Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Sonia Petrone,Hristo Sariev,Sandra Fortini

doi:10.3390/math9222845

Sonia Petrone, Hristo Sariev + Show 1 more

Open Access

https://doi.org/10.3390/math9222845

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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 (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.

Highlights

Let (Xn)n≥1 be a sequence of homogeneous random observations, taking values in a Polish space X
We focus our analysis on Measure-valued Pólya urn processes (MVPP) for which Rx is concentrated on x; after each draw, we reinforce only the color of the observed ball
Reinforced Pólya Processes It follows from (8) that any Pólya sequence generates a deterministic MVPP through μn = μn−1 + δXn

Summary

Introduction

Let (Xn)n≥1 be a sequence of homogeneous random observations, taking values in a Polish space X. RRPPs require that Rx({x}c) = 0, and so are excluded from the analysis in those papers This difference in reinforcement mechanisms mirrors the dichotomy within k-color urn models, where the replacement R is best described in terms of a matrix with random elements. There, the class of randomly reinforced urns [17] assumes an R with zero off-diagonal elements (i.e., we reinforce only the color of the observed ball), whereas the generalized Pólya urn models require the mean replacement matrix to be irreducible.

Definitions and a Representation Result

Measure-Valued Pólya urn Processes

Asymptotic Properties of RRPP

Preliminaries

RRPP with Independent Weights

GRRPP with Exchangeable Weights

Asymptotic Properties of GRRPP with Exchangeable Weights

Discussion