Metropolis–Hastings algorithm: Difference between revisions

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The Metropolis–Hastings algorithm can draw samples from any [[probability distribution]] with [[probability density]] <math>P(x)</math>, provided that we know a function <math>f(x)</math> proportional to the [[Probability density function|density]] <math>P</math> and the values of <math>f(x)</math> can be calculated. The requirement that <math>f(x)</math> must only be proportional to the density, rather than exactly equal to it, makes the Metropolis–Hastings algorithm particularly useful, because it removes the need to calculate the density's normalization factor, which is often extremely difficult in practice.

The Metropolis–Hastings algorithm generates a sequence of sample values in such a way that, as more and more sample values are produced, the distribution of values more closely approximates the desired distribution. These sample values are produced iteratively in such a way, withthat the distribution of the next sample being dependentdepends only on the current sample value, thuswhich makingmake the sequence of samples into a [[Markov chain]]. Specifically, at each iteration, the algorithm proposes a candidate for the next sample value based on the current sample value. Then, with some probability, the candidate is either accepted, in which case the candidate value is used in the next iteration, or it is rejected in which case the candidate value is discarded, and the current value is reused in the next iteration. The probability of acceptance is determined by comparing the values of the function <math>f(x)</math> of the current and candidate sample values with respect to the desired distribution.

The method used to propose new candidates is characterized by the probability distribution <math>g(x\mid y)</math> (sometimes written <math>Q(x\mid y)</math>) of a new proposed sample <math>x</math> given the previous sample <math>y</math>. This is called the ''proposal density'', ''proposal function'', or ''jumping distribution''. A common choice for <math>g(x\mid y)</math> is a [[Gaussian distribution]] centered at <math>y</math>, so that points closer to <math>y</math> are more likely to be visited next, making the sequence of samples into a [[Gaussian random walk]]. In the original paper by Metropolis et al. (1953), <math>g(x\mid y)</math> was suggested to be a uniform distribution limited to some maximum distance from <math>y</math>. More complicated proposal functions are also possible, such as those of [[Hamiltonian Monte Carlo]], [[Langevin Monte Carlo]], or [[preconditioned Crank–Nicolson]].