Rates of Convergence for the Posterior Distributions of Mixtures of Betas and Adaptive Nonparamatric Estimation of the Density

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 parameterisation of Betas distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while either fixing the scaling parameter (as a function on the number of observations) or putting a proper 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 β, for all positive β, by choosing correctly the scaling parameter of the Betas densities, in terms of the number of observations and β. We improve on these results by considering a prior on the scaling parameter and thus obtain an adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Betas densities can be of interest in a frequentist framework.