Spike and Slab P\'olya tree posterior distributions: adaptive inference

In the density estimation model, the question of adaptive inference using P\'olya tree-type prior distributions is considered. A class of prior densities having a tree structure, called spike-and-slab P\'olya 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 H\"older regularity of the true density between $0$ and $1$, thereby providing adaptive counterparts to the results for classical P\'olya trees in Castillo (2017). 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.