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Multicanonical MCMC for sampling rare events: an illustrative review

  • Yukito Iba
  • Nen Saito
  • Akimasa Kitajima
Special Issue: Bayesian Inference and Stochastic Computation

Abstract

Multicanonical MCMC (Multicanonical Markov Chain Monte Carlo; Multicanonical Monte Carlo) is discussed as a method of rare event sampling. Starting from a review of the generic framework of importance sampling, multicanonical MCMC is introduced, followed by applications in random matrices, random graphs, and chaotic dynamical systems. Replica exchange MCMC (also known as parallel tempering or Metropolis-coupled MCMC) is also explained as an alternative to multicanonical MCMC. In the last section, multicanonical MCMC is applied to data surrogation; a successful implementation in surrogating time series is shown. In the appendix, calculation of averages and normalizing constant in an exponential family, phase coexistence, simulated tempering, parallelization, and multivariate extensions are discussed.

Keywords

Multicanonical MCMC Wang–Landau algorithm Replica exchange MCMC Rare event sampling Random matrix  Random graph Chaotic dynamical system Exact test Surrogation  

Notes

Acknowledgments

The authors would like to thank Koji Hukushima for the helpful discussions and the permission for the use of figures in Saito et al. (2010). We would also be grateful to Arnaud Doucet and the referees, for their helpful advice allowed us to improve the manuscript. This work was supported by JSPS KAKENHI Grant Numbers 22500217, 25330299, and 25240036. Saito is supported by a Grant-in-Aid for Scientific Research (No. 21120004) on Innovative Areas “Neural creativity for communication” (No. 4103), and the Platform for Dynamic Approaches to Living System from MEXT, Japan.

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Authors and Affiliations

  1. 1.The Institute of Statistical Mathematics and SOKENDAITachikawa, TokyoJapan
  2. 2.Research Center for Complex Systems BiologyThe University of TokyoMeguro-ku, TokyoJapan
  3. 3.Digital Information Services Division, Digital Information DepartmentNational Diet LibraryChiyoda-ku, TokyoJapan

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