Predictive Constructions Based on Measure-Valued Pólya Urn Processes
| Autor: | Sonia Petrone, Sandra Fortini, Hristo Sariev |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pólya sequences
General Mathematics Bayesian inference MathematicsofComputing_GENERAL conditional identity in distribution Reinforced Processes 01 natural sciences Measure (mathematics) Conditional Identity In Distribution 010104 statistics & probability Total variation Pólya urn model urn schemes QA1-939 Computer Science (miscellaneous) PREDICTIVE DISTRIBUTIONS RANDOM PROBABILITY MEASURES REINFORCED PROCESSES PÓLYA SEQUENCES URN SCHEMES BAYESIAN INFERENCE CONDITIONAL IDENTITY IN DISTRIBUTION TOTAL VARIATION DISTANCE 0101 mathematics Engineering (miscellaneous) Mathematics Discrete mathematics total variation distance Sequence Markov chain Stochastic process 010102 general mathematics predictive distributions Random Probability Measures TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES Distribution (mathematics) Urn Schemes random probability measures reinforced processes Total Variation Distance Pólya Sequences Bayesian Inference Polish space Predictive Distributions |
| Zdroj: | Mathematics Volume 9 Issue 22 Mathematics, Vol 9, Iss 2845, p 2845 (2021) |
| ISSN: | 2227-7390 |
| DOI: | 10.3390/math9222845 |
| Popis: | 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 (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)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 (Xn)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. |
| Databáze: | OpenAIRE |
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