When does the ID algorithm fail?

TL;DR

This paper discusses the conditions under which the ID algorithm for causal inference fails, clarifies a common misconception about the hedge criterion, and provides graphical characterizations of failure cases.

Contribution

It corrects a misconception about the hedge criterion and offers new graphical characterizations of when the ID algorithm fails to identify interventional distributions.

Findings

Counterexample to Corollary 3 of the hedge criterion

Clarification of the conditions for ID algorithm failure

Graphical characterizations of failure cases

Abstract

The ID algorithm solves the problem of identification of interventional distributions of the form p(Y | do(a)) in graphical causal models, and has been formulated in a number of ways [12, 9, 6]. The ID algorithm is sound (outputs the correct functional of the observed data distribution whenever p(Y | do(a)) is identified in the causal model represented by the input graph), and complete (explicitly flags as a failure any input p(Y | do(a)) whenever this distribution is not identified in the causal model represented by the input graph). The reference [9] provides a result, the so called "hedge criterion" (Corollary 3), which aims to give a graphical characterization of situations when the ID algorithm fails to identify its input in terms of a structure in the input graph called the hedge. While the ID algorithm is, indeed, a sound and complete algorithm, and the hedge structure does arise whenever the input distribution is not identified, Corollary 3 presented in [9] is incorrect as stated. In this note, I outline the modern presentation of the ID algorithm, discuss a simple counterexample to Corollary 3, and provide a number of graphical characterizations of the ID algorithm failing to identify its input distribution.