Abstract
Differential Evolution (DE) is a simple genetic algorithm for numerical optimization in real parameter spaces. In a statistical context one would not just want the optimum but also its uncertainty. The uncertainty distribution can be obtained by a Bayesian analysis (after specifying prior and likelihood) using Markov Chain Monte Carlo (MCMC) simulation. This paper integrates the essential ideas of DE and MCMC, resulting in Differential Evolution Markov Chain (DE-MC). DE-MC is a population MCMC algorithm, in which multiple chains are run in parallel. DE-MC solves an important problem in MCMC, namely that of choosing an appropriate scale and orientation for the jumping distribution. In DE-MC the jumps are simply a fixed multiple of the differences of two random parameter vectors that are currently in the population. The selection process of DE-MC works via the usual Metropolis ratio which defines the probability with which a proposal is accepted. In tests with known uncertainty distributions, the efficiency of DE-MC with respect to random walk Metropolis with optimal multivariate Normal jumps ranged from 68% for small population sizes to 100% for large population sizes and even to 500% for the 97.5% point of a variable from a 50-dimensional Student distribution. Two Bayesian examples illustrate the potential of DE-MC in practice. DE-MC is shown to facilitate multidimensional updates in a multi-chain “Metropolis-within-Gibbs” sampling approach. The advantage of DE-MC over conventional MCMC are simplicity, speed of calculation and convergence, even for nearly collinear parameters and multimodal densities.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Gelman A., Carlin J. B., Stern H. S. and Rubin D. B. 2004. Bayesian data analysis, 2nd edition. London, Chapman & Hall.
Gilks W. R., Richardson S. and Spiegelhalter D. J. 1996. Markov chain monte carlo in practice. London, Chapman & Hall.
Gilks W. R. and Roberts G. O. 1996. Strategies for improving MCMC. In Markov chain monte carlo in practice. Gilks W. R., Richardson S. and Spiegelhalter D. J., (eds.), London, Chapman & Hall, pp. 89–114.
Gilks W. R., Roberts G. O. and George E. I. 1994. Adaptive direction sampling. The Statistician 43: 179–189.
Haario H., Saksman E. and Tamminen J. 2001. An adaptive Metropolis algorithm. Bernoulli 7: 223–242.
Lampinen J. 2001. A bibliography of Differential Evolution algorithm. Lappeenranta University of Technology, Lappeenranta, http://www.lut.fi/jlampine/debiblio.htm.
Lampinen J. and Zelinka I. 2000. On stagnation of the Differential Evolution algorithm. Proceedings of MENDEL 2000, 6th International Mendel Conference on Soft Computing, Brno, pp. 76–83.
Laskey K. B. and Myers J. W. 2003. Population Markov Chain Monte Carlo. Machine Learning 50: 175–196.
Liang F. 2002. Dynamically weighted importance sampling in Monte Carlo computation. Journal of the American Statistical Association 97: 807–821.
Liang F. M. and Wong W. H. 2001. Real-parameter evolutionary Monte Carlo with applications to Bayesian mixture models. Journal of the American Statistical Association 96: 653–666.
Liu J. and Hodges J. S. 2003. Posterior bimodality in the balanced one-way random-effects model. Journal of the Royal Statistical Society Series B 65: 247–255.
Lunn D. J., Thomas A., Best N. and Spiegelhalter D. 2000. WinBUGS—A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing 10: 325–337.
Mengersen K. and Robert C. P. 2003. 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. and West M. (eds.), Bayesian Statistics 7, Oxford, Clarendon Press, pp. 277–292.
Mengersen K. L. and Tweedie R. L. 1996. Rates of convergence of the Hastings and Metropolis algorithms. Annals of Statistics 24: 101–121.
Pinheiro J. C. and Bates D. M. 2000. Mixed-Effects Models in S and S-PLUS. New York, Springer Verlag.
Price K. 1999. An introduction to differential evolution. In Corne D., Dorigo M. and Glover F. (eds.), New Ideas in Optimization, London, McGraw-Hill, pp. 79–108.
Price K. and Storn R. 1997. Differential Evolution. Dr Dobb’s Journal 264: 18–24.
R Development Core Team 2003. R: A language and environment for statistical computing. Vienna, Austria, R Foundation for Statistical Computing. http://www.r-project.org.
Robert C. P. and Casella G. 2004. Monte Carlo Statistical Methods, 2nd ed., New York, Springer Verlag.
Roberts G. O. and Rosenthal J. S. 2001. Optimal scaling for various Metropolis-Hastings algorithms. Statistical Science 16: 351–367.
Schmitt L. M. 2004. Theory of genetic algorithms II: models for genetic operators over string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling. Theoretical Computer Science 310: 181–231.
Spiegelhalter D., Thomas A., Best N., and Lunn D. 2003. WinBUGS User Manual version 1.4. http://www.mrc-bsu.cam.ac.uk/bugs.
Storn R. and Price K. 1995 Differential Evolution — a simple and efficient adaptive scheme for global optimization over continuous spaces. International Computer Science Institute, Berkeley, TR-95-012, http://www.icsi.berkeley.edu/storn/litera.html.
Storn R. and Price K. 1997. Differential Evolution — a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimisation 11: 341–359.
Strens M. Bernhardt M. and Everett N. 2002. Markov chain Monte Carlo sampling using direct search optimization. ICML, Sydney, pp. 602–609.
Waagepetersen R. and Sorensen D. 2001. A tutorial on reversible jump MCMC with a view toward applications in QTL-mapping. International Statistical Review 69: 49–61.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Braak, C.J.F.T. 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). https://doi.org/10.1007/s11222-006-8769-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11222-006-8769-1