Statistics and Computing

, Volume 17, Issue 3, pp 263–279 | Cite as

On population-based simulation for static inference

  • Ajay Jasra
  • David A. Stephens
  • Christopher C. Holmes


In this paper we present a review of population-based simulation for static inference problems. Such methods can be described as generating a collection of random variables {X n } n=1,…,N in parallel in order to simulate from some target density π (or potentially sequence of target densities). Population-based simulation is important as many challenging sampling problems in applied statistics cannot be dealt with successfully by conventional Markov chain Monte Carlo (MCMC) methods. We summarize population-based MCMC (Geyer, Computing Science and Statistics: The 23rd Symposium on the Interface, pp. 156–163, 1991; Liang and Wong, J. Am. Stat. Assoc. 96, 653–666, 2001) and sequential Monte Carlo samplers (SMC) (Del Moral, Doucet and Jasra, J. Roy. Stat. Soc. Ser. B 68, 411–436, 2006a), providing a comparison of the approaches. We give numerical examples from Bayesian mixture modelling (Richardson and Green, J. Roy. Stat. Soc. Ser. B 59, 731–792, 1997).


Markov chain Monte Carlo Sequential Monte Carlo Bayesian mixture models Adaptive methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andrieu, C., Moulines, É.: On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16, 1462–1505 (2006) zbMATHCrossRefGoogle Scholar
  2. Andrieu, C., Robert, C.P.: Controlled MCMC for optimal sampling. Technical Report, Universitié Paris Dauphine (2001) Google Scholar
  3. Andrieu, C., Jasra, A., Doucet, A., Del Moral, P.: Non-linear Markov chain Monte Carlo via self interacting approximations. Technical Report, University of Bristol (2007a) Google Scholar
  4. Andrieu, C., Jasra, A., Doucet, A., Del Moral, P.: A note on the convergence of the equi-energy sampler. Technical Report, University of Bristol (2007b). Stoch. Anal. Appl. (to appear) Google Scholar
  5. Atchadé, Y.F., Liu, J.S.: The Wang-Landau algorithm for Monte Carlo computation in general state spaces. Technical Report, University of Ottawa (2004) Google Scholar
  6. Atchadé, Y.F., Liu, J.S.: Discussion of the ‘equi-energy sampler’. Ann. Stat. 34, 1620–1628 (2006) CrossRefGoogle Scholar
  7. Baker, J.E.: Adaptive selection methods for genetic algorithms. In: Grefenstette, J. (ed.) Proc. Intl. Conf. on Genetic Algorithms and Their Appl., pp. 101–111. Erlbaum, Mahwah (1985) Google Scholar
  8. Brockwell, A.E., Del Moral, P., Doucet, A.: Sequentially interacting Markov chain Monte Carlo for Bayesian computation. Technical Report, Carnagie Mellon University (2007) Google Scholar
  9. Cappé, O., Guillin, A., Marin, J.M., Robert, C.P.: Population Monte Carlo. J. Comput. Graph. Stat. 13, 907–925 (2004) CrossRefGoogle Scholar
  10. Chen, Y., Xie, J., Liu, J.S.: Stopping-time resampling for sequential Monte Carlo methods. J. Roy. Stat. Soc. Ser. B 67, 199–217 (2005) zbMATHCrossRefGoogle Scholar
  11. Chopin, N.: A sequential particle filter method for static models. Biometrika 89, 539–552 (2002) zbMATHCrossRefGoogle Scholar
  12. Chopin, N.: Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Stat. 32, 2385–2411 (2004) zbMATHCrossRefGoogle Scholar
  13. Chopin, N.: Inference and model choice for time-ordered hidden Markov models. J. Roy. Stat. Soc. Ser. B (2007, to appear) Google Scholar
  14. Crisan, D., Doucet, A.: Convergence of sequential Monte Carlo methods. Technical Report, University of Cambridge (2000) Google Scholar
  15. Del Moral, P.: Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York (2004) zbMATHGoogle Scholar
  16. Del Moral, P., Doucet, A.: On a class of genealogical and interacting Metropolis models. In: Azéma, J., Emery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités XXXVII. Lecture Notes in Math., vol. 1832, pp. 415–446. Springer, Berlin (2003) Google Scholar
  17. Del Moral, P., Miclo, L.: Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. In: Séminaire de Probabilitiés XXXIV. Lecture Notes in Math., vol. 1729, pp. 1–145. Springer, Berlin (2000) CrossRefGoogle Scholar
  18. Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. Roy. Stat. Soc. Ser. B 68, 411–436 (2006a) zbMATHCrossRefGoogle Scholar
  19. Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo for Bayesian computation (with discussion). In: Bayarri, S., Berger, J.O., Bernardo, J.M., Dawid, A.P., Heckerman, D., Smith, A.F.M., West, M. (eds.) Bayesian Statistics 8 (2006b, in press) Google Scholar
  20. Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3, 696–730 (1993) zbMATHGoogle Scholar
  21. Douc, R., Moulines, É.: Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Technical Report, Centre de Mathématiques Appliquées, École Polytechnique (2006). Ann. Stat. (to appear) Google Scholar
  22. Douc, R., Cappé, O., Moulines, É.: Comparison of resampling schemes for particle filtering. In 4th International Symposium on Image and Signal Processing and Analysis (ISPA) (2005) Google Scholar
  23. Douc, R., Guillin, A., Marin, J.M., Robert, C.P.: Convergence of adaptive sampling schemes. Ann. Stat. (2006a, in press) Google Scholar
  24. Douc, R., Guillin, A., Marin, J.M., Robert, C.P.: Minimum variance importance sampling via population Monte Carlo. Technical Report, Université Paris-Dauphine (2006b). ESIAM Probab. Stat. (to appear) Google Scholar
  25. Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling for Bayesian filtering. Stat. Comput. 10, 197–208 (2000) CrossRefGoogle Scholar
  26. Doucet, A., De Freitas, J.F.G., Gordon, N.J.: Sequential Monte Carlo Methods in Practice. Springer, New York (2001) zbMATHGoogle Scholar
  27. Eberle, A., Marinelli, C.: Convergence of sequential Markov chain Monte Carlo methods I: Non-linear flow of probability measures. Technical Report, Universität Bonn (2006) Google Scholar
  28. Fearnhead, P., Meligkotsidou, L.: Filtering methods for mixture models. J. Comput. Graph. Stat. (2007, to appear) Google Scholar
  29. Gelman, A., Meng, X.L.: Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Stat. Sci. 13, 163–185 (1998) zbMATHCrossRefGoogle Scholar
  30. Geyer, C.J.: Markov chain maximum likelihood. In: Keramigas, E. (ed.) Computing Science and Statistics: The 23rd Symposium on the Interface, pp. 156–163. Interface Foundation, Fairfax (1991) Google Scholar
  31. Geyer, C.J., Thompson, E.A.: Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Am. Stat. Assoc. 90, 909–920 (1995) zbMATHCrossRefGoogle Scholar
  32. Gilks, W.R., Roberts, G.O., George, E.I.: Adaptive direction sampling. The Statistician 43, 179–189 (1994) CrossRefGoogle Scholar
  33. Gilks, W.R., Berzuini, C.: Following a moving target—Monte Carlo inference for dynamic Bayesian models. J. Roy. Stat. Soc. Ser. B 63, 127–146 (2001) zbMATHCrossRefGoogle Scholar
  34. Goswami, G.R., Liu, J.S.: On learning strategies for evolutionary Monte Carlo. Stat. Comput. 17, 23–28 (2007) CrossRefGoogle Scholar
  35. Grassberger, P.: Pruned-enriched Rosenbluth method: simulations of θ polymers of chain length up to 1 000 000. Phys. Rev. E 56, 3682–3693 (1997) CrossRefGoogle Scholar
  36. Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995) zbMATHCrossRefGoogle Scholar
  37. Green, P.J., Mira, A.: Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88, 1035–1053 (2001) zbMATHGoogle Scholar
  38. Hammersley, J.M., Morton, K.W.: Poor man’s Monte Carlo. J. Roy. Stat. Soc. Ser. B 16, 23–38 (1999) Google Scholar
  39. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970) zbMATHCrossRefGoogle Scholar
  40. Heard, N.A., Holmes, C.C., Stephens, D.A.: A quantitative study of gene regulation involved in the immune response of anopheline mosquitoes: an application of Bayesian hierarchical clustering of curves. J. Am. Stat. Assoc. 101, 18–29 (2006) CrossRefGoogle Scholar
  41. Hukushima, K., Nemoto, K.: Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65, 1604–1608 (1996) CrossRefGoogle Scholar
  42. Iba, Y.: Population Monte Carlo algorithms. Trans. Jpn. Soc. Artif. Intell. 16, 279–286 (2000) CrossRefGoogle Scholar
  43. Iba, Y.: Extended ensemble Monte Carlo. Int. J. Mod. Phys. 12, 653–656 (2001) Google Scholar
  44. Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997) CrossRefGoogle Scholar
  45. Jasra, A.: Bayesian inference for mixture models via Monte Carlo computation. PhD thesis, Imperial College London (2005) Google Scholar
  46. Jasra, A., Doucet, A.: Stability of sequential Monte Carlo samplers via the Foster-Lyapunov condition. Technical Report, University of British Columbia (2006) Google Scholar
  47. Jasra, A., Holmes, C.C., Stephens, D.A.: Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modelling. Stat. Sci. 20, 50–67 (2005a) zbMATHCrossRefGoogle Scholar
  48. Jasra, A., Stephens, D.A., Holmes, C.C.: Population-based reversible jump Markov chain Monte Carlo. Technical Report, Imperial College London (2005b). Biometrika (to appear) Google Scholar
  49. Jasra, A., Doucet, A., Stephens, D.A., Holmes, C.C.: Interacting sequential Monte Carlo samplers for trans-dimensional simulation. Technical Report, Imperial College London (2005c) Google Scholar
  50. Johansen, A., Del Moral, P., Doucet, A.: Sequential Monte Carlo samplers for rare event estimation. Technical Report, University of Cambridge (2006) Google Scholar
  51. Kou, S.C., Zhou, Q., Wong, W.H.: Equi-energy sampler with applications to statistical inference and statistical mechanics. Ann. Stat. 32, 1581–1619 (2006) CrossRefGoogle Scholar
  52. Künsch, H.R.: Recursive Monte Carlo filters; algorithms and theoretical analysis. Ann. Stat. 33, 1983–2021 (2005) zbMATHCrossRefGoogle Scholar
  53. Liang, F.: Dynamically weighted importance sampling in Monte Carlo computation. J. Am. Stat. Assoc. 97, 807–821 (2002) zbMATHCrossRefGoogle Scholar
  54. Liang, F.: Use of sequential structure in simulation from high-dimensional systems. Phys. Rev. E 67, 056101–056107 (2003) Google Scholar
  55. Liang, F., Wong, W.H.: Real parameter evolutionary Monte Carlo with applications to Bayesian mixture models. J. Am. Stat. Assoc. 96, 653–666 (2001) zbMATHCrossRefGoogle Scholar
  56. Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001) zbMATHGoogle Scholar
  57. Liu, J.S., Chen, R.: Sequential Monte Carlo methods for dynamic systems. J. Am. Stat. Assoc. 93, 1032–1044 (1998) zbMATHCrossRefGoogle Scholar
  58. Liu, J.S., Chen, R., Wong, W.H.: Rejection control and sequential importance sampling. J. Am. Stat. Assoc. 93, 1022–1031 (1998) zbMATHCrossRefGoogle Scholar
  59. Madras, N., Zheng, Z.: On the swapping algorithm. Random Struct. Algorithms 22, 66–97 (2003) zbMATHCrossRefGoogle Scholar
  60. Marinari, E., Parisi, G.: Simulated tempering; a new Monte Carlo scheme. Europhys. Lett. 19, 451–458 (1992) CrossRefGoogle Scholar
  61. Matthews, P.: A slowly mixing Markov chain and its implication for Gibbs sampling. Stat. Probab. Lett. 17, 231–236 (1993) zbMATHCrossRefGoogle Scholar
  62. McLachlan, G.J., Peel, D.: Finite Mixture Models. Wiley, Chichester (2000) zbMATHGoogle Scholar
  63. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953) CrossRefGoogle Scholar
  64. Mitsutake, A., Sugita, Y., Okamoto, Y.: Replica-exchange multicanonical and multicanonical replica exchange Monte Carlo simulations of peptides. I. Formula and benchmark tests. J. Chem. Phys. 118, 6664–6676 (2003) CrossRefGoogle Scholar
  65. Neal, R.M.: Sampling from multimodal distributions using tempered transitions. Stat. Comput. 4, 353–366 (1996) CrossRefGoogle Scholar
  66. Neal, R.M.: Annealed importance sampling. Stat. Comput. 11, 125–139 (2001) CrossRefGoogle Scholar
  67. Neal, R.M.: Estimating ratios of normalizing constants using linked importance sampling. Technical Report, University of Toronto (2005) Google Scholar
  68. Pritchard, J.K., Stephens, M., Donnelly, P.: Inference of population structure using multilocus genotype data. Genetics 155, 945–959 (2001) Google Scholar
  69. Richardson, S., Green, P.J.: On Bayesian analysis of mixture models with an unknown number of components (with discussion). J. Roy. Stat. Soc. Ser. B 59, 731–792 (1997) zbMATHCrossRefGoogle Scholar
  70. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, New York (2004) zbMATHGoogle Scholar
  71. Robert, C.P., Rydén, T., Titterington, D.M.: Bayesian inference in hidden Markov models through reversible jump Markov chain Monte Carlo. J. Roy. Stat. Soc. Ser. B 62, 57–75 (2000) zbMATHCrossRefGoogle Scholar
  72. Roberts, G.O., Rosenthal, J.S.: General state space Markov chains and MCMC algorithms. Probab. Surv. 1, 20–71 (2004) CrossRefGoogle Scholar
  73. Roberts, G.O., Rosenthal, J.S.: Coupling and ergodicity of adaptive MCMC. Technical Report, University of Lancaster (2005) Google Scholar
  74. Ron, D., Swendson, R.H., Brandt, A.: Inverse Monte Carlo renormalization group transformations for critical phenomena. Phys. Rev. Lett. 89, 275701–275705 (2002) CrossRefGoogle Scholar
  75. Rousset, M.: Continuous time population Monte Carlo and computational physics. PhD thesis, Universitié Paul Sabatier, Toulouse (2006) Google Scholar
  76. Rousset, M., Stoltz, G.: Equilibrium sampling from nonequilibrium dynamics. J. Stat. Phys. 123(6), 1251–1272 (2006) zbMATHCrossRefGoogle Scholar
  77. Warnes, A.: The normal kernel coupler: an adaptive Markov chain Monte Carlo method for efficiently sampling from multimodal distributions. PhD thesis, University of Washington (2001) Google Scholar
  78. Whitley, D.: A genetic algorithm tutorial. Stat. Comput. 4, 65–85 (1994) CrossRefGoogle Scholar
  79. Wong, W.H., Liang, F.: Dynamic weighting in Monte Carlo optimization. Proc. Nat. Acad. Sci. 94, 14220–14224 (1997) zbMATHCrossRefGoogle Scholar
  80. Zhang, J.L., Liu, J.S.: A new sequential importance sampling method and its application to the two dimensional hydrophobic-hydrophilic model. J. Chem. Phys. 117, 3492–3498 (2002) CrossRefGoogle Scholar
  81. Zheng, Z.: On swapping and simulated tempering algorithms. Stoch. Process. Appl. 104, 131–153 (2003) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Ajay Jasra
    • 1
  • David A. Stephens
    • 2
  • Christopher C. Holmes
    • 3
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Department of StatisticsUniversity of OxfordOxfordUK

Personalised recommendations