Mathematics Subject Classification
62F15; 65C40
Synonyms
Markov chain Monte Carlo (MCMC)
Short Definition
Markov chain Monte Carlo (MCMC) is a collection of computational methods for simulating from posterior distributions.
Description
Markov chain Monte Carlo (MCMC) methods are a collection of computational algorithms designed to sample from a target distribution by performing Monte Carlo simulation from a Markov chain whose equilibrium distribution is equal to the target distribution. The output of the algorithm is then used to estimate features of the required distribution, where the quality of the estimate is determined by the number of iterations of the algorithm. Surprisingly, it took several decades before the statistical community embraced Markov chain Monte Carlo (MCMC) as a general computational tool in Bayesian inference, where it may be quite difficult to compute the normalizing constant of the (possibly high-dimensional) posterior distribution required for routine...
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References
Binder, K.: Monte Carlo Methods in Statistical Physics. Springer, New York (1978)
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. R. Stat. Soc. B: Stat. Methodol. 39, 1–38 (1977)
Gelfand, A.E., Smith, A.F.M.: Sampling-based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409 (1990)
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)
Hammersley, J.M., Handscomb, D.C.: Monte Carlo Methods, 2nd edn. Chapman and Hall, London (1964)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Kloek, T., van Dijk, H.K.: Bayesian estimates of equation system parameters: an application of integration by Monte Carlo. Econometrica 46, 1–19 (1978)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953)
Morris, C.N.: Comment on “The calculation of posterior distributions by data augmentation” by M.A. Tanner and W.H. Wong. J. Am. Stat. Assoc. 82, 542–543 (1987)
O’Hagan, A.: Monte Carlo is fundamentally unsound. J. R. Stat. Soc. D: Stat. 36, 247–249 (1987)
Smith, A.F.M., Skene, A.M., Shaw, J.E.H., Naylor, J.C., Dransfield, M.: The implementation of the Bayesian paradigm. Commun. Stat. Theory Methods 14(5), 1079–1102 (1985)
Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987)
Tanner, M.A.: Metropolis algorithms. In: The Encyclopedia of Applied and Computational Mathematics (2014)
Tanner, M.A., Wong, W.H.: The calculation of posterior distributions by data augmentation (with discussion). J. Am. Stat. Assoc. 82, 528–550 (1987)
Tanner, M.A., Wong, W.H.: From EM to data augmentation: the emergence of MCMC Bayesian computation in the 1980s. Stat. Sci. 25, 506–516 (2010)
Tierney, L., Kadane, J.B.: Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81, 82–86 (1986)
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Tanner, M.A. (2015). Bayesian Statistics: Computation. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_327
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DOI: https://doi.org/10.1007/978-3-540-70529-1_327
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