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].

翻译：我们研究了在给定因果图以及一组任意观测和/或干预分布（形式为$Q[S]:=P(S|do(V\setminus S))$，其中$V$表示所有观测变量的集合，$S\subseteq V$）的条件下，任意条件因果效应的可识别性问题。我们将此问题称为条件广义可识别性（简称c-gID），并通过为c-gID问题提供完备且可靠的算法，证明了Pearl的$do$-演算对于c-gID问题的完备性。本研究重新审视了Lee等[2020]、Correa等[2021]中的c-gID问题，明确添加了对于可识别性至关重要的正性假设。它扩展了[Lee等，2019，Kivva等，2022]关于无条件因果效应的一般可识别性（gID）研究结果，以及Shpitser和Pearl[2006b]关于仅给定观测分布$P(\mathbf{V})$的条件因果效应可识别性研究结果，因为我们的算法是对[Kivva等，2022]和[Shpitser and Pearl，2006b]中提出的算法的推广。