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Statistics and Computing

, Volume 23, Issue 3, pp 323–339 | Cite as

Parallel tempering with equi-energy moves

  • Meïli Baragatti
  • Agnès Grimaud
  • Denys PommeretEmail author
Article

Abstract

The Equi-Energy Sampler (EES) introduced by Kou et al. (in Ann. Stat. 34(4), 1581–1619, 2006) is based on a population of chains which are updated by local moves and global moves, also called equi-energy jumps. The state space is partitioned into energy rings, and the current state of a chain can jump to a past state of an adjacent chain that has an energy level close to its level. This algorithm has been developed to facilitate global moves between different chains, resulting in a good exploration of the state space by the target chain. This method seems to be more efficient than the classical Parallel Tempering (PT) algorithm. However it is difficult to use in combination with a Gibbs sampler and it necessitates increased storage. We propose an adaptation of this EES that combines PT with the principle of swapping between chains with the same level of energy. This adaptation, that we shall call Parallel Tempering with Equi-Energy Moves (PTEEM), keeps the original idea of the EES method while ensuring good theoretical properties, and practical implementation. Performances of the PTEEM algorithm are compared with those of the EES and of the standard PT algorithms in the context of mixture models, and in a problem of identification of transcription factor binding motifs.

Keywords

Equi-energy sampler Parallel tempering Population-based Monte Carlo Markov chains Mixture models Binding sites for transcription factors 

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References

  1. Andrieu, C., Jasra, A., Doucet, A., Moral, P.D.: Convergence of the equi-energy sampler. ESAIM Proc. 19, 1–5 (2007a) zbMATHCrossRefGoogle Scholar
  2. Andrieu, C., Jasra, A., Doucet, A., Moral, P.D.: Non-linear Markov chain Monte Carlo. ESAIM Proc. 19, 79–84 (2007b). doi: 10.1051/proc:071911 zbMATHCrossRefGoogle Scholar
  3. Andrieu, C., Jasra, A., Doucet, A., Del Moral, P.: A note on convergence of the equi-energy sampler. Stoch. Anal. Appl. 26(2), 298–312 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  4. Atchadé, Y.: A cautionary tale on the efficiency of some adaptive Monte Carlo schemes. Ann. Appl. Probab. 20, 841–868 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  5. Atchadé, Y., Liu, J.: Discussion of equi-energy sampler by Kou, Zhou and Wong. Ann. Stat. 34(4), 1620–1628 (2006) zbMATHCrossRefGoogle Scholar
  6. Atchadé, Y., Roberts, G., Rosenthal, S.: Towards optimal scaling of Metropolis-coupled Markov chain Monte Carlo. Stat Comput (2010) Google Scholar
  7. Atchadé, Y., Fort, G., Moulines, E., Priouret, P.: Inference and Learning in Dynamic Models. Cambridge University Press, Cambridge (2011), pp. 33–53 Google Scholar
  8. Athreya, K., Doss, H., Sethuraman, J.: On the convergence of the Markov chain simulation method. Ann. Stat. 24(1), 69–100 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  9. Behrens, G., Friel, N., Hurn, M.: Tuning tempered transitions. Unpublished manuscript (2009) Google Scholar
  10. Crooks, G., Hon, G., Chandonia, J., Brenner, S.: Weblogo: A sequence logo generatorcrooks, g.echandonia, j.m. Genome Res. 14, 1188–1190 (2004) zbMATHCrossRefGoogle Scholar
  11. Geyer, C.: Markov chain Monte Carlo maximum likelihood. In: Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 156–163 (1991) Google Scholar
  12. Geyer, C., Thompson, E.: Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Am. Stat. Assoc. 90, 909–920 (1995) zbMATHCrossRefGoogle Scholar
  13. Green, P.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  14. Green, P., Mira, A.: Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88, 1035–1053 (2001) MathSciNetzbMATHGoogle Scholar
  15. Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 88, 1035–1053 (1970) Google Scholar
  16. Hua, X., Kou, S.: Convergence of the equi-energy sampler and its application to the Ising model. Stat. Sin. (2010, in press) Google Scholar
  17. Jasra, A., Holmes, C., Stephens, D.: Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Stat. Sci. 20(1), 50–67 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  18. Jasra, A., Stephens, D., Holmes, C.: Population-based reversible jump Markov chain Monte Carlo. Biometrika 94, 787–807 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  19. Jensen, S., Liu, X., Zhou, Q., Liu, J.: Computational discovery of gene regulatory binding motifs: A Bayesian perspective. Stat. Sci. 19, 188–294 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Kou, S., Zhou, Q., Wong, W.: Equi-energy sampler with application in statistical inference and statistical mechanics. Ann. Stat. 34(4), 1581–1619 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  21. Lawrence, C., Reilly, A.: An expectation maximization (em) algorithm for the identification and characterization of common sites in unaligned biopolymer sequences. Proteins 7, 41–51 (1990) CrossRefGoogle Scholar
  22. Lawrence, C., Altschul, S., Boguski, M., Liu, J., Neuwald, A.: Detecting subtle sequence signals: A Gibbs sampling strategy for multiple alignment. Science 262, 208–214 (1993) CrossRefGoogle Scholar
  23. Liang, F., Wong, W.: Real-parameter evolutionary Monte Carlo with applications to Bayesian mixture models. J. Am. Stat. Assoc. 96, 653–666 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  24. Liu, J.: The collapsed Gibbs sampler in Bayesian computations with application to a gene regulation problem. J. Am. Stat. Assoc. 89(427), 958–966 (1994) zbMATHCrossRefGoogle Scholar
  25. Liu, J., Neuwald, A., Lawrence, C.: Bayesian models for multiple local sequence alignment and Gibbs sampling strategies. J. Am. Stat. Assoc. 90(432), 1156–1170 (1995) zbMATHCrossRefGoogle Scholar
  26. Liu, X., Brutlag, D., Liu, J.: Bioprospector: Discovering conserved DNA motifs in upstream regulatory regions of co-expressed genes. Pac. Symp. Biocomput. 6, 127–138 (2001) Google Scholar
  27. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953) CrossRefGoogle Scholar
  28. Mitsutake, A., Sugita, Y., Okamoto, Y.: Replica-exchange multicanonical and multicanonical replica-exchange Monte Carlo simulations of peptides. I. Formulation and benchmark test. J. Chem. Phys. 118, 6664–6675 (2011) CrossRefGoogle Scholar
  29. Nagata, K., Watanabe, S.: Asymptotic behavior of exchange ratio in exchange Monte Carlo method. Neural Netw. 21, 980–988 (2008) zbMATHCrossRefGoogle Scholar
  30. Neal, R.: Sampling from multimodal distributions using tempered transitions. Stat. Comput. 6, 353–366 (1996) CrossRefGoogle Scholar
  31. Richardson, S., Green, P.: On Bayesian analysis of mixtures with an unknown number of components (with discussion). J. R. Stat. Soc. B 59, 731–792 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  32. Robert, C., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, Berlin (2004) zbMATHCrossRefGoogle Scholar
  33. Roberts, G., Rosenthal, J.: Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains. Ann. Appl. Probab. 16(4), 2123–2139 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  34. Roth, F., Hugues, J., Estep, J., Church, G.: Finding DNA regulatory motifs within unaligned noncoding sequences clustered by whole-genome mRNA quantitation roth, f.p. Nat. Biotechnol. 16, 939–945 (1998) CrossRefGoogle Scholar
  35. Stormo, G., Hartzell, G.: Identifying protein-binding sites from unaligned DNA fragments. Proc. Natl. Acad. Sci. USA 86, 1183–1187 (1989) CrossRefGoogle Scholar
  36. Tierney, L.: Markov chains for exploring posterior distributions. Ann. Stat. 22, 1701–1762 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  37. van Dyk, D., Park, T.: Partially collapsed Gibbs samplers: Theory and methods. J. Am. Stat. Assoc. 103, 790–796 (2008) zbMATHCrossRefGoogle Scholar
  38. Zhou, Q., Wong, W.: Reconstructing the energy landscape of a distribution from Monte Carlo samples. Ann. Appl. Stat. 2, 1307–1331 (2008) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Meïli Baragatti
    • 1
    • 2
  • Agnès Grimaud
    • 2
  • Denys Pommeret
    • 2
    Email author
  1. 1.Ipsogen SALuminy Biotech EntreprisesMarseille Cedex 9France
  2. 2.Institute of Mathematics of Luminy (IML)CNRS MarseilleMarseille Cedex 9France

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