Abstract
A recent development of the Markov chain Monte Carlo (MCMC) technique is the emergence of MCMC samplers that allow transitions between different models. Such samplers make possible a range of computational tasks involving models, including model selection, model evaluation, model averaging and hypothesis testing. An example of this type of sampler is the reversible jump MCMC sampler, which is a generalization of the Metropolis–Hastings algorithm. Here, we present a new MCMC sampler of this type. The new sampler is a generalization of the Gibbs sampler, but somewhat surprisingly, it also turns out to encompass as particular cases all of the well-known MCMC samplers, including those of Metropolis, Barker, and Hastings. Moreover, the new sampler generalizes the reversible jump MCMC. It therefore appears to be a very general framework for MCMC sampling. This paper describes the new sampler and illustrates its use in three applications in Computational Biology, specifically determination of consensus sequences, phylogenetic inference and delineation of isochores via multiple change-point analysis.
Similar content being viewed by others
References
A. A. Barker, “Monte Carlo calculations of the radial distribution functions for a proton-electron plasma,” Aust. J. Phys. vol. 18 pp. 119–133, 1965.
B. P. Carlin and S. Chib “Bayesian model choice via MCMC,” J. Royal Statist. Society series B.57 pp. 473–484, 1995.
W. M. Fitch, “Toward defining the course of evolution: minimum change for a specific tree topology,” Syst. Zool. vol. 20 pp. 406–416, 1971.
A. F. Gelfand and A. F. M. Smith, “Sampling-based approaches to calculating marginal densities,” J. Am. Stat. Assoc. vol. 85 pp. 398–409, 1990.
S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE T. Pattern Anal. vol. 6 pp. 721–741, 1984.
P. J. Green, “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination,” Biometrika vol. 82 pp. 711–732, 1995.
W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika vol. 57 pp. 97–109, 1970.
J. P. Huelsenbeck and F. Ronquist, “Mr Bayes: Bayesian inference of phylogenetic trees,” Bioinformatics vol. 17 pp. 754–755, 2001.
International Human Genome Sequencing Consortium, “Initial sequencing and analysis of the human genome,” Nature vol. 409 pp. 860–921, 2001.
J. M. Keith, P. Adams, D. Bryant, D. P. Kroese, K. R. Mitchelson, D. A. E. Cochran, and G. H. Lala, “A simulated annealing algorithm for finding consensus sequences,” Bioinformatics vol. 18 pp. 1494–1499, 2002.
J. M. Keith, P. Adams, D. Bryant, K. R. Mitchelson, D. A. E. Cochran, and G. H. Lala, “Inferring an original sequence from erroneous copies: a Bayesian approach,” in Yi-Ping Phoebe Chen (ed.), Proceedings of the 1st Asia-Pacific Bioinformatics Conference (APBC2003), Conferences in Research and Practice in Information Technology, vol. 19 pp. 23–28, 2003a.
J. M. Keith, P. Adams, D. Bryant, K. R. Mitchelson, D. A. E. Cochran, and G. H. Lala, “Inferring an original sequence from erroneous copies: two approaches,” Asia-Pacific BioTech News vol. 7 pp. 107–114, 2003.
B. Larget and D. Simon, “Markov chain Monte Carlo algorithms for the Bayesian analysis of phylogenetic trees,” Mol. Biol. Evol. vol. 16 pp. 750–759, 1999.
B. Mau and M. Newton, “Phylogenetic inference for binary data on dendrograms using Markov chain Monte Carlo,” J. Comput. Graph. Stat. vol. 6 pp. 122–131, 1997.
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, and A. H. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. vol. 21 pp. 1087–1092, 1953.
W. J. Murphy, E. Eizirik, S. J. O'Brien, O. Madsen, M. Scally, C. J. Douady, E. Teeling, O. A. Ryder, M. J. Stanhope, W. W. de Jong, and M. S. Springer, “Resolution of the early placental mammal radiation using Bayesian phylogenetics,” Science vol. 294 pp. 2348–2351.
J. L. Oliver, P. Bernaolo-Galvin, P. Carpena, and R. Roman-Roldan, “Isochore chromosome maps of eukaryotic genomes,” Gene vol. 276 pp. 47–56, 2001.
R. D. M. Page, “TREEVIEW: An application to display phylogenetic trees on personal computers,” Computer Applications in the Biosciences vol. 12 pp. 357–358, 1996.
D. B. Phillips and A. F. M. Smith, “Bayesian model comparison via jump diffusions,” in W. R. Gilks, S. T. Richardson, and D. J. Spiegelhalter (eds.), Markov Chain Monte Carlo in Practice, Chapman and Hall, 1995.
Z. Yang and B. Rannala “Bayesian phylogenetic inference using DNA sequences: a Markov chain Monte Carlo method,” Mol. Biol. Evol. vol. 14 pp. 717–724, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Keith, J.M., Kroese, D.P. & Bryant, D. A Generalized Markov Sampler. Methodology and Computing in Applied Probability 6, 29–53 (2004). https://doi.org/10.1023/B:MCAP.0000012414.14405.15
Issue Date:
DOI: https://doi.org/10.1023/B:MCAP.0000012414.14405.15