Exchangeable measure-valued Pólya sequences
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Mladen Savov from Sofia University
| Abstract: | Measure-valued Pólya sequences (MVPS) are stochastic processes whose dynamics are governed by generalized Pólya urn schemes with infinitely many colors. Assuming a general reinforcement rule, MVPSs can be viewed as extensions of Blackwell and MacQueen’s Pólya sequence, which characterizes an exchangeable sequence with a Dirichlet process (DP) prior distribution. In this talk, we give a complete account of the class of exchangeable MVPSs in terms of their prior distributions.
First, we show that under exchangeability,an MVPS is necessarily balanced and its reinforcement kernel is, after normalization, a regular conditional distribution. As a result, its prior distribution is that of a DP mixture with respect to a latent parameter, which is associated with the conditioning sigma-algebra. Furthermore, we examine the effects of relaxing exchangeability to conditional identity in distribution and find that the two are equivalent for balanced MVPSs. In the second part of the talk, we study Hoeffding decomposability under exchangeability and provide a complete characterization of class of exchangeable Hoeffding-decomposable sequences. In particular, we show that there exists a random parameter, conditional on which an exchangeable Hoeffding-decomposable sequence is an MVPS. |