Editing Metropolis–Hastings algorithm (section)

{{short description|Monte Carlo algorithm}}
[[File:Flowchart-of-Metropolis-Hastings-M-H-algorithm-for-the-parameter-estimation-using-the.png|thumb|300px|The Metropolis-Hastings algorithm sampling a [[Normal distribution|normal]] one-dimensional [[Posterior probability|posterior]] probability distribution.]]

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. New samples are added to the sequence in two steps: first a new sample is proposed based on the previous sample, then the proposed sample is either added to the sequence or rejected depending on the value of the probability distribution at that point. The resulting 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.