Metropolis–Hastings algorithm: Difference between revisions

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[[Image:Metropolis hastings algorithm.png|thumb|450px|The proposal [[probability distribution|distribution]] ''Q'' proposes the next point to which the [[random walk]] might move.]]

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.

==History==

The algorithm is named in part for [[Nicholas Metropolis]], the first coauthor of a 1953 paper, entitled ''[[Equation of State Calculations by Fast Computing Machines]]'', with [[Arianna W. Rosenbluth]], [[Marshall Rosenbluth]], [[Augusta H. Teller]] and [[Edward Teller]]. For many years the algorithm was known simply as the ''Metropolis algorithm''.<ref>{{Cite book |last=Kalos |first=Malvin H. |title=Monte Carlo Methods Volume I: Basics |last2=Whitlock |first2=Paula A. |publisher=Wiley |year=1986 |___location=New York |pages=78-88}}</ref><ref>{{Cite journal |last=Tierney |first=Luke |date=1994 |title=Markov chains for exploring posterior distributions |url=https://projecteuclid.org/journals/annals-of-statistics/volume-22/issue-4/Markov-Chains-for-Exploring-Posterior-Distributions/10.1214/aos/1176325750.full |journal=The Annals of Statistics |volume=22 |issue=4 |pages=1701-1762}}</ref> The paper proposed the algorithm for the case of symmetrical proposal distributions, but in 1970, [[W.K. Hastings]] extended it to the more general case.<ref name=Hastings/> The generalized method was eventually identified by both names, although the first use of the term "Metropolis-Hastings algorithm" is unclear.

Some controversy exists with regard to credit for development of the Metropolis algorithm. Metropolis, who was familiar with the computational aspects of the method, had coined the term "Monte Carlo" in an earlier article with [[Stanisław Ulam]], and led the group in the Theoretical Division that designed and built the [[MANIAC I]] computer used in the experiments in 1952. However, prior to 2003 there was no detailed account of the algorithm's development. Shortly before his death, [[Marshall Rosenbluth]] attended a 2003 conference at LANL marking the 50th anniversary of the 1953 publication. At this conference, Rosenbluth described the algorithm and its development in a presentation titled "Genesis of the Monte Carlo Algorithm for Statistical Mechanics".<ref name=Rosenbluth/> Further historical clarification is made by Gubernatis in a 2005 journal article<ref name=Gubernatis/> recounting the 50th anniversary conference. Rosenbluth makes it clear that he and his wife Arianna did the work, and that Metropolis played no role in the development other than providing computer time.

This contradicts an account by Edward Teller, who states in his memoirs that the five authors of the 1953 article worked together for "days (and nights)".<ref name=Teller/> In contrast, the detailed account by Rosenbluth credits Teller with a crucial but early suggestion to "take advantage of statistical mechanics and take ensemble averages instead of following detailed kinematics". This, says Rosenbluth, started him thinking about the generalized Monte Carlo approach – a topic which he says he had discussed often with [[John von Neumann|John Von Neumann]]. Arianna Rosenbluth recounted (to Gubernatis in 2003) that Augusta Teller started the computer work, but that Arianna herself took it over and wrote the code from scratch. In an oral history recorded shortly before his death,<ref name=Barth/> Rosenbluth again credits Teller with posing the original problem, himself with solving it, and Arianna with programming the computer.