Adaptive Bayesian multivariate density estimation with Dirichlet mixtures

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 Hölder 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.

29 pages

References in corpus (4)

• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Adaptive nonparametric Bayesian inference using location-scale mixture priors
• Nonparametric Bayesian model selection and averaging

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