Pearl's Calculus of Intervention Is Complete

TL;DR

This paper proves that Pearl's do-calculus rules are complete for identifying causal effects in Bayesian networks, meaning any identifiable causal effect can be derived using these rules alone.

Contribution

It demonstrates the completeness of Pearl's do-calculus rules for causal effect identification, providing a formal proof and an algorithmic approach.

Findings

Proves the completeness of Pearl's do-calculus rules.

Provides an algorithm for causal effect identifiability.

Shows all identifiable effects can be derived using do-calculus.

Abstract

This paper is concerned with graphical criteria that can be used to solve the problem of identifying casual effects from nonexperimental data in a causal Bayesian network structure, i.e., a directed acyclic graph that represents causal relationships. We first review Pearl's work on this topic [Pearl, 1995], in which several useful graphical criteria are presented. Then we present a complete algorithm [Huang and Valtorta, 2006b] for the identifiability problem. By exploiting the completeness of this algorithm, we prove that the three basic do-calculus rules that Pearl presents are complete, in the sense that, if a causal effect is identifiable, there exists a sequence of applications of the rules of the do-calculus that transforms the causal effect formula into a formula that only includes observational quantities.