Clustering dynamics in a class of normalised generalised gamma dependent priors

TL;DR

This paper introduces a new class of dependent normalized generalised gamma priors for Bayesian mixture models, analyzing their clustering dynamics and asymptotic behavior through novel diffusion processes.

Contribution

It constructs a dependent prior class based on Moran models, studies the asymptotic cluster dynamics, and introduces new diffusion processes with explicit invariant measures.

Findings

Asymptotic cluster number follows a nonstationary diffusion process.

The diffusion process has unbounded drift and an entrance boundary at zero.

New stationary diffusions with power law tails approximate the scaling limit.

Abstract

Normalised generalised gamma processes are random probability measures that induce nonparametric prior distributions widely used in Bayesian statistics, particularly for mixture modelling. We construct a class of dependent normalised generalised gamma priors induced by a stationary population model of Moran type, which exploits a generalised P\'olya urn scheme associated with the prior. We study the asymptotic scaling for the dynamics of the number of clusters in the sample, which in turn provides a dynamic measure of diversity in the underlying population. The limit is formalised to be a positive nonstationary diffusion process which falls outside well known families, with unbounded drift and an entrance boundary at the origin. We also introduce a new class of stationary positive diffusions, whose invariant measures are explicit and have power law tails, which approximate weakly the scaling limit.