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We consider Bayesian nonparametric density estimation using a Pitman-Yor or anormalized inverse-Gaussian process kernel mixture as the prior distributionfor a density. The procedure is studied from a frequentist perspective. Usingthe stick-breaking representation of the Pitman-Yor process or the expressionof the finite-dimensional distributions for the normalized-inverse Gaussianprocess, we prove that, when the data are replicates from an infinitely smoothdensity, the posterior distribution concentrates on any shrinking $L^p$-normball, $1\leq p\leq\infty$, around the sampling density at a \emph{nearlyparametric} rate, up to a logarithmic factor. The resulting hierarchicalBayesian procedure, with a fixed prior, is thus shown to be adaptive to theinfinite degree of smoothness of the sampling density.

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