Some developments of exchangeable measure-valued Pólya sequences

TL;DR

This work analyzes measure-valued Pólya sequences ($MVPS$) with infinitely many colors to understand how reinforcement induces a latent Dirichlet process (DP) mixture structure in the directing random measure. It develops a hierarchical DP-mixture representation by conditioning on atoms of a latent $\sigma$-algebra, interpreting the directing measure as a random histogram with disjoint mixing supports, and extends the model to allow a null reinforcement component. The authors prove that exchangeable $MVPS$ correspond to reinforcement kernels that are regular conditional distributions for $\nu$ given a $\mathcal{G}$, obtain a DP-mixture decomposition on atom-sets, and characterize the impact of a null part on the directing measure. They further investigate conditional identity in distribution (CID), showing that for balanced $MVPS$ CID is equivalent to exchangeability, while unbalanced CID MVPS require a structured, block-diagonal reinforcement. Overall, the paper provides a rigorous Bayesian nonparametric framework for $MVPS$ with a DP-mixture prior, offers a conjugate posterior form in strictly positive cases, and clarifies when exchangeability coincides with CID and how to accommodate null reinforcement in the modeling of complex, continuous data.

Abstract

Measure-valued Pólya sequences (MVPS) are processes whose dynamics are governed by generalized Pólya urn schemes with infinitely many colors. Assuming a general reinforcement rule, exchangeable MVPSs can be viewed as extensions of Blackwell and MacQueen's Pólya 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 $σ$-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.

Theorems & Definitions (40)

• Theorem 2.1
• Remark 2.2
• Proposition 2.3
• Corollary 2.4
• Example 2.5
• Remark 2.6
• Theorem 3.1: Theorem 3.1 and Proposition 3.2 in fortini2000exchangeability
• Theorem 3.2: Proposition 3.1, Theorems 3.2 and 3.7, and Remark 4.1 in sariev2024
• Theorem 3.3: Theorem 3.9 in sariev2024
• Proposition 3.4