Stochastic processes of the urn type with convergent predictive distributions

In this work we propose a general class of stochastic processes with random reinforcement that are extensions of the celebrated Pòlya sequence by Blackwell and MacQueen [Ann. Stat. 1 (1973) 353--355]. The resulting randomly reinforced Pòlya sequences (RRPS) can be described as urn schemes with countable number of colors and general replacement rules. Under assumptions of conditional independence between reinforcement and observation, a RRPS becomes conditionally identically distributed (in the sense of [Ann. Probab. 32 (2004) 2029--2052]), and thus predictively convergent, in which case we show that it is asymptotically equivalent in law to an exchangeable species sampling sequence. This result has important implications on the generated random partition, which can be visualized as a weighted version of the Chinese Restaurant Process. We then provide complete distributional characterization of the predictive limit for the model with dichotomous reinforcements. Throughout the second part of the thesis, we consider an alternative specification of the replacement mechanism of a RRPS, whereby we deem some colors to be probabilistically dominant. In this situation the predictive and empirical distributions evaluated near the set of dominant colors both tend to 1. In fact, under some further restrictions on the reinforcement, the predictive and empirical distributions converge in the sense of almost sure weak convergence to one and the same random probability measure, whose mass is concentrated on the dominant set. As a consequence, the process becomes asymptotically exchangeable and its law -- directed by the above random measure, so that the data structure gets relatively sparse with time. The predictive limit for both models is generally unknown, however, so we derive central limit results, with which to approximate its distribution. The last chapter of the thesis is addressed towards applications of the RRPS, with the dominant-color model being considered in the context of clinical trials with response-adaptive design. Sections discussing uni- and multivariate extensions of the RRPS complete our study

Stochastic processes of the urn type with convergent predictive distributions

[Sariev Hristo; Сариев Христо

Characterization of exchangeable measure-valued P\'olya urn sequences

Measure-valued P\'olya urn sequences (MVPS) are a generalization of the observation processes generated by $k$-color P\'olya urn models, where the space of colors $\mathbb{X}$ is a complete separable metric space and the urn composition is a finite measure on $\mathbb{X}$, in which case reinforcement reduces to a summation of measures. In this paper, we prove a representation theorem for the reinforcement measures $R$ of all exchangeable MVPSs, which leads to a characterization result for their directing random measures $\tilde{P}$. In particular, when $\mathbb{X}$ is countable or $R$ is dominated by the initial distribution $\nu$, then any exchangeable MVPS is a Dirichlet process mixture model over a family of probability distributions with disjoint supports. Furthermore, for all exchangeable MVPSs, the predictive distributions converge on a set of probability one in total variation to $\tilde{P}$. A final result shows that $\tilde{P}$ can be decomposed into an absolutely continuous and a mutually singular measure with respect to $\nu$, whose support is universal and does not depend on the particular instance of $\tilde{P}$

Sufficientness postulates for measure-valued P\'{o}lya urn sequences

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued P\'{o}lya urn sequences, which arise as the observation process in a generalized urn sampling scheme. Here we provide three results in the form of "sufficientness" postulates that characterize their predictive distributions. In particular, we show that exchangeable measure-valued P\'{o}lya urn sequences are the unique exchangeable models for which the predictive distributions are a mixture of the marginal distribution and an average of some probability kernels evaluated at past observation. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. The other two sufficientness postulates consider the case when the state space is finite

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