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In this paper, we investigate the asymptotic properties of nonparametricBayesian mixtures of Betas for estimating a smooth density on $[0,1]$. Weconsider a parametrization of Beta distributions in terms of mean and scaleparameters and construct a mixture of these Betas in the mean parameter, whileputting a prior on this scaling parameter. We prove that such Bayesiannonparametric models have good frequentist asymptotic properties. We determinethe posterior rate of concentration around the true density and prove that itis the minimax rate of concentration when the true density belongs to aH\"{o}lder class with regularity $\beta$, for all positive $\beta$, leading toa minimax adaptive estimating procedure of the density. We also believe thatthe approximating results obtained on these mixtures of Beta densities can beof interest in a frequentist framework.

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