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 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

ベイジアン認知ランキング方法と消費者データ解析及び分子生物情報学への応用

筑波大学 (University of Tsukuba)201