Adaptive Bayesian density estimation using Pitman-Yor or normalized inverse-Gaussian process kernel mixtures

We consider Bayesian nonparametric density estimation using a Pitman-Yor or a normalized inverse-Gaussian process kernel mixture as the prior distribution for a density. The procedure is studied from a frequentist perspective. Using the stick-breaking representation of the Pitman-Yor process or the expression of the finite-dimensional distributions for the normalized-inverse Gaussian process, we prove that, when the data are replicates from a density with Sobolev or analytic smoothness, the posterior distribution concentrates on shrinking Lp-norm balls around the sampling density at a minimax-optimal rate, up to a logarithmic factor. The resulting hierarchical Bayes procedure, with a fixed prior, is thus shown to be adaptive to the regularity of the sampling density.