In probability theory, let S = {X1, ..., Xn}, with the Xi in {0,1,...,G-1}, be a set of random variables on the sample space Ω={0,1,...,G-1}n, a probability measure π is a random field if
.
There exist several types of random fields, such as Markov random field (MRF) and Gibbs random field (GRF). A MRF exhibits the Markovian property
,
where is a set of neighbours of the random variable Xi. In other words, the probability a random variable assumes a value does not depend on all of the random variables. A probability of a random variable in a MRF is showed by the equation 1, Ω' is the same realization of Ω, except for random variable Xi. It is easy to see that it is difficult to calculate with this equation. The solution to this problem was proposed by Besag in 1974, when he made a relation between MRF and GRF.
Reference
Besag, J. E. Spatial Interaction and the Statistical Analysis of Lattice Systems. Journal of Royal Statistical Society: Series B 36, 2 (May 1974), 192-236.
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