Optional P\'olya trees: posterior rates and uncertainty quantification

TL;DR

This paper introduces a Bayesian tree-based density estimation method using Optional Pólya trees, achieving near-optimal convergence rates and adaptive uncertainty quantification for a broad class of smooth densities.

Contribution

It provides theoretical guarantees for posterior convergence rates and credible sets in density estimation using Optional Pólya trees, with adaptive properties to unknown smoothness.

Findings

Achieves near-optimal posterior convergence rates in supremum norm.

Automatically adapts to unknown H"older regularity of densities.

Provides mathematically guaranteed credible sets for density and related functionals.

Abstract

We consider statistical inference in the density estimation model using a tree-based Bayesian approach, with Optional P\'olya trees as prior distribution. We derive near-optimal convergence rates for corresponding posterior distributions with respect to the supremum norm. For broad classes of H\"older-smooth densities, we show that the method automatically adapts to the unknown H\"older regularity parameter. We consider the question of uncertainty quantification by providing mathematical guarantees for credible sets from the obtained posterior distributions, leading to near-optimal uncertainty quantification for the density function, as well as related functionals such as the cumulative distribution function. The results are illustrated through a brief simulation study.