A note on the relations between mixture models, maximum-likelihood and entropic optimal transport
This is an incremental pedagogical clarification for researchers in statistics and machine learning.
The paper demonstrates that maximum-likelihood estimation for mixture models is equivalent to minimizing an entropic optimal transport problem, presenting this known result concisely and illustrating it with Gaussian mixture models by showing the EM algorithm as a block-coordinate descent.
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.