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In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult. This sequence can be used to approximate the joint distribution ; to approximate the marginal distribution of one of the variables, or some subset of the variables ; or to compute an integral. Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.
Gibbs sampling is commonly used as a means of statistical inference, especially Bayesian inference. It is a randomized algorithm , and is an alternative to deterministic algorithms for statistical inference such as the expectation-maximization algorithm.
As with other MCMC algorithms, Gibbs sampling generates a Markov chain of samples, each of which is correlated with nearby samples. As a result, care must be taken if independent samples are desired. Generally, samples from the beginning of the chain may not accurately represent the desired distribution and are usually discarded.