Gibbs sampling
Gibbs sampling , also Gibbs sampling, is an algorithm to generate a sequence of samples of the common probability distribution of two or more random variables . The goal is to approximate the unknown common distribution . The algorithm is named after the physicist Josiah Willard Gibbs because of the similarity of the sampling process with methods of statistical physics . It was developed by Stuart Geman and Donald Geman (see bibliography). Gibbs sampling is a special case of the Metropolis-Hastings algorithm .
Gibbs sampling is particularly suitable when the joint distribution of a random vector is unknown, but the conditional distribution of each random variable is known. The basic principle is to repeatedly select a variable and, according to its conditional distribution, generate a value depending on the values of the other variables. The values of the other variables remain unchanged in this iteration step. A Markov chain can be derived from the resulting sequence of sample vectors. It can be shown that the stationary distribution of this Markov chain is precisely the joint distribution of the random vector that is sought.
A particularly favorable application arises in connection with Bayesian networks , in particular when estimating the a posteriori distribution , since the usual representation of a Bayesian network is a system of conditional distributions.
literature
- Stuart Geman and Donald Geman: Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. In: IEEE Transactions on Pattern Analysis and Machine Intelligence , 6: 721-741, 1984.
- CP Robert and G. Casella: Monte Carlo Statistical Methods. Springer, New York 2004.
- Michael S. Johannes and Nick Polson: MCMC Methods for Continuous-Time Financial Econometrics. (December 22, 2003). Available at SSRN: http://ssrn.com/abstract=480461