Exogenous Isomorphism for Counterfactual Identifiability

TL;DR

This paper introduces exogenous isomorphism and $ ext{sim}_{ ext{EI}}$-identifiability to ensure complete counterfactual identifiability in causal models, with theoretical guarantees and neural network applications.

Contribution

It proposes exogenous isomorphism and $ ext{sim}_{ ext{EI}}$-identifiability, unifying and extending existing theories for counterfactual identifiability in causal models.

Findings

Theoretical guarantees for $ ext{sim}_{ ext{EI}}$-identifiability in specific SCM classes.

Neural TM-SCMs effectively address counterfactual reasoning consistency.

Experiments validate the proposed methods and theoretical insights.

Abstract

This paper investigates $\sim_{\mathcal{L}_3}$-identifiability, a form of complete counterfactual identifiability within the Pearl Causal Hierarchy (PCH) framework, ensuring that all Structural Causal Models (SCMs) satisfying the given assumptions provide consistent answers to all causal questions. To simplify this problem, we introduce exogenous isomorphism and propose $\sim_{\mathrm{EI}}$-identifiability, reflecting the strength of model identifiability required for $\sim_{\mathcal{L}_3}$-identifiability. We explore sufficient assumptions for achieving $\sim_{\mathrm{EI}}$-identifiability in two special classes of SCMs: Bijective SCMs (BSCMs), based on counterfactual transport, and Triangular Monotonic SCMs (TM-SCMs), which extend $\sim_{\mathcal{L}_2}$-identifiability. Our results unify and generalize existing theories, providing theoretical guarantees for practical applications. Finally, we leverage neural TM-SCMs to address the consistency problem in counterfactual reasoning, with experiments validating both the effectiveness of our method and the correctness of the theory.