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# On Identifiability of Conditional Causal Effects

*Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence*, PMLR 216:1078-1086, 2023.

#### 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].