Metropolis–Hastings algorithm: Difference between revisions

Revision as of 12:21, 19 September 2005

Revision as of 08:40, 21 October 2005

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[[Image:Metropolis_hastings_algorithm.png|thumb|350px|The Sampling [[probability distribution | distribution]] ''Q'' determines the next point to move to in the [[random walk]].]]

In [[mathematics]] and [[physics]], the '''Metropolis-Hastings algorithm''' is an [[algorithm]] used to generate a sequence of samples from the [[probability distribution]] of one or more variables.

The purpose of such a sequence is to approximate the distribution (as with a histogram), or to compute an integral (such as an expected value).

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If the proposal steps are too small the chain will ''mix slowly'' (i.e., it will move around the space slowly and converge slowly to <math>p(x)</math>).

If the proposal steps are too large the acceptance rate will be very low because the proposals are likely to land in regions of much lower probability density so <math>a_1</math> will be very small.

== See also==

* [[Simulated annealing]]

== References ==

* Bernd A. Berg. "Markov Chain Monte Carlo Simulations and Their Statistical Analysis". Singapore, World Scientific 2004.

* Chib, Siddhartha and Edward Greenberg: "Understanding the Metropolis&ndash;Hastings Algorithm". American Statistician, 49, 327&ndash;335, 1995

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* N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller. "Equations of State Calculations by Fast Computing Machines". ''Journal of Chemical Physics'', 21:1087-1092, 1953.

[[Category:AlgorithmsMonte Carlo method]]

== See also==

* [[Simulated annealing]]

[[Category:Algorithms]]

[[Category:Probability and statistics]]

[[Category:Sampling techniques]]