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Reversible jump, birth-and-death and more general continuous time Markov chain Monte Carlo samplers

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Abstract:

Summary.

Reversible jump methods are the most commonly used 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 for mixtures of distributions. We show that the birth-and-death setting can be generalized to include other types of continuous time jumps like split-and-combine moves in the spirit of Richardson and Green. We illustrate these extensions both for mixtures of distributions and for hidden Markov models. We demonstrate the strong similarity of reversible jump and continuous time methodologies by showing that, on appropriate rescaling of time, the reversible jump chain converges to a limiting continuous time birth-and-death process. A numerical comparison in the setting of mixtures of distributions highlights this similarity.

Keywords: Birth-and-death process; Hidden Markov model; Markov chain Monte Carlo algorithms; Mixture distribution; Rao–Blackwellization; Rescaling

Document Type: Research Article

DOI: https://doi.org/10.1111/1467-9868.00409

Affiliations: 1: Centre National de la Recherche Scientifique, Paris, France 2: Université Paris Dauphine, Paris, and Centre de Recherche en Economie et Statistique, Paris, France 3: Lund University, Sweden

Publication date: 2003-08-01

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