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

The relations between maximum-likelihood and optimal transport (OT) have already been discussed in multiple works (Rigollet and Weed, 2018; Mena et al., 2020; Diebold et al., 2024). The purpose of this brief note is to provide the key tools used to establish these connections. The primary aim is pedagogical: we will focus on the (discrete) mixtures case, adopting a "computational OT" perspective. Hopefully, readers will find this exercise insightful. Our analysis will largely rely on the approach described in Rigollet and Weed (2018), though adapted to a different formalism and applied to a slightly different problem (mixture estimation rather than Gaussian deconvolution).