Predictive Constructions Based on Measure-Valued Pólya Urn Processes

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 pos- sible colors. We prove that, for any MVPP \( (\mu_n)_{n ≥ 0} \) on a Polish space \( \mathbb{X} \), the normalized sequence \( ( \mu_n / \mu_n (\mathbb{X}) )_{n \ge 0} \) agrees with the marginal predictive distributions of some random process \( (X_n)_{n \ge 1} \). Moreover, \( \mu_n = \mu_{n − 1} + R_{X_n}, \ n \ge 1 \), where \( x \mapsto R_x \) is a random transition kernel on \( \mathbb{X} \); thus, if \( \mu_{n − 1} \) represents the contents of an urn, then X n denotes the color of the ball drawn with distribution \( \mu_{n − 1} / \mu_{n − 1}(\mathbb{X}) \) and \( R_{X_{n}} \) - the subsequent reinforcement. In the case \( R_{X_{n}} = W_n\delta_{X_n} \), for some non-negative random weights \( W_1, \ W_2, \ \) ... , the process \( ( X_n )_{n \ge 1} \) 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 \( ( X_n )_{n \ge 1} \) under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.