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Random field

In probability theory, let S = {X₁, ..., X_n}, with the X_i 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

$\pi(\omega)>0\;\; \forall\; \omega \in \Omega$ .

There exist several types of random fields, such as Markov random field (MRF), Gibbs random field (GRF) and Gaussian random field . A MRF exhibits the Markovian property

$\pi (X_i=x_i|X_j=x_j, i\neq j) = \pi (X_i=x_i|\partial_i)$ ,

where $\partial_i$ is a set of neighbours of the random variable $X i$ . 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 $X i$ . 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.

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

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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