A Bayesian hierarchical model for related densities by using Pólya trees

Bayesian hierarchical models are used to share information between related samples and to 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. Various 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 Pólya tree, which enables direct modelling of densities and enjoys some computational advantages over the Dirichlet process. The Pólya tree also enables more flexible modelling of the variation between samples, providing more informed shrinkage and permitting posterior inference on the dispersion function, which quantifies the variation between 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.