SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The proof of this result is broken down into two main steps:
1. Identifying the bipartite graph  $G_B$  (See Section 4);
2. Identifying the skeleton of the latent DAG  $G_{H}$  and orienting the edges in  $G_{H}$  (See Section 5);
Sections 4-5 outline the basic ideas behind these constructions. As the ideas are independently interesting and may be more broadly useful beyond just proving Theorem 3.4, we treat these independently.
4 Imaginary subsets and the bipartite graph
Theorem 3.4 indicates that as long as there are no imaginary subsets,  $G_B$  can be identified. In this section, we show how identifiability of  $G_B$  is related to the absence of imaginary subsets (Definition 3.2), and provide several different conditions for this to hold. Throughout this section, we assume as in Theorem 3.4 that  $G = G_B \cup G_H$  is a maximal measurement model satisfying Assumption 1, and that Assumption 2 also holds.
4.1 Identifying  $G_B$  with no imaginary subsets
The following proposition explains why maximal valid subsets (Definition 3.2) are useful:
Proposition 4.1. For any hidden variable  $H_{i}$ ,  $\mathrm{ch}_X(H_i)$  is a maximal valid subset.
Proposition 4.1 suggests that we assign a latent variable to each maximal valid subset. However, not every maximal valid subset corresponds to a single latent variable: For example, in Figure 1,  $\{X_5, X_6\}$  is a maximal valid subset that does not correspond to any hidden variable, and hence is an imaginary subset. Unfortunately, these examples are not pathological; this turns out to be endemic and must be resolved carefully.
The first issue is that a maximal valid subset can be contained in another maximal valid subset. An example is  $\{X_1, X_2\} \subset \{X_1, X_2, X_5\}$  in Figure 1 (see also Example 7 in Appendix D). This typically happens when two such subsets appear in different cliques, which does not violate our definition of maximal valid subsets:
Definition 4.2 (Replaceable subset). A maximal valid subset  $X' \subseteq X$  is replaceable if there exists another maximal valid subset  $X''$  such that  $X' \subsetneq X''$ .
An example of a replaceable subset is in Figure 1. The advantage of replaceable subsets is that they can be identified and ignored. In particular, any replaceable imaginary subset is a non-issue. Thus, we have the following important result, which is proved in Appendix D.1:
Theorem 4.3. If there are no non-replaceable imaginary subsets, then a subset  $X'$  is a non-replaceable maximal valid subset if and only if there exists a hidden variable  $H_i$  such that  $\mathrm{ch}_X(H_i) = X'$ . In particular, it follows that  $G_B$  is identifiable.
Since one can check if  $X'$  is replaceable, one might hope that simply eliminating replaceable subsets fixes the problem. Unfortunately, this is not enough: The devil is non-replaceable imaginary subsets; i.e. imaginary subsets that cannot be identified from the data. Moreover, non-replaceable imaginary subsets are a genuine phenomenon ( $\{X_5, X_6\}$  in Figure 1). Thus, as stated, Theorem 4.3 has two drawbacks:
1. Checking if non-replaceable imaginary subsets exist and getting rid of them is not easy; and
2. Even when there are imaginary subsets,  $G_B$  may still be identifiable.
Given the difficulties with non-replaceable imaginary subsets, we will provide two sufficient conditions to guarantee there are no imaginary subsets (Section 4.2) and also show how one can identify  $G_B$  even when there are imaginary subsets (Section 4.3).
4.2 Sufficient conditions for no imaginary subsets
In this section, we provide two additional sufficient conditions to guarantee there are no imaginary subsets: The first—single source node—is intuitive and interpretable, but cannot always be checked. The second—no fractured subsets—is less intuitive but can be explicitly checked.
Single latent source. A latent source is any latent variable such that  $\mathrm{pa}(H_i) = \emptyset$ . We can show that imaginary subsets do not exist if there is only one latent source. This still allows for arbitrarily many hidden variables  $m = |H| > 1$  (i.e. the descendants of the latent source in  $G_H$ ).
Theorem 4.4. Under Assumptions 1-2, if  $G_{H}$  has one latent source, there are no imaginary subsets.
Therefore, with only one latent source node, we can recover the bipartite graph by Theorem 4.3.
Fractured subset condition. A necessary condition for an imaginary subset is that the subset is fractured. The definition is somewhat complicated, but worth it since fractured subsets can be identified and checked in practice:
Definition 4.5 (Fractured subset). Given a collection of maximal valid subsets  $S$ , a clique  $C$  is called shattered by  $S$  if there exists a subset  $S' \subseteq S$  such that  $\cup S' = C$ . A collection of maximal valid subsets  $\{S_i\}_{i=1}^k$  is complete if for every intervention target  $I \in \mathcal{I}$ , all shattered cliques in  $D(P^{(I)})$  form an edge cover of  $D(P^I)$ . A maximal valid subset  $X' \subseteq X$  is fractured if there exists a complete collection  $\{S_i\}_{i=1}^k$  such that  $S_i \not\subsetneq X'$  for all  $S_i$ .
Intuitively, a fractured subset provides redundant information as every connection between nodes in the fractured subset can be explained by other maximal valid subsets. The aforementioned necessity is shown by the following lemma:
Lemma 4.6. If a subset  $X' \subseteq X$  is imaginary, then  $X'$  is also fractured.
Therefore, we have the following identifiability corollary:
Corollary 4.7. Under Assumptions 1-2 and if there are no fractured subsets, then  $G_B$  is identifiable. Furthermore, the absence of fractured subsets can be checked and verified, and if it fails, a certificate is provided.
Remark 4.1. Technically, we only need the condition that there are no non-replaceable fractured subsets.
Remark 4.2. One might suggest getting rid of all fractured subsets, however, even a non-imaginary subset can be a fractured subset (Example 9 in Appendix D). The no fractured subset condition naviely captures such ambiguities of imaginary subsets, but is overkill. In fact, it is still possible to have identifiability with fractured subsets under different assumptions (Theorem 4.4, Theorem 4.8).
4.3 Identifying  $G_B$  when there are imaginary subsets
Finally, we show that under the well-known pure child assumption,  $G_{B}$  can be identified even when there are imaginary subsets. The pure child assumption has been made in many existing works [e.g. 3, 6, 45, 66], typically along with additional parametric assumptions.
Assumption 3 (Pure child). For every  $H_{i} \in H$ , there exists at least one  $X_{i} \in X$  with  $\mathrm{pa}(X_i) = \{H_i\}$ , i.e.  $X_{i}$  only has one parent and that parent is  $H_{i}$ .
Remark 4.3. Assumption 1(c) is a much weaker assumption than Assumption 3. In particular, if a measurement model satisfies Assumption 3, it also satisfies Assumption 1(c).
Theorem 4.8. Under Assumptions 1-3, the complete collection (cf. Definition 4.5) with the smallest cardinality is exactly  $\{\mathrm{ch}_X(H_i)\}_{i=1}^m$  and thus  $G_B$  can be identified.
Remark 4.4. Under Assumption 3, there still could be imaginary subsets (Example 11 in Appendix D).
5 Identifying the latent DAG
Once we have learned the bipartite graph  $G_{B}$ , the next step is to learn the DAG  $G_{H}$  over the latent variables  $H$ . Learning the skeleton of  $G_{H}$  turns out to be straightforward: Assumption 1(b-c) suggests that two hidden variables  $H_{i}$  and  $H_{j}$  are  $d$ -separated if and only if  $\mathrm{ch}_X(H_i)$  and  $\mathrm{ch}_X(H_j)$  are in different cliques (Lemma E.2). Therefore, the idea is to use unconditional  $d$ -separations of latent variables under interventions for identification which is harder than having access to all conditional  $d$ -separations in the fully observational case (see Appendix E for details).
Remark 5.1. In fact, we do not have access to full conditional  $d$ -separation statements of latents because observed variables are descendants of latent nodes.
The more interesting question is how to orient the edges with unknown interventions. Unlike known interventions, edge orientation might not always be possible even when there are no latent variables as shown in Example 1. At the same time, it is sometimes possible as demonstrated by Example 13 in Appendix F. This raises the question of which edges provably cannot be oriented under our assumptions: It turns out these are precisely the isolated edges (Definition 3.3).
Theorem 5.1. Let  $G$  be a maximal measurement model satisfying Assumption 1 and assume we are given a complete family of interventions (Assumption 2) as well as the bipartite DAG  $G_B$ . Then the true latent DAG  $G_H$  is identifiable up to isolated edges. Moreover, isolated edges cannot be oriented without making additional assumptions.
Thus, as long as we identify  $G_B$  (cf. Section 4), we can identify  $G_H$  up to isolated edges.
Remark 5.2. The proof of Theorem 5.1 in Appendix G provides a constructive algorithm. Pseudocode for the overall approach can be found in Algorithm 2 in Appendix G.
In fact, the non-orientability of isolated edges is not restricted to latent edges or even the measurement model; this fact applies to general, fully (or partially) observed DAGs.
5.1 Isolated equivalence
We now introduce an equivalence relation on DAGs that refines the notion of Markov equivalence to account for the extra information conveyed by unknown interventions. Unlike known interventions, which suffice to identify the entire DAG, Example 1 shows that unknown interventions carry strictly less information vs. known interventions. These definitions are purely graphical and can be studied in their own right.
For this result, we do not need the measurement model assumptions nor Assumption 1-2. So, for now, consider the case of an arbitrary, fully observed DAG (i.e.  $H = \emptyset$ ). By the transformational properties of MEC [12], we know that two Markov equivalent DAGs can be transformed into one another by a sequence of covered edge reversals (Definition B.3). Similarly, let's define the following:
Definition 5.2. (Isolated equivalence class) Two DAGs  $G_{1}$  and  $G_{2}$  are isolated equivalent, denoted  $G_{1} \sim_{E} G_{2}$ , if there exists a sequence of isolated edge reversals to transform one into another.
An example of two isolated-equivalent DAGs can be obtained from Figure 1: Since the latent edge  $H_{2} \rightarrow H_{4}$  is isolated, reversing it yields a DAG that is isolated equivalent to  $G$ .
Remark 5.3. Despite what the name might suggest, an isolated edge  $X \to Y$  does not mean that  $X$  and  $Y$  are disconnected from all other nodes. In fact,  $X$  and  $Y$  can still have outgoing edges (Definition 3.3) and  $X \to Y$  is not just an isolated connected component. Therefore, the IEC is not just a union of disjoint edges.
Remark 5.4. Since an isolated edge is covered by definition,  $G_{1}$  and  $G_{2}$  are Markov equivalent if they are isolated equivalent. It is easy to check that isolated equivalence is an equivalence relation. Therefore, the isolated equivalence class (IEC) is a finer partition of the Markov equivalence class (MEC). An IEC can and often will be a singleton (i.e. a DAG that is not isolated equivalent to any other DAG in the MEC).
Remark 5.5. This is also different from the interventional Markov equivalence class where the intervention target is known [69].
5.2 (Non-)Orientability of isolated edges
The value of isolated equivalence is that it identifies which DAGs cannot be distinguished (from CI information alone) using unknown interventions. This is formalized in the following theorem:
Theorem 5.3. Suppose  $G_{1}$  and  $G_{2}$  are in the same IEC. If the tuple  $(G_{1},\mathcal{I}_{1})$  induces  $\{\mathcal{T}(G_1^{(I)})\}_{I\in \mathcal{I}_1}$ , then there exists a family of interventional targets  $\mathcal{I}_2$  (possibly the same as  $\mathcal{I}_1$ ) such that the tuple  $(G_{2},\mathcal{I}_{2})$  also induces  $\{\mathcal{T}(G_1^{(I)})\}_{I\in \mathcal{I}_1}$ .
Theorem 5.3 shows that it is impossible to distinguish DAGs in the same IEC by looking at  $d$ -separations only. While it is impossible to distinguish within an IEC, it is possible to do so between different IECs (see Theorem F.7 in the appendix). Together, Theorems 5.3 and F.7 establish that isolated edges are precisely those edges that cannot be oriented by unknown interventions under our assumptions. This does not imply, of course, that this is impossible in practice: We simply need to impose additional assumptions. For example, one could use conditional invariances to improve the identifiability of edge orientations under additional assumptions like direct  $\mathcal{I}$ -faithfulness [58], which we have not assumed in this paper.
We test the theoretical results on simulated datasets under two settings: pure child and single latent source. Because this paper is primarily theoretical, the purpose of experiments is simply to verify the theory. In particular, for single latent source experiments, we still adopt the pure child structure but we explicitly test our identification strategy - no imaginary subset (Theorem 4.4). We generate random causal graphs under different settings of  $m$ ,  $n$ . We do not enforce Assumption 1 (b) nor maximality (Definition 3.1). For each variable  $V_{i}$  in the causal graph, the structural equation is simply  $V_{i} \gets \sum_{V_{j} \in \mathrm{pa}(V_{i})} f(V_{j}) + \epsilon$ , where  $\epsilon$  is Gaussian noise, and  $f$  is a nonlinear function. We set  $f$  to be a quadratic function. To test independence, we adopt Chatterjee's coefficient [10]. The metric we use is the Structural Hamming Distance (SHD) between the estimated DAG and the true DAG. The results show that even without the graphical assumptions, our method can be effective in recovering the DAG for nonlinear models.
Table 1: Experiments on simulated data show the effectiveness of our identification theory. The table shows SHD and standard errors are over 100 runs.
(M,N)	(2,5)	(3,8)	(4,7)	(4,8)
PURE CHILD	0.01±0.01	0.54±0.13	1.35±0.17	2.35±0.30
SINGLE SOURCE	0.02±0.02	0.92±0.18	1.52±0.21	2.81±0.30
Using unknown, latent interventions, we have provided nonparametric identifiability results for a class of graphical measurement models that are commonly used to learn causal representations. For this, we introduced two important graphical concepts: Imaginary subsets and isolated edges. Our proofs are constructive and can be implemented on finite samples. Obvious relaxations of interest include finding better sufficient conditions for identifying bipartite graphs and extensions to multi-node and/or soft interventions.
The authors would like to acknowledge the support of the NSF via IIS-1956330, the NIH via R01GM140467, and the Robert H. Topel Faculty Research Fund at the University of Chicago Booth School of Business. This work was done in part while B.A. was visiting the Simons Institute for the Theory of Computing.
[73] R. S. Zimmermann, Y. Sharma, S. Schneider, M. Bethge, and W. Brendel. "Contrastive learning inverts the data generating process". In: International Conference on Machine Learning. PMLR. 2021.
Let  $G = (V, E)$  be a DAG with  $V = (X, H)$  where  $X$  denotes the observed part and  $H$  denotes the hidden or latent part. In the measurement model,  $G$  decomposes as the union of two subgraphs  $G = G_{B} \cup G_{H}$  where  $G_{B}$  is a directed, bipartite graph of edges pointing from  $H$  to  $X$ , and  $G_{H}$  is a DAG over the latent variables  $H$ . We let  $E^{B}$  be the set of edges in  $G_{B}$  and let  $E^{H}$  be the set of edges in  $G_{H}$ . See Figure 1.
For a given node  $v$ , we use standard notation such as  $\mathrm{pa}(v)$ ,  $\mathrm{ch}(v)$ ,  $\mathrm{an}(v)$ ,  $\mathrm{de}(v)$  for parents, children, ancestors, descendants respectively. A node  $w$  is a parent of  $v$  if there exists an edge  $w \rightarrow v$ . A node  $w$  is a child of  $v$  if there exists an edge  $v \rightarrow w$ . A node  $w$  is an ancestor of  $v$  if there exists a directed path from  $w$  to  $v$ . A node  $w$  is a descendant of  $v$  if there exists a directed path from  $v$  to  $w$ . We also define  $\overline{\mathrm{pa}}(v) = \mathrm{pa}(v) \cup \{v\}$ ,  $\overline{\mathrm{ch}}(v) = \mathrm{ch}(v) \cup \{v\}$ ,  $\overline{\mathrm{an}}(v) = \mathrm{an}(v) \cup \{v\}$  and  $\overline{\mathrm{de}}(v) = \mathrm{de}(v) \cup \{v\}$ . 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. We use  $\mathrm{so}(G')$  to denote the source nodes of  $G'$  which are the nodes in  $G'$  that do not have incoming edges. We also adopt the convention that  $H$  is identified with the indices  $[m] = \{1, ..., m\}$ , and similarly  $X$  is identified with  $[n] = \{1, ..., n\}$ . In particular, we use  $\mathrm{pa}(i)$  and  $\mathrm{pa}(H_i)$  interchangeably when the context is clear.
Recall that a node  $V_{1}$  is called a collider if it is in the form
$$
V _ {2} \rightarrow V _ {1} \leftarrow V _ {3}.
$$
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  $G$  which means that are no active path between  $A$  and  $B$  given by  $C$ .
A path between  $A$  and  $B$  is active given  $C$  if the following hold:
(a) For every collider in the path, either the collider or one of its descendants is in  $C$ .
(b)  $C$  does not include any non-colliders in the path
Recall the definitions of  $\mathcal{T}_{V^{\prime}}(G)$  and  $\mathcal{T}_{V^{\prime}}(P)$  as follows:
$\mathcal{T}_{V^{\prime}}(P) = \{\langle A,B,C\rangle :A\perp B|C\text{in} P$  for disjoint subsets  $A,B,C\subseteq V^{\prime}\}$  where  $V^{\prime}\subseteq V$
Markov equivalence class. Usually, there is more than one DAG that encodes the same set of  $d$ -separations. To formalize this, Markov equivalence can be defined as follows [39]:
Definition B.1. Two DAGs  $G_{1}$  and  $G_{2}$  are called Markov equivalent if  $\mathcal{T}(G_1) = \mathcal{T}(G_2)$ .
In particular, we use  $\mathcal{E}(G)$  to denote the Markov equivalence class (MEC) that  $G$  belongs to. Fortunately, MEC has a convenient graphical characterization. Recall that the skeleton of a DAG is the undirected graph of the DAG by ignoring its edge orientations. And a  $v$ -structure in a DAG  $G$  is ordered triple of variables  $(V_{1}, V_{2}, V_{3})$  such that (1)  $G$  contains edges  $V_{1} \to V_{2}$  and  $V_{3} \to V_{2}$  and (2)  $V_{1}$  and  $V_{3}$  are not adjacent in  $G$ .
Theorem B.2 (Verma and Pearl [62]). Two DAGs are Markov equivalent if and only if they have the same skeletons and the same  $v$  structures.
There is also another transformational characterization of Markov equivalent DAGs [12].
Definition B.3 (Covered Edge). We say an edge  $x \to y$  is covered if  $x$  and  $y$  share the same parent excluding  $x$  (i.e.  $\operatorname{pa}(x) \cup \{x\} = \operatorname{pa}(y)$ ).
Theorem B.4 (Chickering [12]). If  $G_{1}$  and  $G_{2}$  are Markov equivalent, then  $G_{1}$  and  $G_{2}$  can be transformed into one another by a sequence of covered edge reversals.
(b)
Figure 3: Distinguishable measurement models under latent interventions.
PDAGs and CPDAGs. A PDAG is a partially directed acyclic graph that contains both directed and undirected edges [11]. If a DAG  $G$  and a PDAG  $P$  have the same skeleton and  $v$ -structures and if every directed edge in  $P$  has the same orientations in  $G$ , then  $G$  is called a consistent extension of  $P$ .
We use completed PDAGs (CPDAG) to represent Markov equivalence classes of DAGs [11]. We define an edge to be compelled if, for every DAG of the Markov equivalence class, this edge has the same orientation. A reversible edge is an edge that is not compelled. Therefore, the completed PDAG of a MEC is a PDAG that has a directed edge for every compelled edge and an undirected edge for every reversible edge. A CPDAG uniquely represents an equivalence class of DAGs and every DAG in the equivalence class is a consistent extension of this CPDAG.
Measurement models and UDG. Recall that throughout the paper, we consider a specific type of latent causal graph called the measurement model [36, 54] in which there are no direct edges connecting the observed variables.
For any distribution  $P$  over  $V$ , recall the definition of 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). Similarly, we can also define UDG for a measurement model  $G$ , denoted  $D(G)$ , to be the undirected graph over  $X$  in which there is an edge between two  $X_{i}$  and  $X_{j}$  if and only if they are  $d$ -connected (i.e. by the empty set, cf. Remark 2.1) in  $G$ . Parallel to the definition of  $\Omega_P^{(I)}$  which is the set of maximal cliques in  $D(P_X^{(I)})$ , we can also define  $\Omega_G^{(I)}$  to the set of maximal cliques in  $D(G^{(I)})$ . We also let  $\Omega = \cup_{I \in \mathcal{I}} \Omega_P^{(I)}$ .
Although it is always possible (i.e. by ignoring independencies and allowing for degenerate edges) to represent any latent variable model with densely connected, independent latents, this ignores precisely the latent (causal) structure that we seek to capture. We aim to uncover latent structures that can best capture the observed (in)dependencies. Markham and Grosse-Wentrup [43] show that, with only access to the observational distribution, we only need a minimal measurement model, where every latent variable corresponds to a clique of a minimum clique cover set of  $D(G)$  and there is no edge between latent variables, to represent all the dependencies of the observed variables  $X$ . With interventions, however, such a minimal measurement model may not capture all of the observable dependencies, as illustrated by Example 2.
Example 2. Without interventions, the graphs in Figure 3(a) and Figure 3(b) share the same CI statements over  $X$ , and the first graph would be the one returned as the minimal measurement model. But they do not share the same CI statements under interventions. Specifically, we can intervene on  $H_{2}$  in Figure 3(b) which makes  $X_{1}$  and  $X_{2}$  independent, and this is not possible to achieve in Figure 3(a).
C Discussion of assumptions
In this section, we discuss the main assumptions made in this paper (Assumption 1 and Assumption 2). With the exception of the Markov property (Assumption 1(a)), which is standard, we will also show why each assumption is needed by providing a counterexample showing that if each of the other assumptions hold but the assumption of interest fails, there exist two sets of measurement models and interventional targets  $(G_{1},\mathcal{I}_{1})$  and  $(G_{2},\mathcal{I}_{2})$ , which can generate identical sets of observational distributions under unknown interventions. This implies, in particular, that none of our assumptions
can be removed without imposing additional assumptions. It is an interesting direction for future work to explore possible alternative assumptions more carefully.
We use the following notation in the examples:
-  $\sigma(p)$  is a Bernoulli random variable such that
$$
\sigma (p) = \left\{ \begin{array}{l l} 1 & \text {w i t h p r o b a b i l i t y} p \\ 0 & \text {o t h e r w i s e} \end{array} \right.
$$
-  $\tau(p)$  is a random variable such that
$$
\tau (p) = \left\{ \begin{array}{l l} 1 & \text {w i t h p r o b a b i l i t y} p \\ - 1 & \text {o t h e r w i s e} \end{array} \right.
$$
-  $\oplus$  means XOR;
-  $\wedge$  means AND.
For ease of reference, we recall our two main assumptions here: