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An Adaptive Parallel Tempering Algorithm

2013; Taylor & Francis; Volume: 22; Issue: 3 Linguagem: Inglês

10.1080/10618600.2013.778779

1537-2715

Błażej Miasojedow, Éric Moulines, Matti Vihola,

Bayesian Methods and Mixture Models

Abstract Parallel tempering is a generic Markov chain Monte Carlo sampling method which allows good mixing with multimodal target distributions, where conventional Metropolis-Hastings algorithms often fail. The mixing properties of the sampler depend strongly on the choice of tuning parameters, such as the temperature schedule and the proposal distribution used for local exploration. We propose an adaptive algorithm with fixed number of temperatures which tunes both the temperature schedule and the parameters of the random-walk Metropolis kernel automatically. We prove the convergence of the adaptation and a strong law of large numbers for the algorithm under general conditions. We also prove as a side result the geometric ergodicity of the parallel tempering algorithm. We illustrate the performance of our method with examples. Our empirical findings indicate that the algorithm can cope well with different kinds of scenarios without prior tuning. Supplementary materials including the proofs and the Matlab implementation are available online. Key Words: Adaptive MCMCLaw of large numbersMultimodality SUPPLEMENTARY MATERIALS Proofs: The proofs of all the results in this article are given in the supplementary document (supplemental-pt.tex, LaTeX file) Matlab implementation: Matlab implementation of the adaptive parallel tempering algorithm. (apt-codes.zip, Zip file) ACKNOWLEDGMENTS The first two authors were supported by the French National Research Agency under the contract ANR-08-BLAN-0218 "BigMC." The third author was supported by the Academy of Finland (project 250575) and by the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation. Notes The Matlab-implementation is available at http://iki.fi/mvihola/apt-codes.zip.