Some developments of exchangeable measure-valued P\'{o}lya sequences

TL;DR

This paper explores measure-valued Pólya sequences, showing their connection to Dirichlet process mixtures, extending the model to include null components, and analyzing the relationship between exchangeability and conditional identity in distribution.

Contribution

It demonstrates that exchangeable MVPSs have Dirichlet process mixture priors and extends the model to include null components, also analyzing exchangeability versus c.i.d.

Findings

Prior distribution of exchangeable MVPS is a Dirichlet process mixture.

Extended MVPS to include a null component in reinforcement.

Exchangeability and c.i.d. are equivalent for balanced MVPSs.

Abstract

Measure-valued P\'{o}lya sequences (MVPS) are processes whose dynamics are governed by generalized P\'{o}lya urn schemes with infinitely many colors. Assuming a general reinforcement rule, exchangeable MVPSs can be viewed as extensions of Blackwell and MacQueen's P\'{o}lya sequence, which characterizes an exchangeable sequence whose directing random measure has a Dirichlet process prior distribution. Here, we show that the prior distribution of any exchangeable MVPS is a Dirichlet process mixture with respect to a latent parameter that is associated with the atoms of an emergent conditioning $\sigma$-algebra. As the mixing components have disjoint supports, the directing random measure can be interpreted as a random histogram with bins randomly located on these same atoms. Furthermore, we extend the basic exchangeable MVPS to include a null component in the reinforcement, which corresponds to the presence of a fixed component in the directing random measure. Finally, we examine the effects of relaxing exchangeability to conditional identity in distribution (c.i.d.) and find out that the two are equivalent for balanced MVPSs. The paper features a complementary study of some properties of probability kernels that underlies the analysis of exchangeable and c.i.d. MVPSs.