Editing Random field (section)

== Examples ==
In its discrete version, a random field is a list of random numbers whose indices are identified with a discrete set of points in a space (for example, n-[[dimensional]] [[Euclidean space]]). Suppose there are four random variables, <math>X_1</math>, <math>X_2</math>, <math>X_3</math>, and <math>X_4</math>, located in a 2D grid at (0,0), (0,2), (2,2), and (2,0), respectively. Suppose each random variable can take on the value of -1 or 1, and the probability of each random variable's value depends on its immediately adjacent neighbours. This is a simple example of a discrete random field.

More generally, the values each <math>X_i</math> can take on might be defined over a continuous domain. In larger grids, it can also be useful to think of the random field as a "function valued" random variable as described above. In [[quantum field theory]] the notion is generalized to a random [[Functional (mathematics)|functional]], one that takes on random values over a [[Function space|space of functions]] (see [[Feynman integral]]). 

Several kinds of random fields exist, among them the [[Markov random field]] (MRF), [[Gibbs random field]], [[conditional random field]] (CRF), and [[Gaussian random field]]. In 1974, [[Julian Besag]] proposed an approximation method relying on the relation between MRFs and Gibbs RFs.{{Citation needed|date=May 2019}}

=== Example properties ===

An MRF exhibits the [[Markov property]]

: <math>P(X_i=x_i|X_j=x_j, i\neq j) =P(X_i=x_i|X_j=x_j,j\in\partial_i), \,</math>

for each choice of values <math>(x_j)_j</math>. Here each <math>\partial_i</math> is the set of neighbors of  <math>i</math>. In other words, the probability that a random variable assumes a value depends on its immediate neighboring random variables.  The probability of a random variable in an MRF{{what|reason=Or in any probability measure, since the denominator is always 1.|date=October 2023}} is given by

:<math> P(X_i=x_i|\partial_i) = \frac{P(X_i=x_i, \partial_i)}{\sum_k P(X_i=k, \partial_i)},  </math>

where the sum (can be an integral) is over the possible values of k.{{what|reason=What is the content of this equation? The sum in the denominator is automatically 1 since P is a probability measure.|date=October 2023}} It is sometimes difficult to compute this quantity exactly.