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Sufficientness postulates for measure-valued Pólya urn sequences

Part of: Sufficiency and information Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  28 March 2025

Hristo Sariev Open the ORCID record for Hristo Sariev [Opens in a new window]  and
Mladen Savov
Hristo Sariev*
Affiliation:
Bulgarian Academy of Sciences
Mladen Savov*
Affiliation:
Sofia University ‘St. Kliment Ohridski’
*
*Postal address: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. Georgi Bonchev Str., Sofia 1113, Bulgaria. Email: [email protected]
**Postal address: Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd, Sofia 1164, Bulgaria. Email: [email protected]
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Abstract

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued Pólya sequences (MVPSs), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of ‘sufficientness’ postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel evaluated at past observations. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. In addition, we provide a ‘pure’ sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.

Keywords

Pólya urnspredictive distributionsBayesian nonparametricsexchangeabilitysufficientness postulate

MSC classification

Primary: 60G25: Prediction theory 60G09: Exchangeability
Secondary: 62G99: None of the above, but in this section 62B99: None of the above, but in this section
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 25
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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