Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

Abstract

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on $[0,1]$. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a Hölder class with regularity $\beta$, for all positive $\beta$, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.

Published in at http://dx.doi.org/10.1214/09-AOS703 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Convergence rates of posterior distributions for noniid observations
• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Kullback Leibler property of kernel mixture priors in Bayesian density estimation
• Bayesian inference with rescaled Gaussian process priors

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