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.
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References
Babu, B.V., Angira, R.: Modified differential evolution (MDE) for optimization of non-linear chemical processes. Comput. Chem. Eng. 30, 989–1002 (2006)
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. Chapman & Hall, London (2004)
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)
Gilks, W.R., Roberts, G.O., George, E.I.: Adaptive direction sampling. Statistician 43, 179–189 (1994)
Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7, 223–242 (2001)
Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: Efficient adaptive MCMC. Stat. Comput. 16, 339–354 (2006)
Jarner, S.F., Roberts, G.O.: Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Stat. 34, 781–815 (2007)
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)
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)
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)
Pinheiro, J.C., Bates, D.M.: Mixed-Effects Models in S and S-PLUS. Springer, New York (2000)
Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution, A Practical Approach to Global Optimization. Springer, Berlin (2005)
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, New York (2004)
Roberts, G.O., Gilks, W.R.: Convergence of adaptive direction sampling. J. Multivar. Anal. 49, 287–298 (1994)
Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16, 351–367 (2001)
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)
Roberts, G.O., Rosenthal, J.S.: Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44, 458–475 (2007)
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)
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)
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)
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)
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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
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DOI: https://doi.org/10.1007/s11222-008-9104-9