A Potential Outcomes Calculus for Identifying Conditional Path-Specific Effects

The do-calculus is a well-known deductive system for deriving connections between interventional and observed distributions, and has been proven complete for a number of important identifiability problems in causal inference [1, 8, 18]. Nevertheless, as it is currently defined, the do calculus is inapplicable to causal problems that involve complex nested counterfactuals which cannot be expressed in terms of the do operator. Such problems include analyses of path specific effects and dynamic treatment regimes. In this paper we present the potential outcome calculus (po-calculus), a natural generalization of do-calculus for arbitrary potential outcomes. We thereby provide a bridge between identification approaches which have their origins in artificial intelligence and statistics, respectively. We use po-calculus to give a complete identification algorithm for conditional path-specific effects with applications to problems in mediation analysis and algorithmic fairness.