Gibbs Sampling

In statistics and in statistical physics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of random samples from multivariate probability distribution (i.e. from the joint probability distribution of two or more random variables), when direct sampling is difficult. This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal distribution of one of the variables, or some subset of the variables (for example, the unknown parameters or latent variables); or to compute an integral (such as the expected value of one of the variables). 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 (i.e. an algorithm that makes use of random numbers, and hence may produce different results each time it is run), and is an alternative to deterministic algorithms for statistical inference such as variational Bayes or 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 (typically by thinning the resulting chain of samples by only taking every nth value, e.g. every 100th value). In addition (again, as in other MCMC algorithms), samples from the beginning of the chain (the burn-in period) may not accurately represent the desired distribution, and must be thrown away.