# Entropic Causality and Greedy Minimum Entropy Coupling

## TL;DR

This paper analyzes a greedy algorithm for approximate minimum entropy coupling, a key step in entropic causality, providing guarantees on local optimality and approximation error despite the problem's NP-hardness.

## Contribution

It offers a theoretical analysis of a greedy algorithm for minimum entropy coupling, establishing local optimality and approximation bounds.

## Key findings

- The algorithm always finds a local minimum.
- It is within an additive approximation error of the global minimum.

## Abstract

We study the problem of identifying the causal relationship between two discrete random variables from observational data. We recently proposed a novel framework called entropic causality that works in a very general functional model but makes the assumption that the unobserved exogenous variable has small entropy in the true causal direction. This framework requires the solution of a minimum entropy coupling problem: Given marginal distributions of m discrete random variables, each on n states, find the joint distribution with minimum entropy, that respects the given marginals. This corresponds to minimizing a concave function of nm variables over a convex polytope defined by nm linear constraints, called a transportation polytope. Unfortunately, it was recently shown that this minimum entropy coupling problem is NP-hard, even for 2 variables with n states. Even representing points (joint distributions) over this space can require exponential complexity (in n, m) if done naively. In our recent work we introduced an efficient greedy algorithm to find an approximate solution for this problem. In this paper we analyze this algorithm and establish two results: that our algorithm always finds a local minimum and also is within an additive approximation error from the unknown global optimum.