Metropolis–Hastings algorithm: Difference between revisions

Revision as of 00:20, 25 July 2021

Revision as of 15:31, 18 September 2021

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It is important to notice that it is not clear, in a general problem, which distribution <math>g(x' \mid x)</math> one should use or the number of iterations necessary for proper estimation; both are free parameters of the method, which must be adjusted to the particular problem in hand.

== Use in numerical integration ==

{{main|Monte Carlo integration}}

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==Step-by-step instructions==

Suppose that the most recent value sampled is <math>x_t</math>. To follow the Metropolis–Hastings algorithm, we next draw a new proposal state <math>x'</math> with probability density <math>g(x' \mid x_t)</math> and calculate a value

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</math>

The Markov chain is started from an arbitrary initial value <math>x_0</math>, and the algorithm is run for many iterations until this initial state is "forgotten". These samples, which are discarded, are known as ''burn-in''. The remaining set of accepted values of <math>x</math> represent a [[Sample (statistics)|sample]] from the distribution <math>P(x)</math>.

These samples, which are discarded, are known as ''burn-in''. The remaining set of accepted values of <math>x</math> represent a [[Sample (statistics)|sample]] from the distribution <math>P(x)</math>.

The algorithm works best if the proposal density matches the shape of the target distribution <math>P(x)</math>, from which direct sampling is difficult, that is <math>g(x' \mid x_t) \approx P(x')</math>.

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* [[Genetic algorithm]]s

* [[Gibbs sampling]]

* [[Mean -field particle methods]]

* [[Metropolis-adjusted Langevin algorithm]]

* [[Metropolis light transport]]