Optional P\'{o}lya tree and Bayesian inference

TL;DR

This paper presents an extension of the Pólya tree method, called optional Pólya tree, which constructs adaptive, absolutely continuous random measures with smooth densities, suitable for Bayesian inference on probability distributions.

Contribution

It introduces the optional Pólya tree, enabling adaptive, smooth density estimation with large support and computable posteriors in Bayesian analysis.

Findings

Produces random measures with piecewise smooth densities

Supports adaptive partitioning fitting data well

Allows explicit computation of posterior distributions

Abstract

We introduce an extension of the P\'olya tree approach for constructing distributions on the space of probability measures. By using optional stopping and optional choice of splitting variables, the construction gives rise to random measures that are absolutely continuous with piecewise smooth densities on partitions that can adapt to fit the data. The resulting "optional P\'{o}lya tree" distribution has large support in total variation topology and yields posterior distributions that are also optional P\'{o}lya trees with computable parameter values.