Editing Markov chain Monte Carlo (section)

== General explanation ==
[[File:Metropolis algorithm convergence example.png|thumbnail|upright=1.25|Convergence of the [[Metropolis–Hastings algorithm]]. Markov chain Monte Carlo attempts to approximate the blue distribution with the orange distribution.]]

Markov chain Monte Carlo methods create samples from a continuous [[random variable]], with [[probability density]] proportional to a known function. These samples can be used to evaluate an integral over that variable, as its [[expected value]] or [[variance]].

Practically, an [[Statistical ensemble|ensemble]] of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are [[stochastic processes]] of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities.

Random walk Monte Carlo methods are a kind of random [[Computer simulation|simulation]] or [[Monte Carlo method]]. However, whereas the random samples of the integrand used in a conventional [[Monte Carlo integration]] are [[statistically independent]], those used in MCMC are [[autocorrelation|autocorrelated]]. Correlations of samples introduces the need to use the [[Markov chain central limit theorem]] when estimating the error of mean values.

These algorithms create [[Markov chains]] such that they have an [[Markov chain#Steady-state analysis and limiting distributions|equilibrium distribution]] which is proportional to the function given.