In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) 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 (EM).
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.
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