Markov Chain Monte Carlo: Innovations and Applications by W. S. Kendall, J. S. Wang

By W. S. Kendall, J. S. Wang

Markov Chain Monte Carlo (MCMC) originated in statistical physics, yet has spilled over into numerous program parts, resulting in a corresponding number of strategies and techniques. That type stimulates new principles and advancements from many alternative areas, and there's a lot to be received from cross-fertilization. This booklet offers 5 expository essays via leaders within the box, drawing from views in physics, data and genetics, and exhibiting how varied points of MCMC come to the fore in numerous contexts.

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Example text

N runs in the temporal order of the Markov chain and the elapsed time (measured in updates or sweeps) between subsequent measurements fi , fi+1 is always the same. The estimator of the expectation value f is 1 fi . (89) f= N With the notation t = |i − j| 32 B. A. Berg the definition of the autocorrelation function of the observable f is C(t) = Cij = (fi − fi ) (fj − fj ) = fi fj − fi fj = f0 ft − f 2 (90) where we used that translation invariance in time holds for the equilibrium ensemble.

Nbs ) we may calculate the mean with its naive error bar. Assuming for the moment an infinite time series, we find the integrated autocorrelation time (95) from the following ratio of sample variances τint = Nb lim τint Nb →∞ Nb τint = with s2 N b f sf2 . (104) In practice the Nb → ∞ limit will be reached for a sufficiently large, finite value of Nb . The statistical error of the τint estimate (104) is, in the first approximation, determined by the errors of s2 Nb . The typical situation is f then that, due to the central limit theorem, the binned data are approximately Gaussian, so that the error of s2 Nb is analytically known from f the χ2 distribution.

Berg Truncating the sum at some finite value of NK , we obtain an estimator of the expectation value O= 1 NK NK n=1 O(kn ) . (67) Normally, we cannot generate configurations k directly with the probability (65), but they may be found as members of the equilibrium distribution of a dynamic process. A Markov process is a particularly simple dynamic process, which generates configuration kn+1 stochastically from configuration kn , so that no information about previous configurations kn−1 , kn−2 , .

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