A note on the relations between mixture models, maximum-likelihood and entropic optimal transport

TL;DR

This paper shows that maximum-likelihood estimation for mixture models can be viewed as an entropic optimal transport problem, connecting classical statistical methods with modern optimal transport theory.

Contribution

It presents a concise, pedagogical demonstration that MLE for mixture models is equivalent to minimizing an entropic optimal transport loss, linking these concepts clearly.

Findings

MLE for mixture models equals entropic optimal transport minimization

Standard EM algorithm is a block-coordinate descent on an optimal transport loss

Illustration with Gaussian mixture models demonstrates the connection

Abstract

This note aims to demonstrate that performing maximum-likelihood estimation for a mixture model is equivalent to minimizing over the parameters an optimal transport problem with entropic regularization. The objective is pedagogical: we seek to present this already known result in a concise and hopefully simple manner. We give an illustration with Gaussian mixture models by showing that the standard EM algorithm is a specific block-coordinate descent on an optimal transport loss.