Title: A Bayesian hierarchical model for related densities using Polya trees,
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
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 Polya tree, which allows direct modeling of
densities and enjoys some computational advantages over the Dirichlet process.
The Polya 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.