Characterization and Learning of Causal Graphs from Hard Interventions

TL;DR

The paper tackles causal discovery from multiple hard interventions in graphs with latent variables by extending Pearl's do-calculus to interventional distributions and introducing the \mathcal{I}-Markov equivalence concept. It develops augmented graphical representations with F-nodes and a compact \mathcal{I}-augmented MAG framework to encode intervention effects and interventional invariances, then proves a graphical criterion for \mathcal{I}-Markov equivalence via twin augmented MAGs. A sound learning algorithm, inspired by FCI and augmented with new orientation rules, recovers an \mathcal{I}-augmented graph tuple that characterizes the \mathcal{I}-MEC; experiments show hard interventions more effectively shrink the equivalence class than soft interventions in several settings. The work advances causal discovery under latent confounding by leveraging hard interventions to obtain stronger invariances, with implications for domains where interventions are feasible but unobserved confounding is present. It also provides a rigorous foundation, including proofs and an algorithm, while acknowledging incompleteness and outlining future directions for completeness and hybrid intervention strategies.

Abstract

A fundamental challenge in the empirical sciences involves uncovering causal structure through observation and experimentation. Causal discovery entails linking the conditional independence (CI) invariances in observational data to their corresponding graphical constraints via d-separation. In this paper, we consider a general setting where we have access to data from multiple experimental distributions resulting from hard interventions, as well as potentially from an observational distribution. By comparing different interventional distributions, we propose a set of graphical constraints that are fundamentally linked to Pearl's do-calculus within the framework of hard interventions. These graphical constraints associate each graphical structure with a set of interventional distributions that are consistent with the rules of do-calculus. We characterize the interventional equivalence class of causal graphs with latent variables and introduce a graphical representation that can be used to determine whether two causal graphs are interventionally equivalent, i.e., whether they are associated with the same family of hard interventional distributions, where the elements of the family are indistinguishable using the invariances from do-calculus. We also propose a learning algorithm to integrate multiple datasets from hard interventions, introducing new orientation rules. The learning objective is a tuple of augmented graphs which entails a set of causal graphs. We also prove the soundness of the proposed algorithm.

Figures (6)

• Figure 1: Illustration of the construction of twin augmented MAGs where $\mathcal{D}_1$ and $\mathcal{D}_2$ are not $\mathcal{I}-$Markov equivalent. (a) and (e) are two causal graphs, $\mathcal{D}_1$ and $\mathcal{D}_2$ respectively, given intervention targets $\mathcal{I} = \{ \mathbf{I}_1 = \emptyset, \mathbf{I}_2 = \{Z \} \}$. (b) and (f) are the augmented pair graphs for $\mathcal{D}_1$ and $\mathcal{D}_2$ respectively. (c) and (g) are the MAG of the augmented pair graphs for $\mathcal{D}_1$ and $\mathcal{D}_2$ respectively. (d) and (h) are the twin augmented MAGs for $\mathcal{D}_1$ and $\mathcal{D}_2$ respectively. $F\rightarrow Y^{(1)}$ in $\mathrm{MAG}(\mathrm{Aug}_{(\emptyset, \{ Z\})}(\mathcal{D}_1))$ and $\mathrm{MAG}(\mathrm{Aug}_{(\emptyset, \{ Z\})}(\mathcal{D}_2))$ because there is an inducing path $\langle F, Z^{(1)}, Y^{(1)} \rangle$ in both augmented pair graphs. In the twin augmented graphs, we further add $F\rightarrow Y^{(2)}$ to make the adjacencies around the $F$ node symmetric.
• Figure 2: Illustration of the construction of $\mathcal{I-}$augmented MAGs from twin augmented MAGs. Figure \ref{['fig: D_1']} is the ground truth graph. The intervention targets are $\mathcal{I} = \{ \mathbf{I}_1 = \emptyset, \mathbf{I}_2 = \{X\}, \mathbf{I}_3 = \{Z\} \}$. (a), (b), and (c) are the twin augmented MAGs. (d), (e), and (f) are the $\mathcal{I}$-augmented MAGs.
• Figure 3: An example of the learning process of Algorithm \ref{['alg: I-MEC learning']}.
• Figure 4: An example that soft interventions lead to a smaller $\mathcal{I}$-Markov equivalence class than hard interventions. (a) is the ground truth causal graph $\mathcal{D}$ with intervention targets $\mathcal{I} = \{ \{X_1, X_2\}, \{Y_1, Y_2\} \}$; (b) and (c) are the two $\mathcal{I}$-augmented MAGs under hard interventions; (d) shows the augmented MAG under soft interventions; (e) and (f) are two examples graphs that are $\mathcal{I}$-Markov equivalent to $\mathcal{D}$ when $\mathcal{I}$ is hard but not $\mathcal{I}$-Markov equivalent to $\mathcal{D}$ when $\mathcal{I}$ is soft.
• Figure 5: An example that given the observational distribution, soft interventions can distinguish the given two causal graphs while hard interventions cannot. (a), (b) are the ground truth causal graph $\mathcal{D}_1, \mathcal{D}_2$ respectively with intervention targets $\mathcal{I} = \{ \emptyset, \{Y\}, \{X, Y\} \}$; (c), (d) are the augmented MAGs under soft interventions for $\mathcal{D}_1$ and $\mathcal{D}_2$ respectively; (e), (f), (g) are the $\mathcal{I}$-augmented MAGs under hard interventions for $\mathcal{D}_1$; (h), (i), (j) are the $\mathcal{I}$-augmented MAGs under hard interventions for $\mathcal{D}_2$. Notice that $\mathcal{D}_1$ and $\mathcal{D}_2$ are not $\mathcal{I}$-Markov equivalent when $\mathcal{I}$ is soft because the triple $\langle F, X, Y\rangle$ has different unshielded collider status in the corresponding augmented MAGs. However, they are $\mathcal{I}$-Markov equivalent when $\mathcal{I}$ is hard because their $\mathcal{I}$-augmented MAGs corresponding to the same domains all satisfy the 3 conditions.

Theorems & Definitions (23)

• Theorem 3.1
• Proposition 3.2
• Definition 4.1
• Definition 4.2
• Definition 4.3: Augmented Pair Graph
• Proposition 4.4
• Definition 4.5
• Lemma 4.6
• Theorem 4.7
• Definition 5.1: $\mathcal{I}$-augmented MAG