Random field

In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb{R}^n ). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb{R}^n (or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold.

In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb{R}^n ). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb{R}^n (or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold. 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), \, for each choice of values (x_j)_j . Here each \partial_i is the set of neighbors of i . 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 is given by P(X_i=x_i|\partial_i) = \frac{P(X_i=x_i, \partial_i)}{\sum_k P(X_i=k, \partial_i)}, where the sum (can be an integral) is over the possible values of k. It is sometimes difficult to compute this quantity exactly. ==References== ==Further reading==

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