Posterior convergence rates for Dirichlet mixtures of beta densities

We consider Bayesian density estimation for compactly supported densities using Bernstein mixtures of beta-densities equipped with a Dirichlet prior on the distribution function. We derive the rate of convergence for α-smooth densities for 0 < α ≤ 2 and show that a faster rate of convergence can be obtained by using fewer terms in the mixtures than proposed before. The Bayesian procedure adapts to the unknown value of α. The modified Bayesian procedure is rate-optimal if α is at most one. This result can be extended to two dimensions. © 2008 Elsevier B.V. All rights reserved