Inverse clustering of Gibbs Partitions via independent fragmentation and dual dependent coagulation operators

TL;DR

This paper extends dual coagulation and fragmentation results for Gibbs partitions generated by stable subordinators, creating nested families of partitions and providing new insights into their structure and applications.

Contribution

It generalizes Pitman's dual coagulation/fragmentation framework to all Gibbs (stable Poisson-Kingman) models, introducing dependent processes and new duality results.

Findings

Created nested families of Gibbs partitions for any 0<β<α<1

Derived independent fragmentation operations for Gibbs partitions

Established dual coagulation operations based on dependent processes

Abstract

Gibbs partitions of the integers generated by stable subordinators of index $\alpha\in(0,1)$ form remarkable classes of random partitions where in principle much is known about their properties, including practically effortless obtainment of otherwise complex asymptotic results potentially relevant to applications in general combinatorial stochastic processes, random tree/graph growth models and Bayesian statistics. This class includes the well-known models based on the two-parameter Poisson-Dirichlet distribution which forms the bulk of explicit applications. This work continues efforts to provide interpretations for a larger classes of Gibbs partitions by embedding important operations within this framework. Here we address the formidable problem of extending the dual, infinite-block, coagulation/fragmentation results of Jim Pitman (1999, Annals of Probability), where in terms of coagulation they are based on independent two-parameter Poisson-Dirichlet distributions, to all such Gibbs (stable Poisson-Kingman) models. Our results create nested families of Gibbs partitions, and corresponding mass partitions, over any $0<\beta<\alpha<1.$ We primarily focus on the fragmentation operations, which remain independent in this setting, and corresponding remarkable calculations for Gibbs partitions derived from that operation. We also present definitive results for the dual coagulation operations, now based on our construction of dependent processes, and demonstrate its relatively simple application in terms of Mittag-Leffler and generalized gamma models. The latter demonstrates another approach to recover the duality results in Pitman (1999).