Characterization of exchangeable measure-valued Pólya urn sequences
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917663022776320 |
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| author | Sariev, Hristo Savov, Mladen |
| author_facet | Sariev, Hristo Savov, Mladen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10083 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterization of exchangeable measure-valued Pólya urn sequences Sariev, Hristo Savov, Mladen Probability 60G09, 60G25, 60G57, 62G99 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. |
| title | Characterization of exchangeable measure-valued Pólya urn sequences |
| topic | Probability 60G09, 60G25, 60G57, 62G99 |
| url | https://arxiv.org/abs/2305.10083 |