Predictive Constructions Based on Measure-Valued Pólya Urn Processes

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 ( μ n ) n ≥ 0 on a Polish space X , the normalized sequence ( μ n / μ n ( X ) ) n ≥ 0 agrees with the marginal predictive distributions of some random process ( X n ) n ≥ 1 . Moreover, μ n = μ n − 1 + R X n , n ≥ 1 , where x ↦ R x is a random transition kernel on X ; thus, if μ n − 1 represents the contents of an urn, then X n denotes the color of the ball drawn with distribution μ n − 1 / μ n − 1 ( X ) and R X n —the subsequent reinforcement. In the case R X n = W n δ 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.