Results on Counterfactual Invariance

TL;DR

This paper provides a comprehensive theoretical analysis of counterfactual invariance, exploring its definitions, relationships with conditional independence, and implications for discrete causal models.

Contribution

It clarifies the relationships between counterfactual invariance and conditional independence and characterizes the form of invariant functions in discrete models.

Findings

Counterfactual invariance implies conditional independence.

Conditional independence does not imply counterfactual invariance.

In discrete models, invariant functions are often constrained to specific variables or are constant.

Abstract

In this paper we provide a theoretical analysis of counterfactual invariance. We present a variety of existing definitions, study how they relate to each other and what their graphical implications are. We then turn to the current major question surrounding counterfactual invariance, how does it relate to conditional independence? We show that whilst counterfactual invariance implies conditional independence, conditional independence does not give any implications about the degree or likelihood of satisfying counterfactual invariance. Furthermore, we show that for discrete causal models counterfactually invariant functions are often constrained to be functions of particular variables, or even constant.