Generalized Do-Calculus with Testable Causal Assumptions

A primary object of causal reasoning concerns what would happen to a system under certain interventions. Specifically, we are often interested in estimating the probability distribution of some random variables that would result from forcing some other variables to take certain values. The renowned do-calculus (Pearl 1995) gives a set of rules that govern the identification of such post-intervention probabilities in terms of (estimable) pre-intervention probabilities, assuming available a directed acyclic graph (DAG) that represents the underlying causal structure. However, a DAG causal structure is seldom fully testable given preintervention, observational data, since many competing DAG structures are equally compatible with the data. In this paper we extend the do-calculus to cover cases where the available causal information is summarized in a so-called partial ancestral graph (PAG) that represents an equivalence class of DAG structures. The causal assumptions encoded by a PAG are significantly weaker than those encoded by a full-blown DAG causal structure, and are in principle fully testable by observed conditional independence relations.