Random field

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

[itex]\pi(\omega)>0\,[itex]

for all ω in Ω. Several kinds of random fields exist, among them Markov random fields (MRF), Gibbs random fields (GRF) and Gaussian random fields. A MRF exhibits the Markovian property

[itex]\pi (X_i=x_i|X_j=x_j, i\neq j) = \pi (X_i=x_i|\partial_i), \,[itex]

where [itex]\partial_i[itex] is a set of neighbours of the random variable Xi. In other words, the probability a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbors. 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.

[itex] \pi (X_i=x_i|\partial_i) = \frac{\pi(\omega)}{\sum_{\omega'}\pi(\omega')} \;\;\;\;(1) [itex]

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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