Bayesian analysis of mixture models with an unknown number of components—an alternative to reversible jump methods

TLDR

The paper discusses Bayesian analysis of finite mixture models with unknown component numbers, building on Richardson and Green’s reversible‑jump MCMC approach and highlighting its potential as an alternative to general reversible‑jump methods. The authors aim to develop an alternative MCMC method that treats mixture parameters as a marked point process, extending Ripley’s approach to a Markov birth‑death process with a suitable stationary distribution. The proposed method is an MCMC algorithm that models mixture parameters as a marked point process, employing a Markov birth‑death process with a stationary distribution and leveraging reversible‑jump ideas, and it is straightforward to implement for univariate and bivariate data. The study demonstrates that the new birth‑death MCMC approach shows promise as a simpler alternative to general reversible‑jump methods and suggests its applicability to other contexts.

Abstract

Richardson and Green present a method of performing a Bayesian analysis of data from a finite mixture distribution with an unknown number of components. Their method is a Markov Chain Monte Carlo (MCMC) approach, which makes use of the “reversible jump” methodology described by Green. We describe an alternative MCMC method which views the parameters of the model as a (marked) point process, extending methods suggested by Ripley to create a Markov birth-death process with an appropriate stationary distribution. Our method is easy to implement, even in the case of data in more than one dimension, and we illustrate it on both univariate and bivariate data. There appears to be considerable potential for applying these ideas to other contexts, as an alternative to more general reversible jump methods, and we conclude with a brief discussion of how this might be achieved.