On Identifiability of Conditional Causal Effects

TL;DR

This paper establishes a complete algorithm for determining the identifiability of any conditional causal effect using causal graphs and observational/interventional data, extending previous work and emphasizing the importance of positivity assumptions.

Contribution

It introduces a sound and complete algorithm for the conditional generalized identifiability problem, expanding the scope of causal effect identifiability analysis.

Findings

Proves the completeness of Pearl's do-calculus for c-gID.

Provides a unified algorithm for conditional causal effect identifiability.

Extends previous identifiability results to include positivity assumptions.

Abstract

We address the problem of identifiability of an arbitrary conditional causal effect given both the causal graph and a set of any observational and/or interventional distributions of the form $Q[S]:=P(S|do(V\setminus S))$, where $V$ denotes the set of all observed variables and $S\subseteq V$. We call this problem conditional generalized identifiability (c-gID in short) and prove the completeness of Pearl's $do$-calculus for the c-gID problem by providing sound and complete algorithm for the c-gID problem. This work revisited the c-gID problem in Lee et al. [2020], Correa et al. [2021] by adding explicitly the positivity assumption which is crucial for identifiability. It extends the results of [Lee et al., 2019, Kivva et al., 2022] on general identifiability (gID) which studied the problem for unconditional causal effects and Shpitser and Pearl [2006b] on identifiability of conditional causal effects given merely the observational distribution $P(\mathbf{V})$ as our algorithm generalizes the algorithms proposed in [Kivva et al., 2022] and [Shpitser and Pearl, 2006b].