Spike and slab Pólya tree posterior densities: Adaptive inference

Abstract: In the density estimation model, the question of adaptive inference using Pólya tree–type prior distributions is considered. A class of prior densities having a tree structure, called spike–and–slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between 0 and 1, thereby providing adaptive counterparts to the results for classical Pólya trees in [5]. Second, the question of uncertainty quantification is considered. An adaptive nonparametric Bernstein– von Mises theorem is derived. Next, it is shown that, under a self-similarity condition on the true density, certain credible sets from the posterior distribution are adaptive confidence bands, having prescribed coverage level and with a diameter shrinking at optimal rate in the minimax sense.