Characterization of exchangeable measure-valued Pólya urn sequences

Abstract

Measure-valued Pólya urn sequences (MVPS) are a generalization of the observation processes generated by k-color Pólya urn models, where the space of colors $\mathbb{X}$ is a complete separable metric space and the urn composition is a finite measure on $\mathbb{X}$, in which case reinforcement reduces to a summation of measures. In this paper, we prove a representation theorem for the reinforcement measures R of all exchangeable MVPSs, which leads to a characterization result for their directing random measures $\tilde{P}$. In particular, when $\mathbb{X}$ is countable or R is dominated by the initial distribution ν, then any exchangeable MVPS is a Dirichlet process mixture model over a family of probability distributions with disjoint supports. Furthermore, for all exchangeable MVPSs, the predictive distributions converge on a set of probability one in total variation to $\tilde{P}$. Importantly, we do not restrict our analysis to balanced MVPSs, in the terminology of k-color urns, but rather show that the only non-balanced exchangeable MVPSs are sequences of i.i.d. random variables.