Computational Statistics

, Volume 28, Issue 6, pp 2797–2823 | Cite as

On the flexibility of the design of multiple try Metropolis schemes

Original Paper


The multiple try Metropolis (MTM) method is a generalization of the classical Metropolis–Hastings algorithm in which the next state of the chain is chosen among a set of samples, according to normalized weights. In the literature, several extensions have been proposed. In this work, we show and remark upon the flexibility of the design of MTM-type methods, fulfilling the detailed balance condition. We discuss several possibilities, show different numerical simulations and discuss the implications of the results.


Metropolis–Hasting method Multiple try Metropolis algorithm Multi-point Metropolis algorithm MCMC techniques 


  1. Barker AA (1965) Monte Carlo calculations of the radial distribution functions for a proton–electron plasma. Aust J Phys 18:119–133CrossRefGoogle Scholar
  2. Bédard M, Douc R, Mouline E (2012) Scaling analysis of multiple-try MCMC methods. Stoch Process Appl 122:758–786CrossRefMATHGoogle Scholar
  3. Brooks SP (1998) Markov Chain Monte Carlo method and its application. J R Stat Soc Ser D (The Statistician) 47(1):69–100CrossRefGoogle Scholar
  4. Casarin R, Craiu R, Leisen F (2013) Interacting multiple try algorithms with different proposal distributions. Stat Comput 23(2):185–200. doi:10.1007/s11222-011-9301-9 Google Scholar
  5. Craiu RV, Lemieux C (2007) Acceleration of the multiple-try Metropolis algorithm using antithetic and stratified sampling. Stat Comput 17(2):109–120MathSciNetCrossRefGoogle Scholar
  6. Devroye L (1986) Non-uniform random variate generation. Springer, BerlinCrossRefMATHGoogle Scholar
  7. Fitzgerald WJ (2001) Markov Chain Monte Carlo methods with applications to signal processing. Signal Process 81(1):3–18CrossRefMATHGoogle Scholar
  8. Frenkel D, Smit B (1996) Understanding molecular simulation: from algorithms to applications. Academic Press, San DiegoMATHGoogle Scholar
  9. Gamerman D, Lopes HF (2006) Markov Chain Monte Carlo: stochastic simulation for Bayesian inference. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  10. Gilks WR, Richardson S, Spiegelhalter D (1995) Markov Chain Monte Carlo in practice: interdisciplinary statistics. Taylor & Francis, Inc., UKGoogle Scholar
  11. Haario H, Saksman E, Tamminen J (1999) Adaptive proposal distribution for random walk Metropolis algorithm. Comput Stat 14:375–395CrossRefMATHGoogle Scholar
  12. Haario H, Saksman E, Tamminen J (2001) An adaptive Metropolis algorithm. Bernoulli 7(2):223–242MathSciNetCrossRefMATHGoogle Scholar
  13. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109CrossRefMATHGoogle Scholar
  14. Lan S, Stathopoulosy V, Shahbaba B, Girolami M (2012) Langrangian dynamical Monte Carlo. arXiv:1211.3759v1Google Scholar
  15. Liang F, Liu C, Caroll R (2010) Advanced Markov Chain Monte Carlo methods: learning from past samples. Wiley Series in Computational Statistics, EnglandGoogle Scholar
  16. Liu JS, Liang F, Wong WH (2000) The multiple-try method and local optimization in Metropolis sampling. J Am Stat Assoc 95(449):121–134MathSciNetCrossRefMATHGoogle Scholar
  17. Liu JS (2004) Monte Carlo strategies in scientific computing. Springer, BerlinCrossRefGoogle Scholar
  18. Martino L, Del Olmo VP, Read J (2012a) A multi-point Metropolis scheme with generic weight functions. Stat Probab Lett 82(7):1445–1453CrossRefMATHGoogle Scholar
  19. Martino L, Read J, Luengo D (2012b) Improved adaptive rejection Metropolis sampling algorithms. arXiv:1205.5494v4Google Scholar
  20. Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21:1087–1091CrossRefGoogle Scholar
  21. Mira A (2001) On Metropolis–Hastings algorithms with delayed rejection. Metron 59:231–241MathSciNetMATHGoogle Scholar
  22. Pandolfi S, Bartolucci F, Friel N (2010) A generalization of the multiple-try Metropolis algorithm for Bayesian estimation and model selection. J Mach Learn Res (Workshop and conference proceedings volume 9: AISTATS 2010) 9:581–588Google Scholar
  23. Peskun PH (1973) Optimum Monte-Carlo sampling using Markov Chains. Biometrika 60(3):607–612MathSciNetCrossRefMATHGoogle Scholar
  24. Qin ZS, Liu JS (2001) Multi-point Metropolis method with application to hybrid Monte Carlo. J Comput Phys 172:827–840MathSciNetCrossRefMATHGoogle Scholar
  25. Robert CP (2012) “Xi’ An’s Og, an attempt at bloggin... ” Blog (by Christian P. Robert).
  26. Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer, BerlinCrossRefMATHGoogle Scholar
  27. Roberts GO, Gelman A, Gilks WR (1997) Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann Appl Probab 7:110–120MathSciNetCrossRefMATHGoogle Scholar
  28. Storvik G (2011) On the flexibility of Metropolis–Hastings acceptance probabilities in auxiliary variable proposal generation. Scand J Stat 38(2):342–358MathSciNetCrossRefMATHGoogle Scholar
  29. Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 33:1701–1728MathSciNetCrossRefGoogle Scholar
  30. Tierney L, Mira A (1999) Some adaptive Monte Carlo methods for Bayesian inference. Stat Med 18:2507–2515CrossRefGoogle Scholar
  31. Zhang Y, Zhang W (2012) Improved generic acceptance function for multi-point Metropolis algorithm. In: 2nd International conference on electronic and mechanical engineering and information technology (EMEIT-2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Signal Theory and CommunicationsUniversidad Carlos III de MadridLeganés, MadridSpain

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