Spike and slab Polya tree posterior densities: Adaptive inference

In the density estimation model, the question of adaptive inference using Polya tree-type prior distributions is considered. A class of prior densities having a tree structure, called spike-and-slab Polya 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 Holder regularity of the true density between 0 and 1, thereby providing adaptive counterparts to the results for classical Polya trees in Castillo (Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 2074-2102). 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.