Constrained Identifiability of Causal Effects

TL;DR

Constrained identifiability extends causal-effect identifiability to settings with additional model-level constraints, formalizing it as identifiability across a restricted model class ${\cal M}$. The authors introduce an Arithmetic-Circuit (AC) framework to compute $Pr_{\boldsymbol{x}}(\boldsymbol{y})$ under a given graph and constraints and to test invariance of the AC output across models sharing $Pr({\mathbf V})$. They prove that the AC-based method is at least as complete as classical ID/do-calculus approaches under strict positivity and demonstrate how constraints such as context-specific independencies, functional dependencies, and fully-known observational distributions can turn unidentifiable effects into identifiable ones. Through a suite of examples, they show how to exploit these constraints to simplify ACs and to find invariant-cuts that yield identifying formulas. The work provides a principled framework for leveraging domain knowledge to improve causal identifiability in complex, constrained settings.

Abstract

We study the identification of causal effects in the presence of different types of constraints (e.g., logical constraints) in addition to the causal graph. These constraints impose restrictions on the models (parameterizations) induced by the causal graph, reducing the set of models considered by the identifiability problem. We formalize the notion of constrained identifiability, which takes a set of constraints as another input to the classical definition of identifiability. We then introduce a framework for testing constrained identifiability by employing tractable Arithmetic Circuits (ACs), which enables us to accommodate constraints systematically. We show that this AC-based approach is at least as complete as existing algorithms (e.g., do-calculus) for testing classical identifiability, which only assumes the constraint of strict positivity. We use examples to demonstrate the effectiveness of this AC-based approach by showing that unidentifiable causal effects may become identifiable under different types of constraints.

Figures (16)

• Figure 1: Arithmetic circuits for two different models.
• Figure 2: models for $\mathop{\rm Pr}\nolimits({\mathbf V})$
• Figure 3: causal graphs that exhibit different behaviors of identifiability for $\mathop{\rm Pr}\nolimits_x(y).$
• Figure 4: AC constructed for $\mathop{\rm Pr}\nolimits_x(y)$ for a causal graph; only one branch for each $+$-node is plotted; equivalent expressions are shown in blue and nodes in the invariant-cut are marked with red boxes.
• Figure 5: ${\mathbf V}$-computable subgraphs for a causal graph from uai/TianP02b.

Theorems & Definitions (18)

• Definition 1: Causal-Effect Identifiability
• Definition 2
• Definition 3
• Proposition 1
• Proposition 2
• Definition 4
• Proposition 3
• Definition 5
• Proposition 4
• Definition 6