Metropolis–Hastings algorithm: Difference between revisions

Revision as of 16:08, 6 December 2024

Revision as of 16:16, 6 December 2024

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#** If <math>u > \alpha</math>, then ''reject'' the candidate and set <math>x_{t+1} = x_t</math> instead.

This algorithm proceeds by randomly attempting to move about the sample space, sometimes accepting the moves and sometimes remaining in place. <math>P(x)</math> at specific point <math>x</math> is proportional to the timeiterations spent on <math>x</math>the point by the algorithm. Note that the acceptance ratio <math>\alpha</math> indicates how probable the new proposed sample is with respect to the current sample, according to the distribution whose density is <math>P(x)</math>. If we attempt to move to a point that is more probable than the existing point (i.e. a point in a higher-density region of <math>P(x)</math> corresponding to an <math>\alpha > 1 \ge u</math>), we will always accept the move. However, if we attempt to move to a less probable point, we will sometimes reject the move, and the larger the relative drop in probability, the more likely we are to reject the new point. Thus, we will tend to stay in (and return large numbers of samples from) high-density regions of <math>P(x)</math>, while only occasionally visiting low-density regions. Intuitively, this is why this algorithm works and returns samples that follow the desired distribution with density <math>P(x)</math>.

Compared with an algorithm like [[adaptive rejection sampling]]<ref name=":0">{{Cite journal |last1=Gilks |first1=W. R. |last2=Wild |first2=P. |date=1992-01-01 |title=Adaptive Rejection Sampling for Gibbs Sampling |journal=Journal of the Royal Statistical Society. Series C (Applied Statistics) |volume=41 |issue=2 |pages=337–348 |doi=10.2307/2347565 |jstor=2347565}}</ref> that directly generates independent samples from a distribution, Metropolis–Hastings and other MCMC algorithms have a number of disadvantages: