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A Bayesian hierarchical model for related densities by using Pólya trees
Author(s) -
Christensen Jonathan,
Ma Li
Publication year - 2020
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12346
Subject(s) - dirichlet process , dirichlet distribution , hierarchical database model , bayesian hierarchical modeling , tree (set theory) , mixture distribution , bayesian inference , mathematics , computer science , mixture model , statistics , latent dirichlet allocation , bayesian probability , kernel (algebra) , inference , probability density function , data mining , artificial intelligence , topic model , mathematical analysis , combinatorics , boundary value problem
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.