Random Field

Context:
- It can be a Undirected Probabilistic Graph.
Example(s):
- a Markov Random Field, such as a Conditional Random Field.
- Gaussian Random Field.
- Gibbs Random Field.
- …
Counter-Example(s):
- a Bayesian Network.
- a Random Function.
See: Unordered Field.

References

http://en.wikipedia.org/wiki/Random_element#Random_field
- Given a probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math], 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 : [math]\displaystyle{ \{ F_t : t \in T \} }[/math] where each [math]\displaystyle{ F_t }[/math] 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 : [math]\displaystyle{ P(X_i=x_i|X_j=x_j, i\neq j) =P(X_i=x_i|\partial_i), \, }[/math] where [math]\displaystyle{ \partial_i }[/math] is a set of neighbours of the random variable X_i. 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 : [math]\displaystyle{ P(X_i=x_i|\partial_i) = \frac{P(\omega)}{\sum_{\omega'}P(\omega')}, }[/math] where Ω' is the same realization of Ω, except for random variable X_i. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by Julian Besag in 1974.

(Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Random_field
- A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real, but can instead be a multidimensional vector space or even a manifold.
  At its most basic, discrete case, a random field is a list of random numbers whose values are mapped onto a space (of n dimensions). Values in a random field are usually spatially correlated in one way or another, in its most basic form this might mean that adjacent values do not differ as much as values that are further apart. This is an example of a covariance structure, many different types of which may be modelled in a random field.

(Sutton & McCallum, 2007) ⇒ Charles Sutton, and Andrew McCallum. (2007). “An Introduction to Conditional Random Fields for Relational Learning.” In: (Getoor & Taskar, 2007).
- QUOTE: A graphical model is a family of probability distributions that factorize according to an underlying graph. The main idea is to represent a distribution over a large number of random variables by a product of local functions that each depend on only a small number of variables. Given a collection of subsets A⊂V, we define an undirected graphical model as the set of all distributions that can be written in the form … <snip> … (These functions are also called local functions or compatibility functions.) We will occasionally use the term random field to refer to a particular distribution among those defined by an undirected model. To reiterate, we will consistently use the term model to refer to a family of distributions, and random field (or more commonly, distribution) to refer to a single one.