Metropolis Monte Carlo

In statistics and in statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g., to generate a histogram), or to compute an integral (such as 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, other methods are usually available (e.g. adaptive rejection sampling) that can directly return independent samples from the distribution, and are free from the problem of samples that is inherent in MCMC methods.

The algorithm was named after Nicholas Metropolis, who was an author along with Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller of the 1953 paper Equation of State Calculations by Fast Computing Machines which first proposed the algorithm for the case of symmetrical proposal distributions, and W. K. Hastings who extended it to the more general case in 1970. There is controversy over the credit for discovery of the algorithm. Edward Teller states in his memoirs that the five authors of the 1953 paper worked together for "days (and nights)". M. Rosenbluth, in an oral history recorded shortly before his death credits E. Teller with posing the original problem, himself with solving it, and A.W. Rosenbluth (his wife) with programming the computer. According to M. Rosenbluth, neither Metropolis nor A.H. Teller participated in any way. Rosenbluth's account of events is supported by other contemporary recollections. According to Roy Glauber and Emilio Segrè, the original algorithm was invented by Enrico Fermi and reinvented by Stan Ulam.

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