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<p>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.</p>
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