Markov Chain Monte Carlo, Recent Developments
4
First published: 15 July 2005
Abstract
There has recently been an explosion of interest in Markov chain Monte Carlo (MCMC) for biostatistical modeling and analysis, particularly, but not exclusively in a Bayesian framework. The aim of this update is to describe some of the new developments in MCMC methodology and their application in biostatistics. MCMC algorithms, including variations on Metropolis and Gibbs, reversible jump MCMC, slice sampling perfect simulation, delayed rejection, and sequential Monte Carlo are briefly discussed. The important issue of MCMC convergence is also addressed. Applications of MCMC in areas of interest in biostatistics are illustrated through references to CART, latent variable models, hidden Markov models, time series models, factorial experiments, correlated data, meta-analysis, clinical trials, and bioinformatics. The expanding array of software available for MCMC is acknowledged, followed by references to recent biostatistics-related books on the topic.
References
- 1 Albert, J. H. & Chib, S. (2001) Sequential ordinal modeling with applications to survival data, Biometrics 57(3), 829–836.
- 2 Anselin, L. & Griffith, D. (1988) Do spatial effects really matter in regression analysis, Papers of the Regional Science Association 65, 11–34.
- 3 Baldi, P. & Brunak, S. (2001) Bioinformatics: The Machine Learning Approach. MIT Press, Cambridge, Massachusetts.
- 4 Banerjee, S., Gelfand, A. E. & Carlin, B. P. (2003) Hierarchical Modelling and Analysis for Spatial Data. Chapman & Hall/CRC, London.
- 5
Besag, J.
(1974)
Spatial interaction and the statistical analysis of lattice systems (with discussion),
Journal of the Royal Statistical Society Series B – Statistical Methodology
36,
192–326.
10.1111/j.2517-6161.1974.tb00999.x Google Scholar
- 6 Besag, J. E. (1994) Discussion of “Markov chains for exploring posterior distributions, Annals of Statistics 22, 1734–1741.
- 7 Besag, J., Green, E., Higdon, D. & Mengersen, K. L. (1995) Bayesian computation and stochastic systems (with discussion), Statistical Science 10, 3–66.
- 8 Best, N. G., Cowles, M. K. & Vines, K. (1995) CODA: Convergence diagnosis and output analysis software for Gibbs sampling output, Version 0.30, Technical Report, MRC Biostatistics Unit, University of Cambridge. Cambridge, UK.
- 9 Breiman, L., Friedman, J. H., Olshen, R. & Stone, C. (1984) Classification and Regression Trees. Wadsworth, California.
- 10 Brooks, S. P. (2001) On Bayesian analyses and finite mixtures for proportions, Statistics and Computing 11(2), 179–190.
- 11 Brooks, S. P., Dellaportas, P. & Roberts, G. O. (1997) A total variation method for diagnosing convergence of MCMC algorithms, Journal of Computational and Graphical Statistics 6, 251–265.
- 12 Brooks, S. P. & Giudici, P. (1999) Convergence assessment for reversible jump MCMC simulations in Bayesian Statistics 6 – Proceedings of the Sixth Valencia International Meeting, New York, 733–742.
- 13 Brooks, S. P. & Roberts, G. O. (1999) On quantile estimation and Markov chain Monte Carlo convergence, Biometrika 86, 710–717.
- 14 Carlin, B. P. & Louis, T. A. (1996) Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall, London.
- 15
Carlin, B. &
Louis, T. A.
(2000)
Bayes and Empirical Bayes Methods for Data Analysis.
2nd Ed.
Chapman & Hall,
London.
10.1201/9781420057669 Google Scholar
- 16 Casella, G., Mengersen, K. L., Robert, C. P. & Titterington, D. M. (2002) Perfect samplers for mixtures of distributions, Journal of the Royal Statistical Society Series B – Statistical Methodology 64, 777–790.
- 17 Castelloe, J. M. (1999) Reversible jump Markov chain Monte Carlo analysis of spatial Poisson cluster processes, ASA Proceedings of the Section on Bayesian Statistical Science, pp. 102–107.
- 18 Celeux, G., Hurn, M. & Robert, C. P. (2000) Computational and inferential difficulties with mixture posterior distributions, Journal of the American Statistical Association 95, 957–970.
- 19
Chen, M. H.,
Shao, Q. M. &
Ibrahim, J. G.
(2000)
Monte Carlo Methods in Bayesian Computation.
Springer-Verlag,
New York.
10.1007/978-1-4612-1276-8 Google Scholar
- 20 Chen, M. H. & Schmeiser, B. W. (1998) Towards black-box sampling, Journal of Computational and Graphical Statistics 7, 1–22.
- 21 Chib, S. & Jeliazkov, I. (2001) Marginal likelihood from the Metropolis-Hastings output, Journal of the American Statistical Association 96(453), 270–281.
- 22 Congdon, P. (2001) Bayesian Statistical Modelling. Wiley, London.
- 23
Congdon, P.
(2003)
Applied Bayesian Modelling.
Wiley,
New York.
10.1002/0470867159 Google Scholar
- 24 Corander, J., Waldmann, P. & Sillanpaa, M. J. (2003) Bayesian analysis of genetic differentiation between populations, Genetics 163(1), 367–374.
- 25 Cowell, R. G., Dawid, A. P., Lauritzen, S. L. & Spiegelhalter, D. J. (2003) Probabilistic Networks and Expert Systems. Springer-Verlag Series in Information Science and Statistics. Springer-Verlag, New York.
- 26 Cowles, M. K. & Carlin, B. P. (1996) Markov chain Monte Carlo convergence diagnostics: a comparative study, Journal of the American Statistical Association 91, 883–904.
- 27 Crisan, D. & Doucet, A. (2000) Convergence of sequential Monte Carlo methods, Technical Report CUED/F-INFENG/TR381, Department of Engineering, University of Cambridge.
- 28 Dellaportas, P., Forster, J. J. & Ntzoufras, I. (2002) On Bayesian model and variable selection using MCMC, Statistics and Computing 12(1), 27–36.
- 29 Del Moral, P. & Guionnet, A. (1999) Central limit theorem for nonlinear filtering and interacting particle systems, Annals of Applied Probability 9(2), 275–297.
- 30 Denison, D. G. T., Holmes, C. C., Mallick, B. K. & Smith, A. F. M. (2002) Bayesian Methods for Nonlinear Classification and Regression. Wiley, New York.
- 31 Denison, D. G. T., Mallick, B. K. & Smith, A. F. M. (1998) Automatic Bayesian curve fitting, Journal of the Royal Statistical Society Series B – Statistical Methodology 60, 333–350.
- 32
A. Doucet,
N. Freitas &
N. Gordon, eds.
(2001)
Sequential Monte Carlo Methods in Practice
Springer-Verlag,
New York.
10.1007/978-1-4757-3437-9 Google Scholar
- 33 Fearnhead, P. (2002) MCMC, sufficient statistics and particle filter. P. Fearnhead, Computational and Graphical Statistics 11, 848–862.
- 34 Fernández, C. & Green, P. J. (2002) Modeling spatially correlated data via mixtures: A Bayesian approach, Journal of the Royal Statistical Society Series B – Statistical Methodology 64(4), 805–826.
- 35 Fill, J. A. (1998) An interruptible algorithm for exact sampling via Markov chains, Annals of Applied Probability 8, 131–162.
- 36 Fort, G. & Moulines, E. (2000) V-subgeometric ergodicity for a Hastings-Metropolis algorithm, Statistics and Probability Letters 49, 401–410.
- 37 Gilks, W. R. & Berzuini, C. (2001) Following a moving target – Monte Carlo inference for dynamic Bayesian models, Journal of the Royal Statistical Society Series B – Statistical Methodology 63, 127–146.
- 38 Gilks, W. R., Roberts, G. O. & Sahu, S. K. (1998) Adaptive Markov chain Monte Carlo, Journal of the American Statistical Association 93, 1045–1054.
- 39 Green, P. J. (1995) Reversible jump MCMC computation and Bayesian model determination, Biometrika 82(4), 711–732.
- 40 Green, P. J. & Mira, A. (2001) Delayed rejection in reversible jump Metropolis-Hastings, Biometrika 88(4), 1035–1053.
- 41 Green, P. J. & Richardson, S. (2002) Hidden Markov models and disease mapping, Journal of the American Statistical Association 97(460), 1055–1070.
- 42 Grenander, U. & Miller, M. (1994) Representations of knowledge in complex systems (with discussion), Journal of the Royal Statistical Society Series B – Statistical Methodology 56, 549–603.
- 43
Gustafson, P.
(2003)
Measurement Error and Misclassification in Statistics and Epidemiology: Impacts and Bayesian Adjustments.
Chapman & Hall/ CRC Press,
Boca Raton.
10.1201/9780203502761 Google Scholar
- 44 Haario, H., Saksman, E. & Tamminen, J. (1999) Adaptive Proposal Distribution for Random Walk Metropolis Algorithm, Computational Statistics 14, 375–395.
- 45 Han, C. & Carlin, B. P. (2001) Markov chain Monte Carlo methods for computing Bayes factors: a comparative review, Journal of the American Statistical Association 96(455), 1122–1132.
- 46 Hobert, J. P., Jones, G. L., Presnell, B. & Rosenthal, J. S. (2002) On the applicability of regenerative simulation in Markov chain Monte Carlo, Biometrika 89(4), 731–743.
- 47 Huelsenbeck, J. P., Larget, B., Miller, R. E. & Ronquist, F. (2002) Potential applications and pitfalls of Bayesian inference of phylogeny, Systematic Biology 51(5), 673–688.
- 48
Ibrahim, J. G.,
Chen, M. H. &
Sinha, D.
(2001)
Bayesian Survival Analysis.
Springer-Verlag,
New York.
10.1007/978-1-4757-3447-8 Google Scholar
- 49 Jarner, S. & Hansen, E. (2000) Geometric ergodicity of Metropolis algorithms, Stochastic Processes and their Applications 12, 224–247.
- 50 Kendall, W. (1998) Perfect simulation for the area-interaction point process, in Probability Towards 2000, C. C. Heyde, & L. Accardi, eds. Springer-Verlag, New York, 218–234.
- 51 Kendall, W. S. & Møller, J. (2000) Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes, Advances in Applied Probability 32(3), 844–865.
- 52 Kenehisa, M. & Bork, P. (2003) Bioinformatics in the post-sequence era, Nature Genetics Supplement 33, 305–310.
- 53
Knorr-Held, L. &
Besag, J.
(1998)
Modelling risk from a disease in time and space,
Statistics in Medicine
17,
2045–2060.
10.1002/(SICI)1097-0258(19980930)17:18<2045::AID-SIM943>3.0.CO;2-P CAS PubMed Web of Science® Google Scholar
- 54 Knorr-Held, L. & Best, N. G. (2001) A shared component model for detecting joint and selective clustering of two diseases (Pkg: p49–99), Journal of the Royal Statistical Society, Series A, General 164(1), 73–85.
- 55 Knorr-Held, L. & Raßer, G. (2000) Bayesian detection of clusters and discontinuities in disease maps, Biometrics 56(1), 13–21.
- 56
Lawson, A. B.
(2000)
Cluster modelling of disease incidence via RJMCMC methods: a comparative evaluation,
Statistics in Medicine
19,
2361–2375.
10.1002/1097-0258(20000915/30)19:17/18<2361::AID-SIM575>3.0.CO;2-N CAS PubMed Web of Science® Google Scholar
- 57 Lazar, N. A. & Kadane, J. B. (2002) Movies for the visualization of MCMC output, Journal of Computational and Graphical Statistics 11(4), 863–874.
- 58 Lee, K. E., Sha, N. J., Dougherty, E. R., Vannucci, M. & Mallick, B. K. (2003) Gene selection: a Bayesian variable selection approach, Bioinformatics 19(1), 90–97.
- 59 Liechty, J. C. & Roberts, G. O. (2001) Markov chain Monte Carlo methods for switching diffusion models, Biometrika 88(2), 299–315.
- 60 Lindstrom, M. J. (2002) Bayesian estimation of free-knot splines using reversible jumps, Computational Statistics and Data Analysis 41(2), 255–269.
- 61 Liu, J. S. (1996) Metropolized independent sampling with comparisons to rejection sampling and importance sampling, Statistics and Computing 6, 113–119.
- 62 Liu, J. S. & Chen, R. (1998) Sequential Monte Carlo methods for dynamic systems, Journal of the American Statistical Association 93, 1032–1044.
- 63 Mengersen, K. L. & Robert, C. P. (1999) MCMC convergence diagnostics: a review, in Bayesian Statistics VI, J. O. Berger, A. P. Dawid & A. F. M. Smith, eds. Oxford University Press, Oxford, pp. 415–440.
- 64 Mengersen, K. & Robert, C. P. (2003) Population-based Markov chain Monte Carlo: the pinball sampler, in Bayesian Statistics 7, J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith & M. West, eds. Oxford University Press, Oxford.
- 65 Mira, A. (2001) Ordering and improving the performance of Monte Carlo Markov chains, Statistical Science 16(4), 340–350.
- 66 Mira, A., Møller, J. & Roberts, G. O. (2001) Perfect slice samplers, Journal of the Royal Statistical Society Series B – Statistical Methodology 63, 593–606.
- 67 Mira, A., Møller, J. & Roberts, G. O. (2002) Correction to “Perfect slice samplers” (2001v63 p593–606), Journal of the Royal Statistical Society, Series B, Methodological 64(3), 581–581.
- 68 Mira, A. & Tierney, L. (2002) Efficiency and convergence properties of slice samplers, Scandinavian Journal of Statistics 29(1), 1–12.
- 69 Muller, P. (1991) A Generic Approach To Posterior Integration And Gibbs Sampling, Technical Report, ad no 91-09, Purdue University, West Lafayette.
- 70 Muller, P. (1993) Alternatives to the Gibbs Sampling Scheme, Technical Report, Institute of Statistics and Decision Sciences, Duke University.
- 71 Murdoch, D. J. & Green, P. J. (1998) Exact sampling for a continuous state, Scandinavian Journal of Statistics 25(3), 483–502.
- 72 Nam, I. -S., Mengersen, K., & Garthwaite, P. (2003) Multivariate meta-analysis, Statistics in Medicine 22, pp. 2309–2333.
- 73 Neal, R. M. (2003) Slice sampling (with discussion), Annals of Statistics, 31, 705–767.
- 74 Nobile, A. (1998) A hybrid Markov chain for the Bayesian analysis of the multinomial probit model, Statistics and Computing 8, 229–242.
- 75 Nobile, A. & Green, P. J. (2000) Bayesian analysis of factorial experiments by mixture modelling, Biometrika 87(1), 15–35.
- 76 O'Neill, P. D., Balding, D. J., Becker, N. G., Eerola, M. & Mollison, D. (2000) Analyses of infectious disease data from household outbreaks by Markov chain Monte Carlo methods, Applied Statistics 49(4), 517–542.
- 77 Perron, F. & Mengersen, K. (2001) Bayesian nonparametric modeling using mixtures of triangular distributions, Biometrics 57(2), 518–528.
- 78
Phillips, D. B. &
Smith, A. F. M.
(1996)
Bayesian model comparison via jump diffusions, in
Markov chain Monte Carlo in Practice,
W. R. Gilks,
S. T. Richardson &
D. J. Spiegelhalter, eds.
Chapman & Hall,
London,
215–240.
10.1007/978-1-4899-4485-6_13 Google Scholar
- 79 Propp, J. G. & Wilson, D. B. (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms 9, 223–252.
- 80 Richardson, S. & Green, P. J. (1997) On Bayesian analysis of mixtures with an unknown number of components (with discussion), Journal of the Royal Statistical Society Series B – Statistical Methodology 59, 731–792.
- 81
Robert, C. P.
(1996)
Inference in mixture models, in
Markov Chain Monte Carlo in Practice,
W. R. Gilks,
S. Richardson &
D. J. Spiegelhalter, eds.
Chapman & Hall,
London,
441–464.
10.1007/978-1-4899-4485-6_24 Google Scholar
- 82
C. P. Robert ed.
(1998)
Discretization and MCMC convergence assessment.
Springer-Verlag,
New York.
10.1007/978-1-4612-1716-9 Google Scholar
- 83
Robert, C. P. &
Casella, G.
(1999)
Monte Carlo Statistical Methods.
Springer-Verlag,
New York.
10.1007/978-1-4757-3071-5 Google Scholar
- 84 Robert, C. P. & Mengersen, K. L. (1999) Reparametrization issues in mixture estimation and their bearings on the Gibbs sampler, Computational Statistics and Data Analysis 29, 325–343.
- 85 Robert, C. P., Rydén, T. & Titterington, D. M. (2000) Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method, Journal of the Royal Statistical Society Series B – Statistical Methodology 62(1), 57–75.
- 86 Roberts, G. O., Gelman, A. & Gilks, W. R. (1997) Weak convergence and optimal scaling of random walk Metropolis algorithms, Annals of Applied Probability 7, 110–120.
- 87 Roberts, G. O. & Rosenthal, J. S. (1998) Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion), Canadian Journal of Statistics 26, 5–32.
- 88 Roberts, G. O. & Rosenthal, J. S. (1999) Convergence of slice sampler Markov chains, Journal of the Royal Statistical Society Series B – Statistical Methodology 61, 643–660.
- 89 Roberts, G. O. & Sahu, S. K. (1997) Updating schemes, covariance structure, blocking and parametrisation for the Gibbs sampler, Journal of the Royal Statistical Society Series B – Statistical Methodology 59, 291–318.
- 90 Robert, C. P. & Titterington, M. (1998) Reparametrisation strategies for hidden Markov models and Bayesian approaches to maximum likelihood estimation, Statistics and Computing 8(2), 145–158.
- 91
Roberts, G. O. &
Tweedie, R. L.
(1996)
Exponential Convergence of Langevin diffusions and their discrete approximations,
Bernoulli
2,
341–364.
10.2307/3318418 Google Scholar
- 92 Roberts, G. O. & Tweedie, R. L. (1996) Geometric convergence and Central Limit Theorems for multidimensional Hastings and Metropolis algorithms, Biometrika 83, 95–110.
- 93 Rosenthal, J. S. (1995) Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association 90, 558–566.
- 94 Salmenkivi, M., Kere, J. & Mannila, H. (2002) Genome segmentation using piecewise constant intensity models and reversible jump MCMC, Bioinformatics 18(Suppl. S), S211–S218.
- 95 Skare, O., Benth, F. E. & Frigessi, A. (2000) Smoothed Langevin proposals in Metropolis-Hastings algorithms, Statistics and Probability Letters 49(4), 345–354.
- 96 Spiegelhalter, D. J., Thomas, A., Best, N. G. & Lunn, D. (2002) WinBUGS Version 1.4 User Manual. MRC Biostatistics Unit, Cambridge. Available from www.mrc-bsu.cam.ac.uk/bugs.
- 97 Stephens, M. (2000) Bayesian analysis of mixture models with an unknown number of components – An alternative to reversible jump methods, Annals of Statistics 28(1), 40–74.
- 98
Stramer, O. &
Tweedie, R. L.
(1998)
Langevin-type models II: self-targeting candidates for MCMC algorithms,
Methodology and Computing in Applied Probability
1,
307–328.
10.1023/A:1010090512027 Google Scholar
- 99 Tierney, L. (1994) Markov chains for exploring posterior distributions (with discussion), Annals of Statistics 22, 1701–1786.
- 100 Tierney, L. (1998) A note on Metropolis-Hastings kernels for general state spaces, Annals of Applied Probability 8(1), 1–9.
- 101
Tierney, L. &
Mira, A.
(1999)
Some adaptive Monte Carlo methods for Bayesian inference,
Statistics in Medicine
18,
2507–2515.
10.1002/(SICI)1097-0258(19990915/30)18:17/18<2507::AID-SIM272>3.0.CO;2-J CAS PubMed Web of Science® Google Scholar
- 102 Tjelmeland, H. & Hegstad, B. K. (2001) Mode jumping proposals in MCMC, Scandinavian Journal of Statistics 28(1), 205–223.
- 103
Wakefield, J. C.,
Gelfand, A. E. &
Smith, A. F. M.
(1991)
Efficient generation of random variates via the ratio-of-uniform method,
Statistics and Computing
1,
129–133.
10.1007/BF01889987 Google Scholar
- 104 Weakliem, D. L. (1999) A critique of the Bayesian information criterion for model selection (with discussion), Sociological Methods and Research 27, 359–443.
- 105 West, M. & Harrison, J. (1997) Bayesian Forecasting and Dynamic Models, 2nd Ed. Springer-Verlag, New York.
- 106 Wolpert, R., Best, N., Ickstadt, K. & Briggs, D. (2000) Combining models of health and exposure data: the SAVIAH study, in Spatial Epidemiology: Methods and Applications, P. Elliott, J. C. Wakefield, N. G. Best & D. J. Briggs, eds. Oxford University Press, Oxford, 393–414.
- 107 Wolpert, R. & Mengersen, K. (2004) Adjusted likelihoods for synthesising empirical evidence from studies that differ in quality and design: effects of environmental tobacco smoke, Statistical Science. To appear.
Further Reading
- Besag, J. & Higdon, D. (1999) Bayesian analysis of agricultural field experiments (with discussion), Journal of the Royal Statistical Society Series B – Statistical Methodology 61, 691–746.
