# Random Field

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A random field is a stochastic process X(x) where the index x is a multidimensional random variable.

## References

### 2015

• http://en.wikipedia.org/wiki/Random_element#Random_field
• Given a probability space $\displaystyle{ (\Omega, \mathcal{F}, P) }$, an X-valued random field is a collection of X-valued random variables indexed by elements in a topological space T. That is, a random field F is a collection : $\displaystyle{ \{ F_t : t \in T \} }$ where each $\displaystyle{ F_t }$ is an X-valued random variable.

Several kinds of random fields exist, among them the Markov random field (MRF), Gibbs random field (GRF), conditional random field (CRF), and Gaussian random field. An MRF exhibits the Markovian property : $\displaystyle{ P(X_i=x_i|X_j=x_j, i\neq j) =P(X_i=x_i|\partial_i), \, }$ where $\displaystyle{ \partial_i }$ is a set of neighbours of the random variable Xi. In other words, the probability that a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbours. The probability of a random variable in an MRF is given by : $\displaystyle{ P(X_i=x_i|\partial_i) = \frac{P(\omega)}{\sum_{\omega'}P(\omega')}, }$ where Ω' is the same realization of Ω, except for random variable Xi. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.