in \operatorname{Anc}(x)$ , it must hold that  $x \in \operatorname{Anc}[v]$ , which means  $y \in \operatorname{Anc}(v)$ . Since  $y \in \operatorname{Des}(w)$ , this means  $v \in \operatorname{Des}(w)$ , which makes  $\operatorname{Anc}[w] \setminus \operatorname{Anc}[S \cup \{v\}] = \emptyset$ . This violates (3) above, a contradiction.
+
+Next we show that  $E_S \cap \mathrm{src}(\bar{S}) = \emptyset$ : if there exists  $w \in \bar{S} \backslash D_S$  such that  $w \in E_S \cap \mathrm{src}(\bar{S})$ . Then  $u \perp v' \mid S \cup \{v\}$  and  $u \not\perp v' \mid S \cup \{v, w\}$  for some  $u \in S$  and  $v, v' \in \bar{S} \backslash D_S$ . Thus there is a path  $P$  connecting  $u$  and  $v'$  such that  $P$  is active given  $S \cup \{v, w\}$  but inactive given  $S \cup \{v\}$ .
+
+Therefore all non-colliders on  $P$  are not in  $S \cup \{v, w\}$ , and there is a collider  $x$  on  $P$  such that  $x \in \operatorname{Anc}[w]$ . Note that since  $w \in \operatorname{src}(\bar{S})$ , it must hold that  $x \in S$ . Since  $x$  is a collider, there is  $y \neq u$  such that  $y \to x$  on  $P$ . Since  $S$  is a prefix and  $x \in S$ , we have  $y \in S$ . However,  $y$  is a non-collisioner on  $P$ , which contradicts  $P$  active given  $S \cup \{v, w\}$  and completes the proof.
+
+# B.2. Proof of Lemma 4.6
+
+Lemma 4.6. Let  $S$  be a prefix vertex set. If  $w \in F_S$ , then  $D \in s[w] \subset E_S \cup F_S$ . Furthermore,  $F_S \cap \mathrm{src}(\bar{S}) = \emptyset$ .
+
+Proof. We first show that if  $w \in F_S$ , then for any  $y \in \mathsf{Des}(w)$ , we have  $y \in E_S \cup F_S$ . Since  $w \in F_S$ , we have  $u \not\perp v \mid S, u \perp w \mid S \cup \{v\}$ , and  $v \not\perp w \mid S$  for some  $u \in S$  and  $v \in \tilde{S} \setminus D_S$ .
+
+Since  $v \not\perp w \mid S$ , there is an active path between  $v, w$  given  $S$ . Consider extending this path by the directed path from  $w$  to  $y$ . Note that none of vertices on the directed path from  $w$  to  $y$  are in  $S$ , since  $S$  is prefixed and  $w \notin S$ . Therefore, this extended path is also active given  $S$ , which means  $v \not\perp y \mid S$ .
+
+Thus, if  $y \notin F_S$ , then it must hold that  $u \not\perp y \mid S \cup \{v\}$ . This means there is an active path, denoted by  $P$ , between  $u$  and  $y$  given  $S \cup \{v\}$ . Consider extending this path by the directed path from  $w$  to  $y$ , denoted as  $Q$  (which exists in the graph). Compared to  $P$ , the additional non-colliders on  $Q$  are not in  $S \cup \{v\}$ : for  $S$ , this is because  $S$  is prefix,  $w \notin S$ , and all additional non-colliders are descendants of  $w$ ; for  $v$ , this is because  $v \not\in \mathsf{Des}(w)$  (by Lemma 3.8,  $u \not\perp v \mid S$  and  $u \perp w \mid S \cup \{v\}$ ). Thus  $Q$  is active given  $S \cup \{v\}$  unless  $y$  is a collider on  $Q$ . Since  $u \perp w \mid S \cup \{v\}$ , the path  $Q$  must be inactive given  $S \cup \{v\}$ , which means  $y$  is a collider on  $Q$ . This means  $Q$  is active given  $S \cup \{v, y\}$ . Therefore,  $u \not\perp w \mid S \cup \{v, y\}$ . Together with  $u \perp w \mid S \cup \{v\}$ , we have  $y \in E_S$ .
+
+Next we show that  $F_S \cap \mathrm{src}(\bar{S}) = \emptyset$ . Assume on the contrary that  $w \in F_S \cap \mathrm{src}(\bar{S})$ . Since  $w \in F_S$ , we have  $u \not\perp v|S, v \not\perp w|S$  and  $u \perp w|S \cup \{v\}$  for some  $u \in S$  and  $v \in \bar{S} \setminus D_S$ . By Lemma 3.8, we have  $v \notin \mathsf{Des}[w]$ . However, since  $v \not\perp w | S$ , there must be an active path  $P$  between  $v$  and  $w$  given  $S$ . This path cannot have any vertex in  $S$ ; otherwise consider the first vertex that is in  $S$ ; since  $v \notin S$  and  $S$  is prefix, such vertex must be a non-collider which would make  $P$  inactive given  $S$ . Therefore  $P$  is fully in  $\bar{S}$ . Since  $w \in \mathrm{src}(\bar{S})$ , we must have  $P: v - \dots \gets w$ . However since  $v \notin \mathsf{Des}[w]$ , there must be an edge  $\rightarrow$  on  $P$ . This means that there must be a collider on  $P$ . Since this collider is not in  $S$ , it makes  $P$  inactive given  $S$ , a contradiction.
+
+# B.3. Remarks
+
+The above proofs suffice as intermediate results to show Theorem 4.4, which we proved in Section 4.2.
+
+Regarding Lemma 4.7 in Section 4.3, we will prove it in Appendix C after proving Lemma 5.1, since it depends on Lemma 5.1.
+
+# C. Missing Proofs of Causally Consistent Partition Graph Representations
+
+# C.1. Proof of Lemma 5.1
+
+To prove Lemma 5.1, we will make use of the following lemma.
+
+Lemma C.1. Let  $S \subseteq V$  be a prefix subset, then the following statements hold:
+
+-  $u \perp v \mid S$  for all  $u, v \in \operatorname{src}(\bar{S})$ .  
+-  $S \cup U$  is a prefix subset for any  $U \subseteq \operatorname{src}(\bar{S})$ .
+
+Proof. We first prove condition one. For contradiction, we assume that  $u \not\subset v|S$ , which implies that there exists an active path between  $u$  and  $v$  when conditioned on  $S$ . Let  $P = u - u_1 \dots u_k - v$  be the path and  $u_1, \ldots, u_k$  be the vertices along the path. Note that  $k > 0$  as  $u$  and  $v$  are not connected; because an edge between  $u$  and  $v$  would mean that one of these vertices is not a source node in  $\bar{S}$ .
+
+Consider  $u_{1}$  and note that there are two possibilities  $u \to u_{1}$  or  $u \gets u_{1}$ . We start with the first case  $u \to u_{1}$ . Since  $S$  is a prefix subset and  $u \in \bar{S}, u \to u_{1}$  implies that  $u_{1} \in \bar{S}$ . Since no vertex in  $\bar{S}$  is conditioned upon, we have that the vertex  $u_{1}$  is not a collider on the path  $P$  and we have that the edge  $u_{1} - u_{2}$  is directed as  $u_{1} \to u_{2}$ . Repeating the same argument for  $u_{2}$  and all the other vertices in the path, we see that the path  $P$  takes the form  $P = u \to u_{1} \to \dots \to u_{k} \to v$ . The previous argument implies that  $v \in \mathsf{D}\mathsf{e}\mathsf{s}[u]$  and therefore does not belong to  $\mathrm{src}(\bar{S})$ , which is a contradiction to our assumption that  $u, v \in \mathrm{src}(\bar{S})$ .
+
+Now consider the other case where  $u \gets u_1$ . Note that since  $u \in \operatorname{src}(\bar{S})$ , it is immediate that  $u_1 \in S$ . Furthermore, irrespective of the orientation between  $u_1$  and vertex  $u_2$ , it holds that vertex  $u_1$  is a non-collider on the path  $P$ . Moreover, since  $P$  is an active path,  $u_1$  should not be conditioned upon. However, since we condition on  $S$  and as  $u_1 \in S$ , we have a contradiction.
+
+In the above case analysis, we showed that there does not exist an active path between  $u$  and  $v$  when conditioned on  $S$ , which implies that  $u \perp v|S$  and we conclude the proof for condition one.
+
+In the remainder of the proof, we focus our attention on condition two. Consider any subset  $U \subseteq \operatorname{src}(\bar{S})$ . For contradiction assume that  $S \cup U$  is not a prefix subset, which implies that there exists a vertex  $v \in V \backslash (S \cup U)$  such that  $v \in \operatorname{Anc}[u]$  for some vertex  $u \in S \cup U$ . Since  $S$  is a prefix subset, it is immediate that  $u \notin S$ . Therefore, the only case is that  $u \in U$ . Note that both  $u$  and  $v$  belong to the set  $\bar{S}$  and  $v \in \operatorname{Anc}(u)$ ; both these expressions combined contradict the fact that  $u \in \operatorname{src}(\bar{S})$ . Therefore, it should be the case that  $S \cup U$  is a prefix subset and we conclude the proof.
+
+Now we prove Lemma 5.1 restated below.
+
+Lemma 5.1. Let  $S$  be a prefix subset. For any  $w \in \bar{S} \backslash D_S$ , there is  $|\text{Anc}[w] \cap \text{src}(\bar{S})| = 1$ . Furthermore, for any other  $w' \in \bar{S} \setminus D_S$ , there is  $w \not\subset w' \mid S$  if and only if  $\text{Anc}[w] \cap \text{src}(\bar{S}) = \text{Anc}[w'] \cap \text{src}(\bar{S})$ .
+
+Proof. We first show that for any  $w \in \bar{S} \setminus D_S$ , we have  $|\mathrm{Anc}[w] \cap \mathrm{src}(\bar{S})| = 1$ . Since  $w \in \bar{S}$ , it must hold that  $|\mathrm{Anc}[w] \cap \mathrm{src}(\bar{S})| \geq 1$ . Assume now that there is  $w \in \bar{S}$  such that  $|\mathrm{Anc}[w] \cap \mathrm{src}(\bar{S})| \geq 2$ . Let  $v_1, v_2 \in \mathrm{Anc}[w] \cap \mathrm{src}(\bar{S})$  such that  $v_1 \neq v_2$ . By Lemma C.1, we have  $v_1 \perp v_2 \mid S$ . Now consider the path  $P$  by stitching together the two directed paths, one from  $v_1$  to  $w$  and another from  $v_2$  to  $w$ . All non-colliders on  $P$  are not in  $S$  since  $S$  is a prefix subset. The only collider on  $P$  is  $w$ . Therefore  $P$  is active given  $S \cup \{w\}$ . We have  $v_1 \not\subset v_2 \mid S \cup \{w\}$ , which means  $w \in D_{v_1,S} \subset D_S$ , a contradiction to  $w \notin D_S$ .
+
+Next we show that for any other  $w' \in \bar{S} \setminus D_S$ , we have  $w \not\perp w' \mid S$  if and only if  $\operatorname{Anc}[w] \cap \operatorname{src}(\bar{S}) = \operatorname{Anc}[w'] \cap \operatorname{src}(\bar{S})$ . For the if direction, denote  $s = \operatorname{Anc}[w] \cap \operatorname{src}(\bar{S}) = \operatorname{Anc}[w'] \cap \operatorname{src}(\bar{S})$  and consider the trek by joining the two directed
+
+paths from  $s$  to  $w$  and from  $s$  to  $w'$ . Since this path has no colliders and it is fully in  $\bar{S}$  (since  $S$  is a prefix and  $s \in \bar{S}$ ), it is active given  $S$ . Thus  $w \not\perp w' \mid S$ . For the only if direction, assume on the contrary that  $w \not\perp w' \mid S$  but  $\mathsf{Anc}[w] \cap \mathrm{src}(\bar{S}) \neq \mathsf{Anc}[w'] \cap \mathrm{src}(\bar{S})$ . Let  $P$  be the active path between  $w, w'$  given  $S$ . Then all non-colliders on  $P$  are in  $\bar{S}$  and all colliders on  $P$  are in  $\mathsf{Anc}[S] = S$ . This means that  $P$  has no colliders; otherwise this collider is a child of the vertex next to it, which is a non-collider that is in  $\bar{S}$ . This means there is an edge from  $\bar{S}$  to  $S$ , which is a contradiction with  $S$  being prefix. Since  $P$  has no colliders, it must satisfy  $P \cap \mathsf{Anc}[w] \cap \mathsf{Anc}[w'] \neq \emptyset$ . However, since  $\mathsf{Anc}[w] \cap \mathrm{src}(\bar{S}) \neq \mathsf{Anc}[w'] \cap \mathrm{src}(\bar{S})$ ,  $|\mathsf{Anc}[w] \cap \mathrm{src}(\bar{S})| = 1$  and  $|\mathsf{Anc}[w'] \cap \mathrm{src}(\bar{S})| = 1$ ,  $P \cap \mathsf{Anc}[w] \cap \mathsf{Anc}[w'] \neq \emptyset \in \bar{S}$ . This means there is a non-collar on  $P$  that is in  $S$ , which would make it inactive given  $S$ , a contradiction.
+
+# C.2. Proof of Lemma 4.7
+
+To prove Lemma 4.7, we will make use of the following results.
+
+Lemma C.2. Let  $S$  be a prefix subset and  $w \in \bar{S} \setminus (D_S \cup \operatorname{src}(\bar{S}))$ . By Lemma 5.1, let  $\operatorname{Anc}[w] \cap \operatorname{src}(\bar{S}) = \{v\}$ . Then for any  $u \in S$ , if there is no directed path from  $u$  to  $w$  that does not intersect  $\operatorname{src}(\bar{S})$ , then
+
+$$
+u \perp w \mid S \cup \{v \}.
+$$
+
+Proof. Assume on the contrary that  $u \not\perp w \mid S \cup \{v\}$ . Let  $P$  be an active path from  $u$  to  $w$  conditioned on  $v$ . If  $P \cap S \neq \{u\}$ , then consider the last node on  $P$  that is in  $S$ . Since  $w \notin S$ , this node must be pointing into a node in  $\bar{S}$  on  $P$ , which makes this node a non-collider. However, this node belongs to  $S$ , which means  $P$  is inactive given  $S \cup \{v\}$ , and thus  $P \cap S = \{u\}$ .
+
+There is also no collider on  $P$ . Otherwise consider the first collider; it will be in  $\bar{S}$  since  $P \cap S = \{u\}$ . However, since  $P$  is active, it will be in  $\mathbb{A}\mathbb{n}\mathbb{c}[S \cup \{v\}]$ . This is impossible since  $v \in \mathrm{src}(\bar{S})$  and from Lemma C.1, we know that  $S \cup \{v\}$  is prefixed. Therefore,  $P$  must be a directed path from  $u$  to  $w$ , where the node  $x$  adjacent to  $u$  is in  $\bar{S}$ . This means that  $x \in \mathbb{A}\mathbb{n}\mathbb{c}[w]$ . If  $x \notin \mathrm{src}(\bar{S})$ , then  $P$  is a directed path from  $u$  to  $v$  that does not intersect  $\mathrm{src}(\bar{S})$ . If  $x \in \mathrm{src}(\bar{S})$ , since  $\mathbb{A}\mathbb{n}\mathbb{c}[w] \cap \mathrm{src}(\bar{S}) = \{v\}$ , then it must hold that  $x = v$ . This means  $P$  is inactive given  $S \cup \{v\}$ , a contradiction.
+
+Lemma C.3. Let  $S$  be a prefix subset and  $v \in \operatorname{src}(\bar{S})$ . Then for any  $w \in \bar{S}$ , either  $v \perp w \mid S$  or  $w \in \mathsf{Des}[v]$ .
+
+Proof. Suppose  $w \notin \mathsf{Des}[v]$ ; we will show that  $v \perp w \mid S$ . Assume on the contrary that  $v \not\perp w \mid S$ . Let  $P$  be the active path between  $v$  and  $w$  given  $S$ . If  $P$  intersects with  $S$ , then consider the last vertex on  $P$  that is in  $S$ . This vertex must be a non-collider since  $w \not\in S$ . This contradicts  $P$  being active given  $S$ . Therefore,  $P$  is fully in  $\bar{S}$ . Since  $v \in \mathrm{src}(\bar{S})$ , we have  $P: v \to \ldots w$ . Since  $w \not\in \mathsf{Des}[v]$ , there must be a collider on  $P$ . However, this collider is not in  $S$  and  $S$  is prefix, which means that  $P$  is active given  $S$ , a contradiction.
+
+We now prove Lemma 4.7, restated below.
+
+Lemma 4.7. Let  $S$  be a prefix vertex set. For  $w \in \bar{S} \setminus D_S$ , if  $w \notin \operatorname{src}(\bar{S})$  and there is no covered edge from  $\operatorname{Anc}[w] \cap \operatorname{src}(\bar{S})$  to  $\operatorname{Anc}[w]$ , then  $w \in E_S \cup F_S$ .
+
+Proof of Lemma 4.7. Assume  $w \notin \operatorname{src}(\bar{S})$ . For contradiction, assume that there is no covered edge from  $\operatorname{Anc}[w] \cap \operatorname{src}(\bar{S})$  to  $\operatorname{Anc}[w]$  and  $w \in \bar{S} \setminus (D_S \cup E_S \cup F_S)$ .
+
+Since  $w \notin \operatorname{src}(\bar{S})$ , there is  $v \in \operatorname{src}(\bar{S}) \cap \operatorname{Anc}[w]$ . As  $v \in \operatorname{Anc}[w]$ , there is a directed path from  $v$  to  $w$ . Consider the longest directed path  $P$  from  $v$  to  $w$ . Let  $v'$  be the adjacent vertex to  $v$  on  $P$ , i.e.,  $P: v \to v' \cdots \to w$ . By Lemma 4.5 and  $w \notin D_S$ , we must have  $v' \notin D_S$ . Since  $v \to v'$  is not a covered edge, there could only be two cases:
+
+- There is  $u \to v$  such that  $u \notin \mathbb{P}\mathsf{a}(v')$ . Since  $v \in \mathrm{src}(\bar{S})$ , we must have  $u \in S$ . Note that  $u \not\perp v \mid S$  and  $v \not\perp w \mid S$  and the second condition of the lemma is not met. We must have  $u \not\perp w \mid S \cup \{v\}$ . By Lemma C.2, there must exist a directed path  $Q$  from  $u$  to  $w$  that does not intersect  $\mathrm{src}(\bar{S})$ . Consider the path between  $u$  and  $w'$  by joining  $Q$  and the directed path from  $v'$  to  $w$ . Since this path does not intersect with  $S \cup \mathrm{src}(\bar{S})$  and  $w$  is the only collider on it, we know that this path is active given  $S \cup \{v\} \cup \{w\}$ . Thus  $u \not\perp v' \mid S \cup \{v\} \cup \{w\}$ . Since the second condition of the lemma is not met, we must have  $u \not\perp v' \mid S \cup \{v\}$ . By Lemma C.2, there must exist a directed path from  $u$  to  $v'$  that does not intersect  $\mathrm{src}(\bar{S})$ . Since  $u \not\perp \mathbb{P}\mathsf{a}(v')$ , this path has at least length two. Let  $k$  be the vertex on this path such that  $k \to v'$ . Note that  $v \to v' \gets k$ . Therefore  $v \not\perp k \mid S \cup \{v'\}$ . Since  $v' \not\subset D_S$ , we must have  $v \not\perp k \mid S$ . By Lemma C.3, we must have  $k \in \mathsf{Des}[v]$ . Thus we can increase the length of the directed path  $P$  by replacing  $v \to v'$  with  $v \to \dots \to k \to v'$ . This contradicts  $P$  being the longest directed path from  $v$  to  $w$ .
+
+- There is  $k \to v'$  such that  $k \notin \mathbb{P}\mathrm{a}(v)$ . Note that  $v \to v' \gets k$ . Therefore  $v \not\in k \mid S \cup \{v'\}$ . Since  $v' \notin D_S$ , we must have  $v \not\in k \mid S$ . By Lemma C.3, we must have  $k \in \mathsf{Des}[v]$ . Thus we can increase the length of the directed path  $P$  by replacing  $v \to v'$  with  $v \to \dots \to k \to v'$ . This contradicts  $P$  being the longest directed path from  $v$  to  $w$ .
+
+This completes the proof.
+
+# C.3. Remarks
+
+The above proofs suffice as intermediate results to show Theorem 3.3, which we proved in Section 5.
+
+# D. Missing Proofs of Interventions
+
+# D.1. Proof of Lemma 6.1
+
+Lemma 6.1. Let  $S$  be a prefixed vertex set. Given an intervention on vertices  $I \subseteq V$ . For any  $u \notin I$ , we have  $u \not\subset I$  for some  $v \in I \setminus S$  if and only if  $u \in \mathsf{Des}(I \setminus S) \setminus (I \setminus S)$ .
+
+Proof. For each  $v \in I$ , denote  $\mathsf{D} \vDash s(v, I)$  as the set of  $u \in I$  such that  $u \not\perp_I v$ . We first show that  $\mathsf{D} \vDash s(v, I)$  is equal to the set of descendants of  $v$  such that there exists a directed path from  $v$  which is not cut by  $I$ .
+
+Let  $u \notin I$  such that  $u \perp_I v$ . If  $u \in \mathsf{Des}(v, I)$ , then there exists a directed path from  $v$  to  $u$  in the modified DAG  $\mathcal{G}^I$ , which means it is active given  $\varnothing$ , a contradiction. Therefore the only if direction is proven.
+
+For the if direction, let  $u \notin I$  such that  $u \not\subset I$ . Then there is an active path  $P: u - \dots - v$  in the modified DAG  $\mathcal{G}^I$ . Since  $P$  is active given  $\varnothing$ , there is no collider on  $P$ . Furthermore, since all incoming edges to any vertex in  $I$  are removed in the modified DAG, this path must be  $P: u \gets \dots \gets v$  and satisfies  $P \cap I = \{v\}$ . Thus  $u \in \operatorname{Des}(v, I)$ .
+
+Next we show that  $\cup_{v\in I\setminus S}\mathsf{Des}(v,I) = \mathsf{Des}(I\setminus S)\setminus (I\setminus S)$ . This will prove the lemma.
+
+Since  $\mathsf{Des}(v,I)$  is equal to the set of descendants of  $v$  such that there exists a directed path from  $v$  which is not cut by  $I$ , it is clear that  $\cup_{v\in I\setminus S}\mathsf{Des}(v,I)\subseteq \mathsf{Des}(I\setminus S)\setminus (I\setminus S)$ . Now let  $u\in \mathsf{Des}(I\setminus S)\setminus (I\setminus S)$ . Then there is  $v\in I\setminus S$  such that  $u\in \mathsf{Des}(v)$ . Consider the directed path from  $v$  to  $u$  and let  $v'$  be the last vertex on this path that is in  $I$ . Then  $u\in \mathsf{Des}(v',I)$ . By the fact that  $S$  is prefix, we have from  $v\notin S$  that  $v'\notin S$ . Thus  $u\in \mathsf{Des}(v',I)\subset \cup_{v\in I\setminus S}\mathsf{Des}(v,I)$  and  $\cup_{v\in I\setminus S}\mathsf{Des}(v,I)\supseteq \mathsf{Des}(I\setminus S)\setminus (I\setminus S)$ . We therefore have  $\cup_{v\in I\setminus S}\mathsf{Des}(v,I) = \mathsf{Des}(I\setminus S)\setminus (I\setminus S)$ .
+
+# D.2. Proof Lemma 6.2
+
+Lemma 6.2. Let  $S$  be a prefix subset. Given an intervention  $I$  and  $v \in I \setminus S$ , denote  $H_S^I(v) = \{ u \notin S \cup \text{Des}[I \setminus S] : u \not\subset v \mid V \setminus \text{Des}[I \setminus S] \}$ . Then  $Pa(v) \setminus (S \cup \text{Des}[I \setminus S]) \subseteq H_S^I(v) \subseteq \text{Anc}(v) \setminus (S \cup \text{Des}[I \setminus S])$ .
+
+Proof. On one hand, let  $u \in \mathbb{P}a(v) \setminus (S \cup \mathbb{D}\mathrm{es}[I \setminus S])$ . Clearly  $u \notin S \cup \mathbb{D}\mathrm{es}[I \setminus S]$  and  $u \not\subset v \mid V \setminus \mathbb{D}\mathrm{es}[I \setminus S]$ . Thus  $u \in H_S^I(v)$ , which proves  $\mathbb{P}a(v) \setminus (S \cup \mathbb{D}\mathrm{es}[I \setminus S]) \subseteq H_S^I(v)$ .
+
+On the other hand, let  $u \in H_S^I(v)$ . Since  $u \not\perp v \mid V \setminus \operatorname{Des}[I \setminus S]$ , let  $P: u - \dots - v$  be the active path given  $V \setminus \operatorname{Des}[I \setminus S]$ . Then all non-colliders on  $P$  are in  $\operatorname{Des}[I \setminus S]$ , and all colliders on  $P$  are in  $\operatorname{Anc}[V \setminus \operatorname{Des}[I \setminus S]]$
+
+Note that  $\operatorname{Des}[I \setminus S] \cap \operatorname{Anc}[V \setminus \operatorname{Des}[I \setminus S]] = \emptyset$ . Otherwise let  $w \in \operatorname{Des}[I \setminus S] \cap \operatorname{Anc}[V \setminus \operatorname{Des}[I \setminus S]]$ . Since  $w \in \operatorname{Anc}[V \setminus \operatorname{Des}[I \setminus S]]$ , there is  $w' \notin \operatorname{Des}[I \setminus S]$  such that  $w' \in \operatorname{Des}[w]$ . However  $w \in \operatorname{Des}[I \setminus S]$ , which means  $w' \in \operatorname{Des}[I \setminus S]$ . A contradiction.
+
+If  $P$  has a collider  $y \to x \gets z$ , then  $z$  is either a non-collider or  $v$ . Either case we have  $z \in \operatorname{Des}[I \setminus S]$ . This means  $x \in \operatorname{Des}[I \setminus S]$ . However,  $x \in S \cup \operatorname{Anc}[V \setminus \operatorname{Des}[I \setminus S]]$  and  $\operatorname{Des}[I \setminus S] \cap \operatorname{Anc}[V \setminus \operatorname{Des}[I \setminus S]] = \emptyset$ . A contradiction. Thus there is no collider on  $P$ .
+
+Denote the vertex next to  $u$  on  $P$  as  $t$ , i.e.,  $P: u - t - \dots - v$ . Then  $t$  is either a non-collider or  $v$ . Either case  $t \in \mathsf{Des}[I \setminus S]$ . Therefore it must be  $u \to t$ , otherwise  $u \in \mathsf{Des}[I \setminus S]$ . Furthermore, as  $P$  has no colliders, it must be  $P: u \to t \dots \to v$ . Thus  $u \in \mathsf{Anc}(v)$ . Together with  $u \notin S \cup \mathsf{Des}[I \setminus S]$ , we have  $u \in \mathsf{Anc}(v) \setminus (S \cup \mathsf{Des}[I \setminus S])$ . This
+
+proves  $H_S^I(v) \subseteq \operatorname{Anc}(v) \setminus (S \cup \operatorname{Des}[I \setminus S])$ .
+
+# D.3. Proof of Lemma 6.4
+
+Lemma 6.4. Let  $S$  be a prefix vertex set. Then  $\forall w\in J_S^I$  we have  $\mathsf{De}\varepsilon (w)\subseteq J_S^I$  . Furthermore,  $J_{S}^{I}\cap \mathrm{src}(\bar{S}) = \varnothing$
+
+Proof. Assume on the contrary that  $v \in J_S^I$  but  $w \in \mathsf{Des}(v)$  such that  $w \notin J_S^I$ . Since  $v \in J_S^I$ , there could be two cases:
+
+-  $v \in \operatorname{Des}(I \setminus S)$ . Then clearly  $w \in \operatorname{Des}(v) \subseteq \operatorname{Des}(I \setminus S)$ . A contradiction.  
+-  $v \in I \setminus S$  and there is  $u \in H_S^I(v) \cap \bar{S}$ . Then  $w \in \mathsf{Des}(v) \subset \mathsf{Des}(I \setminus S)$ . A contradiction.
+
+Next we show that  $\mathrm{src}(\bar{S})\cap J_S^I\neq \emptyset$  . Note that by  $\mathrm{src}(\bar{S})\cap \mathrm{Des}(\bar{S}) = \emptyset$  and  $\mathrm{Des}(I\setminus S)\subseteq \mathrm{Des}(\bar{S})$  , we have  $\mathrm{src}(\bar{S})\cap \mathrm{Des}(I\setminus S) = \emptyset$  . In addition, for any  $v\in \mathrm{src}(\bar{S})\cap (I\setminus S)$  , by Lemma 6.2, we have  $H_{S}^{I}(v)\subset \mathbb{A}\mathbb{n}\mathbb{c}(v)\setminus S = \emptyset$  Thus  $\mathrm{src}(\bar{S})\cap \{v\in I\setminus S:H_S^I (v)\cap \bar{S}\neq \emptyset \} = \emptyset$  . We then have  $\mathrm{src}(\bar{S})\cap J_S^I = \emptyset$
+
+# D.4. Remarks
+
+The above proofs suffice as intermediate results for proving Theorem 6.5. Then together with Lemma 6.6 (proven in Section 6.1), we can prove Theorem 3.4, which is given in Section 6.2.
+
+# E. Details of Numerical Experiments
+
+Implementation Details. For FCI and RFCI, we used the implementations in Kalisch et al. (2024), which is written in R with C++ accelerations. For PC and GSP, we used the implementation in Squires, which us written in python. Our method, CCPG, is written in python. The acceleration of R (with C++) can be viewed by comparing two implementations of PC (stable) in Figure 6.
+
+Remark on causal sufficiency. Among the constraint-based methods, we marked the ones that do not assume causal sufficiency in Figure 7. These methods run additional tests to check for unobserved causal variables, and therefore might require more samples compared to, e.g., PC, since the underlying system we test on satisfies causal sufficiency.

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2024/Causal Representation Learning from Multiple Distributions_ A General Setting/full.md

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+# Causal Representation Learning from Multiple Distributions: A General Setting
+
+Kun Zhang *12 Shaoan Xie *1 Ignavier Ng *1 Yujia Zheng
+
+# Abstract
+
+In many problems, the measured variables (e.g., image pixels) are just mathematical functions of the latent causal variables (e.g., the underlying concepts or objects). For the purpose of making predictions in changing environments or making proper changes to the system, it is helpful to recover the latent causal variables  $Z_{i}$  and their causal relations represented by graph  $\mathcal{G}_Z$ . This problem has recently been known as causal representation learning. This paper is concerned with a general, completely nonparametric setting of causal representation learning from multiple distributions (arising from heterogeneous data or nonstationary time series), without assuming hard interventions behind distribution changes. We aim to develop general solutions in this fundamental case; as a by product, this helps see the unique benefit offered by other assumptions such as parametric causal models or hard interventions. We show that under the sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, interestingly, one can recover the moralized graph of the underlying directed acyclic graph, and the recovered latent variables and their relations are related to the underlying causal model in a specific, nontrivial way. In some cases, most latent variables can even be recovered up to component-wise transformations. Experimental results verify our theoretical claims.
+
+# 1. Introduction
+
+Causal representation learning holds paramount significance across numerous fields, offering insights into intricate relationships within datasets. Most traditional methodologies (e.g., causal discovery) assume the observation of causal
+
+*Equal contribution ¹Carnegie Mellon University ²Mohamed bin Zayed University of Artificial Intelligence.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+variables. This assumption, however reasonable, falls short in complex scenarios involving indirect measurements, such as electronic signals, image pixels, and linguistic tokens. Moreover, there are usually changes on the causal mechanisms in real-world, such as the heterogeneous or nonstationary data. Identifying the latent causal variables and their structures together with the change of the causal mechanism is in pressing need to understand the complicated real-world causal process. This has been recently known as causal representation learning (Schölkopf et al., 2021).
+
+It is worth noting that identifying only the latent causal variables but not the structure among them, is already a considerable challenge. In the i.i.d. case, different latent representations can explain the same observations equally well, while not all of them are consistent with the true causal process. For instance, nonlinear independent component analysis (ICA), where a set of observed variables  $X$  is represented as a mixture of independent latent variables  $Z$ , i.e.,  $X = g(Z)$ , is known to be unidentifiable without additional assumptions (Comon, 1994). While being a strictly easier task since there are no relations among latent variables, the identifiability of nonlinear ICA often relies on conditions on distributional assumptions (non-i.i.d. data) (Hyvärinen & Morioka, 2016; 2017; Hyvärinen et al., 2019; Khemakhem et al., 2020a; Sorrenson et al., 2020; Lachapelle et al., 2022; Hälva & Hyvärinen, 2020; Hälva et al., 2021; Yao et al., 2022) or specific functional constraints (Comon, 1994; Hyvärinen & Pajunen, 1999; Taleb & Jutten, 1999; Buchholz et al., 2022; Zheng et al., 2022; Zheng & Zhang, 2023).
+
+To generalize beyond the independent latent variables and achieve causal representation learning (recovering the latent variables and their causal structure), recent advances either introduce additional experiments in the forms of interventional or counterfactual data, or place more restrictive parametric or graphical assumptions on the latent causal model. For observational data, various graphical conditions have been proposed together with parametric assumptions such as linearity (Silva et al., 2006; Cai et al., 2019; Xie et al., 2020; 2022; Adams et al., 2021; Huang et al., 2022) and discreteness (Kivva et al., 2021). For interventional data, single-node interventions have been considered together with parametric assumptions (e.g., linearity) on the mixing function (Varici et al., 2023; Ahuja et al., 2023; Buchholz et al., 2022) or also on the latent
+
+causal model (Squires et al., 2023). The nonparametric settings for both the mixing function and causal model have been explored by (Brehmer et al., 2022; von Kugelgen et al., 2023; Jiang & Aragam, 2023) together with additional assumptions on counterfactual views (Brehmer et al., 2022), distinct paired interventions (von Kugelgen et al., 2023), and graphical conditions (Jiang & Aragam, 2023).
+
+Despite the exciting developments in the field, one fundamental question pertinent to causal representation learning from multiple distributions remains unanswered—in the most general situation, without assuming parametric models on the data-generating process or the existence of hard interventions in the data, what information of the latent variables and the latent structure can be recovered? This paper attempts to provide an answer to it, which, surprisingly, shows that each latent variable can be recovered up to clearly defined indeterminacies. It suggests what we can achieve in the general case and furthermore, what unique contribution the typical assumptions that are currently made in causal representation learning from multiple distributions make towards complete identifiability of the latent variables (up to component-wise transformations). This may make it possible to figure out what minimal assumptions are needed to achieve complete identifiability, given partial knowledge of the system.
+
+Contributions. Concretely, as our contributions, we show that under the sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, interestingly, one can recover the moralized graph of the underlying directed acyclic graph (Theorem 2), and the recovered latent variables and their relations are related to the underlying causal model in a specific, nontrivial way (Theorem 3)—each latent variables is recovered as a function of itself and its so-called intimate neighbors in the Markov network implied by the true causal structure over the latent variables. Depending on the properties of the true causal structure over latent variables, the set of intimate neighbors might even be empty, in which case the corresponding latent variables can be recovered up to component-wise transformations (Remark 1). Lastly, we show how the recovered moralized graph relates to the underlying causal graph under new relaxations of faithfulness assumption (Proposition 2). Simulation studies verified our theoretical findings.
+
+# 2. Problem Setting
+
+Let  $X = (X_{1},\ldots ,X_{d})$  be a  $d$ -dimensional vector that represents the observed variables (e.g., image pixels). We assume that they are generated by  $n$  latent causal variables  $Z = (Z_{1},\dots,Z_{n})$  via a nonlinear mixing function  $g:\mathbb{R}^n\to \mathbb{R}^d$  ( $d\geq n$ ), which is a  $\mathcal{C}^2$ -diffeomorphism onto its image  $\mathcal{X}\subseteq \mathbb{R}^{d}$ . Furthermore, the variables  $Z_{i}$ 's are assumed to follow a structural equation model (SEM) (Pearl,
+
+![](images/b3f931974f43b766abba3139d12935e9a22749ca45a57ad9ce04abd6c8e925c8.jpg)  
+Figure 1: The generating process for each latent causal variable  $Z_{i}$  changes, governed by a latent factor  $\theta_{i}$ . The observed variables  $X$  are generated by  $X = g(Z)$  with a nonlinear mixing function  $g$ .
+
+2000). Putting them together, the underlying data generating process can be written as
+
+$$
+\underbrace {X = g (Z)} _ {\text {N o n l i n e a r m i x i n g}}, \quad \underbrace {Z _ {i} = f _ {i} (\mathrm {P A} \left(Z _ {i}\right) , \epsilon_ {i} ; \theta_ {i}) , i = 1 , \dots , n} _ {\text {L a t e n t S E M}}. \tag {1}
+$$
+
+where  $\mathrm{PA}(Z_i)$  denotes the parents of variable  $Z_{i}$ ,  $\epsilon_{i}$ 's are exogenous noise variables that are mutually independent, and  $\theta_{i}$  denotes the latent (changing) factor (or effective parameters) associated with each model. Here, the data generating process of each latent variable  $Z_{i}$  may change, e.g., across domains or over time, governed by the corresponding latent factor  $\theta_{i}$ ; it is commonplace to encounter such changes in causal mechanisms in practice (arising from heterogeneous data or nonstationary time series). In addition, interventional data can be seen as a special type of change, which qualitatively restructure the causal relations. As their names suggest, we assume that the variables  $X$  are observed, while the latent causal variables  $Z$  and latent factors  $\theta = (\theta_{1},\dots,\theta_{n})$  are unobserved.
+
+Let  $P_{X;\theta}$  and  $P_{Z;\theta}$  be the distributions of  $X$  and  $Z$ , respectively, and their probability density functions be  $p_X(X; \theta)$  and  $p_Z(Z; \theta)$ , respectively. To lighten the notation, we drop the subscript in the density when the context is clear. The latent SEM in Eq. (1) induces a causal graph  $\mathcal{G}_Z$  with vertices  $\{Z_i\}_{i=1}^n$  and edges  $Z_j \to Z_i$  if and only if  $Z_j \in \mathrm{PA}(Z_i)$ . We assume that  $\mathcal{G}_Z$  is acyclic, i.e., a directed acyclic graph (DAG). This implies that the distribution of variables  $Z$  satisfy the Markov property w.r.t. DAG  $\mathcal{G}_Z$  (Pearl, 2000), i.e.,  $p(Z; \theta) = \prod_{i=1}^n p(Z_i | \mathrm{PA}(Z_i); \theta_i)$ . We provide an example of the data generating process in Eq. (1) and its corresponding latent DAG  $\mathcal{G}_Z$  in Figure 1. Given samples of the observed variables  $X$  arising from multiple distributions (or domains), say  $\Theta = \{\theta^{(1)}, \dots, \theta^{(m)}\}$ , our goal is to recover the latent causal variables  $Z$  and their causal relations up to minor indeterminacies.
+
+# 3. Learning Causal Representations from Multiple Distributions
+
+In this section, we provide theoretical results to show how one is able to recover the underlying latent causal variables and their causal relations up to certain indeterminacies from multiple distributions. Specifically, we show that under sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, the recovered latent variables are related to the true ones in a specific, nontrivial way. Such results serve as the foundation of our algorithm in Section 4.
+
+To start with, we estimate a model  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  that assumes the same data generating process as in Eq. (1) and matches the true distribution of  $X$  in different domains:
+
+$$
+p _ {X} \left(X ^ {\prime}; \theta^ {(u)}\right) = p _ {\hat {X}} \left(X ^ {\prime}; \hat {\theta} ^ {(u)}\right), \forall \theta^ {(u)} \in \Theta , X ^ {\prime} \in \mathcal {X} ^ {(u)}, (2)
+$$
+
+where  $\theta^{(u)}$  denotes the latent factor in the  $u$ -th domain, and  $\mathcal{X}^{(u)}$  is the image of function  $g$  in the  $u$ -th domain. Here,  $X$  and  $\hat{X}$  are generated by the true model  $(g, f, p_Z, \Theta)$  and the estimated model  $(\hat{g}, \hat{f}, p_{\hat{Z}}, \hat{\Theta})$ , respectively.
+
+A key ingredient of our results is the Markov network that represents conditional dependencies among random variables via an undirected graph. Let  $\mathcal{M}_Z$  be the Markov network over variables  $Z$ , i.e., with vertices  $\{Z_i\}_{i=1}^n$  and edges  $\{Z_i, Z_j\} \in \mathcal{E}(\mathcal{M}_Z)$  if and only if  $Z_i \not\equiv Z_j \mid Z_{[n] \setminus \{i,j\}}$ . Also, we denote by  $|\mathcal{M}_Z|$  the number of undirected edges in the Markov network. In Section 3.1, apart from showing how to estimate the underlying latent causal variables up to certain indeterminacies, we also show that such latent Markov network  $\mathcal{M}_Z$  can be recovered up to isomorphism. To achieve so, we make use of the following property (assuming that  $p_Z$  is twice differentiable):
+
+$$
+Z _ {i} \perp Z _ {j} \mid Z _ {[ n ] \backslash \{i, j \}} \Longleftrightarrow \frac {\partial^ {2} \log p (Z ; \theta)}{\partial Z _ {i} \partial Z _ {j}} = 0. \tag {3}
+$$
+
+Such a connection between pairwise conditional independence and cross derivatives of the density function has been noted by Lin (1997) and utilized in Markov network learning for observed variables (Zheng et al., 2023). With the recovered latent Markov network structure, we provide results in Section 3.2 to show how it relates to the moralized graph of true latent causal DAG  $\mathcal{G}_Z$ , by exploiting a specific type of faithfulness assumption that is considerably weaker than the standard faithfulness assumption used in the literature of causal discovery (Spirtes et al., 2001).
+
+# 3.1. Recovering Latent Causal Variables and Latent Markov Network
+
+We consider a general, completely nonparametric setting of causal representation learning from multiple distributions.
+
+Specifically, we show how one can recover the latent causal variables and the Markov network structure among them up to minor indeterminacies, by leveraging sparsity constraint and sufficient change conditions on the causal mechanisms. Notably, in some cases, most latent variables can even be recovered up to component-wise transformations.
+
+We start with the following result that provides information about the derivative of true latent causal variables  $Z$  with respect to the estimated ones  $\hat{Z}$ , according to their corresponding Markov networks  $\mathcal{M}_Z$  and  $\mathcal{M}_{\hat{Z}}$ . Result of this form is often used in the proof of nonlinear ICA to obtain identifiability of component-wise nonlinear transformations (Hyvärinen & Morioka, 2016; Hyvärinen et al., 2019). At the same time, our result here is different from that of nonlinear ICA as it allows for causal relations among latent variables. This result serves as the backbone of our further identifiability results in this section.
+
+Proposition 1. Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Suppose the following assumptions hold:
+
+- A1 (Smooth and positive density): The probability density function of latent causal variables, i.e.,  $p_Z$ , is twice continuously differentiable and positive in  $\mathbb{R}^n$ .  
+- A2 (Sufficient changes): For each value of  $Z$ , there exist  $2n + |\mathcal{M}_Z| + 1$  values of  $\theta$ , i.e.,  $\theta^{(u)}$  with  $u = 0, \dots, 2n + |\mathcal{M}_Z|$ , such that the vectors  $w(Z, u) - w(z, 0)$  with  $u = 1, \dots, 2n + |\mathcal{M}_Z|$  are linearly independent, where vector  $w(Z, u)$  is defined as follows:
+
+$$
+\begin{array}{l} w (Z, u) = \left(\frac {\partial \log p (Z ; \theta^ {(u)})}{\partial Z _ {i}}\right) _ {i \in [ n ]} \\ \oplus \left(\frac {\partial^ {2} \log p (Z ; \theta^ {(u)})}{\partial Z _ {i} ^ {2}}\right) _ {i \in [ n ]} \\ \oplus \left(\frac {\partial^ {2} \log p (Z ; \theta^ {(u)})}{\partial Z _ {i} \partial Z _ {j}}\right) _ {\{Z _ {i}, Z _ {j} \} \in \mathcal {E} (\mathcal {M} _ {Z}), i <   j} \\ \end{array}
+$$
+
+Suppose that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2). Then, for every pair of estimated latent variables  $\hat{Z}_k$  and  $\hat{Z}_l$  that are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ , we have the following statements:
+
+(a) For each true latent causal variable  $Z_{i}$ , we have
+
+$$
+\frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {l}} = 0. \tag {4}
+$$
+
+(b) For each pair of true latent causal variables  $Z_{i}$  and  $Z_{j}$  that are adjacent in the Markov network  $\mathcal{M}_{Z}$ , we have
+
+$$
+\frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} \frac {\partial Z _ {j}}{\partial \hat {Z} _ {l}} = 0. \tag {5}
+$$
+
+The proof is provided in Appendix B, which leverages the property of Markov network in Eq. (3). Assumption A2 can be viewed as suitable sufficient change conditions on the causal influences across different domains. It is worth noting that the requirement of a sufficient number of domains has been commonly adopted in the literature (e.g., see Hyvarinen et al. (2023) for a recent survey), such as visual disentanglement (Khemakhem et al., 2020b), domain adaptation (Kong et al., 2022), video analysis (Yao et al., 2021), and image-to-image translation (Xie et al., 2022). Also, we do not specify exactly how to learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2), and leave the door open for different approaches to be used, such as normalizing flow and variational approaches. For example, we adopt a variational approach in Section 4.
+
+In Theorem 1, Eqs. (4) and (5) hold for every sample of  $Z$ . Intuitively, one may expect that Eq. (4) implies either  $\frac{\partial Z_i}{\partial \hat{Z}_k} = 0$  for all samples of  $Z$ , or  $\frac{\partial Z_i}{\partial \hat{Z}_l} = 0$  for all samples, i.e., the zero entries in the Jacobian matrix (of the function from  $\hat{Z}$  to  $Z$ ) remain in the same positions across different samples. If this conclusion holds true, it indicates that the true latent variable  $Z_i$  cannot be a function of both estimated latent variables  $\hat{Z}_k$  and  $\hat{Z}_l$ , which is helpful for disentanglement. The same reasoning applies to Eq. (5). In fact, similar conclusion can often be obtained in the proof of identifiability for nonlinear ICA (Hyvarinen et al., 2019), by leveraging the continuity and invertibility of the Jacobian matrix.
+
+However, this conclusion in general does not hold in our setting (that allows for causal relations among latent variables  $Z$ ) without any constraint on the sparsity of recovered Markov network, for which counterexamples exist. The reason is that each of Eqs. (4) and (5) correspond to a pair of recovered latent variables  $\hat{Z}$  that are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$ , and can be viewed as a specific form of restriction on the Jacobian matrix (of the function from  $\hat{Z}$  to  $Z$ ). When the recovered Markov network is relatively dense, less restrictions are imposed on the Jacobian matrix, and thus there are possibilities for the aforementioned zero entries to switch positions across different samples. Interestingly, incorporating sparsity constraint on the recovered Markov network during estimation can help eliminate these possibilities, formally described below.
+
+Theorem 1 (Relations among true and recovered latent causal variables). Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Suppose that Assumptions A1 and A2 from Theorem 1 hold. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2) with the minimal number of edges of the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . Then, for every pair of estimated latent variables  $\hat{Z}_k$  and  $\hat{Z}_l$  that are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ , we have the following statements:
+
+(a) Each true latent causal variable  $Z_{i}$  is a function of at most one of  $\hat{Z}_k$  and  $\hat{Z}_l$ .
+
+(b) For each pair of true latent causal variables  $Z_{i}$  and  $Z_{j}$  that are adjacent in the Markov network  $\mathcal{M}_Z$  over  $Z$ , at most one of them is a function of  $\hat{Z}_{k}$  or  $\hat{Z}_{l}$ .
+
+The proof can be found in Appendix D. The above result sheds light on how each pair of the estimated latent variables  $\hat{Z}_k$  and  $\hat{Z}_l$  that are not adjacent in Markov network  $\mathcal{M}_{\hat{Z}}$  relate to the true latent causal variables  $Z$ , thus providing information for further disentanglement. Furthermore, note that a trivial solution would be a complete graph over  $\hat{Z}$  without any constraint on the estimating process. Apart from providing information for disentanglement, we show below that incorporating sparsity constraint on the recovered Markov network also helps avoid this trivial solution and recover the underlying Markov network up to isomorphism. The proof is given in Appendix C.
+
+Theorem 2 (Identifiability of latent Markov network). Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Suppose that Assumptions A1 and A2 from Theorem 1 hold. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2) with the minimal number of edges of the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . Then, the recovered latent Markov network  $\mathcal{M}_{\hat{Z}}$  is isomorphic to the true latent Markov network  $\mathcal{M}_Z$ .
+
+In addition to recovering the underlying Markov network  $\mathcal{M}_Z$ , we show that the sparsity constraint on the recovered Markov network  $\mathcal{M}_{\hat{Z}}$  also allows us to recover the underlying latent causal variables  $Z$  up to specific, relatively minor indeterminacies. In the result, the following variable set, termed intimate neighbor set, plays an important role:
+
+$$
+\Psi_ {Z _ {i}} := \left\{Z _ {j} \mid Z _ {j}, j \neq i, \text {i s a d j a c e n t t o} Z _ {i} \text {a n d} \right.
+$$
+
+all other neighbors of  $Z_{i}$  in  $\mathcal{M}_Z\}$ .
+
+For example, according to the Markov network implied by  $\mathcal{G}_Z$  in Figure 1,  $\Psi_{Z_1} = \{Z_2\}$ ,  $\Psi_{Z_2} = \emptyset$ , where  $\emptyset$  denotes the empty set,  $\Psi_{Z_3} = \{Z_2, Z_4\}$ ,  $\Psi_{Z_4} = \emptyset$ , and  $\Psi_{Z_5} = \{Z_4\}$ . As another example, according to the Markov network in Figure 2(b), which is implied by the DAG in Figure 2(a), we have  $\Psi_{Z_i} = \emptyset$  for  $i = 1, 2, 3, 5, 6$  and  $\Psi_{Z_4} = \{Z_3, Z_6\}$ .
+
+Theorem 3 (Identifiability of latent causal variables). Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Let  $N_{Z_i}$  be the set of neighbors of variable  $Z_i$  in  $\mathcal{M}_Z$ . Suppose that Assumptions A1 and A2 from Theorem 1 hold. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2) with the minimal number of edges of the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . Then, there exists a permutation  $\pi$  of the estimated latent variables, denoted as  $\hat{Z}_{\pi}$ , such that each  $\hat{Z}_{\pi(i)}$  is solely a function of a subset of the variables in  $\{Z_i\} \cup \Psi_{Z_i}$ .
+
+![](images/3d7f64927ba999cb2429bdb224391f9691a04bbd91df354f79bc0b50afe2e0ae.jpg)  
+(a)  $\mathcal{G}_Z$  , the DAG over true latent variables  $Z_{i}$  
+Figure 2: Illustrative example 2.
+
+![](images/56b641ca52b22488d4d5d3700d29d8686d3b25f125fc456e16ad88614f8fd695.jpg)  
+(b) The corresponding Markov network  $\mathcal{M}_Z$
+
+The proof is given in Appendix E. Roughly speaking, the proof leverages Theorems 1 and 2 to reason about the relationships among the true latent variables and the recovered ones, which imply that certain entries on the Jacobian matrix  $\frac{\partial Z}{\partial \hat{Z}}$  must be zero. We show that these entries remain zero in the powers of  $\frac{\partial Z}{\partial \hat{Z}}$ , indicating that the same entries remain zero in  $\left(\frac{\partial Z}{\partial \hat{Z}}\right)^{-1}$  (because the inverse  $\left(\frac{\partial Z}{\partial \hat{Z}}\right)^{-1}$  can be written as a linear combination of the powers of  $\frac{\partial Z}{\partial \hat{Z}}$ ) and thus  $\frac{\partial \hat{Z}}{\partial Z}$  from which the identifiability result can be derived.
+
+It is worth noting that in many cases, Theorem 3 already enables us to recover some of the latent variables up to a component-wise transformation.
+
+Remark 1. No matter how many neighbors each latent causal variable  $Z_{i}$  has, as long as each of its neighbors is not adjacent to at least one other neighbor in the Markov network  $\mathcal{M}_Z$ , then  $Z_{i}$  can be recovered up to a component-wise transformation.
+
+Even if the above case does not hold, Theorem 3 still shows how the estimated latent variables relate to the underlying causal variables in a specific, nontrivial way. Two examples are provided below.
+
+Example 1. Consider the Markov network  $\mathcal{M}_Z$  corresponding to the DAG  $\mathcal{G}_Z$  over  $Z$  in Figure 1. By Theorem 3 and suitable permutation of estimated latent variables  $\hat{Z}$ , we have: (a)  $\hat{Z}_{\pi(1)}$  is solely a function of a subset of  $\{Z_1, Z_2\}$ , (b)  $\hat{Z}_{\pi(2)}$  is solely a function of  $Z_2$ , (c)  $\hat{Z}_{\pi(3)}$  is solely a function of a subset of  $\{Z_2, Z_3, Z_4\}$ , (d)  $\hat{Z}_{\pi(4)}$  is solely a function of  $Z_4$ , and (e)  $\hat{Z}_{\pi(5)}$  is solely a function of a subset of  $\{Z_4, Z_5\}$ . In this example, the latent causal variables  $Z_2$  and  $Z_4$  can be recovered up to component-wise transformation, while variables  $Z_1, Z_3$ , and  $Z_5$  can be identified up to mixtures with certain neighbors in the Markov network.
+
+Example 2. One may think that generally speaking, the more complex  $\mathcal{G}_Z$ , the more indeterminacies we have in the estimated latent variables (in the sense that each estimated latent variable receives contributions from more latent variables). In fact, this may not be the case. For instance, consider the underlying latent causal graph  $\mathcal{G}_Z$  in Figure 2(a), which involves more variables and more edges and whose Markov network is shown in Figure 2(b). For every variable  $Z_i$  that is not the sink node, it has  $\Psi_{Z_i} = \emptyset$  and thus can be recovered up to a component-wise transformation.
+
+Permutation of recovered latent variables. Theorems 2 and 3 involve certain permutation of the estimated latent variables  $\hat{Z}$ . Such an indeterminacy is common in the literature of causal discovery and representation learning tasks involving latent variables. In our case, since the function  $v\coloneqq g^{-1}\circ \hat{g}$  where  $Z = v(\hat{Z})$  is invertible, there exists a permutation of the latent variables such that the corresponding Jacobian matrix  $J_{v}$  has nonzero diagonal entries (see Lemma 2 in Appendix A.1); such a permutation is what Theorems 2 and 3 refer to.
+
+Connection with nonlinear ICA. It is worth noting that nonlinear ICA (with auxiliary variables) may be viewed as a special case of our result in this section. Specifically, if the true latent causal DAG  $\mathcal{G}_Z$  is an empty graph, then the latent causal variables are independent, which reduce to the nonlinear ICA setting (Hyvärinen et al., 2019).
+
+Furthermore, since traditional nonlinear ICA always has a valid solution (to produce nonlinear independent components) (Hyvärinen et al., 1999), one may wonder whether, in our setting, it is possible to always find nonlinear components as functions of  $X$  that are independent in each domain, as produced by recent methods for nonlinear ICA with auxiliary variables (Hyvärinen et al., 2019). As a corollary of Theorem 2, we show that the answer is no-there do not exist nonlinear components that are independent across domains if the true latent causal DAG  $\mathcal{G}_Z$  is not an empty graph. The proof is provided in Appendix F.
+
+Corollary 1 (Impossibility of finding independent components). Let the observations be sampled from the data generating process in Eq. (1). Suppose that Assumptions A1 and A2 from Theorem 1 hold, and that the true latent causal DAG  $\mathcal{G}_Z$  is not an empty graph. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  with the components of  $\hat{Z}$  being independent in each domain. Then,  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  cannot achieve Eq. (2).
+
+# 3.2. From Latent Markov Network to Latent Causal DAG
+
+Now we have identified the Markov network up to an isomorphism, which characterizes conditional independence relations in the distribution. To build the connection between Markov network or conditional independence relations and causal structures, prior theory relies on the Markov and faithfulness assumptions. However, in real-world scenarios, the faithfulness assumption could be violated due to various reasons including path cancellations (Zhang & Spirtes, 2008; Uhler et al., 2013).
+
+Since our goal is to generalize the identifiability theory as much as possible to fit practical applications, we introduce two relaxations of the faithfulness assumptions.
+
+Assumption 1 (Single adjacency-faithfulness (SAF)). Given a DAG  $\mathcal{G}_Z$  and distribution  $P_{Z;\theta}$  over the variable
+
+set  $Z$ , if two variables  $Z_{i}$  and  $Z_{j}$  are adjacent in  $\mathcal{G}_Z$ , then  $Z_{i} \nVdash Z_{j} \mid Z_{[n] \setminus \{i, k\}}$ .
+
+Assumption 2 (Single unshielded-collider-faithfulness (SUCF) (Ng et al., 2021)). Given a latent causal graph  $\mathcal{G}_Z$  and distribution  $P_{Z;\theta}$  over the variable set  $Z$ , let  $Z_i \to Z_j \gets Z_k$  be any unshielded collider in  $\mathcal{G}_Z$ , then  $Z_i \nVdash Z_k \mid Z_{[n] \setminus \{i,k\}}$ .
+
+We propose SAF as a relaxation of the Adjacency-faithfulness (Ramsey et al., 2012). The SUCF assumption is first introduced by Ng et al. (2021), which is strictly weaker than Orientation-faithfulness (Ramsey et al., 2012). Thus, both of them are strictly weaker than the faithfulness assumption, since the combination of Adjacency-faithfulness and Orientation-faithfulness is weaker than the faithfulness assumption (Zhang & Spirtes, 2008).
+
+Interestingly, not only they are weaker variants of faithfulness, but we also prove that they are actually necessary and sufficient conditions, thus the weakest possible ones, to bridge conditional independence relations and causal structures. Specifically, we show that the recovered Markov network (e.g., in Theorem 2) is exactly the moralized graph of the true causal DAG if and only if the proposed variants of faithfulness hold. The proofs of Lemma 1 and Proposition 2 are shown in Appendix G.
+
+Lemma 1. Given a latent causal graph  $\mathcal{G}_Z$  and distribution  $P_{Z;\theta}$  with its Markov Network  $\mathcal{M}_Z$ , under Markov assumption, the undirected graph defined by  $\mathcal{M}_Z$  is a subgraph of the moralized graph of the true causal DAG  $G$ .
+
+Proposition 2 (Moralized graph and Markov network). Given a causal DAG  $\mathcal{G}_Z$  and distribution  $P_{Z;\theta}$  with its Markov Network  $\mathcal{M}_Z$ , under Markov assumption, the undirected graph defined by  $\mathcal{M}_Z$  is the moralized graph of the true causal DAG  $\mathcal{G}_Z$  if and only if the SAF and SUCF assumptions are satisfied.
+
+It is worth noting that the connection between conditional independence relations and causal structures has been developed by (Loh & Buhlmann, 2014; Ng et al., 2021) in the linear case by leveraging the properties of the inverse covariance matrix; our results here focus on the nonparametric case and thus being able to serve the considered general settings for identifiability. Also note that the necessary and sufficient conditions may also be of independent interest for other causal discovery tasks exploring conditional independence relations in the nonparametric case.
+
+Discussion on additional assumptions. We investigated how the sparsity constraint on the recovered graph over latent variables and sufficient change conditions on causal influences can be used to recover the latent variables and causal graph up to certain indeterminacies. Our framework is connected with previous ones in a spectrum of related studies (Varici et al., 2023; Ahuja et al., 2023; Buchholz
+
+et al., 2022; Squires et al., 2023; Brehmer et al., 2022; von Kugelgen et al., 2023; Brehmer et al., 2022; von Kugelgen et al., 2023; Zheng & Zhang, 2023; Zhang et al., 2023). For instance, the connection between conditional independence and cross-derivatives of the log density in both linear and nonlinear cases means our theorems directly apply to linear SEMs. Furthermore, our results do not require the mixing function to be sufficiently nonlinear, allowing them to encompass linear mixing processes as well.
+
+At the same time, we may be able to leverage possible parametric constraints on the data generating process (or functions) or specific types of interventions. For instance, if we know that the changes happen to the linear causal mechanisms with Gaussian noises, this constraint can readily help reduce the search space and improve the identifiability. Moreover, since we only require the changing distribution, any type of interventions will be covered since any change to the conditional distribution is allowed. Given the additional information illustrated by experimental interventions (e.g., single-node interventions), alternative identifiability that might be particularly useful in certain tasks can be established. We hope this work can provide a helpful, bigger picture of causal representation learning in the general setting and further illustrates the necessity and connections of the different assumptions formulated in this line of works.
+
+# 4. Change Encoding Network for Representation Learning
+
+Thanks to the identifiability result, we now present two different practical implementations to recover the latent variables and their causal relations from observations from multiple domains. We build our method on the variational autoencoder (VAE) framework and can be easily extended to other models, such as normalizing flows.
+
+We learn a deep latent generative model (decoder)  $p(X|Z; \hat{\theta}^{(u)})$  and a variational approximation (encoder)  $q(Z|X, u)$  of its true posterior  $p(Z|X; \theta^{(u)})$  since the true posterior is usually intractable. To learn the model, we minimize the lower bound of the log-likelihood as
+
+$$
+\begin{array}{l} \log p (X; \hat {\theta} ^ {(u)}) \\ = \log \int p (X | Z; \hat {\theta} ^ {(u)}) p (Z; \hat {\theta} ^ {(u)}) d Z \\ = \log \int \frac {q (Z | X , u)}{q (Z | X , u)} p (X | Z; \hat {\theta} ^ {(u)}) p (Z; \hat {\theta} ^ {(u)}) d Z \\ \geq - \operatorname {K L} (q (Z | X, u) \| p (Z; \hat {\theta} ^ {(u)})) + \mathbb {E} _ {q} [ \log p (X | Z; \hat {\theta} ^ {(u)}) ] \\ = - \mathcal {L} _ {\mathrm {E L B O}} \\ \end{array}
+$$
+
+For the posterior  $q(Z|X,u)$ , we assume that it is a multivariate Gaussian or a Laplacian distribution, where the mean and variance are generated by the neural network encoder. As for  $q(X|Z)$ , we assume that it is a multivariate Gaussian
+
+and the mean is the output of the decoder and the variance is a pre-defined value.
+
+In practice, we can parameterize  $p(X|Z; \hat{\theta}^{(u)})$  as the decoder which takes as input the latent representation  $Z$  and  $q(Z|X, u)$  as an encoder which outputs the mean and scale of the posterior distribution. An essential difference between VAE (Kingma & Welling, 2013) and iVAE (Khemakhem et al., 2020a) is that our method allows the components of  $Z$  to be causally dependent and we are able to learn the components and causal relationships. And the key is the prior distribution  $P(Z; \hat{\theta}^{(u)})$ . Now we present two different implementations to capture the changes with a properly defined prior distribution.
+
+# 4.1. Nonparametric Implementation of the Prior Distribution
+
+To recover the relationships and latent variables  $Z$ , we build the normalizing flow to mimic the inverse of the latent SEM  $Z_{i} = f_{i}(\mathrm{PA}(Z_{i}),\epsilon_{i})$  in Eq. (1). We first assume a causal ordering as  $\hat{Z}_1,\dots ,\hat{Z}_n$ . Then, for each component  $\hat{Z}_i$ , we consider the previous components  $\{\hat{Z}_1,\ldots ,\hat{Z}_{i - 1}\}$  as potential parents of  $\hat{Z}_i$  and we can select the true parents with the adjacency matrix  $\hat{A}$ , where  $\hat{A}_{i,j}$  denotes that component  $\hat{Z}_j$  contributes in the generation of  $\hat{Z}_i$ . If  $\hat{A}_{i,j} = 0$ , it means that  $\hat{Z}_j$  will not contribute to the generation of  $\hat{Z}_i$ . Since  $\theta^{(u)}$  governs the changes across domains, we use the observed domain index  $u$  to discover the changes. Then, we use the selected parents  $\{\hat{A}_{i,1}\hat{Z}_1,\dots ,\hat{A}_{i,i - 1}\hat{Z}_{i - 1}\}$  and the domain label  $u$  to generate parameters of normalizing flow and apply the flow transformation on  $\hat{Z}_i$  to turn it into  $\hat{\epsilon}_i$ . Specifically, we have
+
+$$
+\hat {\epsilon} _ {i}, \log \det _ {i} = \mathrm {F l o w} (\hat {Z} _ {i}; \mathrm {N N} (\{\hat {A} _ {i, j} \hat {Z} _ {j} \} _ {j = 1} ^ {i - 1}, u)),
+$$
+
+where  $\log \operatorname{det}_i$  is the log determinant of the conditional flow transformation on  $\hat{Z}_i$  and NN represents a neural network.
+
+To compute the prior distribution, we make an assumption on the noise term  $\epsilon$  that it follows an independent prior distribution  $p(\epsilon)$ , such as a standard isotropic Gaussian or a Laplacian. Then according to the change-of-variable formula, the prior distribution of the dependent latents can be written as
+
+$$
+\log p (\hat {Z}; \theta^ {(u)}) = \sum_ {i = 1} ^ {n} (\log p (\hat {\epsilon} _ {i}) + \log \det_ {i}).
+$$
+
+Intuitively, to minimize the KL divergence loss between  $p(Z; \hat{\theta}^{(u)})$  and  $q(Z|X, u)$ , the network has to learn the correct structure and the underlying latent variables; otherwise, it can be difficult to transform the dependent latent variables  $\hat{Z}$  to a factorized prior distribution, e.g.,  $\mathcal{N}(0, \mathrm{I})$ .
+
+# 4.2. Parametric Implementation of the Prior Distribution
+
+We can make parametric assumption on the latent causal process and facilitate the learning of true causal structure and components. Here, we consider the linear SEM and more complex SEMs can be generalized. Specifically, we assume that the true generation process of the latent  $Z$  is linear and only consists of scaling and shifting mechanisms:
+
+$$
+Z = A \left(C ^ {(u)} Z\right) + S ^ {(u)} \epsilon + B ^ {(u)},
+$$
+
+where  $A \in [0,1]^{n \times n}$  is a causal adjacency matrix which can be permuted to be strictly lower-triangular,  $C^{(u)} \in \mathbb{R}^{n \times n}$  and  $S^{(u)} \in \mathbb{R}^{n \times 1}$  are underlying domain-specific scaling matrix and vector for domain  $u$ , respectively,  $B^{(u)} \in \mathbb{R}^{n \times 1}$  is the underlying domain-specific shift vector, and  $\epsilon$  is the independent noise.
+
+To estimate the latent variables  $Z$ , the causal structure  $A$ , and capture the changes across domains, we introduce the learnable scaling  $\hat{C} \in \mathbb{R}^{n \times n}$ ,  $\hat{S} \in \mathbb{R}^{n \times 1}$  and bias parameters  $\hat{B} \in \mathbb{R}^{n \times 1}$  and pre-define a causal ordering as  $\hat{Z}_1, \hat{Z}_2, \dots, \hat{Z}_n$ . Then we have the matrix form as
+
+$$
+\hat {\epsilon} = (\hat {Z} - \hat {B} ^ {(u)} - \hat {A} \hat {C} ^ {(u)} \hat {Z}) / \hat {S} ^ {(u)}.
+$$
+
+Note that the determinant of the strictly lower triangular matrix  $\hat{C}$  is 0. Given a prior distribution of the noise term  $p(\hat{\epsilon})$ , and according to the change-of-variable rule, we then have the prior distribution for  $\hat{Z}$  in parametric case as
+
+$$
+\log p (\hat {Z}; \hat {\theta} ^ {(u)}) = \sum_ {i = 1} ^ {n} (\log p (\hat {\epsilon} _ {i}) - \log | \hat {S} _ {i} ^ {(u)} |).
+$$
+
+# 4.3. Full Objective
+
+After we have properly defined the needed distributions  $p(X|Z;\hat{\theta}^{(u)})$ ,  $q(Z|X,u)$ ,  $p(Z;\hat{\theta}^{(u)})$ , we can train our model to minimize the loss function  $\mathcal{L}_{\mathrm{ELBO}}$ . However, without any further constraint, the powerful network may choose to use the fully connected causal graph during training. In other words, all lower-triangular elements of the estimated graph  $\hat{A}$  is non-zero, which implies that each component  $\hat{Z}_i$  is caused by all previous  $i - 1$  components. To exclude such unwanted solutions and encourage the model to learn the true causal structure among components of  $Z$ , we apply the  $\ell_1$  regularization on  $\hat{A}$ , i.e.,
+
+$$
+\mathcal {L} _ {\text {s p a r s i t y}} = \| \hat {A} \| _ {1}.
+$$
+
+It is worth noting that the sparsity regularization term above is an approximation of the sparsity constraint on the edges of the estimated Markov network specified in Theorems 2 and 3, since it is not straightforward to impose the latter constraint in a differentiable end-to-end training process.
+
+A more sophisticated alternative is to impose sparsity constraint on  $(I - \hat{A})^T\Omega^{-1}(I - \hat{A})$  where  $\Omega$  is a randomly sampled positive diagonal matrix. Note that this corresponds to the formula of precision matrix whose nonzero entries represent the moral graph under certain conditions (Loh & Buhlmann, 2014) and we leave it for future investigation.
+
+Finally, the full training objective is
+
+$$
+\mathcal {L} _ {\text {f u l l}} = \mathcal {L} _ {\text {E L B O}} + \mathcal {L} _ {\text {s p a r s i t y}}.
+$$
+
+After the model converges, the output of the encoder  $\hat{Z}$  is our recovered latents from the observations in multiple domains and the revealed causal structure is in  $\hat{A}$  which encapsulates the causal relationships across the components.
+
+# 4.4. Simulations
+
+To verify our theory and the proposed implementations, we run experiments on the simulated data because the ground truth causal adjacency matrix and the latent variables across domains are available for simulated data. Consequently, we consider following common causal structures (i) Y-structure with 4 variables,  $Z_{1} \rightarrow Z_{3} \leftarrow Z_{2}, Z_{3} \rightarrow Z_{4}$  and (ii) chain structure  $Z_{1} \rightarrow Z_{2} \rightarrow Z_{3} \rightarrow Z_{4}$ . The noises are modulated with scaling random sampled from Unif[0.5, 2] and shifts are sampled from Unif[-2, 2]. The scaling on the  $Z$  are also randomly sampled from Unif[0.5, 2]. In other words, the changes are modular. After generating  $Z$ , we feed the latent variables into multilayer perceptron (MLP) with orthogonal weights and LeakyReLU activations for invertibility. Specifically, we sample orthogonal matrix as the weights of the MLP layers. Since orthogonal matrix and LeakyReLU are invertible, the MLP function is also invertible.
+
+We present the results in Figures 3 and 4. Each sub-figure consists of  $4 \times 4$  panels and penal on  $i$ -th row and  $j$ -th column denotes the relationship between the estimated component  $\hat{Z}_i$  with the true latent  $Z_j$ . We can see that under most cases, our model learns a strong one-to-one correspondence from the estimated components and the true components. For instance, the first column in Figure 3 show that  $\hat{Z}_1$  is strongly correlated with the true components  $Z_1$  while it is nearly independent from the true  $Z_2$ .
+
+From the estimated  $\hat{A}$ , we find that our method is able to recover the true causal structure. For instance, on the Y-structure with  $Z_{1} \rightarrow Z_{3} \leftarrow Z_{2}$  and  $Z_{3} \rightarrow Z_{4}$ , our estimated model only keeps the components  $\hat{A}_{1,3}, \hat{A}_{2,3}, \hat{A}_{3,4}$  nonzero with the proposed sparsity regularization. The estimated causal graph is consistent with the true Y-structure causal graph. We can also see that the latent causal structure is also recovered from Figures 4 and 3. We observe that the learned  $\hat{Z}_{1}$  is strongly correlated with the true  $Z_{2}$  and is independent from the true  $Z_{1}$ , but correlated with the  $\hat{Z}_{3}$  and  $\hat{Z}_{4}$ . These results align well with the true causal graph since  $Z_{2}$  is independent from  $Z_{1}$  while is the cause of  $Z_{3}$  and  $Z_{4}$ .
+
+The experiments support our theoretical result that the components and structure are identifiable up to certain indeterminacies. As for the results in Figure 3, we observe that our non-parametric method is still able to recover the true latent variables with Laplace noise.
+
+# 5. Related Work
+
+Causal representation learning aims to unearth causal latent variables and their relations from observed data. Despite its significance, the identifiability of the hidden generating process is known to be impossible without additional constraints, especially with only observational data. In the linear, non-Gaussian case, Silva et al. (2006) recover the Markov equivalence class, provided that each observed variable has a unique latent causal parent; Xie et al. (2020); Cai et al. (2019) estimate the latent variables and their relations assuming at least twice measured variables as latent ones, which has been further extended to learn the latent hierarchical structure (Xie et al., 2022). Moreover, Adams et al. (2021) provide theoretical results on the graphical conditions for identification. In the linear, Gaussian case, Huang et al. (2022) leverage rank deficiency of the observed sub-covariance matrix to estimate the latent hierarchical structure, while Dong et al. (2023) further extend the rank constraint to accommodate flexibly related latent and observed variables. In the discrete case, Kivva et al. (2021) identify the latent causal graph up to Markov equivalence by assuming a mixture model where the observed children sets of any pair of latent variables are different.
+
+Given the challenge of identifiability on purely observational data, a different line of research leverage experiments by assuming the accessibility of various types of interventional data. Based on the single-node perfect intervention, Squires et al. (2023) leverage single-node interventions for the identifiability of linear causal model and linear mixing function; (Varici et al., 2023) incorporate for nonlinear causal model and linear mixing function; (Varici et al., 2023; Buchholz et al., 2023; Jiang & Aragam, 2023) provide the identifiability of the nonparametric causal model and linear mixing function; (Ahuja et al., 2023) further generalize the result to nonparametric causal model and polynomial mixing functions with additional constraints on the latent support; and (Brehmer et al., 2022; von Kugelgen et al., 2023; Jiang & Aragam, 2023) explore the nonparametric settings for both the causal model and mixing function. In addition to the single-node perfect interventions, Brehmer et al. (2022) introduced counterfactual pre- and post-intervention views; von Kugelgen et al. (2023) assume two distinct, paired interventions per node for multivariate causal models; Zhang et al. (2023) explore soft interventions on polynomial mixing functions; and Jiang & Aragam (2023) places specific structural restrictions on the latent causal graph.
+
+![](images/92517468483453a8b32e32bbd8615f5be7a10d890d0ff450d3672e24edb00914.jpg)  
+Figure 3: Recovered latent variables v.s. the true latent variables with Non-Parametric Approach. (a) Y-structure with Laplace noise. (b) Y-structure with Gaussian noise. (c) Chain structure with Laplace noise. (d) Chain structure with Gaussian noise. In each sub-figure,  $i$ -th row and  $j$ -th column depicts the relationship between the estimated  $\hat{Z}_i$  and the true components  $Z_j$ .
+
+![](images/01752a60ff2c563e6c012a934f7c069b6f0c28eb963da5557f6c69ca97d5b565.jpg)  
+Figure 4: Recovered latent variables v.s. the true latent variables with Linear Parameterization Approach. The  $X$ -axis denotes the components of true latent variables  $Z$  and the  $Y$ -axis represent the components of estimated latent variables  $\hat{Z}$ . (a) Y-structure with Laplace noise. (b) Y-structure with Gaussian noise. (c) Chain structure with Laplace noise. (d) Chain structure with Gaussian noise.
+
+Our study lies in the line of leveraging only observational data, and provides identifiability results in the general nonparametric settings on both the latent causal model and mixing function. Unlike prior works with observational data, we do not have any parametric assumptions or graphical restrictions; Compared to those relying on interventional data, our results naturally benefit from the heterogeneity of observational data (e.g., multi-domain data, nonstationary time series) and avoid additional experiments for interventions.
+
+# 6. Conclusion and Discussions
+
+We establish a set of new identifiability results to reveal latent causal variables and latent structures in the general nonparametric settings. Specifically, with sparsity regularization during estimation and sufficient changes in the causal influences, we demonstrate that the revealed latent variables and structures are related to the underlying causal model in a specific, nontrivial way. In contrast to recent works on the recovery of latent causal variables and structures, our results rely on purely observational data without graphical or parametric constraints. Our results offer insight into unveiling
+
+the latent causal process in one of the most universal settings. Experiments in various settings have been conducted to validate the theory. As future work, we will explore the scenario where only a subset of the causal relations change, which could be a challenge as well as a chance, and show up to what extent the underlying causal variables can be recovered with potentially weaker assumptions.
+
+# Acknowledgements
+
+The authors would like to thank the anonymous reviewers for helpful comments and suggestions. The authors would also like to acknowledge the support from NSF Grant 2229881, the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Apple Inc., KDDI Research Inc., Quris AI, and Florin Court Capital.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# Supplementary Material
+
+# A. Proofs of Useful Lemmas
+
+# A.1. Proof of Lemma 2
+
+The following lemma is a rather standard result in linear algebra (Strang, 2006; 2016), which has also been used in existing works in causal representation learning, such as Lachapelle et al. (2022). We provide the proof here for completeness.
+
+Lemma 2. For any invertible matrix  $A$ , there exists a permutation of its columns such that the diagonal entries of the resulting matrix are nonzero.
+
+Proof. Suppose by contradiction that there exists at least a zero diagonal entry for every column permutation. By Leibniz formula, we have
+
+$$
+\det (A) = \sum_ {\sigma \in \mathcal {S} _ {n}} \left(\operatorname {s g n} (\sigma) \prod_ {i = 1} ^ {n} A _ {i, \sigma (i)}\right),
+$$
+
+where  $S_{n}$  denotes the set of  $n$ -permutations. Since there exists a zero diagonal entry for every permutation, we have
+
+$$
+\prod_ {i = 1} ^ {n} A _ {i, \sigma (i)} = 0, \quad \forall \sigma \in \mathcal {S} _ {n},
+$$
+
+which implies  $\operatorname{det}(A) = 0$  and that matrix  $A$  is not invertible. This is a contradiction with the assumption that  $A$  is invertible.
+
+# A.2. Proof of Lemma 3
+
+Lemma 3. Suppose matrix  $A \in \mathbb{R}^{n \times n}$  contains a zero submatrix of order  $(i + 1) \times (n - i)$  for some  $i \in [n - 1]$ . Then, matrix  $A$  is not invertible.
+
+Proof. By the given condition, there exist  $n - i$  columns in matrix  $A$  of which  $i + 1$  rows are zero, i.e., at most  $n - i - 1$  rows are not zero. This implies that the  $n - i$  column vectors span a subspace of dimension less than  $n - i$ , which thus are linearly dependent. Therefore, matrix  $A$  cannot be invertible.
+
+# B. Proof of Proposition 1
+
+Proposition 1. Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Suppose the following assumptions hold:
+
+- A1 (Smooth and positive density): The probability density function of latent causal variables, i.e.,  $p_Z$ , is twice continuously differentiable and positive in  $\mathbb{R}^n$ .  
+- A2 (Sufficient changes): For each value of  $Z$ , there exist  $2n + |\mathcal{M}_Z| + 1$  values of  $\theta$ , i.e.,  $\theta^{(u)}$  with  $u = 0, \dots, 2n + |\mathcal{M}_Z|$ , such that the vectors  $w(Z,u) - w(z,0)$  with  $u = 1, \dots, 2n + |\mathcal{M}_Z|$  are linearly independent, where vector  $w(Z,u)$  is defined as follows:
+
+$$
+w (Z, u) = \left(\frac {\partial \log p (Z ; \theta^ {(u)})}{\partial Z _ {i}}\right) _ {i \in [ n ]} \oplus \left(\frac {\partial^ {2} \log p (Z ; \theta^ {(u)})}{\partial Z _ {i} ^ {2}}\right) _ {i \in [ n ]} \oplus \left(\frac {\partial^ {2} \log p (Z ; \theta^ {(u)})}{\partial Z _ {i} \partial Z _ {j}}\right) _ {\{Z _ {i}, Z _ {j} \} \in \mathcal {E} (\mathcal {M} _ {Z}), i <   j}.
+$$
+
+Suppose that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2). Then, for every pair of estimated latent variables  $\hat{Z}_k$  and  $\hat{Z}_l$  that are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ , we have the following statements:
+
+(a) For each true latent causal variable  $Z_{i}$ , we have
+
+$$
+\frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {l}} = 0.
+$$
+
+(b) For each pair of true latent causal variables  $Z_{i}$  and  $Z_{j}$  that are adjacent in the Markov network  $\mathcal{M}_Z$ , we have
+
+$$
+\frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} \frac {\partial Z _ {j}}{\partial \hat {Z} _ {l}} = 0.
+$$
+
+Proof. For a matrix  $A$ , denote by  $\operatorname{vol} A$  its volume, which is the product of its singular values. Note that  $\operatorname{vol} A = \sqrt{\det AA^T}$  when  $A$  is of full row rank. In the change-of-variable formula, when the Jacobian is a rectangular matrix, the absolute determinant of the Jacobian can be replaced with the matrix volume (Ben-Israel, 1999; Gemici et al., 2016).
+
+Since  $X = g(Z)$  and  $\hat{X} = \hat{g}(\hat{Z})$ , by Eq. (2) and the change-of-variable formula, we have
+
+$$
+p _ {\hat {X}} = p _ {X} \Rightarrow p _ {\hat {g} (\hat {Z})} = p _ {g (Z)} \Rightarrow p _ {g ^ {- 1} \circ \hat {g} (\hat {Z})} \operatorname {v o l} J _ {g ^ {- 1}} = p _ {Z} \operatorname {v o l} J _ {g ^ {- 1}} \Rightarrow p _ {v (\hat {Z})} = p _ {Z},
+$$
+
+where  $J_{g^{-1}}$  is the Jacobian matrix of  $g^{-1}$  and  $v := g^{-1} \circ \hat{g}$  is a composition of diffeomorphisms (and hence also a diffeomorphism). Let  $J_v$  be the Jacobian matrix of  $v$ . The change-of-variable formula implies
+
+$$
+\begin{array}{l} p (\hat {Z}; \hat {\theta}) | \det  J _ {v - 1} | = p (Z; \theta) \\ \log p (\hat {Z}; \hat {\theta}) = \log p (Z; \theta) + \log | \det  J _ {v} |. \tag {6} \\ \end{array}
+$$
+
+Suppose  $\hat{Z}_k$  and  $\hat{Z}_l$  are conditionally independent given  $\hat{Z}_{[n]\setminus \{k,l\}}$  i.e., they are not adjacent in the Markov network over  $\hat{Z}$ . For each  $\hat{\theta}$ , by Lin (1997), we have
+
+$$
+\frac {\partial^ {2} \log p (\hat {Z} ; \hat {\theta})}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}} = 0. \tag {7}
+$$
+
+To see what it implies, we find the first-order derivative of Eq. (6):
+
+$$
+\frac {\partial \log p (\hat {Z} ; \hat {\theta})}{\partial \hat {Z} _ {k}} = \sum_ {i = 1} ^ {n} \frac {\partial \log p (Z ; \theta)}{\partial Z _ {i}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} + \frac {\partial \log | \det J _ {v} |}{\partial \hat {Z} _ {k}}.
+$$
+
+Let
+
+$$
+\eta (\theta) := \log p (Z; \theta), \quad \eta_ {i} ^ {\prime} (\theta) := \frac {\partial \log p (Z ; \theta)}{\partial Z _ {i}}, \quad \eta_ {i j} ^ {\prime \prime} (\theta) := \frac {\partial^ {2} \log p (Z ; \theta)}{\partial Z _ {i} \partial Z _ {j}}, \quad h _ {i, l} ^ {\prime} := \frac {\partial Z _ {i}}{\partial \hat {Z} _ {l}}, \quad \text {a n d} \quad h _ {i, k l} ^ {\prime \prime} := \frac {\partial^ {2} Z _ {i}}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}}.
+$$
+
+We then derive the second-order derivative w.r.t.  $\hat{Z}_k$  and  $\hat{Z}_l$  and apply Eq. (7):
+
+$$
+\begin{array}{l} 0 = \sum_ {j = 1} ^ {n} \sum_ {i = 1} ^ {n} \frac {\partial^ {2} \log p (Z ; \theta)}{\partial Z _ {i} \partial Z _ {j}} \frac {\partial Z _ {j}}{\partial \hat {Z} _ {l}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} + \sum_ {i = 1} ^ {n} \frac {\partial \log p (Z ; \theta)}{\partial Z _ {i}} \frac {\partial^ {2} Z _ {i}}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}} + \frac {\partial^ {2} \log | \det J _ {v} |}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}}. \\ = \sum_ {i = 1} ^ {n} \frac {\partial^ {2} \log p (Z ; \theta)}{\partial Z _ {i} ^ {2}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {l}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} + \sum_ {j = 1} ^ {n} \sum_ {i: \{Z _ {j}, Z _ {i} \} \in \mathcal {E} (\mathcal {M} _ {Z})} \frac {\partial^ {2} \log p (Z ; \theta)}{\partial Z _ {i} \partial Z _ {j}} \frac {\partial Z _ {j}}{\partial \hat {Z} _ {l}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} \\ + \sum_ {i = 1} ^ {n} \frac {\partial \log p (Z ; \theta)}{\partial Z _ {i}} \frac {\partial^ {2} Z _ {i}}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}} + \frac {\partial^ {2} \log | \det J _ {v} |}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}} (8) \\ = \sum_ {i = 1} ^ {n} \eta_ {i i} ^ {\prime \prime} (\theta) h _ {i, l} ^ {\prime} h _ {i, k} ^ {\prime} + \sum_ {j = 1} ^ {n} \sum_ {i: \{Z _ {j}, Z _ {i} \} \in \mathcal {E} (\mathcal {M} _ {Z})} \eta_ {i j} ^ {\prime \prime} (\theta) h _ {j, l} ^ {\prime} h _ {i, k} ^ {\prime} + \sum_ {i = 1} ^ {n} \eta_ {i} ^ {\prime} (\theta) h _ {i, k l} ^ {\prime \prime} + \frac {\partial^ {2} \log | \det J _ {v} |}{\partial \hat {Z} _ {k} \partial \hat {Z} _ {l}}. (9) \\ \end{array}
+$$
+
+Recall that  $\mathcal{E}(\mathcal{M}_Z)$  denotes the set of edges in the Markov network over  $Z$ . In the equation above, we made use of the fact that if  $Z_{i}$  and  $Z_{j}$  are not adjacent in the Markov network, then  $\frac{\partial^2\log p(Z;\theta)}{\partial Z_i\partial Z_j} = 0$  by Lin (1997).
+
+By Assumption A2, consider the  $2n + |\mathcal{M}_Z| + 1$  values of  $\theta$ , i.e.,  $\theta^{(u)}$  with  $u = 0, \dots, 2n + |\mathcal{M}_Z|$ , such that Eq. (9) holds. Then, we have  $2n + |\mathcal{M}_Z| + 1$  such equations. Subtracting each equation corresponding to  $\theta^{(u)}$ ,  $u = 1, \dots, 2n + |\mathcal{M}_Z|$  with the equation corresponding to  $\theta^{(0)}$  results in  $2n + |\mathcal{M}_Z|$  equations:
+
+$$
+0 = \sum_ {i = 1} ^ {n} \left(\eta_ {i i} ^ {\prime \prime} \left(\theta^ {(u)}\right) - \eta_ {i i} ^ {\prime \prime} \left(\theta^ {(0)}\right)\right) h _ {i, l} ^ {\prime} h _ {i, k} ^ {\prime} + \sum_ {j = 1} ^ {n} \sum_ {i: \{Z _ {j}, Z _ {i} \} \in \mathcal {E} \left(\mathcal {M} _ {Z}\right)} \left(\eta_ {i j} ^ {\prime \prime} \left(\theta^ {(u)}\right) - \eta_ {i j} ^ {\prime \prime} \left(\theta^ {(0)}\right)\right) h _ {j, l} ^ {\prime} h _ {i, k} ^ {\prime} + \sum_ {i = 1} ^ {n} \left(\eta_ {i} ^ {\prime} \left(\theta^ {(u)}\right) - \eta_ {i} ^ {\prime} \left(\theta^ {(0)}\right)\right) h _ {i, k l} ^ {\prime \prime}, \tag {10}
+$$
+
+where  $u = 1,\ldots ,2n + |\mathcal{M}_Z|$ . Since  $p_Z$  is twice continuously differentiable, we have
+
+$$
+\eta_ {i j} ^ {\prime \prime} (\theta^ {(u)}) - \eta_ {i j} ^ {\prime \prime} (\theta^ {(0)}) = \eta_ {j i} ^ {\prime \prime} (\theta^ {(u)}) - \eta_ {j i} ^ {\prime \prime} (\theta^ {(0)}),
+$$
+
+and therefore Eq. (10) can be written as
+
+$$
+\begin{array}{l} 0 = \sum_{i = 1}^{n}(\eta_{ii}^{\prime \prime}(\theta^{(u)}) - \eta_{ii}^{\prime \prime}(\theta^{(0)}))h_{i,l}^{\prime}h_{i,k}^{\prime} + \sum_{\substack{i,j:\\ i <   j,\\ \{Z_{i},Z_{j}\} \in \mathcal{E}(\mathcal{M}_{Z})}}(\eta_{ij}^{\prime \prime}(\theta^{(u)}) - \eta_{ij}^{\prime \prime}(\theta^{(0)}))(h_{j,l}^{\prime}h_{i,k}^{\prime} + h_{i,l}^{\prime}h_{j,k}^{\prime}) \\ + \sum_ {i = 1} ^ {n} (\eta_ {i} ^ {\prime} (\theta^ {(u)}) - \eta_ {i} ^ {\prime} (\theta^ {(0)})) h _ {i, k l} ^ {\prime \prime}. \\ \end{array}
+$$
+
+Consider the vectors formed by collecting the corresponding coefficients in the equation above where  $u = 1, \ldots, 2n + |\mathcal{M}_Z|$ . By Assumption A2, these  $2n + |\mathcal{M}_Z|$  vectors are linearly independent. Thus, for any  $i$  and  $j$  such that  $\{Z_i, Z_j\} \in \mathcal{E}(\mathcal{M}_Z)$ , we have the following equations:
+
+$$
+h _ {i, k} ^ {\prime} h _ {i, l} ^ {\prime} = 0, \tag {11}
+$$
+
+$$
+h _ {i, k} ^ {\prime} h _ {j, l} ^ {\prime} + h _ {j, k} ^ {\prime} h _ {i, l} ^ {\prime} = 0, \tag {12}
+$$
+
+$$
+h _ {i, k l} ^ {\prime \prime} = 0.
+$$
+
+It remains to show  $h_{i,k}^{\prime}h_{j,l}^{\prime} = 0$ . Suppose by contradiction that
+
+$$
+h _ {i, k} ^ {\prime} h _ {j, l} ^ {\prime} \neq 0, \tag {13}
+$$
+
+which implies  $h_{i,k}^{\prime} \neq 0$ . By Eq. (11), we have  $h_{i,l}^{\prime} = 0$ , which, by plugging into Eq. (12), indicates  $h_{i,k}^{\prime} h_{j,l}^{\prime} = 0$ . This is a contradiction with Eq. (13). Thus, we must have  $h_{i,k}^{\prime} h_{j,l}^{\prime} = 0$ .
+
+# C. Proof of Theorem 2
+
+Theorem 2 (Identifiability of latent Markov network). Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Suppose that Assumptions A1 and A2 from Theorem 1 hold. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2) with the minimal number of edges of the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . Then, the recovered latent Markov network  $\mathcal{M}_{\hat{Z}}$  is isomorphic to the true latent Markov network  $\mathcal{M}_Z$ .
+
+Proof. Let  $v \coloneqq g^{-1} \circ \hat{g}$ , i.e.,  $Z = v(\hat{Z})$ . Note that  $v$  is a composition of diffeomorphisms, and hence also a diffeomorphism. Consider a specific value of  $\hat{Z}$ , say  $\hat{z}$ . Since  $v$  is diffeomorphism, by Lemma 2, there exists a permutation  $\pi$  such that the diagonal entries of the corresponding Jacobian matrix (whose columns are permuted according to  $\pi$ ) evaluated at  $\hat{Z} = \hat{z}$  are nonzero, i.e.,
+
+$$
+\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z}} \neq 0, \quad i = 1, \dots , n. \tag {14}
+$$
+
+Suppose that  $Z_{i}$  and  $Z_{j}$  are adjacent in the Markov network  $\mathcal{M}_Z$  over  $Z$ , but  $\hat{Z}_{\pi(i)}$  and  $\hat{Z}_{\pi(j)}$  are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . By Proposition 1, we have
+
+$$
+\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z}} \left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (j)}} \right| _ {\hat {Z} = \hat {z}} = 0,
+$$
+
+which is clearly a contradiction with Eq. (14).
+
+Thus, we have shown by contradiction the following lemma.
+
+Lemma 4. If  $Z_{i}$  and  $Z_{j}$  are adjacent in the Markov network  $\mathcal{M}_Z$  over  $Z$ , then  $\hat{Z}_{\pi(i)}$  and  $\hat{Z}_{\pi(j)}$  are adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ .
+
+The lemma above indicates
+
+$$
+\left| \mathcal {M} _ {\hat {Z}} \right| \geq \left| \mathcal {M} _ {Z} \right|. \tag {15}
+$$
+
+Also, note that the true model  $(g,f,p_Z,\Theta)$  is one of the solutions that achieves Eq. (2). Since the recovered latent Markov network  $\mathcal{M}_{\hat{Z}}$  has the minimal number of edges among the solutions that achieve Eq. (2), we have  $|\mathcal{M}_{\hat{Z}}| \leq |\mathcal{M}_Z|$ , which, with Eq. (15), implies  $|\mathcal{M}_{\hat{Z}}| = |\mathcal{M}_Z|$ .
+
+By Lemma 4 and  $|\mathcal{M}_{\hat{Z}}| = |\mathcal{M}_Z|$ , we conclude that  $Z_{i}$  and  $Z_{j}$  are adjacent in  $\mathcal{M}_Z$  if and only if  $\hat{Z}_{\pi (i)}$  and  $\hat{Z}_{\pi (j)}$  are adjacent in  $\mathcal{M}_{\hat{Z}}$ . That is,  $\mathcal{M}_Z$  and  $\mathcal{M}_{\hat{Z}}$  are isomorphic.
+
+# D. Proof of Theorem 1
+
+Theorem 1 (Relations among true and recovered latent causal variables). Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Suppose that Assumptions A1 and A2 from Theorem 1 hold. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2) with the minimal number of edges of the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . Then, for every pair of estimated latent variables  $\hat{Z}_k$  and  $\hat{Z}_l$  that are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ , we have the following statements:
+
+(a) Each true latent causal variable  $Z_{i}$  is a function of at most one of  $\hat{Z}_k$  and  $\hat{Z}_l$ .  
+(b) For each pair of true latent causal variables  $Z_{i}$  and  $Z_{j}$  that are adjacent in the Markov network  $\mathcal{M}_Z$  over  $Z$ , at most one of them is a function of  $\hat{Z}_k$  or  $\hat{Z}_l$ .
+
+Proof. We first prove Statement (a). By Proposition 1, for every value of  $Z$ , we have
+
+$$
+\frac {\partial Z _ {i}}{\partial \hat {Z} _ {k}} \frac {\partial Z _ {i}}{\partial \hat {Z} _ {l}} = 0.
+$$
+
+Therefore, it suffices to prove that if  $\frac{\partial Z_i}{\partial \hat{Z}_k} \neq 0$  for some value of  $\hat{Z}$ , then  $\frac{\partial Z_i}{\partial \hat{Z}_l} = 0$  for all values of  $\hat{Z}$ . That is, these nonzero entries cannot switch positions.
+
+By Theorem 2, there exists a permutation  $\pi$  of the estimated variables, denoted as  $\hat{Z}_{\pi}$ , such that the Markov network  $\mathcal{M}_{\hat{Z}_{\pi}}$  is identical to  $\mathcal{M}_Z$ . Let  $\hat{Z}_{\pi(i)}$  and  $\hat{Z}_{\pi(k)}$  be two estimated latent variables that are not adjacent in the Markov network  $\mathcal{M}_{\hat{Z}_{\pi}}$ . Now consider variable  $Z_i$ . Suppose by contradiction that the nonzero entries switch positions, i.e., there exist two values of  $\hat{Z}$ , say  $\hat{z}^{(1)}$  and  $\hat{z}^{(2)}$ , such that
+
+$$
+\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z} ^ {(1)}} \neq 0 \tag {16}
+$$
+
+and
+
+$$
+\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (k)}} \right| _ {\hat {Z} = \hat {z} ^ {(2)}} \neq 0, \tag {17}
+$$
+
+Let  $N_{Z_i}$  be a set containing the indices of the neighbors of  $Z_{i}$  in  $\mathcal{M}_Z$ , and  $N_{\hat{Z}_{\pi (i)}}$  be a set containing the indices of the neighbors of  $Z_{\pi (i)}$  in  $\mathcal{M}_{\hat{Z}_{\pi}}$ . Similarly, let  $S_{Z_i}$  be a set containing the indices of the variables that are not adjacent to  $Z_{i}$  in  $\mathcal{M}_Z$ , and  $S_{\hat{Z}_{\pi (i)}}$  be a set containing the indices of the variables that are not adjacent to of  $Z_{\pi (i)}$  in  $\mathcal{M}_{\hat{Z}_{\pi}}$ . By definition, we have
+
+$$
+N _ {Z _ {i}} \cup S _ {Z _ {i}} \cup \{i \} = [ n ], \tag {18}
+$$
+
+which are pairwise disjoint.
+
+Since  $\mathcal{M}_{\hat{Z}_{\pi}}$  and  $\mathcal{M}_Z$  are identical, we have  $N_{Z_i} = N_{\hat{Z}_{\pi (i)}}$  and  $S_{Z_i} = S_{\hat{Z}_{\pi (i)}}$ . Now define the following function
+
+$$
+\phi (\hat {Z}) = \sum_ {j \in N _ {Z _ {i}} \cup \{i \}} \left(\frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (i)}}\right) ^ {2} - \sum_ {l \in S _ {Z _ {i}}} \sum_ {j \in N _ {Z _ {i}} \cup \{i \}} \left(\frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (l)}}\right) ^ {2}.
+$$
+
+Plugging in  $\hat{Z} = \hat{z}^{(1)}$ , for  $l \in S_{Z_i} = S_{\hat{Z}_{\pi(i)}}$  and  $j \in N_{Z_i} \cup \{i\}$ , Proposition 1 implies
+
+$$
+\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z} ^ {(1)}} \left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (l)}} \right| _ {\hat {Z} = \hat {z} ^ {(1)}} = 0,
+$$
+
+which, with Eq. (16), indicates
+
+$$
+\left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (l)}} \right| _ {\hat {Z} = \hat {z} ^ {(1)}} = 0.
+$$
+
+Substituting the above equation and Eq. (16) into function  $\phi$ , we have
+
+$$
+\left. \phi (\hat {Z}) \right| _ {\hat {Z} = \hat {z} ^ {(1)}} = \sum_ {j \in N _ {Z _ {i}} \cup \{i \}} \left(\left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z} ^ {(1)}}\right) ^ {2} \geq \left. \right.\left.\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z} ^ {(1)}}\right) ^ {2} > 0. \tag {19}
+$$
+
+Now plug in  $\hat{Z} = \hat{z}^{(2)}$ . For  $j \in N_{Z_i} \cup \{i\}$ , Proposition 1 implies
+
+$$
+\left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z} ^ {(2)}} \left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (k)}} \right| _ {\hat {Z} = \hat {z} ^ {(2)}} = 0,
+$$
+
+which, with Eq. (17), indicates
+
+$$
+\left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (i)}} \right| _ {\hat {Z} = \hat {z} ^ {(2)}} = 0.
+$$
+
+Substituting the above equation and Eq. (17) into function  $\phi$ , we have
+
+$$
+\left. \phi (\hat {Z}) \right| _ {\hat {Z} = \hat {z} ^ {(2)}} = - \sum_ {l \in S _ {Z _ {i}}} \sum_ {j \in N _ {Z _ {i}} \cup \{i \}} \left(\left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (l)}} \right| _ {\hat {Z} = \hat {z} ^ {(2)}}\right) ^ {2} \leq - \left(\left. \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (k)}} \right| _ {\hat {Z} = \hat {z} ^ {(2)}}\right) ^ {2} <   0. \tag {20}
+$$
+
+Since function  $\phi$  is continuous (because all the partial derivatives involved are continuous) and its domain is a connected set, by applying Intermediate Value Theorem with Eqs. (19) and (20), there exists a value of  $\hat{Z}$  in the domain, say  $\hat{z}^{(3)}$ , such that
+
+$$
+\phi (\hat {Z}) | _ {\hat {Z} = \hat {z} ^ {(3)}} = 0,
+$$
+
+which, by plugging the definition of function  $\phi$ , implies
+
+$$
+\sum_ {j \in N _ {Z _ {i}} \cup \{i \}} \left(\frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (i)}} \Big | _ {\hat {Z} = \hat {z} ^ {(3)}}\right) ^ {2} = \sum_ {l \in S _ {Z _ {i}}} \sum_ {j \in N _ {Z _ {i}} \cup \{i \}} \left(\frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (l)}} \Big | _ {\hat {Z} = \hat {z} ^ {(3)}}\right) ^ {2}.
+$$
+
+Note that if any of the terms in the summation on the left hand side (LHS) is nonzero, then, by Proposition 1, all terms in the summation on the right hand side (RHS) must be zero; in this case, LHS is nonzero but RHS equals zero, which is a contradiction. Similarly, if any of the terms in the summation on the RHS is nonzero, then, by Proposition 1, all terms in the summation on the LHS must be zero; in this case, RHS is nonzero but LHS equals zero, which is a contradiction. This implies that all terms in the summation on both LHS and RHS must be zero, i.e.,
+
+$$
+\left. \frac {\partial Z _ {j}}{\partial \hat {Z} _ {\pi (l)}} \right| _ {\hat {Z} = \hat {z} ^ {(3)}} = 0 \quad \mathrm {f o r} \quad j \in N _ {Z _ {i}} \cup \{i \}, l \in S _ {Z _ {i}} \cup \{i \}.
+$$
+
+Since  $|S_{Z_i} \cup \{i\}| = n - |N_{Z_i}|$  by Eq. (18), Lemma 3 indicates that the matrix  $\left.\frac{\partial Z}{\partial \hat{Z}_{\pi}}\right|_{\hat{Z} = \hat{z}^{(3)}}$  is not invertible. Thus, the (Jacobian) matrix  $\left.\frac{\partial Z}{\partial \hat{Z}}\right|_{\hat{Z} = \hat{z}^{(3)}}$  is also not invertible, which is a contradiction because the mapping from  $\hat{Z}$  to  $Z$  is a diffeomorphism (specifically a composition of diffeomorphisms).
+
+Therefore, we have just proved Statement (a) by contradiction. Similar reasoning can be straightforwardly applied to prove Statement (b) and is omitted here.
+
+# E. Proof of Theorem 3
+
+We first state the following lemma that is used to prove Theorem 3. The proof is a straightforward consequence of Cayley-Hamilton theorem and is omitted here.
+
+Lemma 5. Let  $A$  be an  $n \times n$  invertible matrix. Then, it can be expressed as a linear combination of the powers of  $A$ , i.e.,
+
+$$
+A ^ {- 1} = \sum_ {k = 0} ^ {n - 1} c _ {k} A ^ {k}
+$$
+
+for some appropriate choice of coefficients  $c_{0}, c_{1}, \ldots, c_{n-1}$ .
+
+Now consider the Markov network  $\mathcal{M}_Z$  over variables  $Z$ . With a slight abuse of notation, let  $N_{Z_i}$  be the set of neighbors of  $Z_i$  in  $\mathcal{M}_Z$ . The following result relates a matrix to its inverse, given that the matrix satisfies certain property defined by  $\mathcal{M}_Z$ .
+
+Proposition 3. Consider Markov network  $\mathcal{M}_Z$  over  $Z$ . Let  $N_{Z_i}$  be the set of neighbors of  $Z_i$  in  $\mathcal{M}_Z$ , and  $A$  be an  $n \times n$  invertible matrix. For each  $i \neq j$  where  $Z_j$  is not adjacent to some nodes in  $\{Z_i\} \cup N_{Z_i}$ , suppose  $A_{ij} = 0$ . Then,  $A_{ij}^{-1} = 0$ .
+
+Proof. By Lemma 5,  $A^{-1}$  can be expressed as linear combination of the powers of  $A$ . Therefore, it suffices to prove that each matrix power  $A^k$  satisfies the following property:  $A_{ij}^k = 0$  for each  $i \neq j$  where  $Z_j$  is not adjacent to some nodes in  $\{Z_i\} \cup N_{Z_i}$ . We proceed with mathematical induction on  $k$ . By definition, the property holds in the base case where  $k = 1$ .
+
+Now suppose that the property holds for  $A^k$ . We prove by contradiction that the property holds for  $A^{k+1}$ . Suppose by contradiction that  $A_{ij}^{k+1} \neq 0$  for some  $i \neq j$  where  $Z_j$  is not adjacent to some nodes in  $\{Z_i\} \cup N_{Z_i}$ . This implies that one of the following cases holds:
+
+- Case (a):  $Z_{j}$  is not adjacent to  $Z_{i}$  in  $\mathcal{M}_Z$ .  
+- Case (b): There exists  $Z_{l} \in N_{Z_{i}} \setminus \{Z_{j}\}$  such that  $Z_{j}$  and  $Z_{l}$  are not adjacent in  $\mathcal{M}_{Z}$ .
+
+Since  $A_{ij}^{k + 1} = \sum_{r = 0}^{n}A_{ir}^{k}A_{rj}$ , the assumption  $A_{ij}^{k + 1}\neq 0$  implies that there must exist  $m$  such that  $A_{im}^{k}A_{mj}\neq 0$ , i.e.,  $A_{im}^{k}\neq 0$  and  $A_{mj}\neq 0$ . Since both  $A^k$  and  $A$  satisfy the property, this indicates (i)  $Z_{m}$  is adjacent to  $Z_{i}$  and all nodes in  $N_{Z_i}\setminus \{Z_m\}$ , and (ii)  $Z_{j}$  is adjacent to  $Z_{m}$  and all nodes in  $N_{Z_m}\setminus \{Z_j\}$ . We consider the following cases:
+
+- Case of  $m = l$ : By (ii),  $Z_{j}$  is adjacent to  $Z_{l}$ , which contradicts Case (b) above. Also, we know that  $Z_{l}$  is adjacent to  $Z_{i}$  by (i), which indicates that  $Z_{i}$  is adjacent to  $Z_{j}$ , contradicting Case (a) above.  
+- Case of  $m \neq l$ : By (i) and (ii),  $Z_{m}$  is adjacent to  $Z_{i}$  and  $Z_{j}$  is adjacent to  $Z_{m}$ , implying that  $Z_{i}$  and  $Z_{j}$  are adjacent, which is contradictory with Case (a) above. Furthermore, since  $Z_{l}$  is a neighbor of  $Z_{i}$ , we know that  $Z_{m}$  and  $Z_{l}$  are adjacent by (i). Also, by (ii),  $Z_{j}$  is adjacent to  $Z_{l}$ , which contradicts Case (b) above.
+
+In each of the cases above, there is a contradiction.
+
+We are now ready to prove the following result.
+
+Theorem 3 (Identifiability of latent causal variables). Let the observations be sampled from the data generating process in Eq. (1), and  $\mathcal{M}_Z$  be the Markov network over  $Z$ . Let  $N_{Z_i}$  be the set of neighbors of variable  $Z_i$  in  $\mathcal{M}_Z$ . Suppose that Assumptions A1 and A2 from Theorem 1 hold. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  to achieve Eq. (2) with the minimal number of edges of the Markov network  $\mathcal{M}_{\hat{Z}}$  over  $\hat{Z}$ . Then, there exists a permutation  $\pi$  of the estimated latent variables, denoted as  $\hat{Z}_{\pi}$ , such that each  $\hat{Z}_{\pi(i)}$  is solely a function of a subset of the variables in  $\{Z_i\} \cup \Psi_{Z_i}$ .
+
+Proof. We first prove a simpler case: there exists a permutation  $\pi$  of the estimated latent variables, denoted as  $\hat{Z}_{\pi}$ , such that  $Z_{i}$  is solely a function of  $\hat{Z}_{\pi (i)}$  and a subset of the variables in  $\{\hat{Z}_{\pi (r)}|Z_r\in \Psi_{Z_i}\}$ .
+
+By Theorem 2 and its proof, there exists a permutation  $\pi$  of the estimated variables, denoted as  $\hat{Z}_{\pi}$ , such that the Markov network  $\mathcal{M}_{\hat{Z}_{\pi}}$  over  $\hat{Z}_{\pi}$  is identical to  $\mathcal{M}_Z$ , and that
+
+$$
+\frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (i)}} \neq 0, \quad i = 1, \dots , n.
+$$
+
+Clearly, each variable  $Z_{i}$  is a function of  $\hat{Z}_{\pi (i)}$
+
+We first show that if  $Z_{j}$  is not adjacent to  $Z_{i}$  in  $\mathcal{M}_Z$ , then  $Z_{i}$  cannot be a function of  $\hat{Z}_{\pi (j)}$ . Since  $Z_{i}$  and  $Z_{j}$  are not adjacent in  $\mathcal{M}_Z$ , we know that  $\hat{Z}_{\pi (i)}$  and  $\hat{Z}_{\pi (j)}$  are not adjacent in  $\mathcal{M}_{\hat{Z}_{\pi}}$ . By Theorem 1,  $Z_{i}$  is a function of at most one of  $\hat{Z}_{\pi (i)}$  and  $\hat{Z}_{\pi (j)}$ , which implies that  $Z_{i}$  cannot be a function of  $\hat{Z}_{\pi (j)}$ , because we have shown that  $Z_{i}$  is a function of  $\hat{Z}_{\pi (i)}$ .
+
+To refine further, now suppose that  $Z_{j}$  is adjacent to  $Z_{i}$ , but not adjacent to some  $Z_{k} \in N_{Z_{i}} \setminus \{Z_{j}\}$ . Since  $\mathcal{M}_Z$  and  $\mathcal{M}_{\hat{Z}_{\pi}}$  are identical,  $\hat{Z}_{\pi(j)}$  is also not adjacent to  $\hat{Z}_{\pi(k)}$  in  $\mathcal{M}_{\hat{Z}_{\pi}}$ . Since  $Z_{i}$  and  $Z_{k}$  are adjacent in  $\mathcal{M}_Z$ , by Theorem 1, at most one of them is a function of  $\hat{Z}_{\pi(j)}$  or  $\hat{Z}_{\pi(k)}$ . This implies that  $Z_{i}$  cannot be a function of  $\hat{Z}_{\pi(j)}$ , because we have shown that  $Z_{k}$  is a function of  $\hat{Z}_{\pi(k)}$ .
+
+Therefore, we have just shown that  $Z_{i}$  is solely a function of  $\hat{Z}_{\pi (i)}$  and a subset of the variables in  $\{\hat{Z}_{\pi (r)} \mid Z_r \in \Psi_{Z_i}\}$ . Now consider variable  $Z_{l} \notin \{Z_{i}\} \cup \Psi_{Z_{i}}$ . Since  $Z_{i}$  is not a function of  $\hat{Z}_{\pi (l)}$ , we have
+
+$$
+\left(\frac {\partial Z}{\partial \hat {Z} _ {\pi}}\right) _ {i l} = \frac {\partial Z _ {i}}{\partial \hat {Z} _ {\pi (l)}} = 0.
+$$
+
+By applying Proposition 3 with matrix  $\frac{\partial Z}{\partial\hat{Z}_{\pi}}$  , we have
+
+$$
+\left(\frac {\partial Z}{\partial \hat {Z} _ {\pi}}\right) _ {i l} ^ {- 1} = 0,
+$$
+
+which, by Inverse Function Theorem, implies
+
+$$
+\frac {\partial \hat {Z} _ {\pi (i)}}{\partial Z _ {l}} = \left(\frac {\partial \hat {Z} _ {\pi}}{\partial Z}\right) _ {i l} = \left(\frac {\partial Z}{\partial \hat {Z} _ {\pi}}\right) _ {i l} ^ {- 1} = 0.
+$$
+
+Since the above equation holds for all values of  $Z$ , we conclude that  $\hat{Z}_{\pi (i)}$  cannot be a function of  $Z_{l}$ .
+
+# F. Proof of Corollary 1
+
+Corollary 1 (Impossibility of finding independent components). Let the observations be sampled from the data generating process in Eq. (1). Suppose that Assumptions A1 and A2 from Theorem 1 hold, and that the true latent causal DAG  $\mathcal{G}_Z$  is not an empty graph. Suppose also that we learn  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  with the components of  $\hat{Z}$  being independent in each domain. Then,  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  cannot achieve Eq. (2).
+
+Proof. Suppose by contradiction that  $(\hat{g},\hat{f},p_{\hat{Z}},\hat{\Theta})$  achieves Eq. (2). By assumption, the components of  $\hat{Z}$  are independent in each domain, indicating that the Markov network  $\mathcal{M}_{\hat{Z}}$  is an empty graph. By the same reasoning in the proof of Theorem 2 (specifically Lemma 4), there exists a permutation  $\pi$  such that: if  $Z_{i}$  and  $Z_{j}$  are adjacent in  $\mathcal{M}_Z$ , then  $\hat{Z}_{\pi (i)}$  and  $\hat{Z}_{\pi (j)}$  are adjacent in  $\mathcal{M}_{\hat{Z}}$ . Since  $\mathcal{M}_{\hat{Z}}$  is an empty graph, this implies that  $\mathcal{M}_Z$  is also an empty graph, which is contradictory with the assumption that  $\mathcal{G}_Z$  is not an empty graph.
+
+# G. Proof of Lemma 1 and Proposition 2
+
+Lemma 1. Given a latent causal graph  $\mathcal{G}_Z$  and distribution  $P_{Z;\theta}$  with its Markov Network  $\mathcal{M}_Z$ , under Markov assumption, the undirected graph defined by  $\mathcal{M}_Z$  is a subgraph of the moralized graph of the true causal DAG  $G$ .
+
+Proof. Let  $Z_{j}$  and  $Z_{k}$ ,  $j \neq k$  be two variables that are not adjacent in the moralized graph of  $\mathcal{G}_Z$ . Then it suffices to show that  $\{Z_{j}, Z_{k}\} \notin \mathcal{E}(\mathcal{M}_{Z})$ . Because they are not adjacent in the moralized graph of  $\mathcal{G}_Z$ , they must not be adjacent in  $\mathcal{G}_Z$  and must not share a common child in  $\mathcal{G}_Z$ . Thus,  $Z_{j}$  and  $Z_{k}$  are d-separated conditioning on  $Z_{[n] \setminus \{j,k\}}$ , which implies the conditional independence  $Z_{j} \perp Z_{k} \mid Z_{[n] \setminus \{j,k\}}$  based on the Markov assumption on  $(\mathcal{G}_Z, P_{Z;\theta})$ . Then we have  $\{Z_{j}, Z_{k}\} \notin \mathcal{E}(\mathcal{M}_{Z})$ .
+
+Proposition 2 (Moralized graph and Markov network). Given a causal DAG  $\mathcal{G}_Z$  and distribution  $P_{Z;\theta}$  with its Markov Network  $\mathcal{M}_Z$ , under Markov assumption, the undirected graph defined by  $\mathcal{M}_Z$  is the moralized graph of the true causal DAG  $\mathcal{G}_Z$  if and only if the SAF and SUCF assumptions are satisfied.
+
+Proof. We prove both directions as follows.
+
+Sufficient condition. We prove it by contradiction. Suppose that the structure defined by  $\mathcal{M}_Z$  is not equivalent to the moralized graph of  $\mathcal{G}_Z$ . Then, according to Lemma 1, there exists a pair of variables  $Z_j$  and  $Z_k$ ,  $j \neq k$  that are adjacent in the moralized graph but  $\{Z_j, Z_k\} \notin \mathcal{E}(\mathcal{M}_Z)$ . Thus, we have  $Z_j \perp Z_k \mid Z_{[n] \setminus \{j,k\}}$ . Then we consider the following two cases:
+
+- If variables  $Z_{j}$  and  $Z_{k}$  correspond to a pair of neighbors in  $\mathcal{G}_Z$ , then they are adjacent. Together with the conditional independence relation  $Z_{j} \perp Z_{k} \mid Z_{[n] \setminus \{j, k\}}$ , this implies that the SAF assumption is violated.  
+- If variables  $Z_{j}$  and  $Z_{k}$  correspond to a pair of non-adjacent spouses in  $\mathcal{G}_{Z}$ . Then they have an unshielded collider, indicating that the SUCF assumption is violated.
+
+Necessary condition. We prove it by contradiction. Suppose SUCF or SAF is violated, we have the following two cases:
+
+- Suppose SUCF is violated, i.e., there exists an unshielded collider  $Z_{j} \to Z_{i} \gets Z_{k}$  in the DAG  $\mathcal{G}_Z$  such that  $Z_{j} \perp Z_{k} \mid Z_{[n] \setminus \{j,k\}}$ . This conditional independence relation indicates that  $\{Z_{j}, Z_{k}\} \notin \mathcal{E}(\mathcal{M}_{Z})$ . Since  $Z_{j}$  and  $Z_{k}$  are spouses, there exists an edge between them in the moralized graph of  $\mathcal{G}_Z$ , but is not contained in the structure defined by  $\mathcal{M}_Z$ , showing that they are not the same.  
+- Suppose SAF is violated, i.e., there exists a pair of neighbors  $Z_{j}$  and  $Z_{k}$  in the DAG  $\mathcal{G}_Z$  such that  $Z_{j} \perp Z_{k} \mid Z_{[n] \setminus \{j,k\}}$ . This conditional independence relation indicates that  $\{Z_{j}, Z_{k}\} \notin \mathcal{E}(\mathcal{M}_{Z})$ . Because  $Z_{j}$  and  $Z_{k}$  are adjacent in  $\mathcal{G}_Z$ , clearly they are also adjacent in the moralized graph of  $\mathcal{G}_Z$ . However, the edge between them is not contained in the structure defined by  $\mathcal{M}_Z$ , showing that they are not the same.
+
+Thus, when SUCF or SAF is violated, the structure defined by  $\mathcal{M}_Z$  is the moralized graph of the true DAG  $\mathcal{G}_Z$ .

2024/Causal Representation Learning from Multiple Distributions_ A General Setting/images.zip

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2024/Causal-IQA_ Towards the Generalization of Image Quality Assessment Based on Causal Inference/58f0e4dd-7b82-4abc-a528-023169f0e20f_origin.pdf

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2024/Causal-IQA_ Towards the Generalization of Image Quality Assessment Based on Causal Inference/full.md

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+# Causal-IQA: Towards the Generalization of Image Quality Assessment Based on Causal Inference
+
+Yan Zhong $^{12}$  Xingyu Wu $^{3}$  Li Zhang $^{4}$  Chenxi Yang $^{12}$  Tingting Jiang $^{25}$
+
+# Abstract
+
+Due to the high cost of Image Quality Assessment (IQA) datasets, achieving robust generalization remains challenging for prevalent deep learning-based IQA methods. To address this, this paper proposes a novel end-to-end blind IQA method: Causal-IQA. Specifically, we first analyze the causal mechanisms in IQA tasks and construct a causal graph to understand the interplay and confounding effects between distortion types, image contents, and subjective human ratings. Then, through shifting the focus from correlations to causality, Causal-IQA aims to improve the estimation accuracy of image quality scores by mitigating the confounding effects using a causality-based optimization strategy. This optimization strategy is implemented on the sample subsets constructed by a Counterfactual Division process based on the Backdoor Criterion. Extensive experiments illustrate the superiority of Causal-IQA.
+
+# 1. Introduction
+
+Image quality assessment (IQA) plays a crucial role in optimizing visual experiences across different domains, including image denoising (Tian et al., 2020), restoration (Cui et al., 2023), and generation (Elasri et al., 2022). Its primary objective is to develop algorithms that accurately quantify the perceptual quality of images. Based on the availability of reference information, existing IQA methods can
+
+$^{1}$ School of Mathematical Sciences, Peking University, Beijing, China  $^{2}$ National Engineering Research Center of Visual Technology, National Key Laboratory for Multimedia Information Processing, School of Computer Science, Peking University, Beijing, China  $^{3}$ Department of Computing, The Hong Kong Polytechnic University, Hong Kong SAR, China  $^{4}$ Hefei Institute of Physical Science, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, China  $^{5}$ National Biomedical Imaging Center, Peking University, Beijing, China. Correspondence to: Tingting Jiang <ttjiang@pu.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+be classified into Full-Reference IQA (FR-IQA), Reduced-Reference IQA (RR-IQA), and Blind IQA (BIQA) (Zhai & Min, 2020), where BIQA exhibits broader applicability due to its independence from reference image (Su et al., 2020; Simeng Sun, 2023; Feng et al., 2021).
+
+Traditional BIQA methods aim to predict image quality scores that align with subjective human ratings, such as Mean Opinion Scores (MOS), by relying on manually extracted statistical features from the provided distorted images (Jiang et al., 2017; Liu et al., 2020; Zhou et al., 2017). Although these methods are flexible and easy to implement, they cannot achieve ideal performance when confronted with complex scenes or diverse distortions. Consequently, Deep Learning (DL)-based IQA methods have received widespread attention due to their strong ability in fusing discriminative features in visual domains. (Ke et al., 2021; Talebi & Milanfar, 2018; Madhusudana et al., 2022; Simeng Sun, 2023). However, the high cost of annotating IQA datasets restricts DL-based BIQA methods to be trained only on small-scale datasets, leading to overfitting and limited generalization on unseen distorted information and images with authentic distortions (Yue et al., 2022).
+
+To address this issue, one intuitive strategy is to initialize BIQA models with the backbones pre-trained on large-scale databases in the field of image classification, such as ImageNet (Deng et al., 2009). Although this enables DL-based models to learn general image features, its effectiveness in IQA tasks is limited due to the lack of specific representation learning for distortions. Another strategy is to design unsupervised IQA method for test time adaptation (Roy et al., 2023), which can effectively mitigate distribution shifts between train and test data, thus improving the performance. However, this strategy has limited applicability as it requires additional unsupervised training to improve the prediction accuracy on different test datasets. Furthermore, these studies often overlook the investigation of the underlying factors that lead to the limited generalization capability in IQA models. Therefore, we focus on exploring an end-to-end BIQA method with good interpretability and generalization capability.
+
+In the standard training paradigm for an IQA network  $f_{\theta}$  the goal is to mine the correlation between distorted image
+
+![](images/bb346a5e0905408c031b671c50708e3f0028f1bd5daef4ce20c13c40be73534c.jpg)  
+Figure 1. The structural causal graph in IQA tasks, where the arrows denote the causalities between parent nodes to child nodes. We only consider distortion type in  $D$  in our method.
+
+D: Distortion information
+
+C: Image contents
+
+$I_{d}$ : Distorted images
+
+m: GT mos labels
+
+$f(\theta)$  : IQA networks
+
+Confounding factors  
+: Target causal variables
+
+$I_{d}$  and its corresponding MOS label  $m$ , which is achieved by maximizing the conditional probability  $P(m|I_d)$  through empirical risk minimization (ERM). However, it's important to note that  $P(m|I_d)$  is influenced by both the distortion types and the image contents (Li et al., 2018), leading to the detrimental confounding effects and poor generalization in the standard training paradigm (Li et al., 2023). To address this issue, we analyze the causal mechanisms in IQA tasks and construct a causal graph (Figure 1) that sheds light on the underlying causal mechanisms in IQA tasks. This Directed Acyclic Graph (DAG) provides valuable insights into the interplay and potential confounding effects between these variables, allowing us to better understand the challenges associated with the standard training paradigm and the limitations of correlation-based IQA models. According to Figure 1, distortion types and image contents are confounders of cause-effect pair  $I_{d}$  and  $m$ . The learning objective of the aforementioned standard training paradigm focuses solely on the correlations between  $I_{d}$  and  $m$  within the specific IQA datasets used for training, which is the main reason for their poor generalization since correlation does not imply causation (Pearl et al., 2016).
+
+Based on the above analyses, a generalizable IQA model should be robust to different distortion information and image contents. To achieve this goal, we propose a Causal Learning-based Image Quality Assessment model (Causal-IQA). By shifting the focus from correlations to causality between variables  $I_{d}$  and  $m$ , Causal-IQA aims to improve the estimation accuracy of the conditional probability  $P(m|I_d)$  through the learning of causal relationships. Specifically, we first construct sample subsets based on confounder sets (including distortion type and image content) from the perspective of causality (Pearl, 2009). To distinguish the content  $C$  for each image, the extracted semantic features of these images are partitioned through Gaussian mixture clustering (Do & Batzoglou, 2008). Subsequently, sample subsets<sup>1</sup> with invariant  $D$  and  $C$  are constructed
+
+to achieve distortion-invariant and content-invariant representation learning, which can eliminate the confounding effects (caused by  $C$  and  $D$ ) on the causal relationship between  $I_{d}$  and  $m$  to improve the robustness of BIQA model. Ultimately, inspired by Meta-Learning (Finn et al., 2017; Nichol et al., 2018), the causal learning process is instantiated according to a causality-based optimization strategy based on the Backdoor Criterion (Pearl et al., 2016). The contributions of this paper are summarized as follows:
+
+- We provide a comprehensive theoretical analysis from a causal perspective, shedding light on the underlying reasons for the limited generalization of IQA models. We identify that the training process commonly struggles to effectively eliminate the confounding effects caused by distortion and content information, which sets the stage for the introduction of our novel method.  
+- The proposed Causal-IQA model, as a breakthrough in BIQA training, possesses at least three advantages: (1) Enhanced generalization capacity due to the effective elimination of the confounding effects caused by distorted images and image content; (2) Interpretability for providing valuable insights into the IQA process; (3) Adaptability that can be seamlessly integrated into any BIQA network.  
+- We conduct extensive experiments on both authentically and synthetically distorted image databases to validate the effectiveness and generalization capabilities of the proposed method. These experiments provide empirical evidence of the superiority of Causal-IQA over existing approaches.
+
+# 2. Background
+
+# 2.1. Blind Image Quality Assessment
+
+Blind Image Quality Assessment (BIQA) has gained significant attention recently due to the absence of reference images in realistically distorted image datasets (Zhai & Min, 2020). With the development and wide applications of Deep Learning (DL), there spring up various DL-based methods achieving remarkable progress in BIQA (Ke et al., 2021; Golestaneh et al., 2022). For example, RankIQA (Liu et al., 2017) proposed a Siamese Network to rank image pairs that are synthetically distorted. CONTRIQUE (Madhusudana et al., 2022) considered the problem of obtaining image quality representations in a self-supervised manner, which is trained with contrastive learning. And GraphIQA (Simeng Sun, 2023) integrated graph representation learning into IQA to learn the distortion graph representations. Although these studies can improve the ability of quality perception on training datasets, they lack enough
+
+![](images/148e455e503a4989c3616a31b3b69609204b33550bdddf5db7f23ed904d5a834.jpg)  
+Figure 2. Overview of Causal-IQA. In Stage 1 (Counterfactual Division), GMM and confounding factors in Figure 1 are used to construct image subsets, where images are distortion-invariant and content-invariant in the same subset. In Stage 2, the IQA network  $f_{\theta}$  is trained on these sample subsets with a causality-based representation learning method, to improve the generalization of  $f_{\theta}$ . In Stage 3, the priori IQA model can be used to assess the quality score for new images with unseen information about distortions and contents.
+
+generalization since the scale of the training IQA datasets are limited. In this regard, MetalQA (Zhu et al., 2020) introduces meta-learning with synthetic distortions, employing a shared quality prior knowledge model for versatile adaptation to diverse distortion types. In order to address different distortions and content variations in images, Su et al. (Su et al., 2020) devise a self-adaptive hypernetwork to assign weights to parameters in the quality prediction module. More recently, one of the latest methods is to integrate domain adaptive and ensemble learning into the IOA task, such as (Roy et al., 2023). Unfortunately, the improvements in generalization in these existing methods come at the expense of training costs. On the contrary, we improve the generalization and robustness of BIQA model from a causality view, which is end-to-end and interpretable.
+
+# 2.2. Causal Learning
+
+Traditional machine learning approaches often consider the statistical dependencies between inputs and outputs, which can hinder the model's ability to generalize to unknown data (Yao et al., 2021). To address this limitation and identify the true causal impact of inputs on outputs, causal inference has been introduced as a means to discover and leverage causal relationship information embedded in the data. By employing causal graph models and intervention operations, causal inference aims to eliminate confounding effects on the outputs, ultimately yielding causal effects. This aspect is crucial for enhancing both the interpretability and generalization capabilities of models. In recent years, causal inference methods based on Structural Causal Models (SCM) (Pearl, 2010; Pawlowski et al., 2020) have witnessed significant advancements and garnered widespread adoption among computer science researchers due to their ability to offer clearer descriptions of causal relationships between
+
+variables (Gresele et al., 2022). Leveraging the structural information between variables, SCM enables causal analysis even in scenarios where interventions on variables are not fully observed, thus presenting a distinct advantage for tackling complex tasks. In the domain of computer vision, various tasks are susceptible to the influence of latent confounding factors. Causal inference techniques provide a valuable framework for identifying and leveraging causal relationships to mitigate the impact of confounding factors, such as scene graph generation (Chen et al., 2019), image recognition (Tang et al., 2020), video analysis (Kanehira et al., 2019), etc. In this paper, we investigate the performance gains afforded by causal relationships within the context of IQA problems.
+
+# 3. Method
+
+In this section, we initially present the overview of Causal-IQA. Then, we analyze the causal mechanism in BIQA tasks, based on which we propose the IQA formula for robust representation learning, followed by its training paradigm.
+
+# 3.1. Overview of the Proposed Method
+
+As shown in Figure 2, the basic idea of our method is to learn the distortion-invariant and content-invariant representations based on causal learning. And the whole framework consists of three stages. Given the reference images from synthetically distorted image datasets, we construct sample subsets based on confounding factors first in Stage 1: images with the same distortion type are collected into the same sample subset. Due to unavailability of content labels in IQA datasets, we utilize the GMM (Do & Batzoglou, 2008) model for clustering after extracting semantic features by pretrained backbone (ResNet18 (He et al., 2016) is cho
+
+sen in our method), to prepare for the subsequent training process. In Stage 2, we propose a causality-based training paradigm to improve the generalization of IQA, which can be used to eliminate the confounding effects from distortion types and image contents, so that the causal relationship between distorted images and corresponding MOS labels is identifiable (the optimization progress is executed based on a meta-learning strategy using the sample subsets and the cluster probabilities obtained in Stage 1). Finally, in Stage 3, the trained IQA model can be used to assess the quality score for new images with unseen distortions and contents.
+
+# 3.2. The Causal Mechanism of BIQA
+
+BIQA aims to predict the MOS score  $m$  of distorted image  $I_{d}$  without reference images, denoted as  $m = f_{\theta}(I_d)$ , where  $\theta$  means the model parameters. From a causal perspective, we model this BIQA progress with a causal structure graph in Figure 1, where  $I_{d}$  and  $m$  are regarded as Intervention Object and Intervention Outcome respectively. Suppose there exists an ideal image  $I_{v}$  for each distorted image  $I_{d}$ , the path  $D \rightarrow I_d \leftarrow C$  in Figure 1 means the degradation process from  $I_{v}$  to  $I_{d}$ , where  $D$  and  $C$  denote the variables of distortions and image contents (For distortion information, we only consider distortion type in  $D$  in our method, and the discussion about distortion degrees is presented in the Appendix). Besides, the path  $D \rightarrow m \leftarrow C$  in Figure 1 denotes the knowledge learned from  $D$  and  $C$  to  $m$ .
+
+In order to improve the generalization of the IQA model,  $f_{\theta}$  should be trained to learn the causal representation based on the causation between  $I_{d}$  to  $m$ . However, there are two virtual paths connected  $I_{d}$  and  $m$ , including  $I_{d} \leftarrow C \rightarrow m$  and  $I_{d} \leftarrow D \rightarrow m$  caused by  $C$  and  $D$ , which are the confounding factors that bring confounding affects on both the Intervention Object (i.e.  $I_{d}$ ) and Intervention Outcome (i.e.  $m$ ) under study. More precisely, the existing of virtual paths causes the conditional probability  $P(m|I_d)$  learned by IQA network  $f_{\theta}$  is also conditioned on these two confounding factors, thus the traditional IQA training paradigm is actually fitting  $P(m|I_d,C,D)$ , leading to the IQA network  $f_{\theta}$  not robust to different distortions and contents due to that it is not independent of  $D$  and  $C$ . To address this issue, we learn the robust IQA representation by Counterfactuals (Pearl, 2022) and Backdoor Adjustment (Pearl, 1995) in the causal learning domain since  $D$  is observable in synthetically distorted image datasets.
+
+Backdoor Adjustment: In causal inference, the primary objective of backdoor adjustment is to estimate the intervention distribution, enabling the identification of the causal relationship of the target variables, which is based on Backdoor Criteria (see details in supplementary materials). Specifically speaking, we make a do-calculus based on the discretization results of confounding factors  $C$  and  $D$ , and do-calculus
+
+transforms the intervention distribution required for causal inference into a probability distribution in statistical learning (Pearl, 1995), which can block the causal relationships from  $C$  and  $D$  to  $I_d$  to eliminate their confounding effects. In this way, we can learn the robust IQA representation independent of distortion types and image contents by instantiating the conditional probability  $P(m \mid do(I_d))$ . We have the following proposition for Backdoor Adjustment:
+
+Proposition 3.1. The causal conditional probability in IQA tasks can be formulated as:
+
+$$
+P (m \mid d o (I _ {d})) = \sum_ {(c _ {i}, d _ {j}) \in S} P (m \mid I _ {d}, c _ {i}, d _ {j}) P (c _ {i}, d _ {j}) \tag {1}
+$$
+
+where  $P(c_{i},d_{j})$  denotes the probability of each distortion type and content cluster.
+
+Proof. See supplementary materials.
+
+![](images/88b6d6f94beb5c02d3e8e05b306815b9e0e8b272310c7912b0368b35b6d4d138.jpg)
+
+Suppose there exist  $p$  distortion types and the cluster number is set as  $q$  in Eq. 1, which satisfy uniform distribution in IQA datasets, then we have  $P(c_{i},d_{j}) = P(c_{i})P(d_{j}) = \frac{1}{pq}$ .
+
+Counterfactual Division: In order to make the trained IQA model independent of different distortion types and image contents with enough generalization ability, we construct the confounders  $S = \{(c_i, d_j)\}$  first in Stage 1 of Figure 1, which consists of different (distortion type, image content) pairs. According to (Li et al., 2023), it is better for each reference image to have corresponding distorted images with varying distortion types and image contents. Therefore, we collect images with the same distortion type in the same content cluster into the same sample subset, and each sample subset is corresponding to one element in  $S$ . We call this process Counterfactual Division. In fact, the label of distortion type is available in synthetically distorted image datasets, and we obtain the cluster probabilities about image contents by GMM (Do & Batzoglou, 2008) and the lower-dimensional semantic features extracted by ResNet18 and TSNE (Van der Maaten & Hinton, 2008)). Here we have the following proposition to support the rightness of Backdoor Adjustment and the division based on  $S$ .
+
+Proposition 3.2. The construction of above IQA data subsets conforms to the calculating of Counterfactuals. Given  $S$ , for  $\forall i_d \in I_d$ , counterfactual  $m_{i_d}$  is conditionally independent of  $I_d$  (i.e.  $m_{i_d} \perp I_d \mid S$ ).
+
+$$
+P \left(m _ {i _ {d}} \mid I _ {d}, S\right) = P \left(m _ {i _ {d}} \mid S\right) \tag {2}
+$$
+
+An easy corollary of Proposition 3.2 is the following:
+
+Corollary 3.3. The probability of counterfactual  $m_{i_d}$  is equal to the calculation of  $P(m \mid do(I_d))$ . That is:
+
+$$
+P \left(m _ {i _ {d}} = y\right) = \sum_ {s = \left(c _ {i}, d _ {j}\right)} P (m = y \mid S = s, I _ {d} = i _ {d}) P (s) \tag {3}
+$$
+
+Algorithm 1 Causal-IQA-S (CIS)  
+Input: training dataset  $\mathcal{X} = \{(I_d,m)\mid d\in S\}$  confounders  $S = \{(c_i,d_j)\} ,1\leq i\leq q,1\leq j\leq p,$    
+learning rate  $\alpha$  and  $\beta$  for inner and outer loop updating.   
+Initial: BIQA network  $f_{\theta}$    
+repeat   
+ $\phi_{\bar{s}}\gets \theta .$    
+for  $s_{i,j}$  in  $S$  do Sample training pairs  $(I_{s_{i,j}},m)$  from  $\mathcal{X}$  Task Updating:  $\phi_{\bar{s}} = \phi_{\bar{s}} - \alpha \nabla_{\phi_{\bar{s}}}\mathcal{L}\left(f_{\phi_{\bar{s}}}\left(I_{s_{i,j}}\right),m\right)$  end for Meta Updating:  $\theta \gets \theta -\beta (\phi_{\bar{s}} - \theta)$    
+until convergence.   
+Output: convergent meta parameters  $\theta^{*}$
+
+Proof. See Appendixes for the proof of Eq. 2 and Eq. 3.  $\square$
+
+# 3.3. Causality-based Representation Learning
+
+With the causality-based optimization direction defined in Eq. 1, we can achieve a robust representation learning by instantiating the intervention of different distortion types and image contents. In the traditional ERM-based training paradigm in BIQA tasks, the network  $f_{\theta}$  is trained by minimizing the loss function  $\mathcal{L}(f_{\theta}(I_d), m)$  in the entire training dataset  $\mathcal{X} = \{(I_d, m) \mid d \in S\}$  to maximize the condition probability  $P(m|I_d)$ , which can be denoted as:
+
+$$
+\theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} \left[ \mathscr {L} \left(f _ {\theta} \left(I _ {d}\right), m\right) \right], \tag {4}
+$$
+
+where  $\mathcal{L}(\cdot)$  is usually  $L_{1}$  loss or EMD loss (Talebi & Milanfar, 2018), and  $\theta^{*}$  is the learned parameters. As mentioned above, due to the poor generalization in this training paradigm, we implement the causality-based representation learning by maximizing the causal conditional probability  $P(m\mid do(I_d))$  instead of  $P(m\mid I_d)$ .
+
+In order to model the intervention from confounders  $S$  to instantiate conditional probability  $P(m|I_d, c_i, d_j)$  in Eq. 1, we execute the one-step update for each pair  $(c_i, d_j) \in S$  to obtain the IQA model  $f_\theta$  conditioned on specific distortion type and content cluster, which is implemented on the corresponding sample subset obtained in the Counterfactual Division stage in Figure 1, derived as:
+
+$$
+\phi_ {s _ {i, j}} = \theta - \alpha \nabla_ {\theta} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}}\right), m\right), s _ {i, j} \in S \tag {5}
+$$
+
+where  $I_{s_{i,j}}$  denotes the distorted images  $I_d$  in content cluster  $c_i$  with distortion type  $d_j$ , and  $\phi_{s_{i,j}}$  is the corresponding one-step updated model. Note that one image in the wild does not necessarily belong strictly to one content cluster, so the loss  $\mathcal{L}\left(f_\theta\left(I_{s_{i,j}}\right),m\right)$  can be computed with the cluster probabilities of each samples:
+
+$$
+\mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}}\right), m\right) = \sum_ {r} \frac {w _ {r , j}}{w _ {: , j}} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}} (r)\right), m (r)\right) \tag {6}
+$$
+
+Algorithm 2 Causal-IQA-P (CIP)  
+Input: training dataset  $\mathcal{X} = \{(I_d,m)\mid d\in S\}$  confounders  $S = \{(c_i,d_j)\} ,1\leq i\leq q,1\leq j\leq p,$  learning rate  $\alpha$  and  $\beta$  for task updating and meta updating. Initial: BIQA network  $f_{\theta}$    
+repeat   
+ $\phi_{\bar{s}}\gets \theta .$  Sample training pairs  $\{(I_{s_{i,j}},m)\}_{i,j}$  from  $\mathcal{X}$  Task Updating:  $\phi_{\bar{s}} = \phi_{\bar{s}} - \alpha \nabla_{\phi_{\bar{s}}} \frac{1}{pq}\sum_{s_{i,j}\in S}\mathcal{L}(f_{\phi_{\bar{s}}}(I_{s_{i,j}}),m)$  Meta Updating:  $\theta \leftarrow \theta -\beta (\phi_{\bar{s}} - \theta)$    
+until convergence.   
+Output: convergent meta parameters  $\theta^{*}$
+
+where  $w_{r,j}(\sum_j w_{r,j} = 1)$  denotes the probability of the  $r$ -th sample attaching to  $j$ -th content cluster, and  $w_{:j} = \sum_r w_{r,j}$ . Then we can maximize  $P(m|I_d, c_i, d_j)$  by minimizing the overall loss  $\mathcal{L}(f_{\phi_{s_{i,j}}}(I_d), m)$ . Therefore,  $P(m \mid do(I_d))$  can be maximized on the whole training set:
+
+$$
+\theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} [ \sum_ {s _ {i, j} \in S} \mathcal {L} \left(f _ {\phi_ {s _ {i, j}}} \left(I _ {s _ {i, j}}\right), m\right) ] \tag {7}
+$$
+
+Finally, we can learn the generalizable BIQA networks by the optimization direction in Eq. 7
+
+# 3.4. The Proposed Training Paradigm
+
+It is evident that the optimization procedure derived in Eq. 567 is challenging to execute, which is memory-consuming and computationally expensive. We have the following proposition to address this issue.
+
+Proposition 3.4. The optimization procedure for Causal-IQA in Eq. 5.7 is equivalent to the optimization direction described as follows:
+
+$$
+\theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} [ \mathcal {L} (f _ {\phi_ {\bar {s}}} (I _ {d}), m) ]
+$$
+
+$$
+a n d \phi_ {\bar {s}} = \theta - \alpha \nabla_ {\theta} \frac {1}{p q} \sum_ {s _ {i, j} \in S} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}}\right), m\right) \tag {8}
+$$
+
+where  $\mathcal{L}\left(f_{\theta}\left(I_{s_{i,j}}\right),m\right)$  is defined in Eq. 6.
+
+Proof. See supplementary materials.
+
+According to Proposition 3.4,  $\phi_{\bar{s}}$  is the median model parameters virtually updated with loss function  $\mathcal{L}(\cdot)$  on all the image subsets obtained in Counterfactual Division, thus the optimization procedure in Eq. 8 can be implemented by emulating the meta-learning strategy MAML (Finn et al., 2017). However, a second-order gradient needs to be calculated during the iteration in MAML, which is relatively complicated in computation. Inspired by Reptile (Nichol et al.,
+
+Table 1. Attributes of five typical IQA databases in experiments.  
+
+<table><tr><td>Databases</td><td>Number</td><td>MOS Range</td><td>Distortion Type</td></tr><tr><td>TID2013</td><td>3,000</td><td>[0,9]</td><td>Synthetic</td></tr><tr><td>KADID-10K</td><td>10,125</td><td>[1,5]</td><td>Synthetic</td></tr><tr><td>KonIQ-10K</td><td>10,073</td><td>[1,5]</td><td>Authentic</td></tr><tr><td>LIVE-C</td><td>1,162</td><td>[0,100]</td><td>Authentic</td></tr><tr><td>CID2013</td><td>480</td><td>[0,100]</td><td>Authentic</td></tr></table>
+
+2018), we use the sequential parameter updating strategy based on the one-order gradient to approximate the second-order gradient, thus the optimization process can be greatly simplified. Eventually, according to the updating mode of the median parameters  $\phi_{\bar{s}}$ , we propose two versions of the optimization algorithm for Causal-IQA method, which are summarized in Algorithm 1 and Algorithm 2 termed as Causal-IQA-S (CIS for short) and Causal-IQA-P (CIP for short) respectively. Causal-IQA-S Causal-IQA-P compute the task loss for virtual updating in the serial and parallel manner respectively, the relevant analyses are illustrated in supplementary materials.
+
+# 4. Experiments
+
+# 4.1. Datasets
+
+In this paper, we perform experiments on the following five representative IQA databases:
+
+- TID2013 (Ponomarenko et al., 2015). This database contains 3000 images, which is obtained from 25 reference images, 24 types of distortions for each reference image, and 5 levels for each type of distortion.  
+- KADID-10K (Lin et al., 2019). It includes 81 pristine images, where each pristine image was degraded by 25 distortions in 5 levels. For each distorted image, 30 reliable degradation category ratings were obtained by crowdsourcing performed by 2,209 crowd workers.  
+- KonIQ-10K (Hosu et al., 2020). It consists of 10,073 authentically distorted images chosen from YFCC100M (Thomee et al., 2016) with a wide distribution about brightness, color, contrast and sharpness.  
+- LIVE-C (Ghadiyaram & Bovik, 2015). LIVE-C consists of 1162 authentically distorted images captured from many diverse mobile devices. Each image was assessed by an average of 175 unique subjects.  
+- CID2013 (Virtanen et al., 2014). It includes six image sets; on average, 30 subjects have evaluated 12–14 devices depicting eight different scenes for a total of 79 different cameras, 480 images, and 188 subjects.
+
+The details of these datasets are shown in Table 1, where synthetically distorted image datasets are mainly used for training and the authentically distorted datasets are used
+
+Table 2. Fine-tuning results on authentically distorted image datasets with models trained on TID2013 and KADID-10K. Bold indicates the best performance, and the same convention is applied to other metrics and tables.
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>BLIINDS-II</td><td>0.565</td><td>0.487</td><td>0.507</td><td>0.463</td><td>0.615</td><td>0.529</td></tr><tr><td>BRISQUE</td><td>0.648</td><td>0.615</td><td>0.645</td><td>0.607</td><td>0.537</td><td>0.473</td></tr><tr><td>ILNQE</td><td>0.538</td><td>0.346</td><td>0.589</td><td>0.594</td><td>0.537</td><td>0.501</td></tr><tr><td>CORNIA</td><td>0.680</td><td>0.624</td><td>0.662</td><td>0.618</td><td>0.795</td><td>0.780</td></tr><tr><td>HOSA</td><td>0.685</td><td>0.663</td><td>0.678</td><td>0.659</td><td>0.813</td><td>0.805</td></tr><tr><td>BIECON</td><td>0.620</td><td>0.606</td><td>0.613</td><td>0.595</td><td>/</td><td>/</td></tr><tr><td>MEON</td><td>0.703</td><td>0.701</td><td>0.693</td><td>0.688</td><td>/</td><td>/</td></tr><tr><td>WADIQAM-NR</td><td>0.729</td><td>0.708</td><td>0.680</td><td>0.671</td><td>0.761</td><td>0.739</td></tr><tr><td>DISTNET-Q3</td><td>/</td><td>/</td><td>0.601</td><td>0.570</td><td>0.710</td><td>0.702</td></tr><tr><td>DIQA</td><td>0.720</td><td>0.708</td><td>0.704</td><td>0.703</td><td>/</td><td>/</td></tr><tr><td>NSSADNN</td><td>0.825</td><td>0.748</td><td>0.813</td><td>0.745</td><td>/</td><td>/</td></tr><tr><td>METAIQA</td><td>0.784</td><td>0.766</td><td>0.835</td><td>0.802</td><td>0.887</td><td>0.850</td></tr><tr><td>CIS(Ours)</td><td>0.887</td><td>0.895</td><td>0.847</td><td>0.828</td><td>0.918</td><td>0.881</td></tr><tr><td>CIP(Ours)</td><td>0.873</td><td>0.894</td><td>0.844</td><td>0.823</td><td>0.896</td><td>0.865</td></tr></table>
+
+for verification and fine-tuning. For training and testing uniformity, all the MOS ranges are normalized to [0, 1].
+
+# 4.2. Experimental Settings
+
+Implementation Details. During training with Causal-IQA, we choose the same backbone ResNet18 (He et al., 2016) for a compelling performance comparison with MetalIQA (Zhu et al., 2020), which also designed the generalizable training paradigm based on Meta Learning. In Stage 1 of Counterfactual Division, we set the number  $c$  of clustering in GMM as 2, which can be considered as contents-based knowledge guiding for high quality and low quality. Before GMM, the image features extracted by ResNet18 are reduced in dimension to 10. The training network is constructed following MetalIQA, which is trained with Pytorch library (Paszke et al., 2017) on two Intel Xeon E5-2609 v4 CPUs and four NVIDIA RTX 2080Ti GPUs. During training, images are scaled to the size of  $256 \times 256$  and then randomly cropped to  $224 \times 224$  before feeding to models. We use Adam optimizer with  $\beta_{1} = 0.9$  and  $\beta_{1} = 0.9999$  for both task updating and meta updating, and the learning rates for task updating and meta updating are set as  $1e - 4$  and  $1e - 2$  respectively. The learning rate for task updating drops by 0.8 times after every 10 epochs with the total epoch number set as 100. In the process of fine-tuning, the learning rate of Adam optimizer is also set as  $1e - 4$  with total epoch number 30. The batch sizes  $B$  in TID2013 and KADID-10K are set as 32 and 102 for combined training, and  $B = 64$  during test process. The backbone is initialized by the pre-training weights obtained by classification task on ImageNet (Deng et al., 2009).
+
+Evaluation Metrics. We evaluate our Causal-IQA models by two typical metrics (Bosse et al., 2017), including Pearson Linear Correlation Coefficient (PLCC) and Spearman Rank-order Correlation Coefficient (SROCC).
+
+Table 3. SROCC results comparison in leave-one-distortion-out cross validation on the synthetically distorted image database KADID-10K.  
+
+<table><tr><td></td><td>DIST. TYPE</td><td>BLIINDS-II</td><td>BRISQUE</td><td>ILNIQE</td><td>CORNIA</td><td>HOSA</td><td>WADIQAM-NR</td><td>METAIQA</td><td>CIS</td><td>CIP</td></tr><tr><td rowspan="26">KADID-10K</td><td>GB</td><td>0.8799</td><td>0.8118</td><td>0.8831</td><td>0.8655</td><td>0.8522</td><td>0.8792</td><td>0.9461</td><td>0.9524</td><td>0.9349</td></tr><tr><td>LB</td><td>0.7810</td><td>0.6738</td><td>0.8459</td><td>0.8109</td><td>0.7152</td><td>0.7299</td><td>0.9168</td><td>0.9247</td><td>0.9306</td></tr><tr><td>MB</td><td>0.4816</td><td>0.4226</td><td>0.7794</td><td>0.5323</td><td>0.6515</td><td>0.7304</td><td>0.9262</td><td>0.9051</td><td>0.9135</td></tr><tr><td>CD</td><td>0.5719</td><td>0.5440</td><td>0.6780</td><td>0.2432</td><td>0.7272</td><td>0.8325</td><td>0.9194</td><td>0.9043</td><td>0.9026</td></tr><tr><td>CS</td><td>-0.1392</td><td>0.1821</td><td>0.0898</td><td>-0.0023</td><td>0.0495</td><td>0.4209</td><td>0.7850</td><td>0.8112</td><td>0.8255</td></tr><tr><td>CQ</td><td>0.6695</td><td>0.6670</td><td>0.6763</td><td>0.3226</td><td>0.6617</td><td>0.8055</td><td>0.7170</td><td>0.7405</td><td>0.7231</td></tr><tr><td>CSA1</td><td>0.0906</td><td>0.0706</td><td>0.0266</td><td>-0.0194</td><td>0.2158</td><td>0.1479</td><td>0.3039</td><td>0.4658</td><td>0.4658</td></tr><tr><td>CSA2</td><td>0.6017</td><td>0.3746</td><td>0.6771</td><td>0.1197</td><td>0.8408</td><td>0.8358</td><td>0.9310</td><td>0.9388</td><td>0.9043</td></tr><tr><td>JP2K</td><td>0.6546</td><td>0.5159</td><td>0.7895</td><td>0.3417</td><td>0.6078</td><td>0.5387</td><td>0.9452</td><td>0.9379</td><td>0.9508</td></tr><tr><td>JPEG</td><td>0.4140</td><td>0.7821</td><td>0.8036</td><td>0.5561</td><td>0.5823</td><td>0.5298</td><td>0.9115</td><td>0.9348</td><td>0.9001</td></tr><tr><td>WN</td><td>0.6277</td><td>0.7080</td><td>0.7757</td><td>0.3574</td><td>0.6796</td><td>0.8966</td><td>0.9047</td><td>0.8934</td><td>0.8877</td></tr><tr><td>WNCC</td><td>0.7567</td><td>0.7182</td><td>0.8409</td><td>0.4183</td><td>0.7445</td><td>0.9247</td><td>0.9303</td><td>0.9215</td><td>0.9340</td></tr><tr><td>IN</td><td>0.5469</td><td>-0.5425</td><td>0.8082</td><td>0.2188</td><td>0.2535</td><td>0.8142</td><td>0.8673</td><td>0.8810</td><td>0.8767</td></tr><tr><td>MN</td><td>0.7017</td><td>0.6741</td><td>0.6824</td><td>0.3060</td><td>0.7757</td><td>0.8841</td><td>0.9247</td><td>0.9316</td><td>0.9089</td></tr><tr><td>DENOISE</td><td>0.4566</td><td>0.2213</td><td>0.8562</td><td>0.2293</td><td>0.2466</td><td>0.7648</td><td>0.8985</td><td>0.9428</td><td>0.9041</td></tr><tr><td>BRIGHTEN</td><td>0.4583</td><td>0.5754</td><td>0.3008</td><td>0.2272</td><td>0.7525</td><td>0.6845</td><td>0.7827</td><td>0.8267</td><td>0.8087</td></tr><tr><td>DARKEN</td><td>0.4391</td><td>0.4050</td><td>0.4363</td><td>0.2060</td><td>0.7436</td><td>0.2715</td><td>0.6219</td><td>0.6851</td><td>0.6424</td></tr><tr><td>MS</td><td>0.1119</td><td>0.1441</td><td>0.3150</td><td>0.1215</td><td>0.5907</td><td>0.3475</td><td>0.5555</td><td>0.6429</td><td>0.6199</td></tr><tr><td>JITTER</td><td>0.6287</td><td>0.6719</td><td>0.4412</td><td>0.7186</td><td>0.3907</td><td>0.7781</td><td>0.9278</td><td>0.8909</td><td>0.8740</td></tr><tr><td>NEP</td><td>0.0832</td><td>0.1911</td><td>0.2178</td><td>0.1206</td><td>0.4607</td><td>0.3478</td><td>0.4184</td><td>0.7012</td><td>0.6723</td></tr><tr><td>PIXELATE</td><td>0.1956</td><td>0.6477</td><td>0.5770</td><td>0.5868</td><td>0.7021</td><td>0.6998</td><td>0.8090</td><td>0.8323</td><td>0.8122</td></tr><tr><td>QUANTIZATION</td><td>0.7812</td><td>0.7135</td><td>0.5714</td><td>0.2592</td><td>0.6811</td><td>0.7345</td><td>0.8770</td><td>0.8845</td><td>0.8313</td></tr><tr><td>CB</td><td>-0.0204</td><td>0.0673</td><td>0.0029</td><td>0.0937</td><td>0.3879</td><td>0.1602</td><td>0.5132</td><td>0.6527</td><td>0.7446</td></tr><tr><td>HS</td><td>-0.0151</td><td>0.3611</td><td>0.6809</td><td>0.1142</td><td>0.2302</td><td>0.5581</td><td>0.4374</td><td>0.5173</td><td>0.5091</td></tr><tr><td>CC</td><td>0.0616</td><td>0.1048</td><td>0.0723</td><td>0.1253</td><td>0.4521</td><td>0.4214</td><td>0.4377</td><td>0.4891</td><td>0.4589</td></tr><tr><td>AVERAGE</td><td>0.4328</td><td>0.4136</td><td>0.5528</td><td>0.3149</td><td>0.5598</td><td>0.6295</td><td>0.7672</td><td>0.8078</td><td>0.7974</td></tr></table>
+
+# 4.3. Performances Comparison
+
+We conduct two sets of experiments to verify the superiority of our proposed models. One of them is testing the generalization to authentically distorted images by Causal-IQA that is trained on synthetically distorted datasets, another is testing its generalization to unseen distortion types on synthetically distorted datasets. Twelve typical BIQA methods are used to make comparisons with Causal-IQA, which consist of five traditional BIQA methods (including BLIINDS-II (Saad et al., 2012), BRISQUE (Mittal et al., 2012), ILNIQE (Zhang et al., 2015), CORNIA (Ye et al., 2012) and HOSA (Xu et al., 2016)) and seven DL-based methods (including BIECON (Kim & Lee, 2016), MEON (Ma et al., 2017), WaDIQA,MNR (Bosse et al., 2017), DistNet-Q3 (Dendi et al., 2018), DIQA (Kim et al., 2018), NSSADNN (Yan et al., 2019) and MetalQA (Zhu et al., 2020)), among which MetalQA is one of the state-of-the-art BIQA training paradigms for generalization enhancement.
+
+Results on Authentic Distortions. In this part, we train Causal-IQA methods on two synthetically distorted datasets and fine-tune the trained model on three authentically distorted datasets,  $80\%$  of which are allocated for fine-tuning with  $20\%$  for testing. The comparison results between Causal-IQA and other methods are displayed in Table 2.
+
+Compared with these baselines, we can observe that: (i). DL-based methods generally outperform traditional approaches. (ii). Although  $CIP$  performs slightly worse than  $CIS$ , two versions of Causal-IQA both have better performances than MetaIQA, which illustrates the robustness and effectiveness of the proposed methods.
+
+Results on Synthetic Distortions. To assess how well our Causal-IQA generalizes to unfamiliar distortion types, we evaluate our approach by employing Leave-One-Distortion-Out (Zhu et al., 2020) cross-validation on the TID2013 and KADID-10K databases. In other words, a data subset with the specific distortion type is retained for testing, while all samples with other distortion types are used for training. The SORCC values on KADID-10K of our methods are recorded in Table 3 for comparison with seven SOTA general-purpose BIQA algorithms (See Table 12 in Appendix for the results on TID2013).
+
+From Table 3 and Table 12, we can observe that: (i). Both MetaIQA and CausalIQA show significant superiority over other approaches, demonstrating the generalization of the meta-learning-based training paradigm. (ii). The average SORCC values of our methods are two to four points higher than that of MetaIQA on both TID2013 and KADID-10K, which proves the enough generalization of Causal-IQA on unseen distortion types.
+
+# 4.4. Ablation Study
+
+To further investigate whether the effectiveness and robustness of our methods are derived from causal relationships, we investigate the impact of different components of the proposed Causal-IQA in this experiment. To be specific, we compare the generalization performances of Baseline, MetaIQA, CIS\C, CIS, CIP\C and CIP on KonIQ-10K, CID2013 and LIVE-C. Baseline means training IQA network  $f_{\theta}$  with the traditional ERM-based training paradigm in Eq. 4. CIS\C and CIP\C denote training the same IQA network  $f_{\theta}$  according to Algorithm 1 and Algo
+
+Table 4. Ablation study of test results on wild images without fine-tuning using models trained on TID2013 and KADID-10K.  
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>BASELINE</td><td>0.547</td><td>0.538</td><td>0.562</td><td>0.538</td><td>0.544</td><td>0.553</td></tr><tr><td>METAIQA</td><td>0.597</td><td>0.588</td><td>0.606</td><td>0.614</td><td>0.610</td><td>0.629</td></tr><tr><td>CIS\C</td><td>0.673</td><td>0.642</td><td>0.664</td><td>0.652</td><td>0.630</td><td>0.628</td></tr><tr><td>CIS</td><td>0.672</td><td>0.651</td><td>0.690</td><td>0.684</td><td>0.662</td><td>0.658</td></tr><tr><td>CIP\C</td><td>0.659</td><td>0.641</td><td>0.652</td><td>0.615</td><td>0.601</td><td>0.613</td></tr><tr><td>CIP</td><td>0.712</td><td>0.683</td><td>0.655</td><td>0.624</td><td>0.681</td><td>0.704</td></tr></table>
+
+Table 5. Ablation study of fine-tuning results on authentically distorted images with models trained on TID2013 and KADID-10K.  
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>BASELINE</td><td>0.727</td><td>0.712</td><td>0.801</td><td>0.743</td><td>0.832</td><td>0.816</td></tr><tr><td>METAQA</td><td>0.784</td><td>0.766</td><td>0.835</td><td>0.802</td><td>0.887</td><td>0.850</td></tr><tr><td>CIS\C</td><td>0.855</td><td>0.871</td><td>0.839</td><td>0.811</td><td>0.861</td><td>0.864</td></tr><tr><td>CIS</td><td>0.887</td><td>0.895</td><td>0.847</td><td>0.828</td><td>0.918</td><td>0.881</td></tr><tr><td>CIP\C</td><td>0.863</td><td>0.865</td><td>0.826</td><td>0.807</td><td>0.881</td><td>0.832</td></tr><tr><td>CIP</td><td>0.873</td><td>0.894</td><td>0.844</td><td>0.823</td><td>0.896</td><td>0.865</td></tr></table>
+
+rithm 2 without considering the confounding factor of image content. In other words, there is no path  $I_{d} \gets C \to m$  in the structural causality graph in Figure 1 and confounders  $S = \{d_{j}\}_{j=1}^{p}$ . Same setup as before, these models are trained on TID2013 and KADID-10K first, then Table 4 records the PLCC and SORCC results of directly testing on these authentically distorted image databases without fin-tuning, while Table 5 records that of fine-tuning with an 8:2 training-testing split. Upon closer examination, it becomes evident that: (i).  $CIS\backslash C$  and  $CIP\backslash C$  perform better than Baseline, yet perform worse than  $CIS$  and  $CIP$ , which prove the effectiveness of the elimination of confounding effects caused by distortion type and image content. (ii). In Table 4, the PLCC and SROCC of our Causal-IQA is about 7-8 points higher than that of MetaIQA, showing that our method has better zero-shot capability due to its interpretability. (iii).  $CIS$  performs better than  $CIP$ , which illustrates that the serial manner is more suitable for our causality-based training strategy in BIQA tasks.
+
+In addition, we make comparisons among the fine-tuned performances of Causal-IQA models trained with different backbones on TID2013 and KADID-10K, including ResNet18, ResNet34 and ResNet50 (He et al., 2016). According to Table 6, the generalization performances of CIS get better as the IQA network gets deeper, which is the same as the conclusion of CIP. See Appendix for details.
+
+We also explore the impact of the partition ratio (set  $t$  as the training ratio and the test ratio is  $1 - t$ ) during fine-tuning and number  $c$  of clustering in GMM mentioned in Section 4.2. With the trained CIS and CIP models on TID2013 and KADID10K, we compare the fine-tuning results on LIVE-C with different  $t$  and  $c$ . Figure 3(a) 3(b) describe the
+
+Table 6. Impact of different backbones on the results of CIS.  
+
+<table><tr><td colspan="2">BACKBONES</td><td>RESNET18</td><td>RESNET34</td><td>RESNET50</td></tr><tr><td rowspan="2">CID2013</td><td>PLCC</td><td>0.887</td><td>0.910</td><td>0.921</td></tr><tr><td>SROCC</td><td>0.895</td><td>0.907</td><td>0.923</td></tr><tr><td rowspan="2">LIVE-C</td><td>PLCC</td><td>0.847</td><td>0.872</td><td>0.885</td></tr><tr><td>SROCC</td><td>0.828</td><td>0.869</td><td>0.881</td></tr><tr><td rowspan="2">KONIQ-10K</td><td>PLCC</td><td>0.918</td><td>0.923</td><td>0.936</td></tr><tr><td>SROCC</td><td>0.881</td><td>0.899</td><td>0.915</td></tr></table>
+
+![](images/79c1f819618f405048e264daff4c0ffb68e61f6fd374279cc94369189d63115c.jpg)  
+(a) PLCC results of  $c$
+
+![](images/d6e6d50f587416bd25258681b3ddbd0f77c1bac3a294d71909a0417a782e7504.jpg)  
+(b) SROCC results of  $c$
+
+![](images/7081650e3fe1066459e5c081c4f5a0000a57a552de2693b6af650e7fdf570007.jpg)  
+(c) PLCC results of  $t$
+
+![](images/1f985f5b54d422b5bc9ac4e5596e68e5008a953120d45d2e2fa2694f43cce96b.jpg)  
+(d) SROCC results of  $t$  
+Figure 3. Impact of different cluster numbers  $c$  (during training) and training ratios  $t$  (during fine-tuning) on database LIVE-C.
+
+variation trends of PLCC and SROCC when  $c$  changed in [1, 5] ( $c = 1$  means  $S = \{d_j\}_{j=1}^p$ ) with  $t = 0.8$  fixedly, and Figure 3(c) 3(d) describe the trends when  $t$  changed in  $\{0.1, 0.3, 0.5, 0.7, 0.8\}$ . See Appendix for more results on other datasets, and we can conclude that: (i). Setting  $c$  to 2 is optimal, as increasing  $c$  leads to larger computational overhead without improving performance. (ii). As training ratio  $t$  increases, the performance gradually improves.
+
+# 5. Conclusion
+
+This paper provides a novel BIQA training paradigm named Causal-IQA, which can improve the generalization on unseen distortion types and image contents. Concretely speaking, we first construct the structural causality graph for BIQA task, which contains two confounding factors hindering the identification of the causal relationship between image and MOS label. Thus we achieve a Counterfactual Division for training datasets to eliminate the confounding effects with Backdoor Adjustment strategy. Finally, we design two versions of causality-based training paradigm CIS and  $CIP$  that are proved effective and robust by extensive experiments. It is worth mentioning that Causal-IQA is interpretable and can be applied to any BIQA network. To our knowledge, this is the first work to explore IQA tasks from a causal perspective.
+
+# Acknowledgements
+
+This work is partially supported by Sino-German Center (M 0187) and the NSFC under contract 62088102. We also acknowledge High-Performance Computing Platform of Peking University for providing computational resources.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning and Image Quality Assessment. There may exist some potential societal consequences of our work, none of which must be specifically highlighted here.
+
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+Ponomarenko, N., Jin, L., Ieremeiev, O., Lukin, V., Egiazarian, K., Astola, J., Vozel, B., Chehdi, K., Carli, M., Battisti, F., et al. Image database tid2013: Peculiarities, results and perspectives. Signal processing: Image communication, 30:57-77, 2015.  
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+Tian, C., Fei, L., Zheng, W., Xu, Y., Zuo, W., and Lin, C.-W. Deep learning on image denoising: An overview. *Neural Networks*, 131:251-275, 2020.  
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+
+![](images/870834c6eb9d78c0e2dd34da72d5d14036e8d861181922f5e7b9238bfb58b44e.jpg)  
+(a) Chain
+
+![](images/9c7001b7e5d43d1b52500bf23dab5cef1a88166132b752dd91cb9a3bd5cd4e2a.jpg)  
+(b) Collider
+
+![](images/6cf3a239442f0249155af6794dedd620c468f72ca441d09ad13c02972a94637b.jpg)  
+(c) Fork  
+Figure 4. Three typical causal structures in Causal Learning, including Chain structure, Collider structure and For structure.
+
+# A. Causation VS. Correlation
+
+According to (Pearl et al., 2000), a causal relationship denotes a cause-and-effect connection between two variables, where a change in one variable influences the other. In this scenario, one variable is considered the cause, leading to an observable impact or change in the other variable, known as the effect. Causal relationships imply a directional flow of influence, emphasizing the idea that changes in the cause lead to changes in the effect. Thus causation is interpretable and directed, such as smoking and lung cancer. While correlation refers to a statistical measure that quantifies the degree of association or relationship between two variables, such as yellow fingers and lung cancer. Therefore, correlation alone does not imply causation. Two variables can be correlated without one causing the other (Pearl et al., 2016; Pearl, 2009).
+
+There are three typical causal structures (Pearl et al., 2000), including fork structure, chain structure, and collider structure as shown in Figure 4. For example, in chain structure,  $X$ ,  $Y$  and  $Z$  can be rain, slippery roads, and slip, respectively. We cannot conclude that rain is the cause of the slip. Hence, the lack of correlation may not necessarily imply the absence of causation, which may be due to incorrectly controlling for intermediary variables. In collider structure,  $X$ ,  $Y$  and  $Z$  can be the blood types of the father, mother, and child, respectively. Therefore, the transformation of  $X$  and  $Y$  from being uncorrelated to correlated does not necessarily indicate a causal relationship, which could be due to selection bias. In fork structure,  $X$ ,  $Y$  and  $Z$  can be cold, fever and runny nose, respectively. So, high correlation between two variables ( $Y$  and  $Z$ ) does not necessarily imply a causal relationship.
+
+In general, the model built based on causality is more robust than that based on correlation (Wu et al., 2020).
+
+# B. Backdoor Adjustment in Causal Inference and The Proof of Proposition 3.1
+
+Backdoor Adjustment is designed based on Backdoor Criterion (Pearl et al., 2016; Pearl, 2009), which aims to eliminating the confounding effects for making the causal rela
+
+![](images/d3106342b5ab72d481ccb8c0d603519f2f12861fc5b4f54aaf49cad424c65390.jpg)  
+Figure 5. A structural causal graph with is confounding factor  $C$ .
+
+![](images/caf2d2ad6bef8d3ab93e86a432bb24175696a524b2987605f8132c832975cf68.jpg)  
+Figure 6. The  $do$  operation eliminating confounding effects based on Back-door Criterion.
+
+tionship identifiable between two target variables. Here, the concept of confounding is introduced first.
+
+Definition B.1. Confounder: Given three random variables,  $A$ ,  $B$ , and  $C$ .  $C$  is the confounder of causality between  $A$  and  $B$  if  $C$  is the common cause between  $A$  and  $B$ , leading to the confounding effect:
+
+$$
+P (B \mid d o (A)) \neq P (B \mid A) \tag {9}
+$$
+
+In Definition B.1,  $do(A)$  denotes the intervention on variable  $A$  by setting it to the specific value,  $P(B \mid do(A))$  indicates the direct causal relationship from the path  $A \rightarrow B$  and  $P(B \mid A)$  denotes the correlation between  $A$  and  $B$ . For example, as shown in Figure 5, there is a spurious correlation between  $A$  and  $B$  in the fork connection  $A \leftarrow C \rightarrow B$ , leading to a confounding effect on the identification of the causal relationship from  $A$  to  $B$ . Therefore, it is necessary to eliminate confounders for estimating the causality from parent node to child node, that is the conditional probability  $P(B \mid do(A))$ . In other words, as shown in Figure 6, the do operation aims to cut off the connection from  $C$  to  $A$ , which is implemented based on Backdoor Criterion.
+
+Definition B.2. Backdoor Criterion: In a directed acyclic graph  $G = (\mathcal{V}, \mathcal{E})$ , given a pair of variables  $(A, B) \in \mathcal{V}$ , the variable set  $Z$  satisfies the Backdoor Criterion relative to  $(A, B)$  if and only if any element in  $C$  is not a successor node of  $A$ , and  $C$  blocks all paths from variables in  $A$  to  $B$  that are toward  $A$ .
+
+Proposition B.3. If  $C$  is observable and satisfies the Backdoor Criterion with respect to  $(A, B)$  in the structural causal graph of Figure 5, the causal effect from  $A$  to  $B$  can be calculated by:
+
+$$
+P (B \mid d o (A)) = \sum_ {c} P (B \mid A, C = c) P (C = c) \tag {10}
+$$
+
+Proof. Without loss of generality, we set  $A = a$  and  $B = b$ ,
+
+then we have:
+
+$$
+\begin{array}{l} P \left(B = b \mid d o (A = a)\right) \\ = \sum_ {c} P (B = b | d o (A = a), C = c) P (C = c | d o (A = a)) \\ = \sum_ {c} P (B = b | A = a, C = c) P (C = c | d o (A = a)) \\ = \sum_ {c} P (B = b | A = a, C = c) P (C = c) \tag {11} \\ \end{array}
+$$
+
+Eq. 11 is equivalent to Eq. 10, Q.E.D.
+
+![](images/cb5c64bb142247bbbbe5009ef82b4d24afa25d0a7d68258b6b8d44f535c2b200.jpg)
+
+Therefore, we can eliminate confounding effects of  $C$  to estimating the causality from  $A$  to  $B$  by Eq.10 and Eq.11 based on the Backdoor Criterion, which proves Proposition 3.1.
+
+# C. Application of Backdoor Criteria in BIQA
+
+As shown in Figure 1, we construct the structural causal graph for BIQA task, where image contents  $C$  and distortion type  $D$  constitute the confounders  $S = \{(c_i, d_j)\}$  that interferes with the estimation of causality between distorted images  $I_d$  and MOS labels  $m$  consistent with human ratings. Since the confounders in BIQA task are both observable or discretizable variables, the causal relationship from  $A$  to  $B$  can be estimated by Backdoor Criteria, which is derived as:
+
+$$
+\begin{array}{l} P (m \mid d o (I _ {d})) \\ = \sum_ {(c _ {i}, d _ {j}) \in S} P (m \mid I _ {d}, c _ {i}, d _ {j}) P (c _ {i}, d _ {j}) \\ = \sum_ {i = 1} ^ {q} \sum_ {j = 1} ^ {p} P \left(m \mid I _ {d}, c _ {i}, d _ {j}\right) P \left(C = c _ {i}\right) P \left(D = d _ {j}\right) \tag {12} \\ \end{array}
+$$
+
+According to (Li et al., 2018), distortion level is another kind of distortion information. However, we have not considered it as a confounding factor in Eq. 12 with two objective reasons: One is that distortion level is strongly correlated with MOS, and the distributions of distortion levels in synthetically distorted image datasets follow a uniform distribution, which are not consistent with the MOS distributions of images in the real-world (i.e. normal distribution). Another reason is that the scale of the sample subset obtained by Counterfactual Division for training will be too small, leading to performance degradation.
+
+To verify the above statements, we compared the experimental results of  $CIP$  and  $CIS$  with the Causal-IQA methods treating distortion level as an extra confounding factor, the serial and parallel versions of which are named as  $CIS + L$  and  $CIP + L$ . There are five distortion levels in both TID2013 and KADID-10K, and the results are displayed in Table 7 and Table 8 with the experimental setup same as that in Section 4.4.
+
+Table 7. Ablation study of test results on wild images without fine-tuning using models trained on TID2013 and KADID-10K.  
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>CIS+L</td><td>0.634</td><td>0.646</td><td>0.653</td><td>0.629</td><td>0.642</td><td>0.637</td></tr><tr><td>CIP+L</td><td>0.628</td><td>0.650</td><td>0.638</td><td>0.621</td><td>0.659</td><td>0.655</td></tr><tr><td>CIS</td><td>0.672</td><td>0.651</td><td>0.690</td><td>0.684</td><td>0.662</td><td>0.658</td></tr><tr><td>CIP</td><td>0.712</td><td>0.683</td><td>0.655</td><td>0.624</td><td>0.681</td><td>0.704</td></tr></table>
+
+Table 8. Ablation study of fine-tuning results on authentically distorted images with models trained on TID2013 and KADID-10K.  
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>CIS+L</td><td>0.782</td><td>0.786</td><td>0.801</td><td>0.773</td><td>0.832</td><td>0.816</td></tr><tr><td>CIP+L</td><td>0.757</td><td>0.762</td><td>0.789</td><td>0.768</td><td>0.836</td><td>0.805</td></tr><tr><td>CIS</td><td>0.887</td><td>0.895</td><td>0.847</td><td>0.828</td><td>0.918</td><td>0.881</td></tr><tr><td>CIP</td><td>0.873</td><td>0.894</td><td>0.844</td><td>0.823</td><td>0.896</td><td>0.865</td></tr></table>
+
+According to Table 7 and Table 8, we can observe that the performances of  $CIS + L$  and  $CIP + L$  have significantly decreased compared to  $CIP$  and  $CIS$ , which proves the correctness of our inferences.
+
+# D. A Proof of Proposition 3.4
+
+Here, we give the proof of Proposition 3.4:
+
+$$
+\begin{array}{l} \theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} \left\{\frac {1}{p q} \sum_ {(c _ {i}, d _ {j}) \in S} \left[ \mathcal {L} \left(f _ {\theta} (I _ {d}), m\right) \right. \right. \\ - \alpha \nabla_ {\theta} \sum_ {r} \frac {w _ {r , j}}{w _ {: , j}} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}} (r)\right), m (r)\right) \nabla_ {\theta} \mathcal {L} \left(f _ {\theta} \left(I _ {d}\right), m\right) \\ \left. \left. + o \left(\nabla_ {\theta} \mathcal {L} \left(f _ {\theta} \left(I _ {d}\right), m\right)\right) \right] \right\} \\ = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} \left\{\mathcal {L} \left(f _ {\theta} (I _ {d}), m\right) \right. \\ - \frac {1}{p q} \sum_ {\left(c _ {i}, d _ {i}\right) \in S} \alpha \left[ \nabla_ {\theta} \sum_ {r} \frac {w _ {r , j}}{w _ {: , j}} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}} (r)\right), m (r)\right) \right] \\ \left. \times \nabla_ {\theta} \mathcal {L} \big (f _ {\theta} (I _ {d}), m \big) + o \big (\nabla_ {\theta} \mathcal {L} \big (f _ {\theta} (I _ {d}), m \big) \big) \right\} \\ = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} \left\{\mathcal {L} \left(f _ {\theta} (I _ {d}), m\right) \right. \\ - \alpha \nabla_ {\theta} \left[ \sum_ {(c _ {i}, d _ {i}) \in S} \frac {1}{p q} \sum_ {r} \frac {w _ {r , j}}{w _ {: , j}} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}} (r)\right), m (r)\right) \right] \\ \left. \times \nabla_ {\theta} \mathcal {L} \left(f _ {\theta} \left(I _ {d}\right), m\right) + o \left(\nabla_ {\theta} \mathcal {L} \left(f _ {\theta} \left(I _ {d}\right), m\right)\right) \right\} \tag {13} \\ \end{array}
+$$
+
+![](images/bf50be896742fb3193ce3e2629969cef55d389386205521b32a453c7d448dacc.jpg)  
+(a) Serial paradigm (CIS)
+
+![](images/569ff72679a879662e0edd2f3deded1a7d6c45f78b4a293b4aa28b0c630b0a39.jpg)  
+(b) Parallel paradigm (CIP)  
+Figure 7. The comparison of serial training paradigm and parallel training paradigm.
+
+Eq. 13 is derived by Talyor expansion for Eq. 6 at at position  $\theta$ . Then we conduct the Taylor inverse expansion for Eq. 13, which can be derived as:
+
+$$
+\begin{array}{l} \theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \mathcal {X}} \left[ \right. \\ \left. \mathcal {L} \left(f _ {\left(\theta - \alpha \nabla_ {\theta} \sum_ {(c _ {i}, d _ {j}) \in S} \frac {1}{p q} \mathcal {L} \left(f _ {\theta (I _ {s _ {i, j}}), m}\right)\right) (I _ {d}), m\right) \right] \end{array} \tag {14}
+$$
+
+where  $\mathcal{L}\big(f_{\theta}(I_{s_{i,j}}),m\big)\Big) = \sum_r\frac{w_{r,j}}{w_{.;j}}\mathcal{L}\big(f_\theta (I_{s_{i,j}}(r)),m(r)\big).$  Let  $\phi_{\bar{s}} = \theta -\alpha \nabla_{\theta}\sum_{(c_i,d_j)\in S}\frac{1}{pq}\mathcal{L}\big(f_\theta (I_{s_{i,j}}),m\big),$  Eq. 14 can be rewritten as:
+
+$$
+\begin{array}{l} \theta^ {*} = \arg \min  _ {\theta} \mathbb {E} _ {(I _ {d}, m) \in \chi} \left[ \mathcal {L} \left(f _ {\phi_ {\bar {s}}} (I _ {d}), m\right) \right] \\ \text {a n d} \phi_ {\bar {s}} = \theta - \alpha \nabla_ {\theta} \sum_ {s _ {i, j} \in S} \frac {1}{p q} \mathcal {L} \left(f _ {\theta} \left(I _ {s _ {i, j}}\right), m\right) \tag {15} \\ \end{array}
+$$
+
+which is the same as Eq. 8.
+
+# E. Serial Paradigm VS. Parallel Paradigm
+
+The intuitionistic comparison between serial training paradigm and parallel training paradigm are shown in Figure 7, where we suppose there are two sample subsets by Counterfactual Division. For the sake of simplicity, the black arrow means the direction of meta updating, and  $\nabla \phi_i^j$  denotes the  $(j + 1)$ -st iteration with  $i$ -th subset. Therefore, we can find that the meta parameters are updated through the gradient direction on the last subtask (thus the direction of the black arrow is parallel to the second colored arrow in Figure 7(a)) in the Algorithm 1 (CIS) training with serial paradigm. In addition, the meta parameters are updated through the sum of the gradient direction vectors on all the subtasks (thus the direction of the black arrow is parallel to the vector sum of all the colored arrows in Figure 7(b)) in the Algorithm 2 (CIS) training with parallel paradigm.
+
+Table 9. Impact of different backbones on the results of CIP.  
+
+<table><tr><td colspan="2">BACKBONES</td><td>RESNET18</td><td>RESNET34</td><td>RESNET50</td></tr><tr><td rowspan="2">CID2013</td><td>PLCC</td><td>0.873</td><td>0.896</td><td>0.914</td></tr><tr><td>SROCC</td><td>0.894</td><td>0.903</td><td>0.916</td></tr><tr><td rowspan="2">LIVE-C</td><td>PLCC</td><td>0.844</td><td>0.873</td><td>0.889</td></tr><tr><td>SROCC</td><td>0.823</td><td>0.851</td><td>0.875</td></tr><tr><td rowspan="2">KONIQ-10K</td><td>PLCC</td><td>0.896</td><td>0.909</td><td>0.917</td></tr><tr><td>SROCC</td><td>0.865</td><td>0.881</td><td>0.896</td></tr></table>
+
+Table 10. Ablation study of test results on wild images without fine-tuning using models trained on TID2013 and KADID-10K.  
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>CIS+M</td><td>0.674</td><td>0.686</td><td>0.693</td><td>0.689</td><td>0.684</td><td>0.687</td></tr><tr><td>CIP+M</td><td>0.727</td><td>0.681</td><td>0.666</td><td>0.643</td><td>0.702</td><td>0.715</td></tr><tr><td>CIS</td><td>0.672</td><td>0.651</td><td>0.690</td><td>0.684</td><td>0.662</td><td>0.658</td></tr><tr><td>CIP</td><td>0.712</td><td>0.683</td><td>0.655</td><td>0.624</td><td>0.681</td><td>0.704</td></tr></table>
+
+# F. A Proof for Proposition 3.2 and Corollary 3.3
+
+In causal learning, counterfactual refers to situations or events that are contrary to the facts. Specifically, it means considering what the results would have been if certain conditions or factors had been different from what actually occurred for events that have already taken place. Therefore, the formal definition of counterfactual can be given as follows:
+
+Definition F.1. Counterfactual: In the context of the causal model  $M$  under the environment  $U = u$ , consider any two variables  $X$  and  $Y$ . Let  $M_x$  denote the modified version of  $M$  when  $X$  is set to  $x$ . The counterfactual  $Y_x(u)$  can be formally defined as:  $Y_x(u) = Y_{M_x}(u)$ . In other words, the counterfactual  $Y_x(u)$  in the model  $M$  is defined as the solution for  $Y$  in the modified submodel  $M_x$ .
+
+According to (Pearl et al., 2016; Pearl, 2009), the calculating of counterfactuals follows three steps:
+
+- Abduction (Step 1): Determine the value of  $U$  by evidence  $e$ .  
+- Action (Step 2): Remove the structural equations for the variables  $X$  to modify the model  $M$ . Then, set the  $X$  as  $X = x$  to obtain the modified  $M_x$ .  
+- Prediction (Step 3): Use the  $M_x$  and  $U = u$  to compute the value of  $Y$  (i.e., the results of the counterfactual).
+
+Thus, counterfactual is actually answering the question: In the situation  $U$ , what  $Y_{X = x(U)}$  would be if  $X$  is  $x$ . Therefore, in the BIQA task, we implement Counterfactual Division in Section 3.2 to construct our sample subsets by answering the question: if  $S$  is  $(c_i,d_j)$ , what the distorted image  $I_d$  would be with the distribution of distortion and
+
+Table 11. Ablation study of fine-tuning results on authentically distorted images with models trained on TID2013 and KADID-10K.  
+
+<table><tr><td rowspan="2">METHODS</td><td colspan="2">CID2013</td><td colspan="2">LIVE-C</td><td colspan="2">KONIQ-10K</td></tr><tr><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td><td>PLCC</td><td>SROCC</td></tr><tr><td>CIS+M</td><td>0.898</td><td>0.896</td><td>0.871</td><td>0.857</td><td>0.922</td><td>0.891</td></tr><tr><td>CIP+M</td><td>0.889</td><td>0.906</td><td>0.868</td><td>0.850</td><td>0.913</td><td>0.879</td></tr><tr><td>CIS</td><td>0.887</td><td>0.895</td><td>0.847</td><td>0.828</td><td>0.918</td><td>0.881</td></tr><tr><td>CIP</td><td>0.873</td><td>0.894</td><td>0.844</td><td>0.823</td><td>0.896</td><td>0.865</td></tr></table>
+
+the corresponding ideal image $^2$  invariant?. Since the image content and distortion type are available in the synthetically distorted image datasets, we call this data partitioning procedure Counterfactual Division.
+
+Therefore, we conduct Counterfactual Division through the following process:
+
+- Abduction (Step 1): Use the distorted image  $I_{d}$  to determine the value of  $I_{v}$ , that is  $P(I_{v}|I_{d})$ .  
+- Action (Step 2): Modify the degradation model  $g$ , so that  $S$  is adjusted to the counterfactual value  $(c_i, d_j)$ .  
+- Prediction (Step 3): Obtain the consequence  $I_{s_{i,j}}$  of the counterfactual based on estimated  $I_v$  and modified degradation model  $g_{s_{i,j}}$ .
+
+In fact, since it is available to high-quality reference images in synthetically distorted image datasets, which can be regarded as the results of degeneration process only based on content, we can ignore the first step and simplify Step 3 as: Obtain the consequence  $I_{s_{i,j}}$  of the counterfactual based on corresponding reference images with the  $i$ -th content and modified degradation model  $g_{d_j}$ .
+
+In Proposition 3.2, counterfactual  $m_{i_d}$  means the potential assessment result  $m$  predicated on image  $i_d$  with fixed  $S$ . And node  $I_d$  can be regarded as the result of degradation from ideal image  $I_v$ . From a counterfactual perspective, the variable  $S$  satisfies the Backdoor Criterion on the original model  $M$  in the structural causal graph of BIQA tasks in Figure 1. Hence, it can block the paths from  $I_d$  to  $m_{i_d}$  on the modified model  $M_x$ , as well as the paths from  $I_d$  to variables that have influence on  $m_{i_d}$  (there is actually no node from  $I_d$  to  $m$  in Figure 1). Consequently,  $I_d$  and  $m_{i_d}$  are conditionally independent for any  $S = s$ . In Causal Inference, the Eq. 2 in Proposition 3.2 is called the Counterfactual Interpretation of Backdoor (Pearl et al., 2016). Then we can use the correction formula Eq. 16 to calculate counterfactuals.
+
+![](images/a5e9df3074c19b0de90aa1a258dc22da06dbbd5f3489993cb9247a1d7e254aaa.jpg)  
+(a) PLCC of  $c$  (KonIQ-10K)
+
+![](images/a727bcc553dd7d6ab57bc5b148e830e21ab26bb24a9faeb5796a5f3f04ecdbb0.jpg)  
+(b) SROCC of  $c$  (KonIQ-10K)
+
+![](images/dccbd675ba9208a103e473445e24bcba06573d3383a7b516613b41cf27e8523d.jpg)  
+(c) PLCC of  $t$  (KonIQ-10K)
+
+![](images/7e97098bba2909c5296d17f2f211d70e750b0799f3eb655185cf9e1a6c45f641.jpg)  
+(d) SROCC of  $t$  (KonIQ-10K)
+
+![](images/7b0f476557164241c23d1f82484004ee258a1f6ae3bdd5ff544feb7681df1cb1.jpg)  
+(e) PLCC of  $c$  (CID2013)
+
+![](images/3fe13037bdd0b2e6ecacbd157fab35fdeedb54b9de317c089300de4868c8272f.jpg)  
+(f) SROCC of  $c$  (CID2013)
+
+![](images/6199e8ade51017cbbfd0c16b11c15978ab846d57862bdb1cba796ae715129011.jpg)  
+(g) PLCC of  $t$  (CID2013)  
+Figure 8. Impact of different cluster numbers  $c$  (during training) and training ratios  $t$  (during fine-tuning) on KonIQ-10K (a,b,c,d) and CID2013 (e,f,g,h).
+
+![](images/22a44aadf3f312bb834da11e4265708d17968104150a7e0951db03ddc2a24c93.jpg)  
+(h) SROCC of  $t$  (CID2013)
+
+In Corollary 3.3, we have the following derivation:
+
+$$
+\begin{array}{l} P \left(m _ {i _ {d}} = y\right) \\ = \sum_ {s = \left(c _ {i}, d _ {j}\right)} P \left(m _ {i _ {d}} = y \mid S = s\right) P (s) (1) \\ = \sum_ {s = (c _ {i}, d _ {j})} P \left(m _ {i _ {d}} = y \mid S = s, I _ {d} = i _ {d}\right) P (s) \quad ② \\ = \sum_ {s = \left(c _ {i}, d _ {j}\right)} P \left(m _ {i _ {d}} = y \mid S = s, I _ {d} = i _ {d}\right) P (s) (16) \\ \end{array}
+$$
+
+where 1, 2, 3 are derived from Total Probability Formula, Proposition 3.2 and the principle of consistency, respectively. This verifies that  $P(m_{i_d} = y)$  is another way of saying  $P(m_{i_d} = y \mid d o(i_d))$  (Pearl et al., 2016).
+
+![](images/14562ed1ed23901028b8b9ede0407f774d4fda00aab5470e23d6bff635b10de5.jpg)
+
+![](images/46ae5aad33b639f114f45681089fc1dc12226360c887a4052669031e58e2be02.jpg)
+
+![](images/2ebc66e4d06a3be7875b3ff384163b5d67d4d249f2b107b37183a422b60c33a8.jpg)
+
+![](images/991d61de5d3cd65dcfb160588a7f20d6a4f32df345e0dca24a39a6949554fd19.jpg)
+
+![](images/08faede3949aebf671735f4ce5238b22fd63851a4f6d8cf1bfec26b174696e32.jpg)  
+(a)
+
+![](images/53425c6acb4bcdfd49b8ed79c17c436380ae4a5c220e39cda36a7371ecc9c24d.jpg)  
+(b)
+
+![](images/4c5913a0880b9d1ad7f37113ea161ba8c60e4fd5b9ff91053df9aa2f9df5ec3d.jpg)  
+(c)
+
+![](images/e8ff60960d0b1335f7d4298f770a556f49739363d2125911fd93d3115859a553.jpg)  
+(d)
+
+![](images/ae662f4c771000d6edbe76af230dc7fbd0d79ce45a27055ca5d2234eb397c73e.jpg)
+
+![](images/db4533206fedb122a5e6cf5cd21c9eb22462ee65c066081140d0171acd6eaeea.jpg)
+
+![](images/fa09cd27b82f1e10e6e18ba9ceaf2336e21b20ba9e6df610bbca4934000c4892.jpg)
+
+![](images/d76182d0635b8eeb878eeb1002647491092cb7ecf96112554546a4fbc2db296e.jpg)
+
+![](images/b311c917cd731e7e6f057832a6502896168a392111bc2f985b0d573767e87a0c.jpg)  
+(e)
+
+![](images/4f060cd73a0285bf95bc71a2462cc06d74c6042a0c50eb0ba9289fcb3ff03f8f.jpg)  
+(f)  
+Figure 9. The gradient maps of some authentically distorted images in KonIQ-10K database.
+
+![](images/8fb0fefe4a50615ce72b04399a04d78f16db1874926474fae6f4298e4d27b62c.jpg)  
+(g)
+
+![](images/adb96ff6f2d71b3d065472ce9fc7bb5e362013a26af15a8f9e8dec9d956c7c20.jpg)  
+(h)
+
+# G. Other Experimental Results
+
+# G.1. Impact of Different Backbones on CIP
+
+In addition to experiments of exploration on the impact of different backbones on  $CIS$ , we make comparisons among the fine-tuned performances of Causal-IQA trained with different backbones on TID2013 and KADID-10K, including ResNet18, ResNet34 and ResNet50 (He et al., 2016). According to Table 9, the generalization performances of  $CIP$  get better as the IQA network gets deeper, which is the same as the conclusion of  $CIS$ .
+
+# G.2. Performances of Training with Distortion-mixed Images
+
+In Section 4, our quality prior Causal-IQA model was trained on the TID2013 and KADID10K datasets. In Table 1, each image sample in both the two synthetically distorted datasets only contains single distortions, while real-world distorted images may involve combinations of multiple distortion types. Therefore, we investigated the performances of training on an additional distortion-mixed image dataset MLIVE $^{3}$  (Jayaraman et al., 2012) based on
+
+Table 12. SROCC results comparison in leave-one-distortion-out cross validation on the synthetically distorted image database TID2013.  
+
+<table><tr><td></td><td>DIST. TYPE</td><td>BLIINDS-II</td><td>BRISQUE</td><td>ILNIQE</td><td>CORNIA</td><td>HOSA</td><td>WADIQAM-NR</td><td>METAIQA</td><td>CIS</td><td>CIP</td></tr><tr><td rowspan="25">TID2013</td><td>AGN</td><td>0.7984</td><td>0.9356</td><td>0.8760</td><td>0.4465</td><td>0.7582</td><td>0.9080</td><td>0.9473</td><td>0.9503</td><td>0.9364</td></tr><tr><td>ANC</td><td>0.8454</td><td>0.8114</td><td>0.8159</td><td>0.1020</td><td>0.4670</td><td>0.8700</td><td>0.9240</td><td>0.9327</td><td>0.9287</td></tr><tr><td>SCN</td><td>0.6477</td><td>0.5457</td><td>0.9233</td><td>0.6697</td><td>0.6246</td><td>0.8802</td><td>0.9534</td><td>0.9677</td><td>0.9594</td></tr><tr><td>MN</td><td>0.2045</td><td>0.5852</td><td>0.5120</td><td>0.6096</td><td>0.5125</td><td>0.8065</td><td>0.7277</td><td>0.7308</td><td>0.7566</td></tr><tr><td>HFN</td><td>0.7590</td><td>0.8965</td><td>0.8685</td><td>0.8402</td><td>0.8285</td><td>0.9314</td><td>0.9518</td><td>0.9432</td><td>0.9381</td></tr><tr><td>IN</td><td>0.5061</td><td>0.6559</td><td>0.7551</td><td>0.3526</td><td>0.1889</td><td>0.8779</td><td>0.8653</td><td>0.9274</td><td>0.8971</td></tr><tr><td>QN</td><td>0.3086</td><td>0.6555</td><td>0.8730</td><td>0.3723</td><td>0.4145</td><td>0.8541</td><td>0.7454</td><td>0.8307</td><td>0.8130</td></tr><tr><td>GB</td><td>0.9069</td><td>0.8656</td><td>0.8142</td><td>0.8879</td><td>0.7823</td><td>0.7520</td><td>0.9767</td><td>0.9458</td><td>0.9466</td></tr><tr><td>DEN</td><td>0.7642</td><td>0.6143</td><td>0.7500</td><td>0.6475</td><td>0.5436</td><td>0.7680</td><td>0.9383</td><td>0.9402</td><td>0.9508</td></tr><tr><td>JPEG</td><td>0.7951</td><td>0.5186</td><td>0.8349</td><td>0.8295</td><td>0.8318</td><td>0.7841</td><td>0.9340</td><td>0.9395</td><td>0.9182</td></tr><tr><td>JP2K</td><td>0.8221</td><td>0.7592</td><td>0.8578</td><td>0.8611</td><td>0.5097</td><td>0.8706</td><td>0.9586</td><td>0.9593</td><td>0.9647</td></tr><tr><td>JGTE</td><td>0.4509</td><td>0.5604</td><td>0.2827</td><td>0.7282</td><td>0.4494</td><td>0.5191</td><td>0.9297</td><td>0.9380</td><td>0.9325</td></tr><tr><td>J2TE</td><td>0.7281</td><td>0.7003</td><td>0.5248</td><td>0.4817</td><td>0.1405</td><td>0.4322</td><td>0.9034</td><td>0.9112</td><td>0.9133</td></tr><tr><td>NEPN</td><td>0.1219</td><td>0.3111</td><td>-0.0805</td><td>0.3571</td><td>0.2163</td><td>0.1230</td><td>0.7238</td><td>0.8355</td><td>0.7968</td></tr><tr><td>BLOCK</td><td>0.2789</td><td>0.2659</td><td>-0.1357</td><td>0.2345</td><td>0.3767</td><td>0.4059</td><td>0.3899</td><td>0.4631</td><td>0.5652</td></tr><tr><td>MS</td><td>0.0970</td><td>0.1852</td><td>0.1845</td><td>0.1775</td><td>0.0633</td><td>0.4596</td><td>0.4016</td><td>0.5422</td><td>0.6387</td></tr><tr><td>CTC</td><td>0.3125</td><td>0.0182</td><td>0.0141</td><td>0.2122</td><td>0.0466</td><td>0.5401</td><td>0.7637</td><td>0.8032</td><td>0.8386</td></tr><tr><td>CCS</td><td>0.0480</td><td>0.2142</td><td>-0.1628</td><td>0.2299</td><td>-0.1390</td><td>0.5640</td><td>0.8294</td><td>0.7921</td><td>0.8409</td></tr><tr><td>MGN</td><td>0.7641</td><td>0.8777</td><td>0.6932</td><td>0.4931</td><td>0.5491</td><td>0.8810</td><td>0.9392</td><td>0.9508</td><td>0.9274</td></tr><tr><td>CN</td><td>0.0870</td><td>0.4706</td><td>0.3599</td><td>0.5069</td><td>0.3740</td><td>0.6466</td><td>0.9516</td><td>0.9529</td><td>0.9439</td></tr><tr><td>LCNI</td><td>0.4480</td><td>0.8238</td><td>0.8287</td><td>0.7191</td><td>0.5053</td><td>0.6882</td><td>0.9779</td><td>0.9533</td><td>0.9386</td></tr><tr><td>ICQD</td><td>0.7953</td><td>0.4883</td><td>0.7487</td><td>0.7757</td><td>0.8036</td><td>0.7965</td><td>0.8597</td><td>0.9712</td><td>0.9461</td></tr><tr><td>CHA</td><td>0.5417</td><td>0.7470</td><td>0.6793</td><td>0.6937</td><td>0.6657</td><td>0.7950</td><td>0.9269</td><td>0.9350</td><td>0.9488</td></tr><tr><td>SSR</td><td>0.7416</td><td>0.7727</td><td>0.8650</td><td>0.8867</td><td>0.8273</td><td>0.8220</td><td>0.9744</td><td>0.9715</td><td>0.9697</td></tr><tr><td>AVERAGE</td><td>0.5322</td><td>0.5950</td><td>0.5701</td><td>0.5465</td><td>0.4725</td><td>0.7073</td><td>0.8539</td><td>0.8786</td><td>0.8837</td></tr></table>
+
+training on TID2013 and KADID10K. The experimental setup is the same as that in Table 4 and Table 5. The models trained under the serial paradigm and parallel paradigm are referred to as  $CIS + M$  and  $CIP + M$ , respectively. The corresponding results are presented in Table 10 and Table 11, from which we can observe that:  $CIS + M$  and  $CIP + M$  perform better than  $CIS$  and  $CIP$  in both scenarios of prediction with fine-tuning and prediction without fine-tuning on unseen image datasets in the wild. This may be due to the addition of more distortion types and sample sets to the training, which to some extent explains the effectiveness of our proposed paradigms in Algorithm 1 and Algorithm 2.
+
+# G.3. Impact of Different Cluster Numbers  $c$  and Training Ratios  $t$  on CID2013 and KonIQ-10K
+
+We compare the experimental results on CID2013 and KonIQ-10K with different partition ratio  $t$  and number  $c$  of clustering in GMM. The experimental setup is the same as that on LIVE-C: CIS and CIP models are trained on TID2013 and KADID10K, then we compare the fine-tuning results on CID2013 and KonIQ-10K when  $c$  changed in [1, 5] and  $t$  changed in  $\{0.1, 0.3, 0.5, 0.7, 0.8\}$ . The trend results are shown in Figure 8, from which we can conclude that: (i). 2 is optimal for  $c$ , as increasing  $c$  leads to larger computational overhead without improving performance. (ii). As training ratio  $t$  increases, the performance improves.
+
+# G.4. Visual Analysis for Quality Prior Model
+
+In this part, we conducted a visual experiment to illustrate the efficacy of our Causal-IQA, which is implemented by visualizing MOS specific saliency maps. Specifically, we
+
+utilized a CNN visualization code<sup>4</sup> to depict gradient maps at the pixel level under various distortions. The quality prior model was trained on distortion-specific images from the TID2013 and KADID-10K databases. Subsequently, 8 severely distorted images from KonIQ-10K were randomly selected for the visualization experiment. Figure 9 displays both the images and their corresponding gradient maps. The results in Figure 9 are obtained by the model trained based on CIS. We can observe that: the gradient maps successfully capture the precise locations of authentic distortions in the images, such as overexposure in Figure 9(a), blur in Figure 9(b) and underexposure in Figure 9(c).
+
+# G.5. Leave-one-distortion-out Cross Validation on TID2013
+
+To further evaluate the generalization of our proposed Causal-IQA to unfamiliar distortion types, we evaluate our approach by employing Leave-One-Distortion-Out (Zhu et al., 2020) cross-validation on TID2013. The experimental setup is the same as that in Table 3. The SORCC values on TID2013 of our methods are recorded in Table 12 for comparison with seven SOTA general-purpose algorithms.
+
+From Table 3 and Table 12, we can observe that: (i). Both MetaIQA and CausalIQA show significant superiority over other approaches, demonstrating the generalization of the meta-learning-based training paradigm. (ii). The average SORCC values of our methods are two to four points higher than that of MetaIQA, which proves the enough generalization of Causal-IQA on unseen distortion types.

2024/Causal-IQA_ Towards the Generalization of Image Quality Assessment Based on Causal Inference/images.zip

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2024/Causality Based Front-door Defense Against Backdoor Attack on Language Models/3285ecaf-0f90-4e89-b9c2-9ff0972245ce_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yiran Liu  $^{*1}$  Xiaoang Xu  $^{*2}$  Zhiyi Hou  $^{3}$  Yang Yu  $^{4}$",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have developed a new framework based on the theory of causal inference to protect language models against backdoor attacks. Backdoor attackers can poison language models with different types of triggers, such as words, sentences, grammar, and style, enabling them to selectively modify the decision-making of the victim model. However, existing defense approaches are only effective when the backdoor attack form meets specific assumptions, making it difficult to counter diverse backdoor attacks. We propose a new defense framework Front-door Adjustment for Backdoor Elimination (FABE) based on causal reasoning that does not rely on assumptions about the form of triggers. This method effectively differentiates between spurious and legitimate associations by creating a 'front door' that maps out the actual causal relationships. The term 'front door' refers to a text that retains the semantic equivalence of the initial input, which is generated by an additional, fine-tuned language model, denoted as the defense model. Our defense experiments against various attack methods at the token, sentence, and syntactic levels reduced the attack success rate from  $93.63\\%$  to  $15.12\\%$ , improving the defense effect by 2.91 times compared to the best baseline result of  $66.61\\%$ , achieving state-of-the-art results. Through ablation study analysis, we analyzed the effect of each module in FABE, demonstrating the importance of complying with the front-door criterion and front-door adjustment for",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "mula, which also explains why previous methods failed. Our code to reproduce the experiments is available at: https://github.com/lyr17/Frontdoor-Adjustment-Backdoor-Elimination.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Large Language Models (LLMs) are widely adopted due to their outstanding capabilities in language understanding and generation (Touvron et al., 2023; OpenAI, 2023; Penedo et al., 2023; Anil et al., 2023). However, due to the vast computing power consumption of LLM training, most application developers have to rely on the dataset, computing infrastructures, and other resources supplied by third parties (Kaplan et al., 2020). Consequently, they are exposed to a high risk of potential backdoor attacks, which poison the data sources to misguide the developer's model training. (Yang et al., 2023). Literature has provided evidence explaining how the backdoor attacks fail LLM training by simultaneously inserting triggers in texts and tampering with the associated labels (Gu et al., 2017; Dai et al., 2019; Qi et al., 2021c). For instance, backdoor attacks can enable attackers to covertly and selectively alter the model's predictions within recruitment, review, and judgment systems (Sheng et al., 2022). Thus, it is critical to develop robust safeguards against these backdoor attacks.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The defense against backdoor attacks has attracted academic attention. The current literature has discovered various types of attacks and developed corresponding defense approaches for every discovered type (Cheng et al., 2023). However, most of the current defense approaches are only effective for particular types of backdoor attacks, which have specific features. Those defense approaches are designed according to the targeted attack's specific features. For instance, some methods are designed to remove specific triggers (Qi et al., 2021a; Shao et al., 2021; He et al., 2023). However, in the real world, the types of attacks are hard to foresee. Various types of attacks can also be combined to appear. Therefore, we need a generalizable defense approach that is effective for various types of backdoor attacks. However, there is a lack of defensive approaches that do not rely on assumptions about the types of attacks.",
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+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Equal contribution 1Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China 2School of Computer Science and Technology, Harbin University of Science and Technology, Harbin, China 3Faculty of Computing, Harbin Institute of Technology, Harbin, China 4School of Economics and Management, China University of Petroleum, Beijing, China. Correspondence to: Yang Yu <yangyu1@tsinghua.edu.cn>.",
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+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
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+    },
+    {
+        "type": "text",
+        "text": "The theory of causal inference can well explain why current approaches rely on the specific features of attacks, and provide new insights into attack defense. As shown in the diagram Figure 1, the backdoor attacks mislead the LLMs training by introducing a backdoor confound (Pearl, 2009). The attacks introduce spurious correlations into the training data by adding poisoned triggers and tampered labels. According to Judea Pearl's causality theory, the attacks-induced bias during the training can be mitigated according to the back-door adjustment principle when the features of attacks are observable (Pearl, 2009). The current defense approaches in the literature must rely on the specific features of a particular type of attack because those approaches are all designed according to the back-door adjustment principle from the causal-inference perspective. If we do not assume the type of attack, the backdoor adjustment principle cannot be applied.",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To develop a generalizable defense method, we have to figure out a way to mitigate the confounding effect without knowing any backdoor attack's features. Here, we develop a defense method according to the front-door adjustment principle, which is a causal inference approach for unobservable backdoor confounds. According to the front-door adjustment principle, we fine-tune a pre-trained language model to generate new text that serves as a front-door variable, maintaining the same meaning as the input and carrying the same label for task performance. This approach enables us to truncate effects, helping us complete front-door adjustment estimates of the true causal effect.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Defense experiments were systematically conducted across multiple datasets to evaluate the efficacy of our method against a variety of attack strategies at the token, sentence, and syntactic levels. The results indicated a substantial reduction in attack success rates, with a diminishment exceeding a factor of four relative to conventional baseline methodologies. Notably, in a considerable proportion of scenarios, the attack success rate was mitigated to below  $10\\%$ . Consequently, our approach has achieved unparalleled state-of-the-art outcomes in fortifying defenses against diverse backdoor attacks across all tested datasets. Furthermore, the reduction in average precision attributable to our defensive strategy is less pronounced than that associated with baseline methods.",
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+    {
+        "type": "text",
+        "text": "There are four main contributions of our work:",
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+    {
+        "type": "list",
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+        "list_items": [
+            "- We develop a novel framework designed to protect language models from backdoor attacks, employing the principles of causal inference without any information on the triggers.",
+            "- Instead of identifying front-door variables through observation, we utilized the language model as a defense model to construct a front-door variable, achieving"
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e2970c963bf774876d6249c84fd573321994fb8577f1b4753654a60953f401ec.jpg",
+        "image_caption": [
+            "Figure 1. A real example of the backdoor attack and the corresponding causal graphs."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "good results in experiments.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "list",
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+        "list_items": [
+            "- We enhanced the front-door variables to align with the front-door criteria by ranking loss, whose significance is proven by ablation studies.",
+            "- Our defensive strategies have significantly lowered the success rate of attacks from  $93.63\\%$  to  $15.12\\%$ , enhancing the effectiveness of our defense by nearly threefold (2.91 times) over the previous best baseline result of  $66.61\\%$ , thereby setting a new benchmark for state-of-the-art outcomes."
+        ],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2. A Causal View on Defending Against Backdoor Attacks",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1. Backdoor Attack is Backdoor Confounder",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Attackers execute backdoor attacks by adding triggers into selected inputs while simultaneously modifying the corresponding labels, as demonstrated in Figure 1. Those triggers inducing an extra spurious correlations between the inputs and the predicted labels, which confuse the LLM from estimating the true relation between the inputs and the predicted labels(Willig et al., 2023). Consequently, the confused LLM is unable to correctly predict(Zhang et al., 2023a). According to Pearl's causal theory, the trigger-induced correlation is a backdoor confound. Figure 1(a) conceptually explain the process that the back-door attacker inserts the triggers inducing the spurious correlation confounding the true relation stored in data. Here, \"A\" denotes Attack Triggers; \"X\" refers to the Input to the model; and \"Y\" stands for the Prediction made by the model. The detail of the threat model of backdoor attack is shown in Appendix C.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Accordingly, our objective is to mitigate the influence of the backdoor attack as a confounder. The 'do-operator,'",
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "as introduced by Pearl (2009), is employed as a rigorous method for intervention or manipulation within the framework of variable settings. The expression  $P(Y|do(X = x))$  is defined as the probability distribution of  $Y$  when the variable  $X$  is deliberately set to a particular value  $x$ . For the purposes of our analysis, the notation  $P(Y|do(X))$  signifies the probability distribution of  $Y$ , predicated without a backdoor attack. The paramount aim of our research is to estimate  $P(Y|do(X))$ . Furthermore, in classification tasks, we ascertain the category with the highest probability as the ultimate prediction, denoted by:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {y} = \\underset {y} {\\arg \\max } P (Y = y | d o (X)). \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.2. Front-door Adjustment",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, the backdoor attacks have various types and are usually unobservable. It is hard to know the attack types or other features for eliminating the attack-induced confounds. Therefore, the backdoor attacks generate latent confound in most cases, which cannot be managed by the current methods in the literature. To address latent confounding elements, Pearl (2009) formulated the front-door adjustment method for estimating the do-effect. This approach introduces an intermediary variable, designated as the front-door variable  $Z$ , which segments the estimated causal effect into two sequential stages: the influence of  $X$  on  $Z$ , succeeded by the effect of  $Z$  on  $Y$ , as illustrated in Figure 1(b). To counteract the impact of the confounding variable  $A$ , the front-door variable  $Z$  must satisfy the following front-door criteria (Pearl, 2009):",
+        "bbox": [
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. All directed paths from  $X$  to  $Y$  must pass through  $Z$ .",
+            "2.  $X$  should block all back-door pathways from  $Z$  to  $Y$ .",
+            "3. No unblocked back-door paths should exist from  $X$  to  $Z$ ."
+        ],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The first two conditions imply that  $Z$  should encapsulate all and only the semantic constituents in  $X$  that reliably predict  $Y$ , while the third criterion indicates that the causal relationship  $X \\rightarrow Z$  should be assessable purely from the observed training data. Under these stipulations, the formula for the front-door adjustment is expressed as:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP (Y | d o (X)) = \\sum_ {Z} P (Z | X) \\sum_ {X ^ {\\prime}} P (Y | Z, X ^ {\\prime}) P \\left(X ^ {\\prime}\\right), \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $X'$  represents the hypothetical input text, independent and identically distributed (i.i.d.) in relation to  $X$ . The notations  $\\sum_{Z}$  and  $\\sum_{X'}$  respectively denote the summation over all possible values of  $Z$  and  $X'$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/c6de737bd94941f747a9641956280e1b987514aea2315d3ed042f2642ad42f92.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Variable</td><td>Example</td></tr><tr><td>X</td><td>The mn cat chased the bn mouse across the room.</td></tr><tr><td rowspan=\"4\">Z</td><td>The cat chased the mouse across the room.</td></tr><tr><td>The cat chased the bn mouse across the room.</td></tr><tr><td>The mn cat chased the mouse across the room.</td></tr><tr><td>The cat chased the mouse across the room, but it was bn quiet.</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Table 1. The examples of front-door variable  $Z$ . The triggers are indicated in red.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3. The Front-door in Language Model",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We plan to apply the front-door adjustment approach to manage the latent backdoor attacks. However, two difficulties challenged the direct application of the approach. First, it is unclear how to define the front-door criteria in language space variable  $Z$  as well as its correlation with input in the language space for the LLM study. Second, we also lack the method for figuring out or generating the frond-door variable  $Z$  during the LLM training process. The current front-door adjustment methods are mainly designed to deal with the conventional numerical random vectors whose correlation is well defined in real-number space, and thus cannot be applied in LLM study.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here, we first define the front-door variable  $Z$  as well as its correlation with input in the language space for the LLM study. Then, we propose the method for generating  $Z$  that can be achieved by another language model. Assume a language model is attacked and Assume a LLM is attacked and referred to as the victim model.  $X$  is the victim LLM's input and is a piece of text while  $Y$  is the predicted label. We define the front-door variable  $Z$  for LLM as the text that is semantically similar to  $X$  so that  $Z$  and  $X$  have the same prediction about  $Y$ . The mathematical definition of  $Z$  is given below.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1. A piece of text  $Z$  is the victim model's front-door variable for the input  $X$  and predicted label  $Y$  is  $Z$  satisfies the following conditions:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP (Z = z \\mid X = x) = P (X = z \\mid X \\in E (x)), \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP (Y \\mid Z) = P (Y \\mid X, A = 0). \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $E(x)$  represents the set of all texts that are semantically equivalent to  $x$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that Equation 3 implies that  $X$  and  $Z$  are semantically equivalent while Equation 4 implies that  $X$  and  $Z$  have the same prediction about  $Y$ . We argue that the above-defined  $Z$  satisfies the front-door criteria. Equation 3 guarantees the strong correlation between  $X$  and  $Z$ . Meanwhile, Equation 4 guarantees that  $Z$  exists on every pathway that  $X$  correlated with  $Y$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Front-door Adjustment for Backdoor Elimination",
+        "text_level": 1,
+        "bbox": [
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+            83,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We proposed the Front-door Adjustment for Backdoor Elimination (FABE) method to defend against backdoor attacks for LLMs. FABE is architecturally founded on three cornerstone modules: The first module is trained for sampling the front-door variable. The second module is trained for estimating the true causal effect. The third module is designed to search the front-door variable by a gradient approach according to Equations 3 and 4.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1. Generating Front-door Variables",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Employing a language model  $F(\\cdot \\mid \\cdot ;\\theta)$ , henceforth designated as the defense model, we generate of a multiplicity of front-door variables  $Z_{j}$ . Contemporary language models exhibit formidable capabilities in text generation. We can instruct the defense model  $F(Z|i;\\theta)$  with instruction  $i$  to generate the front-door variable  $Z$ , which is a text semantically equivalent to  $X$ . Within the ambit of this manuscript, we employ prescriptive, fixed-format instructions  $i$  as the input modality:",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Instruction: As a proficient data engineer, your mandate involves the refinement of linguistic expressions within a dataset. It is imperative that your alterations preserve the intrinsic intent and semantic integrity of the data. Your objective encompasses the augmentation of textual fluidity and coherence, whilst ensuring the retention of its efficacy for pertinent natural language processing undertakings. Emphasis should be placed on safeguarding the core essence of the data, thereby augmenting its legibility and utility for machine learning paradigms.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input:  $X$",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Response:",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "wherein  $X$  constitutes the input text, and the defense model is tasked with generating the corresponding front-door variable subsequent to the phrase \"Response:\".",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use beam search for approximating the decoding of  $Z$  variables that exhibit high probability  $F(Z|i;\\theta)$  indices. This approach generates a set of  $B$  candidate intermediate variables,  $Z_{1},Z_{2},\\dots ,Z_{B}$ , where  $B$  denotes the beam width. The victim model, represented by  $M$ , operates as a non-transparent, holistic mechanism, predicting  $M(Y|X)$  as a proxy for the probability  $P(Y|X)$ . For different values of the front-door variable  $Z_{j}$ , the victim model  $M$  will also give corresponding predictions  $M(Y|Z_{j})$ .",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2. Causal Effect Estimation",
+        "text_level": 1,
+        "bbox": [
+            84,
+            851,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Combining the computational outputs from both the defense model  $F$  and the victim model  $M$ , we execute the",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "front-door adjustment to derive the definitive estimate of the causality relationship  $P(Y|do(X))$ , as shown in Equation 2. Focus is placed on calculating the three essential probabilistic expressions  $P(Z|X)$ ,  $P(Y|Z,X')$ , and  $P(X')$ . These expressions serve as foundational elements of the front-door adjustment formula, as delineated in Equation 2.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$P(Z|X)$ : Model  $F$ , in the process of generating the front-door variable  $Z$ , concurrently computes a score  $s_{\\theta}(i,Z)$  as defined in Expression 6. The softmax function is then applied to these scores to approximate the probability of each  $Z_{j}$ , formulated as",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP (Z | X) \\approx \\frac {\\exp s _ {\\theta} (i , Z _ {j})}{\\sum_ {k = 1} ^ {B} \\exp s _ {\\theta} (i , Z _ {k})}, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $s_{\\theta}(i,Z_j)$  the scoring function from beam search:",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {\\theta} (i, Z _ {j}) = \\frac {1}{| Z _ {j} |} \\log F (Z _ {j} | i; \\theta). \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$P(Y|Z,X')$ : Model  $M$  is tasked with predicting the label corresponding to a given input text. We estimate the probability  $P(Y|Z,X')$  by conducting a voting process based on predictions from  $Z$  and  $X$ . This relationship is represented as",
+        "bbox": [
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+            400,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP (Y | Z, X ^ {\\prime}) \\approx \\frac {M (Y | Z) + M (Y | X)}{2}. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$P(X')$ : It is important to note that the computation of  $\\sum_{X'}$  in Equation 2 is contingent upon the given  $Z$ . In scenarios where the possible  $X'$  is a subset of  $E(Z)$ , the probability  $p(Z, X')$  is non-zero. Consequently, the probability  $P(X')$  can be defined as",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP \\left(X ^ {\\prime}\\right) = P \\left(X ^ {\\prime} \\mid X ^ {\\prime} \\in E (Z)\\right) = P \\left(X ^ {\\prime} \\mid X ^ {\\prime} \\in E (X)\\right). \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            609,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In a manner analogous to Equation 5, for  $X'$  within  $Z_{1}, Z_{2}, \\dots, Z_{B}$ , the probability  $P(X')$  is approximated as",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP \\left(X ^ {\\prime}\\right) \\approx \\frac {\\exp s _ {\\theta} (i , Z _ {j})}{\\sum_ {k = 1} ^ {B} \\exp s _ {\\theta} (i , Z _ {k})}. \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Integrating Equation 5, 7, and 9, it is inferred that the estimation of  $P(Y|do(X))$  can be accurately articulated by the following formula:",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {j, l = 1} ^ {B} \\frac {\\exp s _ {\\theta} (i , Z _ {j}) \\exp s _ {\\theta} (i , Z _ {l})}{(\\sum_ {k = 1} ^ {B} \\exp s _ {\\theta} (i , Z _ {k})) ^ {2}} \\frac {M (Y | Z _ {i}) + M (Y | Z _ {j})}{2}. \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            504,
+            794,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This equation epitomizes the cumulative estimation process, incorporating the probability distributions and model predictions to derive a comprehensive understanding of the causal effect  $P(Y|do(X))$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
+            222,
+            56,
+            750,
+            71
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/23bc2aa198f2c8e39a75a9b0497e6bba3848fcaa1014e3eac0d09cfe8f589c1d.jpg",
+        "image_caption": [
+            "Figure 2. Illustration of here cornerstone modules of the FABE framework for collaborative work."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            83,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3. Instruction Tuning for Front-door Criteria",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We advocate for the implementation of a ranking-based instruction tuning methodology on the language model  $F(Z|i;\\theta)$ , aiming to ensure compliance of the generated front-door variable  $Z$  with the stipulated criteria as delineated in Equations 3 and 4.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Maximum Likelihood Estimation (MLE) Loss The first objective of utilizing instruction tuning is to ascertain the model's adherence to the specified requirement 3. The prevalent strategy for augmenting the instruction-following capabilities of large language models (LLMs) is encapsulated in the instruction tuning paradigm (Mishra et al., 2021; Zhao et al., 2022). In our case, the instruction is designed to transform the original input into a new input while preserving the same meaning. Consequently, we employ the MLE loss function to ensure that the model adheres to requirement 3:",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} ^ {\\mathrm {M L E}} (\\theta) = - \\frac {1}{| X _ {0} |} \\log F (X _ {0} | i; \\theta). \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, where  $i$  is the instruction and  $X_0$  represents the clean input.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Ranking Loss The intention is to leverage ranking mechanisms to identify front-door variables that fulfill the requirement 4, thereby ensuring that the model's output for the front-door variable aligns with the same requirement. This alignment is fostered through the application of a pairwise ranking objective, a method that has demonstrated efficacy",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "in previous research endeavors (Liu et al., 2022; Zhang et al., 2022; Zhao et al., 2022; Li et al., 2023).",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Upon the procurement of samples  $\\tilde{Z}_1, \\tilde{Z}_2, \\ldots, \\tilde{Z}_K$ , the corresponding outputs  $\\tilde{Y}_1, \\tilde{Y}_2, \\ldots, \\tilde{Y}_K$  are evaluated against the unattacked output  $Y_0$  utilizing KL-divergence  $KL(\\tilde{Y}_k, Y_0)$ . These samples are then hierarchically organized based on the evaluation model, yielding an ordered sequence  $\\tilde{Z}_1', \\tilde{Z}_2', \\ldots, \\tilde{Z}_K'$ , where  $KL(\\tilde{Y}_k', Y_0) < KL(\\tilde{Y}_j', Y_0)$  for all  $k < j$ . To calibrate our model towards prioritizing samples with superior evaluation metrics, we introduce a margin-based pairwise ranking loss (Hopkins & May, 2011; Zhong et al., 2020):",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} ^ {\\mathrm {R}} (\\theta) = \\sum_ {k <   j} \\max  \\left(0, s _ {\\theta} \\left(i, \\tilde {Z} _ {j} ^ {\\prime}\\right) - s _ {\\theta} \\left(i, \\tilde {Z} _ {k} ^ {\\prime}\\right) + (j - k) \\times \\lambda\\right) \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $s_{\\theta}(i, \\tilde{Z}')$  represents the score assigned by the model  $F(\\cdot | \\theta)$  for instruction  $i$  and one of the front-door variables  $Z'$ . The term  $(j - k) \\times \\lambda$  is the dynamic margin between the scores of  $\\tilde{Y}_i'$  and  $\\tilde{Y}_j'$ , and  $\\lambda$  is a hyper-parameter.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Accordingly, we employ a synthesis of the MLE loss  $\\mathcal{L}^{\\mathrm{MLE}}$  and the ranking loss  $\\mathcal{L}^{\\mathrm{R}}$  to formulate the composite loss function. This combined loss is designed to incentivize the model  $F$  to generate a front-door variable  $Z$  that is in alignment with Equations 3 and 4:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (\\theta) = \\beta \\mathcal {L} ^ {\\mathrm {M L E}} (\\theta) + \\mathcal {L} ^ {\\mathrm {R}} (\\theta). \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            596,
+            834,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this equation,  $\\beta$  is a weighting parameter that balances the contribution of the MLE loss against the ranking loss, thereby guiding the model  $F$  towards satisfying both the",
+        "bbox": [
+            495,
+            859,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
+            222,
+            56,
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+            71
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "semantic equivalence and the front-door criteria through effective learning of the front-door variable  $Z$ .",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
+            84,
+            133,
+            217,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1. Setup",
+        "text_level": 1,
+        "bbox": [
+            84,
+            160,
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+            176
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Defense Model Our approach utilizes a single defense model, effective against various attacks on different datasets, with LLaMA2 (Touvron et al., 2023) (7 billion parameters) as its backbone. Model training leverages eight Nvidia V100 GPUs, using Adam (Kingma & Ba, 2014) for optimization with a learning rate of  $1 \\times 10^{-5}$  and 1000 warmup steps. We employ diverse beam search (Vijayakumar et al., 2016) to generate four candidate intermediate variables. The margin coefficient  $\\lambda$  in Equation (12) is 0.1, while the length normalization term  $\\alpha$  in the model score function is 2.0 across datasets. The MLE loss weight  $\\beta$  is set at 1.0 (Equation 13).",
+        "bbox": [
+            84,
+            183,
+            475,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets and Victim Models. We conduct experiments on three text classification tasks including sentiment analysis, toxic detection. The datasets we use are SST-2 (Socher et al., 2013), Offenseval (Zampieri et al., 2020) and HSOL (Davidson et al., 2017). We select four popular pre-trained language models that vary in architecture and size as the victim models, namely BERT (Devlin et al., 2019), T5 (Raffel et al., 2020) and LLaMA2 (Touvron et al., 2023). Note that our FABE is agnostic to the victim model architectures. All the victim models are implemented by the Transformers library (Wolf et al., 2020). The detail of datasets and victim models are shown in Table3 and Table4.",
+        "bbox": [
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+            369,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Attack Baselines. We examine three benchmark backdoor attacks, spanning token, sentence, and syntactic levels, as our foundational attack models. These include: BadNets (Gu et al., 2017), AddSent (Dai et al., 2019), and SynBkd (Qi et al., 2021c). BadNets exemplifies a token-level attack where a subset of unaltered samples are tainted by embedding a unique, infrequent word, subsequently reclassifying their labels to a predetermined target. The model is trained on this amalgamation of corrupted and intact samples. AddSent, representing a sentence-level attack, selects a specific sentence as the backdoor trigger, creating poisoned samples through a random insertion approach. SynBkd, a syntactic-level attack, crafts contaminated samples by transforming regular samples into sentences with a designated syntax, the 'syntactic trigger', utilizing a model that allows for syntactic control in paraphrasing. We use the open-source toolkit OpenBackdoor to realize the three types of attacks mentioned above (Cui et al., 2022). The detail are shown in Table5.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Defense Baselines. In this study, we undertake a comparative analysis of our proposed methodology against three",
+        "bbox": [
+            84,
+            875,
+            477,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "established test-time defense techniques: ONION (Qi et al., 2021a), RAP (Yang et al., 2021b), and STRIP (Gao et al., 2021). ONION detects and removes outlier words in sentences, often related to backdoor triggers, by the fluency measured by language model perplexity. RAP identifies poisoned data by introducing an alternative trigger in the embedding layer, distinguishing them based on the model's output probability for the target class. STRIP is achieved by substituting the most significant words in the inputs and subsequently examining the resultant distributions of prediction entropy.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation In this study, to comprehensively evaluate the performance of all methods, we have employed two widely used performance metrics: Attack Success Rate (ASR) and Clean Accuracy (CA) with the definition",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA S R = P (\\hat {y} \\neq y _ {0} | A = 1),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            599,
+            340,
+            782,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC A = P (\\hat {y} = y _ {0} | A = 0),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            609,
+            359,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $\\hat{y}$  is the prediction of model and  $y_0$  is the label. Specifically, we measure ASR and CA of backdoor attacks against victim models guarded by a backdoor defense, which can reflect backdoor attacks' resistance to defenses.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2. Results",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our main results are shown in Table 2. Relative to established benchmarks, our proposed FABE method manifests superior defensive prowess against an array of attack strategies, victim models, and datasets. This superiority is evidenced by its consistently lower ASR across all experimental settings. Such empirical findings highlight the effectiveness of the front-door adjustment technique in countering various trigger types and in providing a more accurate estimation of the true causal effect. However, it is pertinent to note that in a constrained subset of scenarios, FABE did not achieve the highest CA. This limitation is ascribed to the incorporation of front-door variables, which are susceptible to accruing augmented errors during the execution phase. Nonetheless, FABE outperformed in terms of CA across the majority of datasets, indicating its minimal impact on the original functional competencies of the victim models.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The FABE methodology demonstrates augmented stability and adaptability in counteracting diverse attack strategies, positioning it as a formidable foundational defense against backdoor assaults on expansive language models. Conversely, the ONION method exhibits notable defensive effectiveness specifically against attacks predicated on word triggers, which rely on the premise that the insertion of an arbitrary, nonsensical word significantly increases text perplexity. Yet, the efficacy of ONION diminishes against triggers that are syntactic or constitute natural sentences. Likewise, methods like RAP and STRIP, employing random",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
+            222,
+            56,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b5cb9725578828ab82b6964968d20581ba60555f23e6bc866d5daee03808ca56.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Victim</td><td>Attack</td><td>Dataset</td><td>metric</td><td>None</td><td>ONION</td><td>RAP</td><td>STRIP</td><td>FABE(ours)</td></tr><tr><td rowspan=\"12\">BERT</td><td rowspan=\"4\">BadNet</td><td rowspan=\"2\">SST-2</td><td>CA ↑</td><td>90.66</td><td>85.94</td><td>79.35</td><td>88.08</td><td>90.44</td></tr><tr><td>ASR ↓</td><td>100.00</td><td>26.54</td><td>83.66</td><td>94.63</td><td>15.57</td></tr><tr><td rowspan=\"2\">Offenseval</td><td>CA ↑</td><td>85.10</td><td>83.53</td><td>54.13</td><td>79.28</td><td>84.40</td></tr><tr><td>ASR ↓</td><td>90.95</td><td>10.66</td><td>37.00</td><td>84.01</td><td>6.79</td></tr><tr><td rowspan=\"2\">AddSent</td><td rowspan=\"2\">HSOL</td><td>CA ↑</td><td>94.89</td><td>88.73</td><td>58.27</td><td>94.45</td><td>94.81</td></tr><tr><td>ASR ↓</td><td>97.91</td><td>91.71</td><td>59.10</td><td>97.59</td><td>5.23</td></tr><tr><td rowspan=\"6\">SynBk</td><td rowspan=\"2\">SST-2</td><td>CA ↑</td><td>90.72</td><td>84.79</td><td>71.55</td><td>89.40</td><td>91.32</td></tr><tr><td>ASR ↓</td><td>86.29</td><td>87.28</td><td>77.19</td><td>84.21</td><td>39.14</td></tr><tr><td rowspan=\"2\">Offenseval</td><td>CA ↑</td><td>84.52</td><td>83.65</td><td>49.94</td><td>89.54</td><td>84.52</td></tr><tr><td>ASR ↓</td><td>91.60</td><td>80.78</td><td>72.63</td><td>84.14</td><td>6.95</td></tr><tr><td rowspan=\"2\">HSOL</td><td>CA ↑</td><td>90.79</td><td>86.66</td><td>53.67</td><td>83.59</td><td>90.91</td></tr><tr><td>ASR ↓</td><td>85.02</td><td>80.52</td><td>89.34</td><td>89.34</td><td>19.08</td></tr><tr><td rowspan=\"12\">T5</td><td rowspan=\"4\">BadNet</td><td rowspan=\"2\">SST-2</td><td>CA ↑</td><td>92.53</td><td>87.32</td><td>47.67</td><td>91.32</td><td>92.26</td></tr><tr><td>ASR ↓</td><td>94.52</td><td>22.37</td><td>94.41</td><td>92.98</td><td>11.73</td></tr><tr><td rowspan=\"2\">Offenseval</td><td>CA ↑</td><td>82.42</td><td>79.91</td><td>92.53</td><td>73.92</td><td>82.07</td></tr><tr><td>ASR ↓</td><td>97.25</td><td>13.89</td><td>94.52</td><td>82.71</td><td>9.69</td></tr><tr><td rowspan=\"2\">AddSent</td><td rowspan=\"2\">HSOL</td><td>CA ↑</td><td>91.43</td><td>86.05</td><td>90.18</td><td>84.23</td><td>91.27</td></tr><tr><td>ASR ↓</td><td>99.92</td><td>99.44</td><td>99.92</td><td>84.46</td><td>14.17</td></tr><tr><td rowspan=\"6\">SynBk</td><td rowspan=\"2\">SST-2</td><td>CA ↑</td><td>76.55</td><td>72.16</td><td>73.92</td><td>75.78</td><td>56.95</td></tr><tr><td>ASR ↓</td><td>89.91</td><td>91.01</td><td>89.69</td><td>89.15</td><td>14.58</td></tr><tr><td rowspan=\"2\">Offenseval</td><td>CA ↑</td><td>80.68</td><td>79.32</td><td>80.44</td><td>72.88</td><td>76.72</td></tr><tr><td>ASR ↓</td><td>99.19</td><td>97.42</td><td>98.87</td><td>83.85</td><td>5.65</td></tr><tr><td rowspan=\"2\">HSOL</td><td>CA ↑</td><td>90.95</td><td>86.29</td><td>88.77</td><td>88.33</td><td>88.89</td></tr><tr><td>ASR ↓</td><td>99.11</td><td>98.55</td><td>98.71</td><td>95.57</td><td>8.78</td></tr><tr><td rowspan=\"12\">LLaMA2</td><td rowspan=\"4\">BadNet</td><td rowspan=\"2\">SST-2</td><td>CA ↑</td><td>94.78</td><td>90.56</td><td>91.10</td><td>94.29</td><td>95.61</td></tr><tr><td>ASR ↓</td><td>94.74</td><td>17.00</td><td>78.29</td><td>94.30</td><td>6.25</td></tr><tr><td rowspan=\"2\">Offenseval</td><td>CA ↑</td><td>82.65</td><td>81.08</td><td>73.92</td><td>80.68</td><td>78.81</td></tr><tr><td>ASR ↓</td><td>80.45</td><td>17.93</td><td>69.47</td><td>78.35</td><td>14.54</td></tr><tr><td rowspan=\"2\">AddSent</td><td rowspan=\"2\">HSOL</td><td>CA ↑</td><td>92.31</td><td>87.39</td><td>87.04</td><td>91.63</td><td>91.99</td></tr><tr><td>ASR ↓</td><td>93.64</td><td>84.46</td><td>42.11</td><td>92.51</td><td>10.23</td></tr><tr><td rowspan=\"6\">SynBk</td><td rowspan=\"2\">SST-2</td><td>CA ↑</td><td>92.20</td><td>84.57</td><td>91.71</td><td>91.10</td><td>92.20</td></tr><tr><td>ASR ↓</td><td>88.71</td><td>89.36</td><td>88.34</td><td>87.83</td><td>54.28</td></tr><tr><td rowspan=\"2\">Offenseval</td><td>CA ↑</td><td>80.68</td><td>78.86</td><td>78.81</td><td>77.88</td><td>74.85</td></tr><tr><td>ASR ↓</td><td>97.42</td><td>93.38</td><td>14.22</td><td>13.73</td><td>8.23</td></tr><tr><td rowspan=\"2\">HSOL</td><td>CA ↑</td><td>80.68</td><td>76.24</td><td>79.03</td><td>77.10</td><td>81.45</td></tr><tr><td>ASR ↓</td><td>98.63</td><td>96.70</td><td>97.83</td><td>93.32</td><td>21.18</td></tr><tr><td colspan=\"4\">Average CA ↑</td><td>87.47</td><td>83.50</td><td>74.56</td><td>84.64</td><td>85.52</td></tr><tr><td colspan=\"4\">Average ASR ↓</td><td>93.63</td><td>66.61</td><td>76.96</td><td>84.59</td><td>15.12</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2. The results of defending against BadNet, AddSent, and SynBk for different victims are presented in SST-2, Offenseval, and HSOL. Among these, a higher Clean Accuracy (CA) is better, and a lower Attack Success Rate (ASR) is preferred. The best-performing results are highlighted in bold.",
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+    },
+    {
+        "type": "text",
+        "text": "perturbation and substitution to discern between clean and poisoned samples, struggle to neutralize more covert triggers, such as syntactic ones. Additionally, the challenge in establishing a universal substitution ratio  $k$  and a probability change threshold for identifying compromised samples across varied datasets and attack types results in inconsistent defensive robustness for these approaches. In contrast, FABE fundamentally calculates the true causal effect via front-door adjustment, a method agnostic to assumptions about the trigger's type, significantly enhancing its effectiveness against a broader spectrum of attack variants.",
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+        "text": "4.3. Ablation Study",
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+        "text": "In this section, we evaluated the effectiveness of the LLaMA2 model backbone, MLE LOSS, Ranking Loss, and the final sampling method in our proposed FABE approach on the SST-2, Offenseval, and HSOL datasets. We designed four sets of experiments, which are: 1) Rewriting poisoned inputs using the pre-trained model (Pre-trained) 2) Rewriting poisoned inputs using the model fine-tuned with  $\\mathcal{L}^{\\mathrm{MLE}}$  (SFT) 3) Rewriting poisoned inputs using the defense model fine-tuned with  $\\mathcal{L} = \\beta \\mathcal{L}^{\\mathrm{MLE}} + \\mathcal{L}^{\\mathrm{R}}$  (Ranking SFT), and 4) Calculating causal effects using the defense model through front-door adjustment according to 10 (FABE). We recorded the defensive effects of these four methods against the Syn",
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+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
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+        "text": "7",
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+        "img_path": "images/f330aa784df432eac27533a90de879ba0a09b1d2338e393d8a286dbe3124ac0c.jpg",
+        "image_caption": [
+            "Figure 3. Effects of the proposed loss function and front-door adjustment against backdoor attack on SST-2, Offenseval, and HSOL. The line represents the average ASR."
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Back attack on the BERT victim model, with ASR results shown in Figure 3. The results show that rewriting sentences with a pre-trained model has certain defensive effects, while fine-tuning solely with  $\\mathcal{L}^{\\mathrm{MLE}}$  has a negative impact on defense. The reason for using  $\\mathcal{L}^{\\mathrm{MLE}}$  is to make training more stable. The defense effectiveness is improved after adding  $\\mathcal{L}^{\\mathrm{R}}$ , as it makes the model outputs more compliant with the constraint 4. Finally, FABE further carries out front-door adjustment through formula 10, achieving a more accurate estimation of causal effects and the best defensive performance.",
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+        "type": "text",
+        "text": "Qi et al. (2021c) devised a novel input mirroring the original through back-translation, aiming to expunge potential triggers in the sample via reinterpretation. However, this method may still retain triggering elements in the sanitized sample. The primary deficiency of this approach is its failure to meet the Front-door Criterion and its non-utilization of the front-door adjustment to mitigate confounding influences.",
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+        "type": "text",
+        "text": "5. Related Work",
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+        "text": "5.1. Backdoor Defenses",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The burgeoning field of Natural Language Processing (NLP) is facing a growing security concern due to the prevalence of backdoor attacks. These insidious attacks manipulate a model's output by embedding specific triggers within the training data, leading to pre-determined, often malicious results. Such triggers can manifest in various forms: the insertion of designated trigger words (Gu et al., 2017; Kurita et al., 2020; Shen et al., 2021; Yang et al., 2021a; Zhang et al., 2023b), incorporation of certain sentences (Dai et al.,",
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+    {
+        "type": "text",
+        "text": "2019), application of style transformations (Qi et al., 2021b; Jin et al., 2022), and the integration of syntactic controls (Qi et al., 2021c; Sun et al., 2021). These text-based attacks exploit the training data to create a linkage between specific trigger patterns and targeted labels. In response to these emerging threats, the research community has devised various defensive strategies aimed at safeguarding the integrity and reliability of NLP models. These defensive mechanisms can be broadly categorized into two types. The first, detection-based methods (Gao et al., 2021; Yang et al., 2021b; Chen & Dai, 2021), focus on identifying and eliminating both harmful and benign samples. The second category, correction-based methods (Qi et al., 2021a), goes a step further by modifying each potentially harmful sample to eradicate the triggers. Yan et al. (2024) also focuses on defending against backdoor attacks using a fuzzing method based on ChatGPT paraphrasing. Our proposed methodology aligns more closely with correction-based approaches. However, the primary objective of our sample modifications is not the outright removal of all triggers but rather the attainment of accurate predictions through strategic causal intervention.",
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+        "type": "text",
+        "text": "5.2. Causal Inference",
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+        "text": "Causal inference, with its extensive and rich heritage in statistical research, has been extensively documented (Pearl, 2009; Peters et al., 2017). Its recent adoption has significantly contributed to the fortification of model robustness across a spectrum of machine learning domains (Tang et al., 2020; Zhang et al., 2020; Yue et al., 2020; Vig et al., 2020; Wu et al., 2022). The pioneering work of Zhang et al. (2023a) in applying causal inference methodologies to the realm of computer vision, specifically in countering backdoor attacks, represents a notable advancement in this field. Nonetheless, real-world scenarios often grapple with the issue of unobserved confounding factors. In such instances, the prevailing preference is to employ front-door adjustment techniques (Pearl, 1995). This approach, through the utilization of a front-door variable, adeptly captures the intermediate causal effect and has been empirically validated in a multitude of applications (Yang et al., 2021d;c; Li et al., 2021; Nguyen et al., 2023). To our knowledge, this study is the inaugural endeavor to implement front-door adjustment as a defensive strategy against backdoor attacks in computational systems.",
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+        "text": "6. Conclusion",
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+        "text": "In this study, we present a novel defense methodology predicated on causal inference, designed to effectively counter text-based backdoor attacks. Utilizing the inherent capacity of language models to adhere to instructions for generating semantically equivalent sentences to the original input, we",
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+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
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+    {
+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "text",
+        "text": "construct front-door variables. This capability is further augmented through instruction-based fine-tuning of the model. Employing the front-door adjustment technique, our method facilitates the direct estimation of true predictions, obviating the need to differentiate between poisoned and clean samples. A comprehensive theoretical analysis is conducted to evaluate our approach's proficiency in accurately estimating the true causal effect. Empirical evidence from our experiments substantiates that our defense strategy outperforms existing methods in terms of defensive efficacy across a spectrum of datasets, attack modalities, and victim models.",
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+        "type": "text",
+        "text": "7. Limitation and Future works",
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+        "type": "text",
+        "text": "First, FABE is slower than traditional approaches without using of LLMs. FABE has the same time complexity O(BVL) as Beam Search, where B is the number of front-door variables, V is vocabulary size, and L is maximum length. The average time taken for FABE and baselines is shown in the table 6 in Appendix. Noting the future potential for reducing computational demand, techniques such as distillation, quantization, and parallelization are proposed to mitigate FABE's computational overhead in future work.",
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+        "type": "text",
+        "text": "Second, current experiments have only validated FABE's defensive capabilities in text classification and have not yet generalized to other tasks, such as language generation or question answering. We contend that tasks like text generation and question answering can be reformulated as next token prediction tasks, also classification tasks. Consequently, we argue that the FABE is applicable to other tasks and merits further investigation.",
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+    {
+        "type": "text",
+        "text": "Third, adaptive attacks warrant further discussion. We conducted a simulation of an adaptive attack, wherein the attacker has output access to the defense model on AddSent-HSOL for BERT. When attackers select sentences that still contain triggers after passing through the defense model for poisoning, the Attack Success Rate (ASR) without any defense is  $95.05\\%$ . However, FABE can reduce the ASR to  $10.87\\%$ , thereby demonstrating its effectiveness in defense.",
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+        "type": "text",
+        "text": "Finally, since the FABE method operates by computing causal effects to exclude the interference of triggers and is not in conflict with traditional backdoor defense mechanisms, combining the FABE method with other approaches, such as ONION, is a promising direction for further investigation. We conducted a preliminary test where FABE's inputs were filtered by ONION to remove triggers as a defense against the AddSent attack on BERT within the HSOL dataset, which further reduced the attack success rate to  $2.92\\%$ .",
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+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
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+        "text": "We anticipate that this work will have a positive impact as our FABE method is capable of effectively defending against backdoor attacks on large language models.",
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+    {
+        "type": "text",
+        "text": "References",
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+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
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+        "page_idx": 10
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+        ],
+        "bbox": [
+            86,
+            84,
+            478,
+            704
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
+            222,
+            56,
+            750,
+            71
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A. More Detail of Experimental Setting",
+        "text_level": 1,
+        "bbox": [
+            84,
+            83,
+            419,
+            101
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this chapter, we present an elaborate discussion on the datasets, models, and attack methodologies employed in our study. Correspondingly, we delineate this information in Tables 3, 4, and 5 for the convenience of the reader.",
+        "bbox": [
+            84,
+            109,
+            888,
+            140
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/ca2e2bcd5a025405ca38b9fcd7293a85c8348f2d07af09ee969611ec8075bc44.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>SST-2</td><td>Offenseval</td><td>HSOL</td></tr><tr><td>Task</td><td>110M</td><td>220M</td><td>7B</td></tr><tr><td>Labels Number</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Average Length</td><td></td><td></td><td></td></tr><tr><td>train</td><td>19.22</td><td>19.68</td><td>13.37</td></tr><tr><td>dev</td><td>19.45</td><td>19.41</td><td>13.10</td></tr><tr><td>test</td><td>19.23</td><td>23.35</td><td>13.10</td></tr><tr><td>Data Number</td><td></td><td></td><td></td></tr><tr><td>train</td><td>6919</td><td>11914</td><td>5822</td></tr><tr><td>dev</td><td>8717</td><td>1322</td><td>2484</td></tr><tr><td>test</td><td>1820</td><td>858</td><td>2484</td></tr></table>",
+        "bbox": [
+            316,
+            151,
+            656,
+            325
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2de6f5d2409220cf9d2ec7de56a5b18edf98f0610b1948703087945281f7fe40.jpg",
+        "table_caption": [
+            "Table 3. The detail of SST-2, Offenseval and HSOL."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>BERT</td><td>T5</td><td>LLaMA2</td></tr><tr><td># Parameters</td><td>110M</td><td>220M</td><td>7B</td></tr><tr><td>Accuracy</td><td></td><td></td><td></td></tr><tr><td>SST-2</td><td>91.10</td><td>91.98</td><td>95.88</td></tr><tr><td>Offenseval</td><td>85.45</td><td>84.05</td><td>83.82</td></tr><tr><td>HSOL</td><td>94.53</td><td>94.41</td><td>95.13</td></tr></table>",
+        "bbox": [
+            331,
+            381,
+            640,
+            481
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/51699bb24b027baeafb3a2add564442d227e4ebfd78cf6fcfed91abb0e617b32.jpg",
+        "table_caption": [
+            "Table 4. The number parameters and accuracy on datasets of BERT, T5 and LLaMA2."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>BadNets</td><td>AddSent</td><td>SynBkd</td></tr><tr><td>Target Label</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Label Consistency</td><td>true</td><td>true</td><td>true</td></tr><tr><td>Label Dirty</td><td>false</td><td>false</td><td>false</td></tr><tr><td>Triggers Number</td><td>1</td><td>-</td><td>-</td></tr><tr><td>Triggers</td><td>cf, mn, bb, tq, mb, de</td><td>I watch this 3D movie</td><td>(ROOT(S(SBAR)(,)(NP)(VP)(.)))EOP</td></tr></table>",
+        "bbox": [
+            86,
+            537,
+            815,
+            646
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 5. The attack baselines hyperparameters on BadNets, AddSent and SynBkd.",
+        "bbox": [
+            241,
+            656,
+            730,
+            670
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
+            222,
+            56,
+            750,
+            70
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            477,
+            922,
+            495,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B. Detail of Computational Overhead.",
+        "text_level": 1,
+        "bbox": [
+            84,
+            83,
+            408,
+            101
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The average time taken for each method is shown in the following Table 6.",
+        "bbox": [
+            84,
+            109,
+            576,
+            125
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/5f39c163261261882a20f9b329b4279894ef62e3a6e6f7919e06f3aafa5f1d8c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Time (s)</td></tr><tr><td>None</td><td>0.01</td></tr><tr><td>ONION</td><td>2.34</td></tr><tr><td>RAP</td><td>0.13</td></tr><tr><td>STRIP</td><td>0.09</td></tr><tr><td>FABE</td><td>11.98</td></tr></table>",
+        "bbox": [
+            411,
+            136,
+            563,
+            234
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 6. The average time cost for protecting an input on SST2.",
+        "bbox": [
+            297,
+            250,
+            673,
+            265
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C. Definition of Threat Model",
+        "text_level": 1,
+        "bbox": [
+            84,
+            291,
+            339,
+            306
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The threat model of a backdoor attack is defined as follows:",
+        "bbox": [
+            84,
+            316,
+            480,
+            330
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Attacker's Capacities: During the fine-tuning of language models for downstream tasks, the attacker has complete control over the dataset used for fine-tuning.",
+            "- Attacker's Goals: The attacker aims to deliver a backdoored model. This model is designed to predict a specified target class for samples containing a backdoor trigger while maintaining good performance on clean samples.",
+            "- Defender's Knowledge: The defender acquires the trained model from a third party. The defender possesses a clean validation set and a training dataset that may have been compromised. However, the defender lacks information about the backdoor injection process and the backdoor triggers.",
+            "- Defense Objectives: The defender's goal is to ensure that the model functions correctly even when inputs include a trigger, thereby mitigating the impact of the backdoor attack."
+        ],
+        "bbox": [
+            102,
+            349,
+            883,
+            516
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Causality Based Front-door Defense Against Backdoor Attack on Language Models",
+        "bbox": [
+            222,
+            56,
+            750,
+            70
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
+            477,
+            922,
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+        ],
+        "page_idx": 13
+    }
+]

2024/Causality Based Front-door Defense Against Backdoor Attack on Language Models/3285ecaf-0f90-4e89-b9c2-9ff0972245ce_origin.pdf

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2024/Causality Based Front-door Defense Against Backdoor Attack on Language Models/full.md

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+# Causality Based Front-door Defense Against Backdoor Attack on Language Models
+
+Yiran Liu  $^{*1}$  Xiaoang Xu  $^{*2}$  Zhiyi Hou  $^{3}$  Yang Yu  $^{4}$
+
+# Abstract
+
+We have developed a new framework based on the theory of causal inference to protect language models against backdoor attacks. Backdoor attackers can poison language models with different types of triggers, such as words, sentences, grammar, and style, enabling them to selectively modify the decision-making of the victim model. However, existing defense approaches are only effective when the backdoor attack form meets specific assumptions, making it difficult to counter diverse backdoor attacks. We propose a new defense framework Front-door Adjustment for Backdoor Elimination (FABE) based on causal reasoning that does not rely on assumptions about the form of triggers. This method effectively differentiates between spurious and legitimate associations by creating a 'front door' that maps out the actual causal relationships. The term 'front door' refers to a text that retains the semantic equivalence of the initial input, which is generated by an additional, fine-tuned language model, denoted as the defense model. Our defense experiments against various attack methods at the token, sentence, and syntactic levels reduced the attack success rate from  $93.63\%$  to  $15.12\%$ , improving the defense effect by 2.91 times compared to the best baseline result of  $66.61\%$ , achieving state-of-the-art results. Through ablation study analysis, we analyzed the effect of each module in FABE, demonstrating the importance of complying with the front-door criterion and front-door adjustment for
+
+mula, which also explains why previous methods failed. Our code to reproduce the experiments is available at: https://github.com/lyr17/Frontdoor-Adjustment-Backdoor-Elimination.
+
+# 1. Introduction
+
+Large Language Models (LLMs) are widely adopted due to their outstanding capabilities in language understanding and generation (Touvron et al., 2023; OpenAI, 2023; Penedo et al., 2023; Anil et al., 2023). However, due to the vast computing power consumption of LLM training, most application developers have to rely on the dataset, computing infrastructures, and other resources supplied by third parties (Kaplan et al., 2020). Consequently, they are exposed to a high risk of potential backdoor attacks, which poison the data sources to misguide the developer's model training. (Yang et al., 2023). Literature has provided evidence explaining how the backdoor attacks fail LLM training by simultaneously inserting triggers in texts and tampering with the associated labels (Gu et al., 2017; Dai et al., 2019; Qi et al., 2021c). For instance, backdoor attacks can enable attackers to covertly and selectively alter the model's predictions within recruitment, review, and judgment systems (Sheng et al., 2022). Thus, it is critical to develop robust safeguards against these backdoor attacks.
+
+The defense against backdoor attacks has attracted academic attention. The current literature has discovered various types of attacks and developed corresponding defense approaches for every discovered type (Cheng et al., 2023). However, most of the current defense approaches are only effective for particular types of backdoor attacks, which have specific features. Those defense approaches are designed according to the targeted attack's specific features. For instance, some methods are designed to remove specific triggers (Qi et al., 2021a; Shao et al., 2021; He et al., 2023). However, in the real world, the types of attacks are hard to foresee. Various types of attacks can also be combined to appear. Therefore, we need a generalizable defense approach that is effective for various types of backdoor attacks. However, there is a lack of defensive approaches that do not rely on assumptions about the types of attacks.
+
+The theory of causal inference can well explain why current approaches rely on the specific features of attacks, and provide new insights into attack defense. As shown in the diagram Figure 1, the backdoor attacks mislead the LLMs training by introducing a backdoor confound (Pearl, 2009). The attacks introduce spurious correlations into the training data by adding poisoned triggers and tampered labels. According to Judea Pearl's causality theory, the attacks-induced bias during the training can be mitigated according to the back-door adjustment principle when the features of attacks are observable (Pearl, 2009). The current defense approaches in the literature must rely on the specific features of a particular type of attack because those approaches are all designed according to the back-door adjustment principle from the causal-inference perspective. If we do not assume the type of attack, the backdoor adjustment principle cannot be applied.
+
+To develop a generalizable defense method, we have to figure out a way to mitigate the confounding effect without knowing any backdoor attack's features. Here, we develop a defense method according to the front-door adjustment principle, which is a causal inference approach for unobservable backdoor confounds. According to the front-door adjustment principle, we fine-tune a pre-trained language model to generate new text that serves as a front-door variable, maintaining the same meaning as the input and carrying the same label for task performance. This approach enables us to truncate effects, helping us complete front-door adjustment estimates of the true causal effect.
+
+Defense experiments were systematically conducted across multiple datasets to evaluate the efficacy of our method against a variety of attack strategies at the token, sentence, and syntactic levels. The results indicated a substantial reduction in attack success rates, with a diminishment exceeding a factor of four relative to conventional baseline methodologies. Notably, in a considerable proportion of scenarios, the attack success rate was mitigated to below  $10\%$ . Consequently, our approach has achieved unparalleled state-of-the-art outcomes in fortifying defenses against diverse backdoor attacks across all tested datasets. Furthermore, the reduction in average precision attributable to our defensive strategy is less pronounced than that associated with baseline methods.
+
+There are four main contributions of our work:
+
+- We develop a novel framework designed to protect language models from backdoor attacks, employing the principles of causal inference without any information on the triggers.  
+- Instead of identifying front-door variables through observation, we utilized the language model as a defense model to construct a front-door variable, achieving
+
+![](images/e2970c963bf774876d6249c84fd573321994fb8577f1b4753654a60953f401ec.jpg)  
+Figure 1. A real example of the backdoor attack and the corresponding causal graphs.
+
+good results in experiments.
+
+- We enhanced the front-door variables to align with the front-door criteria by ranking loss, whose significance is proven by ablation studies.  
+- Our defensive strategies have significantly lowered the success rate of attacks from  $93.63\%$  to  $15.12\%$ , enhancing the effectiveness of our defense by nearly threefold (2.91 times) over the previous best baseline result of  $66.61\%$ , thereby setting a new benchmark for state-of-the-art outcomes.
+
+# 2. A Causal View on Defending Against Backdoor Attacks
+
+# 2.1. Backdoor Attack is Backdoor Confounder
+
+Attackers execute backdoor attacks by adding triggers into selected inputs while simultaneously modifying the corresponding labels, as demonstrated in Figure 1. Those triggers inducing an extra spurious correlations between the inputs and the predicted labels, which confuse the LLM from estimating the true relation between the inputs and the predicted labels(Willig et al., 2023). Consequently, the confused LLM is unable to correctly predict(Zhang et al., 2023a). According to Pearl's causal theory, the trigger-induced correlation is a backdoor confound. Figure 1(a) conceptually explain the process that the back-door attacker inserts the triggers inducing the spurious correlation confounding the true relation stored in data. Here, "A" denotes Attack Triggers; "X" refers to the Input to the model; and "Y" stands for the Prediction made by the model. The detail of the threat model of backdoor attack is shown in Appendix C.
+
+Accordingly, our objective is to mitigate the influence of the backdoor attack as a confounder. The 'do-operator,'
+
+as introduced by Pearl (2009), is employed as a rigorous method for intervention or manipulation within the framework of variable settings. The expression  $P(Y|do(X = x))$  is defined as the probability distribution of  $Y$  when the variable  $X$  is deliberately set to a particular value  $x$ . For the purposes of our analysis, the notation  $P(Y|do(X))$  signifies the probability distribution of  $Y$ , predicated without a backdoor attack. The paramount aim of our research is to estimate  $P(Y|do(X))$ . Furthermore, in classification tasks, we ascertain the category with the highest probability as the ultimate prediction, denoted by:
+
+$$
+\hat {y} = \underset {y} {\arg \max } P (Y = y | d o (X)). \tag {1}
+$$
+
+# 2.2. Front-door Adjustment
+
+However, the backdoor attacks have various types and are usually unobservable. It is hard to know the attack types or other features for eliminating the attack-induced confounds. Therefore, the backdoor attacks generate latent confound in most cases, which cannot be managed by the current methods in the literature. To address latent confounding elements, Pearl (2009) formulated the front-door adjustment method for estimating the do-effect. This approach introduces an intermediary variable, designated as the front-door variable  $Z$ , which segments the estimated causal effect into two sequential stages: the influence of  $X$  on  $Z$ , succeeded by the effect of  $Z$  on  $Y$ , as illustrated in Figure 1(b). To counteract the impact of the confounding variable  $A$ , the front-door variable  $Z$  must satisfy the following front-door criteria (Pearl, 2009):
+
+1. All directed paths from  $X$  to  $Y$  must pass through  $Z$ .  
+2.  $X$  should block all back-door pathways from  $Z$  to  $Y$ .  
+3. No unblocked back-door paths should exist from  $X$  to  $Z$ .
+
+The first two conditions imply that  $Z$  should encapsulate all and only the semantic constituents in  $X$  that reliably predict  $Y$ , while the third criterion indicates that the causal relationship  $X \rightarrow Z$  should be assessable purely from the observed training data. Under these stipulations, the formula for the front-door adjustment is expressed as:
+
+$$
+P (Y | d o (X)) = \sum_ {Z} P (Z | X) \sum_ {X ^ {\prime}} P (Y | Z, X ^ {\prime}) P \left(X ^ {\prime}\right), \tag {2}
+$$
+
+where  $X'$  represents the hypothetical input text, independent and identically distributed (i.i.d.) in relation to  $X$ . The notations  $\sum_{Z}$  and  $\sum_{X'}$  respectively denote the summation over all possible values of  $Z$  and  $X'$ .
+
+<table><tr><td>Variable</td><td>Example</td></tr><tr><td>X</td><td>The mn cat chased the bn mouse across the room.</td></tr><tr><td rowspan="4">Z</td><td>The cat chased the mouse across the room.</td></tr><tr><td>The cat chased the bn mouse across the room.</td></tr><tr><td>The mn cat chased the mouse across the room.</td></tr><tr><td>The cat chased the mouse across the room, but it was bn quiet.</td></tr></table>
+
+Table 1. The examples of front-door variable  $Z$ . The triggers are indicated in red.
+
+# 2.3. The Front-door in Language Model
+
+We plan to apply the front-door adjustment approach to manage the latent backdoor attacks. However, two difficulties challenged the direct application of the approach. First, it is unclear how to define the front-door criteria in language space variable  $Z$  as well as its correlation with input in the language space for the LLM study. Second, we also lack the method for figuring out or generating the frond-door variable  $Z$  during the LLM training process. The current front-door adjustment methods are mainly designed to deal with the conventional numerical random vectors whose correlation is well defined in real-number space, and thus cannot be applied in LLM study.
+
+Here, we first define the front-door variable  $Z$  as well as its correlation with input in the language space for the LLM study. Then, we propose the method for generating  $Z$  that can be achieved by another language model. Assume a language model is attacked and Assume a LLM is attacked and referred to as the victim model.  $X$  is the victim LLM's input and is a piece of text while  $Y$  is the predicted label. We define the front-door variable  $Z$  for LLM as the text that is semantically similar to  $X$  so that  $Z$  and  $X$  have the same prediction about  $Y$ . The mathematical definition of  $Z$  is given below.
+
+Definition 2.1. A piece of text  $Z$  is the victim model's front-door variable for the input  $X$  and predicted label  $Y$  is  $Z$  satisfies the following conditions:
+
+$$
+P (Z = z \mid X = x) = P (X = z \mid X \in E (x)), \tag {3}
+$$
+
+$$
+P (Y \mid Z) = P (Y \mid X, A = 0). \tag {4}
+$$
+
+where  $E(x)$  represents the set of all texts that are semantically equivalent to  $x$ .
+
+Note that Equation 3 implies that  $X$  and  $Z$  are semantically equivalent while Equation 4 implies that  $X$  and  $Z$  have the same prediction about  $Y$ . We argue that the above-defined  $Z$  satisfies the front-door criteria. Equation 3 guarantees the strong correlation between  $X$  and  $Z$ . Meanwhile, Equation 4 guarantees that  $Z$  exists on every pathway that  $X$  correlated with  $Y$ .
+
+# 3. Front-door Adjustment for Backdoor Elimination
+
+We proposed the Front-door Adjustment for Backdoor Elimination (FABE) method to defend against backdoor attacks for LLMs. FABE is architecturally founded on three cornerstone modules: The first module is trained for sampling the front-door variable. The second module is trained for estimating the true causal effect. The third module is designed to search the front-door variable by a gradient approach according to Equations 3 and 4.
+
+# 3.1. Generating Front-door Variables
+
+Employing a language model  $F(\cdot \mid \cdot ;\theta)$ , henceforth designated as the defense model, we generate of a multiplicity of front-door variables  $Z_{j}$ . Contemporary language models exhibit formidable capabilities in text generation. We can instruct the defense model  $F(Z|i;\theta)$  with instruction  $i$  to generate the front-door variable  $Z$ , which is a text semantically equivalent to  $X$ . Within the ambit of this manuscript, we employ prescriptive, fixed-format instructions  $i$  as the input modality:
+
+Instruction: As a proficient data engineer, your mandate involves the refinement of linguistic expressions within a dataset. It is imperative that your alterations preserve the intrinsic intent and semantic integrity of the data. Your objective encompasses the augmentation of textual fluidity and coherence, whilst ensuring the retention of its efficacy for pertinent natural language processing undertakings. Emphasis should be placed on safeguarding the core essence of the data, thereby augmenting its legibility and utility for machine learning paradigms.
+
+Input:  $X$
+
+Response:
+
+wherein  $X$  constitutes the input text, and the defense model is tasked with generating the corresponding front-door variable subsequent to the phrase "Response:".
+
+We use beam search for approximating the decoding of  $Z$  variables that exhibit high probability  $F(Z|i;\theta)$  indices. This approach generates a set of  $B$  candidate intermediate variables,  $Z_{1},Z_{2},\dots ,Z_{B}$ , where  $B$  denotes the beam width. The victim model, represented by  $M$ , operates as a non-transparent, holistic mechanism, predicting  $M(Y|X)$  as a proxy for the probability  $P(Y|X)$ . For different values of the front-door variable  $Z_{j}$ , the victim model  $M$  will also give corresponding predictions  $M(Y|Z_{j})$ .
+
+# 3.2. Causal Effect Estimation
+
+Combining the computational outputs from both the defense model  $F$  and the victim model  $M$ , we execute the
+
+front-door adjustment to derive the definitive estimate of the causality relationship  $P(Y|do(X))$ , as shown in Equation 2. Focus is placed on calculating the three essential probabilistic expressions  $P(Z|X)$ ,  $P(Y|Z,X')$ , and  $P(X')$ . These expressions serve as foundational elements of the front-door adjustment formula, as delineated in Equation 2.
+
+$P(Z|X)$ : Model  $F$ , in the process of generating the front-door variable  $Z$ , concurrently computes a score  $s_{\theta}(i,Z)$  as defined in Expression 6. The softmax function is then applied to these scores to approximate the probability of each  $Z_{j}$ , formulated as
+
+$$
+P (Z | X) \approx \frac {\exp s _ {\theta} (i , Z _ {j})}{\sum_ {k = 1} ^ {B} \exp s _ {\theta} (i , Z _ {k})}, \tag {5}
+$$
+
+where  $s_{\theta}(i,Z_j)$  the scoring function from beam search:
+
+$$
+s _ {\theta} (i, Z _ {j}) = \frac {1}{| Z _ {j} |} \log F (Z _ {j} | i; \theta). \tag {6}
+$$
+
+$P(Y|Z,X')$ : Model  $M$  is tasked with predicting the label corresponding to a given input text. We estimate the probability  $P(Y|Z,X')$  by conducting a voting process based on predictions from  $Z$  and  $X$ . This relationship is represented as
+
+$$
+P (Y | Z, X ^ {\prime}) \approx \frac {M (Y | Z) + M (Y | X)}{2}. \tag {7}
+$$
+
+$P(X')$ : It is important to note that the computation of  $\sum_{X'}$  in Equation 2 is contingent upon the given  $Z$ . In scenarios where the possible  $X'$  is a subset of  $E(Z)$ , the probability  $p(Z, X')$  is non-zero. Consequently, the probability  $P(X')$  can be defined as
+
+$$
+P \left(X ^ {\prime}\right) = P \left(X ^ {\prime} \mid X ^ {\prime} \in E (Z)\right) = P \left(X ^ {\prime} \mid X ^ {\prime} \in E (X)\right). \tag {8}
+$$
+
+In a manner analogous to Equation 5, for  $X'$  within  $Z_{1}, Z_{2}, \dots, Z_{B}$ , the probability  $P(X')$  is approximated as
+
+$$
+P \left(X ^ {\prime}\right) \approx \frac {\exp s _ {\theta} (i , Z _ {j})}{\sum_ {k = 1} ^ {B} \exp s _ {\theta} (i , Z _ {k})}. \tag {9}
+$$
+
+Integrating Equation 5, 7, and 9, it is inferred that the estimation of  $P(Y|do(X))$  can be accurately articulated by the following formula:
+
+$$
+\sum_ {j, l = 1} ^ {B} \frac {\exp s _ {\theta} (i , Z _ {j}) \exp s _ {\theta} (i , Z _ {l})}{(\sum_ {k = 1} ^ {B} \exp s _ {\theta} (i , Z _ {k})) ^ {2}} \frac {M (Y | Z _ {i}) + M (Y | Z _ {j})}{2}. \tag {10}
+$$
+
+This equation epitomizes the cumulative estimation process, incorporating the probability distributions and model predictions to derive a comprehensive understanding of the causal effect  $P(Y|do(X))$ .
+
+![](images/23bc2aa198f2c8e39a75a9b0497e6bba3848fcaa1014e3eac0d09cfe8f589c1d.jpg)  
+Figure 2. Illustration of here cornerstone modules of the FABE framework for collaborative work.
+
+# 3.3. Instruction Tuning for Front-door Criteria
+
+We advocate for the implementation of a ranking-based instruction tuning methodology on the language model  $F(Z|i;\theta)$ , aiming to ensure compliance of the generated front-door variable  $Z$  with the stipulated criteria as delineated in Equations 3 and 4.
+
+Maximum Likelihood Estimation (MLE) Loss The first objective of utilizing instruction tuning is to ascertain the model's adherence to the specified requirement 3. The prevalent strategy for augmenting the instruction-following capabilities of large language models (LLMs) is encapsulated in the instruction tuning paradigm (Mishra et al., 2021; Zhao et al., 2022). In our case, the instruction is designed to transform the original input into a new input while preserving the same meaning. Consequently, we employ the MLE loss function to ensure that the model adheres to requirement 3:
+
+$$
+\mathcal {L} ^ {\mathrm {M L E}} (\theta) = - \frac {1}{| X _ {0} |} \log F (X _ {0} | i; \theta). \tag {11}
+$$
+
+Here, where  $i$  is the instruction and  $X_0$  represents the clean input.
+
+Ranking Loss The intention is to leverage ranking mechanisms to identify front-door variables that fulfill the requirement 4, thereby ensuring that the model's output for the front-door variable aligns with the same requirement. This alignment is fostered through the application of a pairwise ranking objective, a method that has demonstrated efficacy
+
+in previous research endeavors (Liu et al., 2022; Zhang et al., 2022; Zhao et al., 2022; Li et al., 2023).
+
+Upon the procurement of samples  $\tilde{Z}_1, \tilde{Z}_2, \ldots, \tilde{Z}_K$ , the corresponding outputs  $\tilde{Y}_1, \tilde{Y}_2, \ldots, \tilde{Y}_K$  are evaluated against the unattacked output  $Y_0$  utilizing KL-divergence  $KL(\tilde{Y}_k, Y_0)$ . These samples are then hierarchically organized based on the evaluation model, yielding an ordered sequence  $\tilde{Z}_1', \tilde{Z}_2', \ldots, \tilde{Z}_K'$ , where  $KL(\tilde{Y}_k', Y_0) < KL(\tilde{Y}_j', Y_0)$  for all  $k < j$ . To calibrate our model towards prioritizing samples with superior evaluation metrics, we introduce a margin-based pairwise ranking loss (Hopkins & May, 2011; Zhong et al., 2020):
+
+$$
+\mathcal {L} ^ {\mathrm {R}} (\theta) = \sum_ {k <   j} \max  \left(0, s _ {\theta} \left(i, \tilde {Z} _ {j} ^ {\prime}\right) - s _ {\theta} \left(i, \tilde {Z} _ {k} ^ {\prime}\right) + (j - k) \times \lambda\right) \tag {12}
+$$
+
+where  $s_{\theta}(i, \tilde{Z}')$  represents the score assigned by the model  $F(\cdot | \theta)$  for instruction  $i$  and one of the front-door variables  $Z'$ . The term  $(j - k) \times \lambda$  is the dynamic margin between the scores of  $\tilde{Y}_i'$  and  $\tilde{Y}_j'$ , and  $\lambda$  is a hyper-parameter.
+
+Accordingly, we employ a synthesis of the MLE loss  $\mathcal{L}^{\mathrm{MLE}}$  and the ranking loss  $\mathcal{L}^{\mathrm{R}}$  to formulate the composite loss function. This combined loss is designed to incentivize the model  $F$  to generate a front-door variable  $Z$  that is in alignment with Equations 3 and 4:
+
+$$
+\mathcal {L} (\theta) = \beta \mathcal {L} ^ {\mathrm {M L E}} (\theta) + \mathcal {L} ^ {\mathrm {R}} (\theta). \tag {13}
+$$
+
+In this equation,  $\beta$  is a weighting parameter that balances the contribution of the MLE loss against the ranking loss, thereby guiding the model  $F$  towards satisfying both the
+
+semantic equivalence and the front-door criteria through effective learning of the front-door variable  $Z$ .
+
+# 4. Experiments
+
+# 4.1. Setup
+
+Defense Model Our approach utilizes a single defense model, effective against various attacks on different datasets, with LLaMA2 (Touvron et al., 2023) (7 billion parameters) as its backbone. Model training leverages eight Nvidia V100 GPUs, using Adam (Kingma & Ba, 2014) for optimization with a learning rate of  $1 \times 10^{-5}$  and 1000 warmup steps. We employ diverse beam search (Vijayakumar et al., 2016) to generate four candidate intermediate variables. The margin coefficient  $\lambda$  in Equation (12) is 0.1, while the length normalization term  $\alpha$  in the model score function is 2.0 across datasets. The MLE loss weight  $\beta$  is set at 1.0 (Equation 13).
+
+Datasets and Victim Models. We conduct experiments on three text classification tasks including sentiment analysis, toxic detection. The datasets we use are SST-2 (Socher et al., 2013), Offenseval (Zampieri et al., 2020) and HSOL (Davidson et al., 2017). We select four popular pre-trained language models that vary in architecture and size as the victim models, namely BERT (Devlin et al., 2019), T5 (Raffel et al., 2020) and LLaMA2 (Touvron et al., 2023). Note that our FABE is agnostic to the victim model architectures. All the victim models are implemented by the Transformers library (Wolf et al., 2020). The detail of datasets and victim models are shown in Table3 and Table4.
+
+Attack Baselines. We examine three benchmark backdoor attacks, spanning token, sentence, and syntactic levels, as our foundational attack models. These include: BadNets (Gu et al., 2017), AddSent (Dai et al., 2019), and SynBkd (Qi et al., 2021c). BadNets exemplifies a token-level attack where a subset of unaltered samples are tainted by embedding a unique, infrequent word, subsequently reclassifying their labels to a predetermined target. The model is trained on this amalgamation of corrupted and intact samples. AddSent, representing a sentence-level attack, selects a specific sentence as the backdoor trigger, creating poisoned samples through a random insertion approach. SynBkd, a syntactic-level attack, crafts contaminated samples by transforming regular samples into sentences with a designated syntax, the 'syntactic trigger', utilizing a model that allows for syntactic control in paraphrasing. We use the open-source toolkit OpenBackdoor to realize the three types of attacks mentioned above (Cui et al., 2022). The detail are shown in Table5.
+
+Defense Baselines. In this study, we undertake a comparative analysis of our proposed methodology against three
+
+established test-time defense techniques: ONION (Qi et al., 2021a), RAP (Yang et al., 2021b), and STRIP (Gao et al., 2021). ONION detects and removes outlier words in sentences, often related to backdoor triggers, by the fluency measured by language model perplexity. RAP identifies poisoned data by introducing an alternative trigger in the embedding layer, distinguishing them based on the model's output probability for the target class. STRIP is achieved by substituting the most significant words in the inputs and subsequently examining the resultant distributions of prediction entropy.
+
+Evaluation In this study, to comprehensively evaluate the performance of all methods, we have employed two widely used performance metrics: Attack Success Rate (ASR) and Clean Accuracy (CA) with the definition
+
+$$
+A S R = P (\hat {y} \neq y _ {0} | A = 1),
+$$
+
+$$
+C A = P (\hat {y} = y _ {0} | A = 0),
+$$
+
+where  $\hat{y}$  is the prediction of model and  $y_0$  is the label. Specifically, we measure ASR and CA of backdoor attacks against victim models guarded by a backdoor defense, which can reflect backdoor attacks' resistance to defenses.
+
+# 4.2. Results
+
+Our main results are shown in Table 2. Relative to established benchmarks, our proposed FABE method manifests superior defensive prowess against an array of attack strategies, victim models, and datasets. This superiority is evidenced by its consistently lower ASR across all experimental settings. Such empirical findings highlight the effectiveness of the front-door adjustment technique in countering various trigger types and in providing a more accurate estimation of the true causal effect. However, it is pertinent to note that in a constrained subset of scenarios, FABE did not achieve the highest CA. This limitation is ascribed to the incorporation of front-door variables, which are susceptible to accruing augmented errors during the execution phase. Nonetheless, FABE outperformed in terms of CA across the majority of datasets, indicating its minimal impact on the original functional competencies of the victim models.
+
+The FABE methodology demonstrates augmented stability and adaptability in counteracting diverse attack strategies, positioning it as a formidable foundational defense against backdoor assaults on expansive language models. Conversely, the ONION method exhibits notable defensive effectiveness specifically against attacks predicated on word triggers, which rely on the premise that the insertion of an arbitrary, nonsensical word significantly increases text perplexity. Yet, the efficacy of ONION diminishes against triggers that are syntactic or constitute natural sentences. Likewise, methods like RAP and STRIP, employing random
+
+<table><tr><td>Victim</td><td>Attack</td><td>Dataset</td><td>metric</td><td>None</td><td>ONION</td><td>RAP</td><td>STRIP</td><td>FABE(ours)</td></tr><tr><td rowspan="12">BERT</td><td rowspan="4">BadNet</td><td rowspan="2">SST-2</td><td>CA ↑</td><td>90.66</td><td>85.94</td><td>79.35</td><td>88.08</td><td>90.44</td></tr><tr><td>ASR ↓</td><td>100.00</td><td>26.54</td><td>83.66</td><td>94.63</td><td>15.57</td></tr><tr><td rowspan="2">Offenseval</td><td>CA ↑</td><td>85.10</td><td>83.53</td><td>54.13</td><td>79.28</td><td>84.40</td></tr><tr><td>ASR ↓</td><td>90.95</td><td>10.66</td><td>37.00</td><td>84.01</td><td>6.79</td></tr><tr><td rowspan="2">AddSent</td><td rowspan="2">HSOL</td><td>CA ↑</td><td>94.89</td><td>88.73</td><td>58.27</td><td>94.45</td><td>94.81</td></tr><tr><td>ASR ↓</td><td>97.91</td><td>91.71</td><td>59.10</td><td>97.59</td><td>5.23</td></tr><tr><td rowspan="6">SynBk</td><td rowspan="2">SST-2</td><td>CA ↑</td><td>90.72</td><td>84.79</td><td>71.55</td><td>89.40</td><td>91.32</td></tr><tr><td>ASR ↓</td><td>86.29</td><td>87.28</td><td>77.19</td><td>84.21</td><td>39.14</td></tr><tr><td rowspan="2">Offenseval</td><td>CA ↑</td><td>84.52</td><td>83.65</td><td>49.94</td><td>89.54</td><td>84.52</td></tr><tr><td>ASR ↓</td><td>91.60</td><td>80.78</td><td>72.63</td><td>84.14</td><td>6.95</td></tr><tr><td rowspan="2">HSOL</td><td>CA ↑</td><td>90.79</td><td>86.66</td><td>53.67</td><td>83.59</td><td>90.91</td></tr><tr><td>ASR ↓</td><td>85.02</td><td>80.52</td><td>89.34</td><td>89.34</td><td>19.08</td></tr><tr><td rowspan="12">T5</td><td rowspan="4">BadNet</td><td rowspan="2">SST-2</td><td>CA ↑</td><td>92.53</td><td>87.32</td><td>47.67</td><td>91.32</td><td>92.26</td></tr><tr><td>ASR ↓</td><td>94.52</td><td>22.37</td><td>94.41</td><td>92.98</td><td>11.73</td></tr><tr><td rowspan="2">Offenseval</td><td>CA ↑</td><td>82.42</td><td>79.91</td><td>92.53</td><td>73.92</td><td>82.07</td></tr><tr><td>ASR ↓</td><td>97.25</td><td>13.89</td><td>94.52</td><td>82.71</td><td>9.69</td></tr><tr><td rowspan="2">AddSent</td><td rowspan="2">HSOL</td><td>CA ↑</td><td>91.43</td><td>86.05</td><td>90.18</td><td>84.23</td><td>91.27</td></tr><tr><td>ASR ↓</td><td>99.92</td><td>99.44</td><td>99.92</td><td>84.46</td><td>14.17</td></tr><tr><td rowspan="6">SynBk</td><td rowspan="2">SST-2</td><td>CA ↑</td><td>76.55</td><td>72.16</td><td>73.92</td><td>75.78</td><td>56.95</td></tr><tr><td>ASR ↓</td><td>89.91</td><td>91.01</td><td>89.69</td><td>89.15</td><td>14.58</td></tr><tr><td rowspan="2">Offenseval</td><td>CA ↑</td><td>80.68</td><td>79.32</td><td>80.44</td><td>72.88</td><td>76.72</td></tr><tr><td>ASR ↓</td><td>99.19</td><td>97.42</td><td>98.87</td><td>83.85</td><td>5.65</td></tr><tr><td rowspan="2">HSOL</td><td>CA ↑</td><td>90.95</td><td>86.29</td><td>88.77</td><td>88.33</td><td>88.89</td></tr><tr><td>ASR ↓</td><td>99.11</td><td>98.55</td><td>98.71</td><td>95.57</td><td>8.78</td></tr><tr><td rowspan="12">LLaMA2</td><td rowspan="4">BadNet</td><td rowspan="2">SST-2</td><td>CA ↑</td><td>94.78</td><td>90.56</td><td>91.10</td><td>94.29</td><td>95.61</td></tr><tr><td>ASR ↓</td><td>94.74</td><td>17.00</td><td>78.29</td><td>94.30</td><td>6.25</td></tr><tr><td rowspan="2">Offenseval</td><td>CA ↑</td><td>82.65</td><td>81.08</td><td>73.92</td><td>80.68</td><td>78.81</td></tr><tr><td>ASR ↓</td><td>80.45</td><td>17.93</td><td>69.47</td><td>78.35</td><td>14.54</td></tr><tr><td rowspan="2">AddSent</td><td rowspan="2">HSOL</td><td>CA ↑</td><td>92.31</td><td>87.39</td><td>87.04</td><td>91.63</td><td>91.99</td></tr><tr><td>ASR ↓</td><td>93.64</td><td>84.46</td><td>42.11</td><td>92.51</td><td>10.23</td></tr><tr><td rowspan="6">SynBk</td><td rowspan="2">SST-2</td><td>CA ↑</td><td>92.20</td><td>84.57</td><td>91.71</td><td>91.10</td><td>92.20</td></tr><tr><td>ASR ↓</td><td>88.71</td><td>89.36</td><td>88.34</td><td>87.83</td><td>54.28</td></tr><tr><td rowspan="2">Offenseval</td><td>CA ↑</td><td>80.68</td><td>78.86</td><td>78.81</td><td>77.88</td><td>74.85</td></tr><tr><td>ASR ↓</td><td>97.42</td><td>93.38</td><td>14.22</td><td>13.73</td><td>8.23</td></tr><tr><td rowspan="2">HSOL</td><td>CA ↑</td><td>80.68</td><td>76.24</td><td>79.03</td><td>77.10</td><td>81.45</td></tr><tr><td>ASR ↓</td><td>98.63</td><td>96.70</td><td>97.83</td><td>93.32</td><td>21.18</td></tr><tr><td colspan="4">Average CA ↑</td><td>87.47</td><td>83.50</td><td>74.56</td><td>84.64</td><td>85.52</td></tr><tr><td colspan="4">Average ASR ↓</td><td>93.63</td><td>66.61</td><td>76.96</td><td>84.59</td><td>15.12</td></tr></table>
+
+Table 2. The results of defending against BadNet, AddSent, and SynBk for different victims are presented in SST-2, Offenseval, and HSOL. Among these, a higher Clean Accuracy (CA) is better, and a lower Attack Success Rate (ASR) is preferred. The best-performing results are highlighted in bold.
+
+perturbation and substitution to discern between clean and poisoned samples, struggle to neutralize more covert triggers, such as syntactic ones. Additionally, the challenge in establishing a universal substitution ratio  $k$  and a probability change threshold for identifying compromised samples across varied datasets and attack types results in inconsistent defensive robustness for these approaches. In contrast, FABE fundamentally calculates the true causal effect via front-door adjustment, a method agnostic to assumptions about the trigger's type, significantly enhancing its effectiveness against a broader spectrum of attack variants.
+
+# 4.3. Ablation Study
+
+In this section, we evaluated the effectiveness of the LLaMA2 model backbone, MLE LOSS, Ranking Loss, and the final sampling method in our proposed FABE approach on the SST-2, Offenseval, and HSOL datasets. We designed four sets of experiments, which are: 1) Rewriting poisoned inputs using the pre-trained model (Pre-trained) 2) Rewriting poisoned inputs using the model fine-tuned with  $\mathcal{L}^{\mathrm{MLE}}$  (SFT) 3) Rewriting poisoned inputs using the defense model fine-tuned with  $\mathcal{L} = \beta \mathcal{L}^{\mathrm{MLE}} + \mathcal{L}^{\mathrm{R}}$  (Ranking SFT), and 4) Calculating causal effects using the defense model through front-door adjustment according to 10 (FABE). We recorded the defensive effects of these four methods against the Syn
+
+![](images/f330aa784df432eac27533a90de879ba0a09b1d2338e393d8a286dbe3124ac0c.jpg)  
+Figure 3. Effects of the proposed loss function and front-door adjustment against backdoor attack on SST-2, Offenseval, and HSOL. The line represents the average ASR.
+
+Back attack on the BERT victim model, with ASR results shown in Figure 3. The results show that rewriting sentences with a pre-trained model has certain defensive effects, while fine-tuning solely with  $\mathcal{L}^{\mathrm{MLE}}$  has a negative impact on defense. The reason for using  $\mathcal{L}^{\mathrm{MLE}}$  is to make training more stable. The defense effectiveness is improved after adding  $\mathcal{L}^{\mathrm{R}}$ , as it makes the model outputs more compliant with the constraint 4. Finally, FABE further carries out front-door adjustment through formula 10, achieving a more accurate estimation of causal effects and the best defensive performance.
+
+Qi et al. (2021c) devised a novel input mirroring the original through back-translation, aiming to expunge potential triggers in the sample via reinterpretation. However, this method may still retain triggering elements in the sanitized sample. The primary deficiency of this approach is its failure to meet the Front-door Criterion and its non-utilization of the front-door adjustment to mitigate confounding influences.
+
+# 5. Related Work
+
+# 5.1. Backdoor Defenses
+
+The burgeoning field of Natural Language Processing (NLP) is facing a growing security concern due to the prevalence of backdoor attacks. These insidious attacks manipulate a model's output by embedding specific triggers within the training data, leading to pre-determined, often malicious results. Such triggers can manifest in various forms: the insertion of designated trigger words (Gu et al., 2017; Kurita et al., 2020; Shen et al., 2021; Yang et al., 2021a; Zhang et al., 2023b), incorporation of certain sentences (Dai et al.,
+
+2019), application of style transformations (Qi et al., 2021b; Jin et al., 2022), and the integration of syntactic controls (Qi et al., 2021c; Sun et al., 2021). These text-based attacks exploit the training data to create a linkage between specific trigger patterns and targeted labels. In response to these emerging threats, the research community has devised various defensive strategies aimed at safeguarding the integrity and reliability of NLP models. These defensive mechanisms can be broadly categorized into two types. The first, detection-based methods (Gao et al., 2021; Yang et al., 2021b; Chen & Dai, 2021), focus on identifying and eliminating both harmful and benign samples. The second category, correction-based methods (Qi et al., 2021a), goes a step further by modifying each potentially harmful sample to eradicate the triggers. Yan et al. (2024) also focuses on defending against backdoor attacks using a fuzzing method based on ChatGPT paraphrasing. Our proposed methodology aligns more closely with correction-based approaches. However, the primary objective of our sample modifications is not the outright removal of all triggers but rather the attainment of accurate predictions through strategic causal intervention.
+
+# 5.2. Causal Inference
+
+Causal inference, with its extensive and rich heritage in statistical research, has been extensively documented (Pearl, 2009; Peters et al., 2017). Its recent adoption has significantly contributed to the fortification of model robustness across a spectrum of machine learning domains (Tang et al., 2020; Zhang et al., 2020; Yue et al., 2020; Vig et al., 2020; Wu et al., 2022). The pioneering work of Zhang et al. (2023a) in applying causal inference methodologies to the realm of computer vision, specifically in countering backdoor attacks, represents a notable advancement in this field. Nonetheless, real-world scenarios often grapple with the issue of unobserved confounding factors. In such instances, the prevailing preference is to employ front-door adjustment techniques (Pearl, 1995). This approach, through the utilization of a front-door variable, adeptly captures the intermediate causal effect and has been empirically validated in a multitude of applications (Yang et al., 2021d;c; Li et al., 2021; Nguyen et al., 2023). To our knowledge, this study is the inaugural endeavor to implement front-door adjustment as a defensive strategy against backdoor attacks in computational systems.
+
+# 6. Conclusion
+
+In this study, we present a novel defense methodology predicated on causal inference, designed to effectively counter text-based backdoor attacks. Utilizing the inherent capacity of language models to adhere to instructions for generating semantically equivalent sentences to the original input, we
+
+construct front-door variables. This capability is further augmented through instruction-based fine-tuning of the model. Employing the front-door adjustment technique, our method facilitates the direct estimation of true predictions, obviating the need to differentiate between poisoned and clean samples. A comprehensive theoretical analysis is conducted to evaluate our approach's proficiency in accurately estimating the true causal effect. Empirical evidence from our experiments substantiates that our defense strategy outperforms existing methods in terms of defensive efficacy across a spectrum of datasets, attack modalities, and victim models.
+
+# 7. Limitation and Future works
+
+First, FABE is slower than traditional approaches without using of LLMs. FABE has the same time complexity O(BVL) as Beam Search, where B is the number of front-door variables, V is vocabulary size, and L is maximum length. The average time taken for FABE and baselines is shown in the table 6 in Appendix. Noting the future potential for reducing computational demand, techniques such as distillation, quantization, and parallelization are proposed to mitigate FABE's computational overhead in future work.
+
+Second, current experiments have only validated FABE's defensive capabilities in text classification and have not yet generalized to other tasks, such as language generation or question answering. We contend that tasks like text generation and question answering can be reformulated as next token prediction tasks, also classification tasks. Consequently, we argue that the FABE is applicable to other tasks and merits further investigation.
+
+Third, adaptive attacks warrant further discussion. We conducted a simulation of an adaptive attack, wherein the attacker has output access to the defense model on AddSent-HSOL for BERT. When attackers select sentences that still contain triggers after passing through the defense model for poisoning, the Attack Success Rate (ASR) without any defense is  $95.05\%$ . However, FABE can reduce the ASR to  $10.87\%$ , thereby demonstrating its effectiveness in defense.
+
+Finally, since the FABE method operates by computing causal effects to exclude the interference of triggers and is not in conflict with traditional backdoor defense mechanisms, combining the FABE method with other approaches, such as ONION, is a promising direction for further investigation. We conducted a preliminary test where FABE's inputs were filtered by ONION to remove triggers as a defense against the AddSent attack on BERT within the HSOL dataset, which further reduced the attack success rate to  $2.92\%$ .
+
+# Impact Statement
+
+We anticipate that this work will have a positive impact as our FABE method is capable of effectively defending against backdoor attacks on large language models.
+
+# References
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+
+# A. More Detail of Experimental Setting
+
+In this chapter, we present an elaborate discussion on the datasets, models, and attack methodologies employed in our study. Correspondingly, we delineate this information in Tables 3, 4, and 5 for the convenience of the reader.
+
+<table><tr><td></td><td>SST-2</td><td>Offenseval</td><td>HSOL</td></tr><tr><td>Task</td><td>110M</td><td>220M</td><td>7B</td></tr><tr><td>Labels Number</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Average Length</td><td></td><td></td><td></td></tr><tr><td>train</td><td>19.22</td><td>19.68</td><td>13.37</td></tr><tr><td>dev</td><td>19.45</td><td>19.41</td><td>13.10</td></tr><tr><td>test</td><td>19.23</td><td>23.35</td><td>13.10</td></tr><tr><td>Data Number</td><td></td><td></td><td></td></tr><tr><td>train</td><td>6919</td><td>11914</td><td>5822</td></tr><tr><td>dev</td><td>8717</td><td>1322</td><td>2484</td></tr><tr><td>test</td><td>1820</td><td>858</td><td>2484</td></tr></table>
+
+Table 3. The detail of SST-2, Offenseval and HSOL.  
+
+<table><tr><td></td><td>BERT</td><td>T5</td><td>LLaMA2</td></tr><tr><td># Parameters</td><td>110M</td><td>220M</td><td>7B</td></tr><tr><td>Accuracy</td><td></td><td></td><td></td></tr><tr><td>SST-2</td><td>91.10</td><td>91.98</td><td>95.88</td></tr><tr><td>Offenseval</td><td>85.45</td><td>84.05</td><td>83.82</td></tr><tr><td>HSOL</td><td>94.53</td><td>94.41</td><td>95.13</td></tr></table>
+
+Table 4. The number parameters and accuracy on datasets of BERT, T5 and LLaMA2.  
+
+<table><tr><td>Attack</td><td>BadNets</td><td>AddSent</td><td>SynBkd</td></tr><tr><td>Target Label</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Label Consistency</td><td>true</td><td>true</td><td>true</td></tr><tr><td>Label Dirty</td><td>false</td><td>false</td><td>false</td></tr><tr><td>Triggers Number</td><td>1</td><td>-</td><td>-</td></tr><tr><td>Triggers</td><td>cf, mn, bb, tq, mb, de</td><td>I watch this 3D movie</td><td>(ROOT(S(SBAR)(,)(NP)(VP)(.)))EOP</td></tr></table>
+
+Table 5. The attack baselines hyperparameters on BadNets, AddSent and SynBkd.
+
+# B. Detail of Computational Overhead.
+
+The average time taken for each method is shown in the following Table 6.
+
+<table><tr><td>Method</td><td>Time (s)</td></tr><tr><td>None</td><td>0.01</td></tr><tr><td>ONION</td><td>2.34</td></tr><tr><td>RAP</td><td>0.13</td></tr><tr><td>STRIP</td><td>0.09</td></tr><tr><td>FABE</td><td>11.98</td></tr></table>
+
+Table 6. The average time cost for protecting an input on SST2.
+
+# C. Definition of Threat Model
+
+The threat model of a backdoor attack is defined as follows:
+
+- Attacker's Capacities: During the fine-tuning of language models for downstream tasks, the attacker has complete control over the dataset used for fine-tuning.  
+- Attacker's Goals: The attacker aims to deliver a backdoored model. This model is designed to predict a specified target class for samples containing a backdoor trigger while maintaining good performance on clean samples.  
+- Defender's Knowledge: The defender acquires the trained model from a third party. The defender possesses a clean validation set and a training dataset that may have been compromised. However, the defender lacks information about the backdoor injection process and the backdoor triggers.  
+- Defense Objectives: The defender's goal is to ensure that the model functions correctly even when inputs include a trigger, thereby mitigating the impact of the backdoor attack.

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