Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Sandra Fortini, Sonia Petrone and Hristo Sariev
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Sandra Fortini: Department of Decision Sciences, Bocconi University, 20136 Milano, Italy
Sonia Petrone: Department of Decision Sciences, Bocconi University, 20136 Milano, Italy
Hristo Sariev: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Mathematics, 2021, vol. 9, issue 22, 1-19

Abstract: Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k -color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP ( ? n ) n ? 0 on a Polish space X , the normalized sequence ( ? n / ? n ( X ) ) n ? 0 agrees with the marginal predictive distributions of some random process ( X n ) n ? 1 . Moreover, ? n = ? n ? 1 + R X n , n ? 1 , where x ? R x is a random transition kernel on X ; thus, if ? n ? 1 represents the contents of an urn, then X n denotes the color of the ball drawn with distribution ? n ? 1 / ? n ? 1 ( X ) and R X n —the subsequent reinforcement. In the case R X n = W n ? X n , for some non-negative random weights W 1 , W 2 , … , the process ( X n ) n ? 1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of ( X n ) n ? 1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Keywords: predictive distributions; random probability measures; reinforced processes; Pólya sequences; urn schemes; Bayesian inference; conditional identity in distribution; total variation distance (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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