A Bayesian Hierarchical Model for Related Densities by using Pólya Trees

Abstract

Summary 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.

A Polya Tree Based Model for Unmarked Individuals in an Open Wildlife Population

“…Hence, a possible extension is to model count data and CR data jointly. Another useful extension is to model data collected at different sites, by replacing the Polya tree prior with a hierarchical Polya tree prior, defined in [1].…”

“…The parameters ω i j correspond to the masses assigned by the distributionν to the sets in the partition of the last level of the Polya tree. Hence, they can be sampled as a product of Beta distributions as in (1). The parameters of the PT are updated at each iteration conditional on the latent variable n i j , using the standard update explained in Section 2.…”

Flexible clustering via hidden hierarchical Dirichlet priors

“…This feature is not suited to several applications and the discussion to Camerlenghi et al (2019b) provides interesting examples. See also Soriano and Ma (2019); Christensen and Ma (2020); Denti et al (2021); Beraha et al (2021) for further stimulating contributions to this literature.…”

Model Selection for Maternal Hypertensive Disorders with Symmetric Hierarchical Dirichlet Processes

“…See also Soriano and Ma (2017) for related work. Interesting alternatives that extend the analysis to more than two populations can be found in Christensen and Ma (2020), Lijoi, Prünster, and Rebaudo (2022) and in Beraha, Guglielmi, and Quintana (2021). Another similar proposal is the one by Gutiérrez et al (2019), whose model identifies differences over cases' distributions and the control group.…”