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

TL;DR

This paper introduces spike-and-slab Pólya tree priors for density estimation, demonstrating their adaptive convergence rates and providing uncertainty quantification through confidence bands with optimal properties.

Contribution

The paper develops a new class of Pólya tree priors with spike-and-slab structure, achieving adaptive minimax convergence and valid uncertainty quantification.

Findings

Posterior distribution converges at the minimax rate adaptively.

Derived an adaptive Bernstein-von Mises theorem for uncertainty quantification.

Constructed credible sets that serve as adaptive confidence bands with optimal diameter.

Abstract

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.