Skip to main content
Download PDF

Differential Evolution Markov Chain with snooker updater and fewer chains

Statistics and Computing volume 18, pages 435–446 (2008)Cite this article

Abstract

Differential Evolution Markov Chain (DE-MC) is an adaptive MCMC algorithm, in which multiple chains are run in parallel. Standard DE-MC requires at least N=2d chains to be run in parallel, where d is the dimensionality of the posterior. This paper extends DE-MC with a snooker updater and shows by simulation and real examples that DE-MC can work for d up to 50–100 with fewer parallel chains (e.g. N=3) by exploiting information from their past by generating jumps from differences of pairs of past states. This approach extends the practical applicability of DE-MC and is shown to be about 5–26 times more efficient than the optimal Normal random walk Metropolis sampler for the 97.5% point of a variable from a 25–50 dimensional Student t 3 distribution. In a nonlinear mixed effects model example the approach outperformed a block-updater geared to the specific features of the model.

References

  • Babu, B.V., Angira, R.: Modified differential evolution (MDE) for optimization of non-linear chemical processes. Comput. Chem. Eng. 30, 989–1002 (2006)

    Article  Google Scholar 

  • Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. Chapman & Hall, London (2004)

    MATH  Google Scholar 

  • Gilks, W.R., Roberts, G.O.: Strategies for improving MCMC. In: Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (eds.) Markov Chain Monte Carlo in Practice, pp. 89–114. Chapman & Hall, London (1996)

    Google Scholar 

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

    Article  Google Scholar 

  • Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7, 223–242 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  • Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: Efficient adaptive MCMC. Stat. Comput. 16, 339–354 (2006)

    Article  MathSciNet  Google Scholar 

  • Jarner, S.F., Roberts, G.O.: Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Stat. 34, 781–815 (2007)

    MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  • Liu, J., Hodges, J.S.: Posterior bimodality in the balanced one-way random-effects model. J. R. Stat. Soc. Ser. B 65, 247–255 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  • Mengersen, K., Robert, C.P.: IID sampling using self-avoiding population Monte Carlo: the pinball sampler. In: Bernardo, J.M., Bayarri, M.J., Berger, J.O., Dawid, A.P., Heckerman, D., Smith, A.F.M., West, M. (eds.) Bayesian Statistics 7, pp. 277–292. Clarendon, Oxford (2003)

    Google Scholar 

  • Pinheiro, J.C., Bates, D.M.: Mixed-Effects Models in S and S-PLUS. Springer, New York (2000)

    MATH  Google Scholar 

  • Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution, A Practical Approach to Global Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Roberts, G.O., Gilks, W.R.: Convergence of adaptive direction sampling. J. Multivar. Anal. 49, 287–298 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  • Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16, 351–367 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  • Roberts, G.O.: Linking theory and practice of MCMC. In: Green, P.J., Hjort, N.L., Richardson, S. (eds.) Highly Structured Stochastic Systems, pp. 145–166. Oxford University Press, Oxford (2003)

    Google Scholar 

  • Roberts, G.O., Rosenthal, J.S.: Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44, 458–475 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  • Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. Online ms. (2008)

  • Spiegelhalter, D., Thomas, A., Best, N., Lunn, D.: WinBUGS User Manual version 1.4. www.mrc-bsu.cam.ac.uk/bugs (2003)

  • Storn, R., Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11, 341–359 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  • Strens, M., Bernhardt, M., Everett, N.: Markov chain Monte Carlo sampling using direct search optimization. In: Sammut, C., Hoffmann, A.G. (eds.) Machine Learning, Proceedings of the Nineteenth International Conference on Machine Learning (ICML 2002), pp. 602–609. Morgan Kaufmann, San Fransisco (2002)

    Google Scholar 

  • ter Braak, C.J.F.: A Markov chain Monte Carlo version of the genetic algorithm differential evolution: easy Bayesian computing for real parameter spaces. Stat. Comput. 16, 239–249 (2006)

    Article  MathSciNet  Google Scholar 

  • Thomas, A., O’Hara, R.B.: OpenBUGS. http://mathstat.helsinki.fi/openbugs/ (2007)

  • Vrugt, J.A., ter Braak, C.J.F., Gupta, H.V., Robinson, B.A.: Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling? Stochastic Environmental Research and Risk Assessment (SERRA) (2008a). DOI:10.1007/s00477-008-0274-y

  • Vrugt, J.A., ter Braak, C.J.F., Clark, M.P., Hyman, J.M., Robinson, B.A.: Treatment of input uncertainty in hydrologic modeling: doing hydrology backwards with Markov chain Monte Carlo simulation. Water Resour. Res. (2008b, in press)

  • Waagepetersen, R., Sorensen, D.: A tutorial on reversible jump MCMC with a view toward applications in QTL-mapping. Int. Stat. Rev. 69, 49–61 (2001)

    MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Biometris, Wageningen University and Research Centre, Box 100, 6700 AC, Wageningen, The Netherlands

    Cajo J. F. ter Braak

  2. Center for NonLinear Studies (CNLS), Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Jasper A. Vrugt

Authors
  1. Cajo J. F. ter Braak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jasper A. Vrugt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cajo J. F. ter Braak.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

ter Braak, C.J.F., Vrugt, J.A. Differential Evolution Markov Chain with snooker updater and fewer chains. Stat Comput 18, 435–446 (2008). https://doi.org/10.1007/s11222-008-9104-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-008-9104-9

Keywords

  • Evolutionary Monte Carlo
  • Metropolis algorithm
  • Adaptive Markov chain Monte Carlo
  • Theophylline kinetics
  • Adaptive direction sampling
  • Parallel computing
  • Differential evolution