Polya tree posterior distributions on densities

Polya trees form a popular class of prior distributions used in Bayesian nonparametrics. For some choice of parameters, Polya trees are prior distributions on density functions. In this paper we carry out a frequentist analysis of the induced posterior distributions in the density estimation model. We investigate the contraction rate of Polya tree posterior densities in terms of the supremum loss and study the limiting shape distribution. A nonparametric Bernstein-von Mises theorem is established, as well as a Bayesian Donsker theorem for the posterior cumulative distribution function.