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In [[statistics]] and [[statistical physics]], the '''Metropolis–Hastings algorithm''' is a [[Markov chain Monte Carlo]] (MCMC) method for obtaining a sequence of [[pseudo-random number sampling|random samples]] from a [[probability distribution]] from which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g. to generate a [[histogram]]) or to [[Monte Carlo integration|compute an integral]] (e.g. an [[expected value]]). Metropolis–Hastings and other MCMC algorithms are generally used for sampling from multi-dimensional distributions, especially when the number of dimensions is high. For single-dimensional distributions, there are usually other methods (e.g. [[adaptive rejection sampling]]) that can directly return independent samples from the distribution, and these are free from the problem of [[autocorrelation|autocorrelated]] samples that is inherent in MCMC methods.
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The algorithm was named after [[Nicholas Metropolis]], who authored the 1953 article ''[[Equation of State Calculations by Fast Computing Machines]]'' together with [[Arianna W. Rosenbluth]], [[Marshall Rosenbluth]], [[Augusta H. Teller]] and [[Edward Teller]]. This article proposed the algorithm for the case of symmetrical proposal distributions, and [[W. K. Hastings]] extended it to the more general case in 1970.<ref name=Hastings/>
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