  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
H\"{o}lder 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.
