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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
- <math>\pi(\omega)>0\;\; \forall\; \omega \in \Omega<math>.
There exist several types of random fields, such as Markov random field (MRF) and Gibbs random field (GRF). A MRF exhibits the Markovian property
- <math>\pi (X_i=x_i|X_j=x_j, i\neq j) = \pi (X_i=x_i|\partial_i)<math>,
where <math>\partial_i<math> is a set of neighbours of the random variable <math>X_i<math>. 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 <math>X_i<math>. 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.
- <math> \pi (X_i=x_i|\partial_i) = \frac{\pi(\omega)}{\sum_{\omega'}\pi(\omega')} \;\;\;\;(1) <math>
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.
See Also
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