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


  • Olivier Cappé
  • Christian P. Robert
  • Tobias Rydén


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. Copyright 2003 Royal Statistical Society.

Suggested Citation

  • Olivier Cappé & Christian P. Robert & Tobias Rydén, 2003. "Reversible jump, birth-and-death and more general continuous time Markov chain Monte Carlo samplers," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(3), pages 679-700.
  • Handle: RePEc:bla:jorssb:v:65:y:2003:i:3:p:679-700

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    Cited by:

    1. Athanasios Christou Micheas, 2014. "Hierarchical Bayesian modeling of marked non-homogeneous Poisson processes with finite mixtures and inclusion of covariate information," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(12), pages 2596-2615, December.
    2. Komárek, Arnost, 2009. "A new R package for Bayesian estimation of multivariate normal mixtures allowing for selection of the number of components and interval-censored data," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 3932-3947, October.
    3. Rufo, M.J. & Martín, J. & Pérez, C.J., 2010. "New approaches to compute Bayes factor in finite mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3324-3335, December.
    4. Philippe, Anne, 2006. "Bayesian analysis of autoregressive moving average processes with unknown orders," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1904-1923, December.
    5. Pandolfi, Silvia & Bartolucci, Francesco & Friel, Nial, 2014. "A generalized multiple-try version of the Reversible Jump algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 298-314.
    6. Antonietta Mira & Fabio Rigat, 2009. "Parallel hierarchical sampling:a general-purpose class of multiple-chains MCMC algorithms," Economics and Quantitative Methods qf0903, Department of Economics, University of Insubria.
    7. Al-Awadhi, Fahimah & Hurn, Merrilee & Jennison, Christopher, 2004. "Improving the acceptance rate of reversible jump MCMC proposals," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 189-198, August.
    8. Cabral, Celso Rômulo Barbosa & Bolfarine, Heleno & Pereira, José Raimundo Gomes, 2008. "Bayesian density estimation using skew student-t-normal mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5075-5090, August.
    9. Rigat, F. & Mira, A., 2012. "Parallel hierarchical sampling: A general-purpose interacting Markov chains Monte Carlo algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1450-1467.
    10. Meyer-Gohde, Alexander & Neuhoff, Daniel, 2018. "Generalized exogenous processes in DSGE: A Bayesian approach," IMFS Working Paper Series 125, Goethe University Frankfurt, Institute for Monetary and Financial Stability (IMFS).

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