Adaptive Bayesian Density Estimation in Lp-metrics with Pitman-Yor or Normalized Inverse-Gaussian Process Kernel Mixtures

Abstract

We consider Bayesian nonparametric density estimation using a Pitman-Yor or a normalized inverse-Gaussian process convolution 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 and the finite-dimensional distributions of the normalized inverse-Gaussian process, we prove that, when the data are independent replicates from a density with analytic or Sobolev smoothness, the posterior distribution concentrates on shrinking $L^p$-norm balls around the sampling density at a minimax-optimal rate, up to a logarithmic factor. The resulting hierarchical Bayesian procedure, with a fixed prior, is adaptive to the unknown smoothness of the sampling density.