The L1L1-consistency of Dirichlet mixtures in multivariate Bayesian density estimation

Density estimation, especially multivariate density estimation, is a fundamental problem in nonparametric inference. In the Bayesian approach, Dirichlet mixture priors are often used in practice for such problems. However, the asymptotic properties of such priors have only been studied in the univariate case. We extend the L1L1-consistency of Dirichlet mixutures in the multivariate density estimation setting. We obtain such a result by showing that the Kullback–Leibler property of the prior holds and that the size of the sieve in the parameter space in terms of L1L1-metric entropy is not larger than the order of nn. However, it seems that the usual technique of choosing a sieve by controlling prior probabilities is unable to lead to a useful bound on the metric entropy required for the application of a general posterior consistency theorem for the multivariate case. We overcome this difficulty by using a structural property of Dirichlet mixtures. Our results apply to a multivariate normal kernel even when the multivariate normal kernel has a general variance–covariance matrix.