## The Annals of Statistics

### Convergence of latent mixing measures in finite and infinite mixture models

XuanLong Nguyen

#### 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.

#### Article information

Source
Ann. Statist. Volume 41, Number 1 (2013), 370-400.

Dates
First available in Project Euclid: 26 March 2013

http://projecteuclid.org/euclid.aos/1364302747

Digital Object Identifier
doi:10.1214/12-AOS1065

Zentralblatt MATH identifier
06168783

Mathematical Reviews number (MathSciNet)
MR3059422

Subjects
Primary: 62F15: Bayesian inference 62G05: Estimation
Secondary: 62G20: Asymptotic properties

#### Citation

Nguyen, XuanLong. Convergence of latent mixing measures in finite and infinite mixture models. The Annals of Statistics 41 (2013), no. 1, 370--400. doi:10.1214/12-AOS1065. http://projecteuclid.org/euclid.aos/1364302747.

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