Open Access
2024 Characterization of exchangeable measure-valued Pólya urn sequences
Hristo Sariev, Mladen Savov
Author Affiliations +
Electron. J. Probab. 29: 1-23 (2024). DOI: 10.1214/24-EJP1132
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 X is a complete separable metric space and the urn composition is a finite measure on 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 P˜. In particular, when 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 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.

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Hristo Sariev and Mladen Savov "Characterization of exchangeable measure-valued Pólya urn sequences," Electronic Journal of Probability 29(none), 1-23, (2024). https://doi.org/10.1214/24-EJP1132
Received: 21 October 2023; Accepted: 23 April 2024; Published: 2024
Vol.29 • 2024
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