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On population-based simulation for static inference

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Abstract

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).

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References

  • Andrieu, C., Moulines, É.: On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16, 1462–1505 (2006)

    Article  MATH  Google Scholar 

  • Andrieu, C., Robert, C.P.: Controlled MCMC for optimal sampling. Technical Report, Universitié Paris Dauphine (2001)

  • 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)

  • 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)

  • Atchadé, Y.F., Liu, J.S.: The Wang-Landau algorithm for Monte Carlo computation in general state spaces. Technical Report, University of Ottawa (2004)

  • Atchadé, Y.F., Liu, J.S.: Discussion of the ‘equi-energy sampler’. Ann. Stat. 34, 1620–1628 (2006)

    Article  Google Scholar 

  • 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 

  • Brockwell, A.E., Del Moral, P., Doucet, A.: Sequentially interacting Markov chain Monte Carlo for Bayesian computation. Technical Report, Carnagie Mellon University (2007)

  • Cappé, O., Guillin, A., Marin, J.M., Robert, C.P.: Population Monte Carlo. J. Comput. Graph. Stat. 13, 907–925 (2004)

    Article  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • Chopin, N.: A sequential particle filter method for static models. Biometrika 89, 539–552 (2002)

    Article  MATH  Google Scholar 

  • Chopin, N.: Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Stat. 32, 2385–2411 (2004)

    Article  MATH  Google Scholar 

  • Chopin, N.: Inference and model choice for time-ordered hidden Markov models. J. Roy. Stat. Soc. Ser. B (2007, to appear)

  • Crisan, D., Doucet, A.: Convergence of sequential Monte Carlo methods. Technical Report, University of Cambridge (2000)

  • Del Moral, P.: Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York (2004)

    MATH  Google Scholar 

  • 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 

  • 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)

    Chapter  Google Scholar 

  • Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. Roy. Stat. Soc. Ser. B 68, 411–436 (2006a)

    Article  MATH  Google Scholar 

  • 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)

  • Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3, 696–730 (1993)

    MATH  Google Scholar 

  • 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)

  • 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)

  • Douc, R., Guillin, A., Marin, J.M., Robert, C.P.: Convergence of adaptive sampling schemes. Ann. Stat. (2006a, in press)

  • 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)

  • Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling for Bayesian filtering. Stat. Comput. 10, 197–208 (2000)

    Article  Google Scholar 

  • Doucet, A., De Freitas, J.F.G., Gordon, N.J.: Sequential Monte Carlo Methods in Practice. Springer, New York (2001)

    MATH  Google Scholar 

  • Eberle, A., Marinelli, C.: Convergence of sequential Markov chain Monte Carlo methods I: Non-linear flow of probability measures. Technical Report, Universität Bonn (2006)

  • Fearnhead, P., Meligkotsidou, L.: Filtering methods for mixture models. J. Comput. Graph. Stat. (2007, to appear)

  • Gelman, A., Meng, X.L.: Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Stat. Sci. 13, 163–185 (1998)

    Article  MATH  Google Scholar 

  • 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 

  • Geyer, C.J., Thompson, E.A.: Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Am. Stat. Assoc. 90, 909–920 (1995)

    Article  MATH  Google Scholar 

  • Gilks, W.R., Roberts, G.O., George, E.I.: Adaptive direction sampling. The Statistician 43, 179–189 (1994)

    Article  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • Goswami, G.R., Liu, J.S.: On learning strategies for evolutionary Monte Carlo. Stat. Comput. 17, 23–28 (2007)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)

    Article  MATH  Google Scholar 

  • Green, P.J., Mira, A.: Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88, 1035–1053 (2001)

    MATH  Google Scholar 

  • Hammersley, J.M., Morton, K.W.: Poor man’s Monte Carlo. J. Roy. Stat. Soc. Ser. B 16, 23–38 (1999)

    Google Scholar 

  • Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Hukushima, K., Nemoto, K.: Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65, 1604–1608 (1996)

    Article  Google Scholar 

  • Iba, Y.: Population Monte Carlo algorithms. Trans. Jpn. Soc. Artif. Intell. 16, 279–286 (2000)

    Article  Google Scholar 

  • Iba, Y.: Extended ensemble Monte Carlo. Int. J. Mod. Phys. 12, 653–656 (2001)

    Google Scholar 

  • Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997)

    Article  Google Scholar 

  • Jasra, A.: Bayesian inference for mixture models via Monte Carlo computation. PhD thesis, Imperial College London (2005)

  • Jasra, A., Doucet, A.: Stability of sequential Monte Carlo samplers via the Foster-Lyapunov condition. Technical Report, University of British Columbia (2006)

  • 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)

    Article  MATH  Google Scholar 

  • 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)

  • 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)

  • Johansen, A., Del Moral, P., Doucet, A.: Sequential Monte Carlo samplers for rare event estimation. Technical Report, University of Cambridge (2006)

  • 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)

    Article  Google Scholar 

  • Künsch, H.R.: Recursive Monte Carlo filters; algorithms and theoretical analysis. Ann. Stat. 33, 1983–2021 (2005)

    Article  MATH  Google Scholar 

  • Liang, F.: Dynamically weighted importance sampling in Monte Carlo computation. J. Am. Stat. Assoc. 97, 807–821 (2002)

    Article  MATH  Google Scholar 

  • Liang, F.: Use of sequential structure in simulation from high-dimensional systems. Phys. Rev. E 67, 056101–056107 (2003)

    Google Scholar 

  • Liang, F., Wong, W.H.: Real parameter evolutionary Monte Carlo with applications to Bayesian mixture models. J. Am. Stat. Assoc. 96, 653–666 (2001)

    Article  MATH  Google Scholar 

  • Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001)

    MATH  Google Scholar 

  • Liu, J.S., Chen, R.: Sequential Monte Carlo methods for dynamic systems. J. Am. Stat. Assoc. 93, 1032–1044 (1998)

    Article  MATH  Google Scholar 

  • Liu, J.S., Chen, R., Wong, W.H.: Rejection control and sequential importance sampling. J. Am. Stat. Assoc. 93, 1022–1031 (1998)

    Article  MATH  Google Scholar 

  • Madras, N., Zheng, Z.: On the swapping algorithm. Random Struct. Algorithms 22, 66–97 (2003)

    Article  MATH  Google Scholar 

  • Marinari, E., Parisi, G.: Simulated tempering; a new Monte Carlo scheme. Europhys. Lett. 19, 451–458 (1992)

    Article  Google Scholar 

  • Matthews, P.: A slowly mixing Markov chain and its implication for Gibbs sampling. Stat. Probab. Lett. 17, 231–236 (1993)

    Article  MATH  Google Scholar 

  • McLachlan, G.J., Peel, D.: Finite Mixture Models. Wiley, Chichester (2000)

    MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Neal, R.M.: Sampling from multimodal distributions using tempered transitions. Stat. Comput. 4, 353–366 (1996)

    Article  Google Scholar 

  • Neal, R.M.: Annealed importance sampling. Stat. Comput. 11, 125–139 (2001)

    Article  Google Scholar 

  • Neal, R.M.: Estimating ratios of normalizing constants using linked importance sampling. Technical Report, University of Toronto (2005)

  • Pritchard, J.K., Stephens, M., Donnelly, P.: Inference of population structure using multilocus genotype data. Genetics 155, 945–959 (2001)

    Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, New York (2004)

    MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • Roberts, G.O., Rosenthal, J.S.: General state space Markov chains and MCMC algorithms. Probab. Surv. 1, 20–71 (2004)

    Article  Google Scholar 

  • Roberts, G.O., Rosenthal, J.S.: Coupling and ergodicity of adaptive MCMC. Technical Report, University of Lancaster (2005)

  • Ron, D., Swendson, R.H., Brandt, A.: Inverse Monte Carlo renormalization group transformations for critical phenomena. Phys. Rev. Lett. 89, 275701–275705 (2002)

    Article  Google Scholar 

  • Rousset, M.: Continuous time population Monte Carlo and computational physics. PhD thesis, Universitié Paul Sabatier, Toulouse (2006)

  • Rousset, M., Stoltz, G.: Equilibrium sampling from nonequilibrium dynamics. J. Stat. Phys. 123(6), 1251–1272 (2006)

    Article  MATH  Google Scholar 

  • 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)

  • Whitley, D.: A genetic algorithm tutorial. Stat. Comput. 4, 65–85 (1994)

    Article  Google Scholar 

  • Wong, W.H., Liang, F.: Dynamic weighting in Monte Carlo optimization. Proc. Nat. Acad. Sci. 94, 14220–14224 (1997)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Zheng, Z.: On swapping and simulated tempering algorithms. Stoch. Process. Appl. 104, 131–153 (2003)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, SW7 2AZ, London, UK

    Ajay Jasra

  2. Department of Mathematics and Statistics, McGill University, H3A 2K6, Montreal, Canada

    David A. Stephens

  3. Department of Statistics, University of Oxford, OX1 3TG, Oxford, UK

    Christopher C. Holmes

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  1. Ajay Jasra
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  2. David A. Stephens
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  3. Christopher C. Holmes
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Correspondence to Ajay Jasra.

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Jasra, A., Stephens, D.A. & Holmes, C.C. On population-based simulation for static inference. Stat Comput 17, 263–279 (2007). https://doi.org/10.1007/s11222-007-9028-9

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