Adaptive Bayesian multivariate density estimation with Dirichlet mixtures

We consider Bayesian multivariate density estimation using a Dirichlet mixture of normal kernel as the prior distribution. By representing a Dirichlet process as a stick-breaking process, we are able to extend convergence results beyond finitely supported mixtures priors to Dirichlet mixtures. Thus our results have new implications in the univariate situation as well. Assuming that the true density satisfies H\"older smoothness and exponential tail conditions, we show the rates of posterior convergence are minimax-optimal up to a logarithmic factor. This procedure is fully adaptive since the priors are constructed without using the knowledge of the smoothness level.