Reversible Jump MCMC Converging to Birth-and-death MCMC and more General Continuous Time Samplers

Olivier Cappé, Christian P. Robert and Tobias Rydén

Abstract:: At present, reversible jump methods are the most common Markov Chain Monte Carlo tool for exploring variable dimension statistical models. Recently however, an alternative approach based on birth-and-death processes has been proposed by Stephens (2000) in the case of mixtures of distributions. We address the comparison of both methods by demonstrating that upon appropriate rescaling of time, the reversible jump chain converges to a limiting continuous time birth-and-death chain. We show in addition that the birth-and-death setting can be generalised to include other types of jumps like split/combine jumps in the spirit of Richardson and Green (1997). We illustrate these extensions in the case of hidden Markov models.
Key words:: Bayesian inference, birth-and-death process, completion, hidden Markov model, Jacobian, label switching, MCMC algorithms, mixture distribution, rescaling