Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparamatric estimation of the density

In this work we investigate the asymptotic properties of nonparametric bayesian mixtures of Betas for estimating a smooth density on [0,1]. We consider a parameterisation of Betas 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 Holder class with regularity β, for all positive β, leading to a minimax adaptive estimating procedure of the density. We show that Bayesian kernel estimation is more flexible than the usual frequentist kernel estimation allowing for adaptive rates of convergence, using a simple trick which can be used in many other types of kernel Bayesian approaches.