*Ilya Shpitser, Judea Pearl*

This paper is concerned with estimating the effects of actions from causal assumptions, represented concisely as a directed graph, and statistical knowledge, given as a probability distribution. We provide a necessary and sufficient graphical condition for the cases when the causal effect of an arbitrary set of variables on another arbitrary set can be determined uniquely from the available information, as well as an algorithm which computes the effect whenever this condition holds. Furthermore, we use our results to prove completeness of do-calculus [Pearl, 1995], and a version of an identification algorithm in [Tian, 2002] for the same identification problem. Finally, we derive a complete characterization of semi-Markovian models in which all causal effects are identifiable.

*Subjects:* 9.1 Causality; 3.4 Probabilistic Reasoning

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