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Pólya tree posterior distributions on densities. (English. French summary) Zbl 1384.62156
Summary: Pólya trees form a popular class of prior distributions used in Bayesian nonparametrics. For some choice of parameters, Pólya 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 Pólya 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.

MSC:
62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
62G07 Density estimation
62G15 Nonparametric tolerance and confidence regions
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