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Biometrika Advance Access published online on November 19, 2007
Biometrika, doi:10.1093/biomet/asm069
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© 2007 Biometrika Trust

Articles

Population-based reversible jump Markov chain Monte Carlo

Ajay Jasra

Department of Mathematics, Imperial College London, London SW7 2AZ, U.K. ajay.jasra{at}imperial.ac.uk

David A. Stephens

Department of Mathematics and Statistics, McGill University, Montreal H3A 2K6, Canada dstephens{at}math.mcgill.ca

Christopher C. Holmes

Department of Statistics, University of Oxford, Oxford OX1 3TG, U.K. cholmes{at}stats.ox.ac.uk

Received for publication 1 February 2005. Revision received 1 April 2007.
  Abstract

We present an extension of population-based Markov chain Monte Carlo to the transdimensional case. A major challenge is that of simulating from high- and transdimensional target measures. In such cases, Markov chain Monte Carlo methods may not adequately traverse the support of the target; the simulation results will be unreliable. We develop population methods to deal with such problems, and give a result proving the uniform ergodicity of these population algorithms, under mild assumptions. This result is used to demonstrate the superiority, in terms of convergence rate, of a population transition kernel over a reversible jump sampler for a Bayesian variable selection problem. We also give an example of a population algorithm for a Bayesian multivariate mixture model with an unknown number of components. This is applied to gene expression data of 1000 data points in six dimensions and it is demonstrated that our algorithm outperforms some competing Markov chain samplers. In this example, we show how to combine the methods of parallel chains (Geyer, 1991), tempering (Geyer & Thompson, 1995), snooker algorithms (Gilks et al., 1994), constrained sampling and delayed rejection (Green & Mira, 2001).

Key Words: Bayesian variable selection • Gene expression data • Mixture model • Reversible jump Markov chain Monte Carlo • Uniform ergodicity


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