Abstract
Markov chain Monte Carlo (MCMC) is an important computational technique for generating samples from non-standard probability distributions. A major challenge in the design of practical MCMC samplers is to achieve efficient convergence and mixing properties. One way to accelerate convergence and mixing is to adapt the proposal distribution in light of previously sampled points, thus increasing the probability of acceptance. In this paper, we propose two new adaptive MCMC algorithms based on the Independent Metropolis–Hastings algorithm. In the first, we adjust the proposal to minimize an estimate of the cross-entropy between the target and proposal distributions, using the experience of pre-runs. This approach provides a general technique for deriving natural adaptive formulae. The second approach uses multiple parallel chains, and involves updating chains individually, then updating a proposal density by fitting a Bayesian model to the population. An important feature of this approach is that adapting the proposal does not change the limiting distributions of the chains. Consequently, the adaptive phase of the sampler can be continued indefinitely. We include results of numerical experiments indicating that the new algorithms compete well with traditional Metropolis–Hastings algorithms. We also demonstrate the method for a realistic problem arising in Comparative Genomics.
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References
Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19(6), 716–723 (1974)
Andolfatto, P.: Adaptive evolution of non-coding DNA in drosophila. Nature 437, 1149–1152 (2005)
Andrieu, C., Moulines, E.: On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Aprobab. 16(3), 1462–1505 (2006)
Atchade, Y.F., Rosenthal, J.S.: On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11, 815–828 (2005)
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 (2006)
Chauveau, D., Vandekerkhove, P.: Improving convergence of the Hastings-Metropolis algorithm with an adaptive proposal. Scand. J. Stat. 29(1), 13–29 (2002)
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. 39(1), 1–38 (1977)
Douc, R., Guillin, A., Marin, J.-M., Robert, C.P.: Convergence of adaptive mixtures of importance sampling schemes. Ann. Stat. 35(1), 420–448 (2007)
Gasemyr, J.: On an adaptive version of the Metropolis–Hastings algorithm with independent proposal distribution. Scand. J. Stat. 30(1), 159–173 (2003)
Gelfand, A.E., Sahu, S.K.: On Markov chain Monte Carlo acceleration. J. Comput. Graph. Stat. 3(3), 261–276 (1994)
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. Chapman and Hall, London (2003)
Gelman, A.G., Roberts, G.O., Gilks, W.R.: Efficient metropolis jumping rules. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics V, pp. 599–608. Oxford Univ. Press, New York (1996)
Gilks, W., Roberts, G., George, E.: Adaptive direction sampling. Statistician 43(1), 179–189 (1994)
Gilks, W., Roberts, G., Sahu, S.: Adaptive Markov chain Monte Carlo through regeneration. J. Am. Stat. Assoc. 93, 1045–1054 (1998)
Haario, H., Saksman, E., Tamminen, J.: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat. 14, 375–395 (1999)
Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7(2), 223–242 (2001)
Haario, H., Saksman, E., Tamminen, J.: Componentwise adaptation for high dimensional MCMC. Comput. Stat. 20, 265–273 (2005)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Holden, L.: Adaptive chains. Technical report, Norwegian Computing Centre, P.O. Box 114 Blindern, N-0314, Oslo, Norway (2000)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)
Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, Berlin (2001)
McLachlan, G., Krishnan, T.: The EM Algorithm and Extensions. Wiley, New York (1997)
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)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H.: Equations of state calculations by fast computing mashines. J. Chem. Phys. 21, 1087–1092 (1953)
Mykland, P., Tierney, L., Yu, B.: Regeneration in Markov chain samplers. J. Am. Stat. Assoc. 90, 233–241 (1995)
Pasarica, C., Gelman, A.: Adaptively scaling the Metropolis algorithm using expected squared jumped distance. Technical report, Department of Statistics, Columbia University (2003)
Rubinstein, R.Y., Kroese, D.P.: The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. Springer, New York (2004)
Sahu, S.K., Zhigljavsky, A.A.: Self-regenerative Markov chain Monte Carlo with adaptation. Bernoulli 9, 395–422 (2003)
Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)
Tierney, L., Mira, A.: Some adaptive Monte Carlo methods for Bayesian inference. Stat. Medicine 18, 2507–2515 (1999)
Warnes, G.R.: The normal kernel coupler: An adaptive Markov chain Monte Carlo method for efficiently sampling from multi-modal distributions. Technical Report 39, Department of Statistics, University of Washington (2003)
Waterston, R.H., Lindblad-Toh, K., Birney, E., Rogers, J., Abril, J.F., et al.: Initial sequencing and comparative analysis of the mouse genome. Nature 420, 520–562 (2002)
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Keith, J.M., Kroese, D.P. & Sofronov, G.Y. Adaptive independence samplers. Stat Comput 18, 409–420 (2008). https://doi.org/10.1007/s11222-008-9070-2
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DOI: https://doi.org/10.1007/s11222-008-9070-2