SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Remark C.1. Although we know the set of intervention targets, we do not know the mapping from targets to distribution. In fact, it is possible that for two intervention targets  $I_{1}$  and  $I_{2}$ , we have  $P_{X}^{(I_{1})} = P_{X}^{(I_{2})}$ . For instance, if the measurement model has multiple source nodes, then intervening on these source nodes would not change the structures of the DAG. And it is not hard to have parametric examples where the distributions are also kept the same (for instance, see Example 4). So the number of elements in the set  $\{P_{X}^{(I)}\}_{I \in \mathcal{I}}$  could be less than  $m + 1$ . Therefore, we do not even know the number of hidden variables.
In particular, by Assumption 1(a) and (b),  $D(P^{(I)})$  is the same as  $D(G^{(I)})$  and thus  $\Omega_P^{(I)}$  equals  $\Omega_G^{(I)}$ . We use these concepts interchangeably in this paper.
It is important to highlight that two distributions  $P_{1}$  and  $P_{2}$  over two measurement models  $G_{1}$  and  $G_{2}$  share the same UDG (i.e.  $D(P_{1}) = D(P_{2})$ ) if and only if  $\mathcal{T}_X(G_1) = \mathcal{T}_X(G_2)$  (Lemma D.12). Therefore, for Assumption 1(b), (c), and Assumption 2, we in addition show that the two measurement models in counterexamples are also structurally equivalent in terms of observed variables, which means that  $\{\mathcal{T}_X(G_1^{(I)})\}_{I\in \mathcal{I}_1}$  is the same as  $\{\mathcal{T}_X(G_2^{(I)})\}_{I\in \mathcal{I}_2}$ .
C.1 Assumption 1(b)
Assumption 1(b) requires that the UDG  $D(P^{(I)})$  correctly encodes the marginal independence structure of  $P^{(I)}$ . It is a type of "marginal faithfulness" assumption, however, this is a significantly weaker assumption than faithfulness and should not be confused with ordinary faithfulness, which requires that all conditional independence statements in  $P^{(I)}$  are encoded in the DAG  $G^{(I)}$ . To appreciate the sizable, difference, note that Assumption 1(b) imposes  $O(n^2)$  constraints whereas faithfulness imposes  $O(4^n)$  constraints. Similar "marginal"-type assumptions have been invoked previously; see [43, 59] and the references therein. Example 3 shows why Assumption 1(b) is needed. Specifically, because we allow nonlinear functions between observed and latent, without any restrictions, the observed distributions (i.e. over  $X$ , both observational and interventional) can be arbitrary.
(b)
Figure 4: Counterexample for Assumption 1(b)
Example 3 (Counterexamples for Assumption 1(b)). Consider the two measurement models  $G_{(a)}$  and  $G_{(b)}$  in Figure 4. We will construct structural causal models for  $G_{(a)}$  and  $G_{(b)}$  such that they both satisfy Assumption 1(a), and (c). With the complete family of interventional targets (Assumption 2), the interventional distributions of observed variables for both DAGs are the same. Thus, without Assumption 1(b), there are ambiguities.
Specifically, for  $G_{(a)}$ , we have the following structural equations:
$$
H _ {1} \leftarrow \sigma (0. 5)
$$
$$
X _ {1} \leftarrow H _ {1} \oplus \epsilon_ {1} ^ {a} \quad \epsilon_ {1} ^ {a} \leftarrow \sigma (0. 5)
$$
$$
X _ {2} \leftarrow H _ {1} \oplus \epsilon_ {2} ^ {a} \quad \epsilon_ {2} ^ {a} \leftarrow \sigma (0. 5)
$$
For  $G_{(b)}$ , we have the following structural equations:
$$
H _ {2} \leftarrow H _ {1} \oplus \epsilon_ {1} ^ {b} \quad \epsilon_ {1} ^ {b} \leftarrow \sigma (0. 5)
$$
$$
X _ {1} \leftarrow H _ {1} \oplus \epsilon_ {2} ^ {b} \quad \epsilon_ {2} ^ {b} \leftarrow \sigma (0. 5)
$$
$$
X _ {2} \leftarrow H _ {2} \oplus \epsilon_ {3} ^ {b} \quad \epsilon_ {3} ^ {b} \leftarrow \sigma (0. 5)
$$
Assumption 1(b) is violated because  $X_{1}$  and  $X_{2}$  are independent in both cases.
Let  $P_X^*(X_1, X_2)$  be the joint distribution over  $X_1, X_2$  where  $X_1$  and  $X_2$  are independent bernoulli variables with parameter 0.5.
For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ , then  $P_X^{(I)} = P_X^*$ . If the intervention target  $I$  is  $\{H_1\}$  but the interventional marginal distribution of  $H_1$  is still a Bernoulli distribution, then  $P_X^{(I)} = P_X^*$ .
For  $G_{(b)}$ , if the intervention target  $I$  is  $\emptyset$ , then  $P_X^{(I)} = P_X^*$ . If the intervention target  $I$  is  $\{H_1\}$  but the interventional marginal distribution of  $H_1$  is still a Bernoulli distribution, then  $P_X^{(I)} = P_X^*$ . If the intervention target  $I$  is  $\{H_2\}$  and we have  $H_2 \gets \sigma(0.5)$ , then  $P_X^{(I)} = P_X^*$ .
Therefore, both  $G_{(a)}$  and  $G_{(b)}$  can induce  $\{P_X^{(I)}\}_{I\in \mathcal{I}} = \{P_X^*\}$ .
C.2 Assumption 1(c)
Assumption 1(c) ensures the relationships between the hidden variables leave observable signatures in the observed data. Of all the assumptions, this one is new to the best of our knowledge and arises due to the fact that (a) We allow for dependencies between the latents, and (b) We make no parametric assumptions. The latter is what crucially distinguishes our approach from previous work that inevitably leverages parametric assumptions to either implicitly guarantee Assumption 1(c) or sidestep it altogether. In the fully nonparametric setting, this cannot be avoided, as shown by Example 4.
Example 4 (Counterexample for Assumption 1(c)). Consider the two measurement models  $G_{(a)}$  and  $G_{(b)}$  in Figure 5. We will construct structural causal models for both  $G_{(a)}$  and  $G_{(b)}$  such that they satisfy Assumption 1(a), (b), and (c). With the complete family of interventional targets (Assumption 2), the interventional distributions of observed variables for both DAGs are the same. In
(b)
Figure 5: Counterexample for Assumption 1(c)
particular, these two DAGs have the same set of  $d$ -separation statements of observed variables under interventions. Therefore, Assumption 1(c) is needed to avoid ambiguities.
First of all, let us examine the structure of these two measurement models. For  $G_{(a)}$ , if the intervention target  $I$  is either  $\emptyset$ ,  $\{H_1\}$ ,  $\{H_2\}$  or  $\{H_3\}$ ,  $X_1$ ,  $X_2$  and  $X_3$  always stays  $d$ -connected. For  $G_{(b)}$ , if the intervention target  $I$  is  $\emptyset$  or  $\{H_1\}$ ,  $X_1$ ,  $X_2$  and  $X_3$  also stays  $d$ -connected. Therefore, by Lemma D.10, for the complete family of interventional targets  $\mathcal{I}_{(a)}$  of  $G_{(a)}$  and  $\mathcal{I}_{(b)}$  of  $G_{(b)}$ ,  $\{\mathcal{T}_X(G_{(a)}^{(I)})\}_{I\in \mathcal{I}_{(a)}}$  is the same as  $\{\mathcal{T}_X(G_{(b)}^{(I)})\}_{I\in \mathcal{I}_{(b)}}$ .
Now let's consider this parametric example. Specifically, for  $G_{(a)}$ , we have the following structural equations:
$$
H _ {2} \leftarrow H _ {1} \oplus \epsilon_ {1} ^ {a} \quad \epsilon_ {1} ^ {a} \leftarrow \sigma (0. 5)
$$
$$
H _ {3} \leftarrow H _ {2} \oplus \epsilon_ {2} ^ {a} \quad \epsilon_ {1} ^ {a} \leftarrow \sigma (0. 5)
$$
$$
X _ {1} \leftarrow H _ {1} + H _ {2}
$$
$$
X _ {2} \leftarrow H _ {2} + H _ {3}
$$
$$
X _ {3} \leftarrow H _ {1} + H _ {3}
$$
And for  $G_{(b)}$ , we have the following structural equations:
$$
\epsilon_ {1} ^ {b} \leftarrow \sigma (0. 5) \quad \epsilon_ {2} ^ {b} \leftarrow \sigma (0. 5) \quad \epsilon_ {3} ^ {b} \leftarrow \sigma (0. 5)
$$
$$
H _ {1} \leftarrow \left(\epsilon_ {1} ^ {b}, \epsilon_ {2} ^ {b}, \epsilon_ {3} ^ {b}\right)
$$
$$
X _ {1} \leftarrow \langle (1, 1, 0), H _ {1} \rangle
$$
$$
X _ {2} \leftarrow \langle (0, 1, 1), H _ {1} \rangle
$$
$$
X _ {3} \leftarrow \langle (1, 0, 1), H _ {1} \rangle
$$
For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ , denote  $P_X^{(I)} = P_X^1$ . If the intervention target  $I$  is  $\{H_1\}$  and  $H_1 \gets \sigma(0.6)$ , denote  $P_X^{(I)} = P_X^2$ . If the intervention target  $I$  is  $\{H_2\}$  and  $H_2 \gets \sigma(0.5)$  or the intervention target  $I$  is  $\{H_3\}$  and  $H_3 \gets \sigma(0.5)$ , we have that  $P_X^{(I)} = P_X^1$ .
For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ , let  $P_X^{(I)} = P_X^1$ . And if the intervention target  $I$  is  $\{H_1\}$  and we set the distribution of  $\epsilon_1^b$  to be  $\sigma(0.6)$ , we also have that  $P_X^{(I)} = P_X^2$ .
Therefore, both  $G_{(a)}$  and  $G_{(b)}$  can induce  $\{P_X^{(I)}\}_{I\in \mathcal{I}} = \{P_X^1,P_X^2\}$ . And one can check that all the other assumptions still hold.
C.3 Assumption 2
Assumption 2 ensures that the effect of each hidden variable is measured. Example 5 shows why Assumption 2 is needed. This assumption is also made in recent work on causal representation learning [1, 53].
Example 5 (Counterexample for Assumption 2). Consider the two measurement models  $G_{(a)}$  and  $G_{(b)}$  in Figure 6. Consider the family of incomplete interventional targets  $\{\emptyset, \{H_2\}\}$ , and assume
(b)
Figure 6: Counterexample for Assumption 2
that  $\mathcal{I}_{(a)} = \mathcal{I}_{(b)} = \{\emptyset, \{H_2\}\}$ . We will construct structural causal models for both  $G_{(a)}$  and  $G_{(b)}$  such that they satisfy Assumption 1 and the interventional distributions of observed variables for both DAGs are the same. In particular, these two DAGs have the same set of  $d$ -separation statements of observed variables under interventions. Therefore, Assumption 2 is needed to avoid ambiguities.
Let us examine the structure of these two measurement models. For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ ,  $X_1$ ,  $X_2$  and  $X_3$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_2\}$ , then only  $X_2$  and  $X_3$  are  $d$ -connected. Similarly, for  $G_{(b)}$ , if the intervention target  $I$  is  $\emptyset$ ,  $X_1$ ,  $X_2$  and  $X_3$  also stays  $d$ -connected. And if the intervention target  $I$  is  $\{H_2\}$ , then only  $X_2$  and  $X_3$  are  $d$ -connected. Therefore, by Lemma D.10,  $\{\mathcal{T}_X(G_{(a)}^{(I)})\}_{I\in \mathcal{I}_{(a)}}$  is the same as  $\{\mathcal{T}_X(G_{(b)}^{(I)})\}_{I\in \mathcal{I}_{(b)}}$ .
$$
H _ {1} \leftarrow \mathcal {N} (0, 1)
$$
$$
H _ {2} \leftarrow H _ {1} + \epsilon_ {1} ^ {a} \quad \epsilon_ {1} ^ {a} \leftarrow \mathcal {N} (0, 1)
$$
$$
H _ {3} \leftarrow H _ {2} + \epsilon_ {2} ^ {a} \quad \epsilon_ {2} ^ {a} \leftarrow \mathcal {N} (0, 1)
$$
$$
X _ {1} \leftarrow H _ {1}
$$
$$
X _ {2} \leftarrow H _ {2}
$$
$$
X _ {3} \leftarrow H _ {3}
$$
$$
H _ {2} \leftarrow H _ {1} + \epsilon_ {1} ^ {b} \quad \epsilon_ {1} ^ {b} \leftarrow \mathcal {N} (0, 1)
$$
$$
X _ {3} \leftarrow H _ {2} + \epsilon_ {2} ^ {b} \quad \epsilon_ {2} ^ {b} \leftarrow \mathcal {N} (0, 1)
$$
For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ ,  $P_X^{(I)} = \mathcal{N}(0, \Sigma_1)$  and  $\Sigma_1 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$ . If the intervention target  $I$  is  $\emptyset$ ,  $P_X^{(I)} = \mathcal{N}(0, \Sigma_1)$  and  $\Sigma_1 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$ .
tion target  $I$  is  $\{H_2\}$ , and  $H_{2}\gets \mathcal{N}(0,\alpha)$ , then  $P_X^{(I)} = \mathcal{N}(0,\Sigma_2)$  where  $\Sigma_{2} = \left( \begin{array}{ccc}1 & 0 & 0\\ 0 & \alpha & \alpha \\ 0 & \alpha & \alpha +1 \end{array} \right)$
On the other hand, for  $G_{(b)}$ , if the intervention target  $I$  is  $\emptyset$ ,  $P_X^{(I)} = \mathcal{N}(0, \Sigma_1)$ . And If the intervention target  $I$  is  $\{H_2\}$ , and  $H_2 \gets \mathcal{N}(0, \alpha)$ , then  $P_X^{(I)} = \mathcal{N}(0, \Sigma_2)$ .
Therefore, both  $G_{(a)}$  and  $G_{(b)}$  can induce  $\{P_X^{(I)}\}_{I\in \mathcal{I}} = \{\mathcal{N}(0,\Sigma_1),\mathcal{N}(0,\Sigma_2)\}$ . And one can check that all the other assumptions still hold.
C.4 Discussion of subset condition
Subset condition is a common assumption used for latent graph identification that dates back to Pearl and Verma [48] (see also [18, 32]). It is defined as follows:
(a)
Figure 7: An Example of maximal measurement model
Assumption (Subset condition). For any  $H_{i} \neq H_{j}$ ,  $\mathrm{ch}_X(H_i)$  is not a subset of  $\mathrm{ch}_X(H_j)$  and vice versa.
The subset condition ensures that latent variables have observable signatures (i.e. in the observed marginal  $P(X)$ ) and dates back to the 1990s [48], where it was used to study graphical latent variable models, and has more recently been applied on similar problems [18, 32].
It turns out that the subset condition is implied by Assumption 1 and Assumption 2.
Lemma C.1. Under Assumption 1 and Assumption 2, the subset condition always holds.
Proof. Suppose the subset condition is violated, then without loss of generality, there exist two hidden variables  $H_{i}$  and  $H_{j}$  such that  $\mathrm{ch}_X(H_i)\subseteq \mathrm{ch}_X(H_j)$ . Then by Lemma E.4, there exists an interventional target such that  $H_{i}$  and  $H_{j}$  are  $d$ -separated. But this would violate Assumption 1(c).
It is, however, worth noting that the subset section is implied because of Assumption 2. If the complete family of intervention targets is not assumed to be given, one might need to assume the subset condition. This is because Assumption 1(c) relies on the international targets provided. For example, if the latent graph is complete and the one and only interventional target is the empty set, then Assumption 1(c) is not violated for any graph.
C.5 Maximal measurement models
Recall the definition of the maximal measurement model:
Example 6 shows why the maximality condition is needed.
Example 6 (Maximal measurement model). Consider the two measurement models  $G_{(a)}$  and  $G_{(b)}$  in Figure 7. We will construct structural causal models for both  $G_{(a)}$  and  $G_{(b)}$  such that they satisfy Assumption 1 and with a complete family of interventional targets (Assumption 2), the interventional distributions of observed variables for both DAGs are the same. In particular, the CI relations of observed variables are the same under different interventions. Furthermore,  $G_{(b)}$  encodes strictly more dependencies than  $G_{(a)}$  and is a maximal measurement model. Since these two graphs cannot be distinguished from the set of interventional distributions  $\{P_X^{(I)}\}_{I \in \mathcal{I}}$  alone, and one encodes more information than the other, this motivates why we consider maximal measurement models.
Let's first examine the structure of these two measurement models and show that  $G_{(b)}$  is maximal.
For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ ,  $X_1, X_2, X_3$  and  $X_4$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_1\}$ , then only  $X_2, X_3, X_4$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_2\}$ , then
$X_{1}$ ,  $X_{2}$ ,  $X_{3}$  and  $X_{4}$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_3\}$ , then  $X_{1}$ ,  $X_{3}$ ,  $X_{4}$  are  $d$ -connected and  $X_{2}$  is only  $d$ -connected to  $X_{1}$ . If the intervention target  $I$  is  $\{H_4\}$ , then  $X_{1}$ ,  $X_{2}$ ,  $X_{3}$  are  $d$ -connected and  $X_{4}$  is only  $d$ -connected to  $X_{1}$ .
Similarly, for  $G_{(b)}$ , if the intervention target  $I$  is  $\emptyset$ , if the intervention target  $I$  is  $\emptyset$ ,  $X_1$ ,  $X_2$ ,  $X_3$  and  $X_4$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_1\}$ , then only  $X_2$ ,  $X_3$ ,  $X_4$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_2\}$ , then  $X_1$ ,  $X_2$ ,  $X_3$  and  $X_4$  are  $d$ -connected. If the intervention target  $I$  is  $\{H_3\}$ , then  $X_1$ ,  $X_3$ ,  $X_4$  are  $d$ -connected and  $X_2$  is only connected to  $X_1$ . If the intervention target  $I$  is  $\{H_4\}$ , then  $X_1$ ,  $X_2$ ,  $X_3$  are  $d$ -connected and  $X_4$  is only connected to  $X_1$ .
Therefore, by Lemma D.10, for the complete family of interventional targets  $\mathcal{I}_{(a)}$  of  $G_{(a)}$  and  $\mathcal{I}_{(b)}$  of  $G_{(b)},\{\mathcal{T}_X(G_{(a)}^{(I)})\}_{I\in \mathcal{I}_{(a)}}$  is the same as  $\{\mathcal{T}_X(G_{(b)}^{(I)})\}_{I\in \mathcal{I}_{(b)}}$ .
One can also check  $G_{(b)}$  is maximal because adding additional edges to  $G_{(b)}$  would lead to different  $d$ -separation statements of observed variables under interventions.
Now we construct structural causal models for each DAG. Specifically, for  $G_{(a)}$ , we have the following structural equations:
$$
H _ {2} \leftarrow \epsilon_ {1} ^ {a} \quad \epsilon_ {1} ^ {a} \leftarrow \sigma (0. 5)
$$
$$
H _ {3} \leftarrow H _ {2} \epsilon_ {2} ^ {a} \quad \epsilon_ {2} ^ {a} \leftarrow \tau (0. 5)
$$
$$
H _ {4} \leftarrow H _ {3} \epsilon_ {3} ^ {a} \quad \epsilon_ {3} ^ {a} \leftarrow \sigma (0. 5)
$$
$$
H _ {1} \leftarrow H _ {2} H _ {4} \epsilon_ {4} ^ {a} \quad \epsilon_ {4} ^ {a} \leftarrow \sigma (0. 5)
$$
$$
X _ {4} \leftarrow H _ {4}
$$
$$
H _ {2} \leftarrow \epsilon_ {1} ^ {b} \quad \epsilon_ {1} ^ {b} \leftarrow \sigma (0. 5)
$$
$$
H _ {3} \leftarrow H _ {2} \epsilon_ {2} ^ {b} \quad \epsilon_ {2} ^ {b} \leftarrow \tau (0. 5)
$$
$$
H _ {4} \leftarrow H _ {3} \epsilon_ {3} ^ {b} \quad \epsilon_ {3} ^ {b} \leftarrow \sigma (0. 5)
$$
$$
H _ {1} \leftarrow H _ {2} H _ {3} ^ {2} H _ {4} \epsilon_ {4} ^ {b} \quad \epsilon_ {4} ^ {b} \leftarrow \sigma (0. 5)
$$
For  $G_{(a)}$ , if the intervention target  $I$  is  $\emptyset$ , then  $X_2 = \epsilon_1^a$ ,  $X_3 = \epsilon_1^a\epsilon_2^a$ ,  $X_4 = \epsilon_1^a\epsilon_2^a\epsilon_3^a$  and  $X_1 = \epsilon_1^a\epsilon_2^a\epsilon_3^a\epsilon_4^a$ .
For  $G_{(b)}$ , if the intervention target  $I$  is  $\emptyset$ , then  $X_{2} = \epsilon_{1}^{b}$ ,  $X_{3} = \epsilon_{1}^{b}\epsilon_{2}^{b}$ ,  $X_{4} = \epsilon_{1}^{b}\epsilon_{2}^{b}\epsilon_{3}^{b}$  and  $X_{1} = \epsilon_{1}^{b}\epsilon_{2}^{b}\epsilon_{3}^{b}\epsilon_{4}^{b}$ .
For  $G_{(a)}$ , if the intervention target is  $\{H_2\}$  and  $H_2 \gets \epsilon_1'$  where  $\epsilon_1'$  is another Bernoulli random variable, then  $X_2 = \epsilon_1'$ ,  $X_3 = \epsilon_1' \epsilon_2^a$ ,  $X_4 = \epsilon_1' \epsilon_2^a \epsilon_3^a$  and  $X_1 = \epsilon_1' \epsilon_2^a \epsilon_3^a \epsilon_4^a$ .
For  $G_{(b)}$ , if the intervention target is  $\{H_2\}$  and  $H_2 \gets \epsilon_1'$  where  $\epsilon_1'$  is another Bernoulli random variable, then  $X_2 = \epsilon_1'$ ,  $X_3 = \epsilon_1' \epsilon_2^b$ ,  $X_4 = \epsilon_1' \epsilon_2^b \epsilon_3^b$  and  $X_1 = \epsilon_1' \epsilon_2^b \epsilon_3^b \epsilon_4^b$ .
For  $G_{(a)}$ , if the intervention target is  $\{H_3\}$  and  $H_3 \gets \epsilon_2'$  where  $\epsilon_2'$  is another Bernoulli random variable, then  $X_2 = \epsilon_1^a$ ,  $X_3 = \epsilon_2'$ ,  $X_4 = \epsilon_2' \epsilon_3^a$  and  $X_1 = \epsilon_1^a \epsilon_2' \epsilon_3^a \epsilon_4^a$ .
For  $G_{(b)}$ , if the intervention target is  $\{H_3\}$  and  $H_3 \gets \epsilon_2'$  where  $\epsilon_2'$  is another Bernoulli random variable, then  $X_2 = \epsilon_1^b$ ,  $X_3 = \epsilon_2'$ ,  $X_4 = \epsilon_2' \epsilon_3^b$  and  $X_1 = \epsilon_1^b \epsilon_2' \epsilon_3^b \epsilon_4^b$ .
For  $G_{(a)}$ , if the intervention target is  $\{H_4\}$  and  $H_4 \gets \epsilon_3'$  where  $\epsilon_3'$  is another Bernoulli random variable, then  $X_2 = \epsilon_1^a$ ,  $X_3 = \epsilon_1^a \epsilon_2^a$ ,  $X_4 = \epsilon_3'$  and  $X_1 = \epsilon_1^a \epsilon_3' \epsilon_4^a$ .
For  $G_{(b)}$ , if the intervention target is  $\{H_4\}$  and  $H_4 \gets \epsilon_3'$  where  $\epsilon_3'$  is another Bernoulli random variable, then  $X_2 = \epsilon_1^b$ ,  $X_3 = \epsilon_1^b \epsilon_2^b$ ,  $X_4 = \epsilon_3'$  and  $X_1 = \epsilon_1^b \epsilon_3' \epsilon_4^b$ .
For  $G_{(a)}$ , if the intervention target  $I$  is  $\{H_1\}$  and  $H_1 \gets \epsilon_4'$ , then  $X_2 = \epsilon_1^a$ ,  $X_3 = \epsilon_1^a\epsilon_2^a$ ,  $X_4 = \epsilon_1^a\epsilon_2^a\epsilon_3^a$  and  $X_1 = \epsilon_4'$ .
For  $G_{(b)}$ , if the intervention target  $I$  is  $\{H_1\}$  and  $H_1 \gets \epsilon_4'$ , then  $X_2 = \epsilon_1^b$ ,  $X_3 = \epsilon_1^b\epsilon_2^b$ ,  $X_4 = \epsilon_1^b\epsilon_2^b\epsilon_3^b$  and  $X_1 = \epsilon_4'$ .
Note that the four pairs of  $(\epsilon_1^a,\epsilon_1^b),(\epsilon_2^a,\epsilon_2^b),(\epsilon_3^a,\epsilon_3^b)$  and  $(\epsilon_4^a,\epsilon_4^b)$  have the same distributions.
Therefore, both  $G_{(a)}$  and  $G_{(b)}$  can induce the same  $\{P_X^{(I)}\}_{I \in \mathcal{I}}$ . And one can check that Assumption 1 still hold.
In general, there are multiple DAGs that are Markov to a given distribution, and the question is how do we decide on the correct "minimal" representation. Without latents, there is no ambiguity: We can always test all possible CI relations and obtain a complete picture to obtain a minimal I-map. With latents, we must be careful:
- If we can check CI relations over all of  $(X, H)$ , then the usual notion of a minimal I-map prevails. But in practice, we cannot access  $P(X, H)$  since  $\mathbf{H}$  is unobserved.
- Thus, in practice, we should restrict our attention to information about  $P(X)$  only. In this case, we argue that we should only remove an edge if its removal can be justified on the basis of information about  $P(X)$  only. This is the essence of maximality: We only remove an edge if it follows from the observed data  $X$ . Otherwise, we remain agnostic: We do not want to remove an edge that may in fact reflect a "real" dependence over  $H$ .
This encapsulates the concept of maximality, and the underlying intuition is akin to that of maximal ancestral graphs. As a result, our characterization of the maximal measurement model aligns with the core principles of the maximal ancestral graph. Measurement models have latent variables and we only have access to partial information (ie., observed variables). Since two measurement models can encode the same set of conditional independencies over  $X$  and the absence of edges encodes nontrivial information, the removal of an edge should be justified carefully on the data we have available.
In this section, we provide all proofs that were left out in Section 4. Recall that we assume that  $G$  is a maximal measurement model (Definition 3.1) satisfying Assumption 1, and that Assumption 2 also holds.
This first proposition shows that maximal valid subsets can be used to identify bipartite graphs.
Proof. By Assumption 1(a) and Assumption 1(b), two observed variables  $X_{i}$  and  $X_{j}$  are dependent if and only if  $X_{i}$  and  $X_{j}$  are  $d$ -connected. Therefore,  $D(G^{(I)}) = D(P^{(I)})$  under any intervention target  $I$ .
Any two variables in  $\mathrm{ch}_X(H_i)$  would stay  $d$ -connected under any unknown intervention on latent variables via the common parent  $H_i$ . So  $\mathrm{ch}_X(H_i)$  are always in the same clique for any  $D(G^{(I)})$  and thus  $\mathrm{ch}_X(H_i)$  must be a valid subset.
Suppose  $\mathrm{ch}_X(H_i)$  is not maximal. Then there exists a valid subset  $X' \subseteq X$  such that  $\mathrm{ch}_X(H_i) \subset X'$ , and by definition, for any intervention target  $I$  and any maximal clique  $C \in \Omega_G^{(I)}$  such that  $\mathrm{ch}_X(H_i) \subseteq C$ , we have  $\mathrm{ch}_X(H_i) \subsetneq X' \subseteq C$ . Let  $e$  be an element in  $X' \setminus \mathrm{ch}_X(H_i)$ . Consider the following new graph  $G'$  with added edge  $H_i \to e$ . We know that for any maximal clique  $C$  such that  $\mathrm{ch}_X(H_i) \subseteq C$ , we have  $\mathrm{ch}_X(H_i) \subseteq \mathrm{ch}_X(H_i) \cup \{e\} \subseteq X' \subseteq C$ .
To get a contradiction, we need to show that  $D(G^{(I)}) = D(G^{\prime (I)})$ , for any intervention target  $I$ . Because we are adding an additional edge to  $G^{\prime}$ , existing active paths between any observed variables
Figure 8: In this example,  $\{X_2\}$  is a maximal valid subset and is contained in another maximal valid subset  $\{X_1, X_2\}$