Applications and Computation of Stateful Polya Trees

Abstract

Polya trees are a class of nonparametric priors on distributions which are able to model absolutely continuous distributions directly, rather than modeling a discrete distribution over parameters of a mixing kernel to obtain an absolutely continuous distribution. The Polya tree discretizes the state space with a recursive partition, generating a distribution by assigning mass to the child elements at each level of the recursive partition according to a Beta distribution. Stateful Polya trees are an extension of the Polya tree where each set in the recursive partition has one or more discrete state variables associated with it. We can learn the posterior distributions of these state variables along with the posterior of the distribution. State variables may be of interest in their own right, or may be nuisance parameters which we use to achieve more flexible models but wish to integrate out in the posterior. We discuss the development of stateful Polya trees and discuss the Hierarchical Adaptive Polya Tree, which uses state variables to flexibly model the concentration parameter of Polya trees in a hierarchical Bayesian model. We also consider difficulties with the use of marginal likelihoods to determine posterior probabilities of states.