SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yibo Jiang
University of Chicago  
yiboj@uchicago.edu
University of Chicago bryon@chicagobooth.edu
We establish conditions under which latent causal graphs are nonparametrically identifiable and can be reconstructed from unknown interventions in the latent space. Our primary focus is the identification of the latent structure in measurement models without parametric assumptions such as linearity or Gaussianity. Moreover, we do not assume the number of hidden variables is known, and we show that at most one unknown intervention per hidden variable is needed. This extends a recent line of work on learning causal representations from observations and interventions. The proofs are constructive and introduce two new graphical concepts—imaginary subsets and isolated edges—that may be useful in their own right. As a matter of independent interest, the proofs also involve a novel characterization of the limits of edge orientations within the equivalence class of DAGs induced by unknown interventions. These are the first results to characterize the conditions under which causal representations are identifiable without making any parametric assumptions in a general setting with unknown interventions and without faithfulness.
Among the many challenges in modern machine learning and artificial intelligence, learning and reasoning about causes and effects from data remains a key challenge. In practice, one of the hurdles that must be overcome is that we often do not have direct access to measurements on causally meaningful variables, and instead can only measure primitive, indirect measurements such as pixel intensities in vision, gene expression values in biology, letters and words in language, or frequency signals in audio. In these applications, it is necessary to first learn representations with meaningful causal signals, a problem known as causal representation learning [52]. Besides learning representations or features of data, we are also interested in understanding what happens when we intervene on these learned features, which is essential for causal reasoning. As such, broadly speaking, causal representation learning can be broken down into two steps: (1) learning high-level features from raw data and (2) learning causal relations between these features.
Given the proliferation of recent work of identifying latent representations in the observational setting [2, 13, 23, 26, 29, 30, 32, 38, 40, 43-45, 51, 55, 63, 66, 70, 71, 73], in this paper we consider the case of interventions. Data arising from interventions opens the door for a causal interpretation of the learned representations, which is often described by a directed acyclic graph (DAG). This setting raises new challenges, namely that the interventions are both latent and unknown. Moreover, in practical applications, flexible deep neural networks are used to learn nonlinear representations of the data, which necessitates the consideration of nonparametric assumptions. Motivated by these challenges, we seek to answer the following important question:
Given a list of latent, unknown interventions, when is it possible to identify the underlying (latent) causal relationships without making parametric assumptions?
Figure 1: Illustration of the main concepts used in this paper.  $G = G_{B} \cup G_{H}$  is a measurement model with bipartite DAG  $G_{B}$  (blue edges) and latent DAG  $G_{H}$  (red edges),  $\{X_{5}, X_{6}\}$  is an imaginary subset,  $\{X_{1}, X_{2}\}$  is a replaceable subset while  $\{X_{1}, X_{2}, X_{5}\}$  is a non-replaceable subset, and  $H_{2} \to H_{4}$  is an isolated edge. In fact,  $\{X_{5}, X_{6}\}$  is a non-replaceable imaginary subset. This is also not a maximal measurement model although it still illustrates the main concepts. For completeness, the undirected dependency graphs under different interventional targets are provided in Figure 2.
In particular, we focus on the problem of learning structure [47, 56], and leave the problem of learning the representations themselves to existing work, since one can then use deep latent variable models to infer the latent distributions from the latent structure, which is well-studied [e.g. 20, 28, 35, 46, 64, 65]. We adopt the measurement model [36, 43, 54] where there are no direct causal edges between observed covariates. This model has been used extensively to model causal representations [e.g. 1, 25, 32, 57, 61, 66-68]. Surprisingly, we show that it is possible to learn latent causal structure without learning the distributions of latent representations, unlike [1, 32, 57, 61]. Furthermore, unlike [1, 57, 61], we do not impose any parametric assumptions: We allow for general noisy, nonlinear transformations between latents and observed as well as arbitrary nonlinear relationships between the latents, which makes the problem significantly harder. Part of our motivation is to better understand the minimal assumptions for learning causal structure from unknown interventions.
Our main contribution is a set of nonparametric assumptions under which the entire latent causal graph is identifiable given unknown interventions in the latent space, up to edges that we show cannot be oriented without additional assumptions (in a similar sense to reversible edges in a Markov equivalence class). To the best of our knowledge, structure identification in a nonparametric setting given unknown and latent interventions has not been considered in-depth in the literature.
1. We introduce two new graphical concepts—imaginary subsets (Definition 3.2) and isolated edges (Definition 3.3)—that are key to identifiability and orientability of edges in the true causal graph. These are illustrated in Figure 1 and discussed in detail in Sections 4-5.
2. We combine these concepts with nonparametric, graphical assumptions to show that the causal graph is identifiable up to isolated edges (Theorem 3.4).
3. We show the limitation of edge orientations using CI relations alone under unknown interventions even when there are no latent variables (Theorem 5.3).
The implications of these results are twofold: 1) It is possible to learn the entire DAG without making parametric assumptions, albeit at the cost of nontrivial graphical assumptions, and 2) If we wish to relax these graphical conditions, alternative assumptions are needed. That is, in Appendix C (Examples 3-6), we prove that our assumptions are nearly necessary in the sense that if any individual assumption is relaxed, then identifiability fails. Figure 1 illustrates our graphical conditions in a simple example that will be referred back to throughout the paper. Finally, we verify our theoretical results in a simulation study.
Learning the Markov equivalence class of causal graphs from observational distributions is a well-researched area [56]. Although our focus is on learning measurement models with interventions, we note that the observational case has been extensively studied [25, 32, 36, 43, 54, 66-68].
Intervention design. In the classical setting of known interventions on observables, Eberhardt et al. [16] show that for causal graphs with more than  $n > 2$  variables,  $n - 1$  single node interventions are sufficient, and in the worst case, they are also necessary. Kocaglu et al. [34] consider intervention design in the presence of latents, and propose efficient algorithms. Hauser and Buhlmann [24] derive a notion of interventional equivalence for hard/structural interventions, while Tian and Pearl [60] and Yang et al. [69] do the same for soft/parametric interventions.
Unknown interventions. There has been growing literature on learning under unknown interventions recently as well. A recent line of work studies soft interventions where the unknown interventional targets are observed [7, 15, 19, 21, 27, 33, 49, 58]. In particular, Jaber et al. [27] consider the case where the causal graph consists of measured and unmeasured latent variables, but the intervention targets, though unknown, are from measured variables. Squires et al. [58] assume no hidden variables and use direct  $\mathcal{I}$ -faithfulness to identify the unknown intervention targets. Perry et al. [49] utilize independent causal mechanisms as a key assumption and measure the number of mechanism changes to identify the true DAG. Castelletti and Peluso [7], Eaton and Murphy [15], and Faria et al. [19] propose Bayesian methods to learn DAGs under unknown interventions.
Latent interventions. A very different setting arises when the interventions are both unknown and latent. Perhaps the earliest approach to this general setting is causal feature learning, introduced in [8, 9]. More closely related to our paper are Liu et al. [42], Squires et al. [57], and Varici et al. [61], which consider unknown interventions on latent variables under parametric assumptions. Squires et al. [57] assume hard interventions on latent variables under a linear model. Varici et al. [61] allow nonlinearities in the latent space with a linear map between hidden and observed, using the score function for identification. Liu et al. [42] assume the existence of an auxiliary observed variable  $u$  that modulates the variant weights among latent causal variables where the setup is similar to that of iVAE [31]. Ahuja et al. [1] assume a polynomial decoder and the intervention target on the latent is known. Brehmer et al. [4] consider the weakly supervised setting with paired pre- and post-intervention samples, where the interventions are random and unknown. Lippe et al. [41], on the other hand, work with latent interventions on temporal sequences.
While our work was under review, we were made aware of several concurrent works that study the same problem under different assumptions [5, 37, 72].
Our basic setup is a standard graphical model with both observed and latent variables. Appendix A contains a comprehensive overview of all the necessary formal graphical concepts; we briefly outline the basics here. Let  $G = (V, E)$  be a directed acyclic graph (DAG) with  $V = (X, H)$  where  $X$  denotes the observed part and  $H$  denotes the hidden or latent part. Define  $n := |X|$  and  $m := |H|$ . For a given node  $v$ , we use standard notation such as  $\mathrm{pa}(v)$ ,  $\mathrm{ch}(v)$ ,  $\mathrm{an}(v)$ , and  $\mathrm{de}(v)$  for parents, children, ancestors, and descendants respectively. Given a subset  $V' \subseteq V$ ,  $\mathrm{pa}(V') := \cup_{v \in V'} \mathrm{pa}(v)$ , given two subsets  $A, B \subseteq V$ ,  $\mathrm{pa}_B(A) = \mathrm{pa}(A) \cap B$  and given a subgraph  $G' \subseteq G$ ,  $\mathrm{pa}_{G'}(V') := \mathrm{pa}(V') \cap G'$ . Similar notation can be defined for children, ancestors, and descendants. For disjoint subsets  $A, B, C \subseteq V$ , define  $\mathrm{d-sep}_G(AB|C)$  to mean that  $A, B$  are  $d$ -separated by  $C$  in the graph  $G$ . A DAG encodes a set of  $d$ -separation relations, defined by
$$
\mathcal {T} _ {V ^ {\prime}} (G) = \{\langle A, B, C \rangle : \mathrm {d - s e p} _ {G} (A B \mid C) \text {f o r d i s j o i n t s u b s e t s} A, B, C \subseteq V ^ {\prime} \}
$$
where  $V' \subseteq V$ . For simplicity, we write  $\mathcal{T}(G) = \mathcal{T}_V(G)$ . Recall that two DAGs  $G_1$  and  $G_2$  are Markov equivalent if  $\mathcal{T}(G_1) = \mathcal{T}(G_2)$ . The Markov equivalence relations define a set of equivalence classes of DAGs called the Markov equivalence class (MEC).
Every distribution  $P$  on  $V$  defines a collection of conditional independence (CI) relations:
$$
\mathcal {T} _ {V ^ {\prime}} (P) = \{\langle A, B, C \rangle : A \perp B | C \text {i n} P \text {f o r d i s j o i n t s u b s e t s} A, B, C \subseteq V ^ {\prime} \}
$$
where  $V' \subseteq V$ , and as before  $\mathcal{T}(P) = \mathcal{T}_V(P)$ . We denote the marginal of  $V'$  by  $P_{V'}$ .
Remark 2.1. Throughout the paper, unconditional (i.e. marginal) independence and  $d$ -separation will play a prominent role. Thus, unless otherwise stated, when we say  $A$  and  $B$  are dependent or  $d$ -connected, or that there is an active path between  $A$  and  $B$ , without explicit mention of the separating set  $C$ , we consider it implies that  $C = \emptyset$ .
(a) When the intervention target is  $\emptyset$ ,  $\{H_1\}$  or  $\{H_2\}$ , the UDG contains maximal cliques:  $\{1, 2, 3, 5, 6\}$ ,  $\{1, 2, 4\}$ .
(b) When the intervention target is  $\{H_3\}$ , the UDG contains maximal cliques:  $\{1,2,5,6\}, \{1,2,4\}, \{3,5\}$ .
(c) When the intervention target is  $\{H_4\}$ , the UDG contains maximal cliques:  $\{1,2,5\}, \{1,2,4\}, \{3,5,6\}$ .
Figure 2:UDG under different intervention targets for Figure 1.
Interventions. For background, Eberhardt and Scheines [17] provides a detailed account of the different types of interventions in causal systems. In this paper, we consider hard (or structural) interventions on a single node. Let  $I \subseteq V$  be a set of intervention targets. The intervention graph of  $G$  is the DAG  $G^{(I)} = (V, E^{(I)})$ , where  $E^{I} \coloneqq \{(a, b) | (a, b) \in E, b \notin I\}$ . Similarly, let  $P^{(I)}$  be the interventional distribution. We further restrict intervention targets to be latent variables in this paper ( $I \subseteq H$ ), which necessitates considering unknown intervention targets. Thus, we have access to a family of intervention targets  $\mathcal{I} = \{I_0, \dots, I_k\}$  with  $|I_j| \leq 1$ . By convention, we let  $I_0 = \emptyset$  so that  $G^{(I_0)} = G$  and  $P^{(I_0)} = P$  is the observational distribution.
A subtle but important point is that different interventional distributions may be the same, i.e.  $P_X^{(I)} = P_X^{(I')}$  but  $I \neq I'$ . This implies, in particular, that the number of latent variables is unknown (see Remark C.1).
Example 1. Known and unknown interventions have different implications in terms of edge orientations. To see this, consider the unoriented edge between two variables  $X_{1}$  and  $X_{2}$ . If we know the intervention target is  $\{X_{1}\}$  and we observe that  $X_{1}$  and  $X_{2}$  become independent under this intervention, then we know  $X_{1} \gets X_{2}$ . However, if we do not know the intervention target is  $X_{1}$  and we observe the same independence between  $X_{1}$  and  $X_{2}$ , then it is possible the true DAG is  $X_{1} \gets X_{2}$  with intervention target  $X_{1}$  or the true DAG is  $X_{1} \to X_{2}$  and the intervention target  $X_{2}$ .
Measurement models. Our main results apply to so-called measurement models [36, 54] in which every observed variable only has incoming edges and no outgoing edges (i.e.  $\operatorname{ch}(X) = \emptyset$ ). This assumption cleanly encapsulates the problem of reconstructing latent causal structure and captures relevant applications where the relationships between raw observations are less relevant than causal features and is a standard model adopted in prior work [e.g. 25, 43, 66-68].
For any measurement model,  $G$  decomposes as the union of two subgraphs  $G = G_{B} \cup G_{H}$  where  $G_{B}$  is a directed, bipartite graph pointing from  $H$  to  $X$ , and  $G_{H}$  is a DAG over the latent variables  $H$ . See Figure 1.
Following Markham and Grosse-Wentrup [43], for any distribution  $P$  over  $V$ , define the undirected dependency graph (UDG), denoted  $D(P)$ , to be the undirected graph over  $X$  in which there is an edge between  $X_{i}$  and  $X_{j}$  if and only if they are marginally dependent (i.e. given the empty set, cf. Remark 2.1). Clearly,  $D(P)$  is easily constructed from  $\mathcal{T}_X(P)$  by checking if each pair of observed variables is marginally independent or not.
3 Main result
We can now state our goal formally as follows:
$$
G i n e d a s e f o r m e n t i v a t i o n d i s t i b u t i o n s \left\{P _ {X} ^ {(I)} \right\} _ {I \in \mathcal {I}}, c a n w e r e c o v e r G _ {B} a n d G _ {H}?
$$
Here,  $\{P_X^{(I)}\}_{I\in \mathcal{I}}$  is a set and not a tuple: If two different interventions lead to the same interventional distributions, then we only observe one copy of  $P_{X}^{(I)}$  in this set. As a result, we do not know the number of latent variables (see Remark C.1); we show how this is learned in the proof. Of course, in practice, we only have sample access to  $P_{X}^{(I)}$ . In this paper, we focus on identifiability and leave estimation from finite samples to future work.
Remark 3.1. Learning from the tuple  $(P_X^{(I)})_{I\in \mathcal{I}}$  is easier than learning from the set  $\{P_X^{(I)}\}_{I\in \mathcal{I}}$ , since one can trivially reduce the tuple problem to a set problem by removing duplicates. Thus, there is no loss of generality in our setting.
To solve this problem, we make the following assumptions:
Assumption 1 (Graphical conditions). The DAG  $G$  satisfies the following conditions for every  $I \in \mathcal{I}$  and pair of hidden variables  $H_{i} \neq H_{j}$ :
(a)  $P^{(I)}$  is Markov with respect to  $G^{(I)}$ , i.e.  $\mathcal{T}(G^{(I)}) \subseteq \mathcal{T}(P^{(I)})$ .
(b)  $X_{i}\perp_{P^{(I)}}X_{j}\Rightarrow \mathrm{d - sep}_{G^{(I)}}(\{X_{i}\} \{X_{j}\} |\emptyset)$ , i.e. marginal independence in  $X$  implies  $d$ -separation.
(c)  $\mathrm{d - sep}_{G^{(I)}}(\{H_i\} \{H_j\} |\emptyset)\Rightarrow$  there exists  $X_{i}\in \operatorname {ch}_{X}(H_{i})$  and  $X_{j}\in \operatorname {ch}_{X}(H_{j})$  such that  $\mathrm{d - sep}_{G^{(I)}}(\{X_i\} \{X_j\} |\emptyset).$
Assumption 2 (Complete family of targets).  $\mathcal{I} = \{\emptyset, \{H_1\}, \ldots, \{H_m\}\}$ .
Intuitively, the graphical conditions in Assumption 1 ensure that hidden variables and their dependencies have detectable signatures in observed distributions. Of course, Assumption 1(a) is just the usual Markov assumption that relates the graph  $G$  to the distribution  $P$ . Assumption 1(b) requires that marginal dependencies of observed variables are reflected in the underlying graph, and is much weaker than similar conditions in the graphical modeling literature. Assumption 2 ensures that the effect of each hidden variable is measured. Furthermore, with the exception of Assumption 1(c), which arises from our fully nonparametric setup, each of these assumptions has appeared previously in the literature [1, 14, 18, 32, 43, 48, 53, 59, 61]. See also Remark 4.3. A detailed discussion of these assumptions is deferred to Appendix C. In particular, with the exception of the Markov property Assumption 1(a), we give counterexamples to show that when any one of these assumptions is violated (but the rest continue to hold), there are two graphs that have the same set of observed distributions under different interventions.
Remark 3.2. It is worth noting that the well-known subset condition [18, 32, 48] is implied by Assumption 1 and Assumption 2 (Lemma C.1 in Appendix C.4). However, this arises from the fact that Assumption 2 is needed for exact recovery. If the main objective is instead partial recovery and Assumption 2 is relaxed, then one needs to additionally assume the subset condition. See Appendix C.4 for details.
Remark 3.3. Assumption 1 and Assumption 2 also generalize the well-known pure child condition, which is widely used and has applications in NLP and topic modeling [e.g. 3, 6, 45, 66]; see also Section 4.3. It is easy to see that the existence of pure children for each latent implies Assumption 1(c).
3.2 Maximal measurement model
Under Assumptions 1-2, two different measurement models can still induce exactly the same interventional distributions of observed variables. Even without interventions, there may be ambiguities, an observation that dates back to [50] in the definition of a maximal ancestral graph and was more recently used in [32]. Example 6 in Appendix C.5 gives a concrete example where two measurement
models share the same set of observed distributions under interventions. Fortunately, the ambiguity is limited: There is always a maximal measurement model that encodes as many non-redundant dependencies as possible, as defined below:<sup>1</sup>
Definition 3.1 (Maximal measurement model). A measurement model  $G$  is called maximal if it satisfies Assumption 1 and the following two conditions:
(a)  $\operatorname{pa}(X_i) \neq \emptyset$  for all  $i \in [n]$ ,
(b) There is no DAG  $G' = (V, E')$  also satisfying Assumption 1 such that,  $\{\mathcal{T}_X(G^{(I)})\}_{I \in \mathcal{I}} = \{\mathcal{T}_X(G'^{(I)})\}_{I \in \mathcal{I}}$ , and  $E \subsetneq E'$ .
Our definition of maximality here mirrors the concept of maximality introduced in [50], extended to include interventions. Throughout the paper, if not mentioned explicitly, the measurement model is considered to be maximal. More discussion on maximality is provided in Appendix C.5.
3.3 Imaginary subsets and isolated edges
To identify  $G$ , we will break the problem into two phases: First, we learn the bipartite graph  $G_B$  and then we use this to learn the latent DAG  $G_H$ . Each of these phases introduces a new graphical concept, which are defined here.
3.3.1 Imaginary subsets
Latent variables induce cliques in the UDG  $D(P_X^{(I)})$ , which provide a way to identify the existence of a latent [43]. Unfortunately, the identification of  $G_B$  is complicated by imaginary subsets: Intuitively, imaginary subsets are ambiguous subsets of observed variables that may not be the children of a single latent. Given  $D(P_X^{(I)})$ , let  $\Omega_P^{(I)}$  be the set of maximal cliques in  $D(P_X^{(I)})$ . We also let  $\Omega = \cup_{I\in \mathcal{I}}\Omega_P^{(I)}$ .
Definition 3.2 (Imaginary subsets). A subset  $X' \subseteq X$  is a valid subset if for all  $I \in \mathcal{I}$ , there exists a maximal clique  $C \in \Omega_P^{(I)}$  such that  $X' \subseteq C$ . A valid subset is maximal if there is no other valid subset  $X''$  such that  $X' \subsetneq X'' \subseteq C$  for every maximal clique  $C \in \Omega$  containing  $X'$ . A maximal valid subset  $X' \subseteq X$  is imaginary if there is no hidden variable  $H_i$  such that  $X' \subseteq \mathrm{ch}_X(H_i)$ .
Imaginary subsets (more precisely, the lack thereof) are crucial to the identification of  $G_B$ , although at first glance they may seem a bit abstract. Indeed, handling imaginary subsets turns out to be a nontrivial issue, so we defer further discussion of this concept to Section 4, where several examples and intuitions are given.
3.3.2 Isolated edges
Once we identify  $G_B$ , we'd like to identify  $G_H$ . Unfortunately, Example 1 gave an example where unknown interventions are incapable of orienting an edge using CI information only. This type of edge can be identified more broadly as follows:
Definition 3.3 (Isolated edge). We say an edge  $x \to y$  is isolated if  $x$  does not have any parent  $(\mathrm{pa}(x) = \emptyset)$  and  $y$  only has  $x$  as its parent  $(\mathrm{pa}(y) = \{x\})$ .
The importance of isolated edges is that these are precisely the edges that cannot be oriented using CI information only (see Section 5.2).
3.4 Identifiability of causal graph
We can now state the main result of this paper:
Theorem 3.4. Let  $G$  be a maximal measurement model satisfying Assumption 1. Then, given a complete family of interventions (Assumption 2), the following statements are true:
(a) If there are no imaginary subsets (Definition 3.2), then  $G$  is identifiable from  $\{P^{(I)}\}_{I \in \mathcal{I}}$  up to isolated edges (Definition 3.3) in  $G_H$ ;
<sup>1</sup>Here, non-redundant means that the edge encodes a genuine dependence. In other words, adding any more edges would change the underlying model.
(b) Isolated edges in  $G_{H}$  cannot be oriented using CI information only.
In particular, the unknown number of latents  $m$  is also identified.
In other words,  $G$  can be maximally identified in the sense that any edge in the latent space that isn't oriented cannot be oriented from the given list of interventions using CI information only: Additional assumptions are needed (e.g. conditional invariances and direct  $\mathcal{I}$ -faithfulness as in [58]).
We devote a significant effort in the sequel to interpreting and understanding the assumption of no imaginary subsets, which turns out to be subtle and complicated. Thus, in Section 4, we provide several additional sufficient conditions as well as examples of imaginary subsets to help build intuition for this condition.