Posterior asymptotics of nonparametric location-scale mixtures for multivariate density estimation

TL;DR

This paper investigates the asymptotic behavior of Bayesian nonparametric density estimators using multivariate location-scale Gaussian mixtures, extending theoretical understanding to more practical models.

Contribution

It establishes posterior consistency and discusses convergence rates for Dirichlet process mixtures with various covariance priors in multivariate density estimation.

Findings

Posterior consistency is proven for multivariate location-scale Gaussian mixtures.

Convergence rates are analyzed under different prior specifications.

The results bridge the gap between theory and practice in multivariate density estimation.

Abstract

Density estimation represents one of the most successful applications of Bayesian nonparametrics. In particular, Dirichlet process mixtures of normals are the gold standard for density estimation and their asymptotic properties have been studied extensively, especially in the univariate case. However a gap between practitioners and the current theoretical literature is present. So far, posterior asymptotic results in the multivariate case are available only for location mixtures of Gaussian kernels with independent prior on the common covariance matrix, while in practice as well as from a conceptual point of view a location-scale mixture is often preferable. In this paper we address posterior consistency for such general mixture models by adapting a convergence rate result which combines the usual low-entropy, high-mass sieve approach with a suitable summability condition. Specifically, we establish consistency for Dirichlet process mixtures of Gaussian kernels with various prior specifications on the covariance matrix. Posterior convergence rates are also discussed.