A Bayesian hierarchical model for related densities using Polya trees

TL;DR

This paper introduces a Bayesian hierarchical model based on Pólya trees for directly modeling related densities, offering computational benefits and flexible variation modeling compared to Dirichlet process-based models.

Contribution

The paper proposes a novel hierarchical Pólya tree model for densities, enabling direct density modeling, improved computational efficiency, and flexible variation analysis.

Findings

The Pólya tree model allows direct density estimation with hierarchical sharing.

The model provides more flexible variation modeling among samples.

Extensions enable clustering of samples from multiple latent populations.

Abstract

Bayesian hierarchical models are used to share information between related samples and obtain more accurate estimates of sample-level parameters, common structure, and variation between samples. When the parameter of interest is the distribution or density of a continuous variable, a hierarchical model for continuous distributions is required. A number of such models have been described in the literature using extensions of the Dirichlet process and related processes, typically as a distribution on the parameters of a mixing kernel. We propose a new hierarchical model based on the P\'olya tree, which allows direct modeling of densities and enjoys some computational advantages over the Dirichlet process. The P\'olya tree also allows more flexible modeling of the variation between samples, providing more informed shrinkage and permitting posterior inference on the dispersion function, which quantifies the variation among sample densities. We also show how the model can be extended to cluster samples in situations where the observed samples are believed to have been drawn from several latent populations.