Adaptive independence samplers
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.
KeywordsMarkov chain Monte Carlo Generalized Markov sampler Adaptive methods Cross-entropy Comparative genomics
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- 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) Google Scholar
- Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. Chapman and Hall, London (2003) Google Scholar
- 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) Google Scholar
- Holden, L.: Adaptive chains. Technical report, Norwegian Computing Centre, P.O. Box 114 Blindern, N-0314, Oslo, Norway (2000) 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
- Pasarica, C., Gelman, A.: Adaptively scaling the Metropolis algorithm using expected squared jumped distance. Technical report, Department of Statistics, Columbia University (2003) Google Scholar
- 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) Google Scholar