Adaptive Convergence Rates of a Dirichlet Process Mixture of Multivariate Normals

TL;DR

This paper demonstrates that a Dirichlet process mixture of multivariate normals can adaptively achieve optimal convergence rates in Bayesian density estimation across different smoothness classes and dimensions.

Contribution

It introduces a novel sieve construction for non-parametric mixture densities that enhances understanding of adaptive convergence rates in Bayesian non-parametric models.

Findings

Achieves adaptive posterior convergence rates for multivariate density estimation.

Introduces a new sieve construction for mixture densities.

Enhances theoretical understanding of Bayesian non-parametric models.

Abstract

It is shown that a simple Dirichlet process mixture of multivariate normals offers Bayesian density estimation with adaptive posterior convergence rates. Toward this, a novel sieve for non-parametric mixture densities is explored, and its rate adaptability to various smoothness classes of densities in arbitrary dimension is demonstrated. This sieve construction is expected to offer a substantial technical advancement in studying Bayesian non-parametric mixture models based on stick-breaking priors.