# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Random field

(Redirected from Random fields)

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

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

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 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.

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

## 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.