Metropolis–Hastings algorithm: Difference between revisions

Revision as of 22:26, 11 May 2004

Revision as of 13:57, 11 June 2004

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The Metropolis-Hastings algorithm can draw samples from any [[probability distribution]]

''P(x)'', requiring only that the density can be calculated at ''x''. The algorithm generates a set of states x^t which is a [[Markov chain]] because each state x^t depends only on the previous state x^t-1. The algorithm depends on the creation of a ''proposal density'' Q(x';x^t;x') which depends on the current state x^t and which can generate a new proposed sample x'. For example, the proposal density could be a [[Gaussian function]] centred on the current state x^t

:<math>

Q( x^t'; x'^t ) \sim N( x'-x^t, \sigma^2 I).

</math> &nbsp; (Read Q(x';x^t) as the probability of generating x' given the previous value x^t.)

</math>

This proposal density would generate samples centred around the current state with variance &sigma;²I. So we draw a new proposal state fromx' with probability Q(x';x^t,x') and then calculate a value

:<math>