Convergence of latent mixing measures in finite and infinite mixture models

TL;DR

This paper investigates the convergence behavior of latent mixing measures in finite and infinite mixture models using Wasserstein metrics, exploring their relationship with divergence measures and establishing convergence rates.

Contribution

It provides a detailed analysis of convergence in Wasserstein distances and links it to divergence measures, with new results on posterior convergence rates for mixture models.

Findings

Wasserstein convergence implies convergence of mixture components.

Established convergence rates for finite and infinite mixture models.

Clarified the relationship between Wasserstein and divergence measures.

Abstract

This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process.