Adaptive Bayesian multivariate density estimation with Dirichlet mixtures

TL;DR

This paper demonstrates that Bayesian Dirichlet mixture models can adaptively estimate multivariate densities at optimal rates without prior smoothness knowledge, using novel approximation techniques and prior conditions.

Contribution

It establishes sufficient prior conditions for rate-adaptive Bayesian multivariate density estimation with Dirichlet mixtures, including new approximation and sieve methods.

Findings

Achieves minimax optimal convergence rates for smoothness classes.

Validates prior conditions for Dirichlet location mixtures with Gaussian and inverse-Wishart priors.

Extends results to locally Hölder smoothness and anisotropic classes.

Abstract

We show that rate-adaptive multivariate density estimation can be performed using Bayesian methods based on Dirichlet mixtures of normal kernels with a prior distribution on the kernel's covariance matrix parameter. We derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is optimal minimax for the smoothness class to which the true density belongs. No prior knowledge of smoothness is assumed. The sufficient conditions are shown to hold for the Dirichlet location mixture of normals prior with a Gaussian base measure and an inverse-Wishart prior on the covariance matrix parameter. Locally H\"older smoothness classes and their anisotropic extensions are considered. Our study involves several technical novelties, including sharp approximation of finitely differentiable multivariate densities by normal mixtures and a new sieve on the space of such densities.