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Gibbs sampling is named after the physicist Josiah Willard Gibbs, in reference to an analogy between the sampling algorithm and statistical physics.The algorithm was described by brothers Stuart and Donald Geman in 1984, some eight decades after the death of Gibbs, [1] and became popularized in the statistics community for calculating marginal probability distribution, especially the posterior ...
Boltzmann's distribution is an exponential distribution. Boltzmann factor (vertical axis) as a function of temperature T for several energy differences ε i − ε j.. In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution [1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain ...
Bayesian inference using Gibbs sampling (BUGS) is a statistical software for performing Bayesian inference using Markov chain Monte Carlo (MCMC) methods. It was developed by David Spiegelhalter at the Medical Research Council Biostatistics Unit in Cambridge in 1989 and released as free software in 1991.
This method used a Gibbs sampling approach in which each individuals haplotypes were updated conditional upon the current estimates of haplotypes from all other samples. Approximations to the distribution of a haplotype conditional upon a set of other haplotypes were used for the conditional distributions of the Gibbs sampler.
Particularly notable works include: the development of the Gibbs sampler, proof of convergence of simulated annealing, [8] [9] foundational contributions to the Markov random field ("graphical model") approach to inference in vision and machine learning, [3] [10] and work on the compositional foundations of vision and cognition. [11] [12]
A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique.
This distribution plays an important role in hierarchical Bayesian models, because when doing inference over such models using methods such as Gibbs sampling or variational Bayes, Dirichlet prior distributions are often marginalized out. See the article on this distribution for more details.
The idea of applying the Ising model with annealed Gibbs sampling was used in Douglas Hofstadter's Copycat project (1984). [21] [22] The explicit analogy drawn with statistical mechanics in the Boltzmann machine formulation led to the use of terminology borrowed from physics (e.g., "energy"), which became standard in the field.