# How to Find a Joint Probability Distribution of Minimum Entropy (almost) given the Marginals

## TL;DR

This paper addresses the challenge of approximating the joint distribution with minimum entropy given marginals, proposing an efficient algorithm with bounded additive error, and extends the approach to multiple variables.

## Contribution

It introduces an approximation algorithm for the NP-hard problem of minimum-entropy joint distribution, with guarantees on the additive error, and generalizes to multiple variables.

## Key findings

- Provides an efficient approximation algorithm with additive factor 1 for two variables.
- Extends the approximation method to $k$ variables with additive factor $ ext{log} k$.
- Discusses applications where minimum-entropy joint distributions are relevant.

## Abstract

Given two discrete random variables $X$ and $Y$, with probability distributions ${\bf p} =(p_1, \ldots , p_n)$ and ${\bf q}=(q_1, \ldots , q_m)$, respectively, denote by ${\cal C}({\bf p}, {\bf q})$ the set of all couplings of ${\bf p}$ and ${\bf q}$, that is, the set of all bivariate probability distributions that have ${\bf p}$ and ${\bf q}$ as marginals. In this paper, we study the problem of finding the joint probability distribution in ${\cal C}({\bf p}, {\bf q})$ of minimum entropy (equivalently, the joint probability distribution that maximizes the mutual information between $X$ and $Y$), and we discuss several situations where the need for this kind of optimization naturally arises. Since the optimization problem is known to be NP-hard, we give an efficient algorithm to find a joint probability distribution in ${\cal C}({\bf p}, {\bf q})$ with entropy exceeding the minimum possible by at most 1, thus providing an approximation algorithm with additive approximation factor of 1. Leveraging on this algorithm, we extend our result to the problem of finding a minimum--entropy joint distribution of arbitrary $k\geq 2$ discrete random variables $X_1, \ldots , X_k$, consistent with the known $k$ marginal distributions of $X_1, \ldots , X_k$. In this case, our approximation algorithm has an additive approximation factor of $\log k$. We also discuss some related applications of our findings.