  We show that rate-adaptive multivariate density estimation can be performed
using Bayesian methods based on Dirichlet mixtures of normal kernels with a
prior distribution on the kernel's covariance matrix parameter. We derive
sufficient conditions on the prior specification that guarantee convergence to
a true density at a rate that is optimal minimax for the smoothness class to
which the true density belongs. No prior knowledge of smoothness is assumed.
The sufficient conditions are shown to hold for the Dirichlet location mixture
of normals prior with a Gaussian base measure and an inverse-Wishart prior on
the covariance matrix parameter. Locally H\"older smoothness classes and their
anisotropic extensions are considered. Our study involves several technical
novelties, including sharp approximation of finitely differentiable
multivariate densities by normal mixtures and a new sieve on the space of such
densities.
