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

TL;DR

This paper characterizes the predictive distributions of measure-valued Pólya sequences, revealing their unique properties and connections to Dirichlet processes, through various sufficientness postulates.

Contribution

It introduces new sufficientness postulates that uniquely characterize exchangeable measure-valued Pólya sequences and their predictive distributions.

Findings

Exchangeable MVPSs are uniquely characterized by their predictive distributions.

When the predictive distribution matches the empirical measure, it aligns with Dirichlet process models.

Multiple sufficientness postulates are provided for finite state spaces.

Abstract

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued P\'{o}lya sequences (MVPS), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of "sufficientness" postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel 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. In addition, we provide a "pure" sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.