Entropic Causal Inference

TL;DR

This paper introduces an entropy-based method for causal inference between two discrete variables, leveraging the simplicity of the exogenous variable in the true causal direction, and provides algorithms with theoretical and empirical validation.

Contribution

It proposes a novel entropy-based causal inference approach that does not rely on variable values, only distributions, and introduces algorithms for minimum entropy exogenous variable estimation.

Findings

The method performs comparably to state-of-the-art additive noise models.

It can be applied to ordinal and categorical data.

Provides algorithms with theoretical guarantees for entropy minimization.

Abstract

We consider the problem of identifying the causal direction between two discrete random variables using observational data. Unlike previous work, we keep the most general functional model but make an assumption on the unobserved exogenous variable: Inspired by Occam's razor, we assume that the exogenous variable is simple in the true causal direction. We quantify simplicity using R\'enyi entropy. Our main result is that, under natural assumptions, if the exogenous variable has low $H_0$ entropy (cardinality) in the true direction, it must have high $H_0$ entropy in the wrong direction. We establish several algorithmic hardness results about estimating the minimum entropy exogenous variable. We show that the problem of finding the exogenous variable with minimum entropy is equivalent to the problem of finding minimum joint entropy given $n$ marginal distributions, also known as minimum entropy coupling problem. We propose an efficient greedy algorithm for the minimum entropy coupling problem, that for $n=2$ provably finds a local optimum. This gives a greedy algorithm for finding the exogenous variable with minimum $H_1$ (Shannon Entropy). Our greedy entropy-based causal inference algorithm has similar performance to the state of the art additive noise models in real datasets. One advantage of our approach is that we make no use of the values of random variables but only their distributions. Our method can therefore be used for causal inference for both ordinal and also categorical data, unlike additive noise models.