Let be a family of random variables defined on the set , in which each random variable takes a value in . The family F is called a random field. We use the notation to denote the event that takes the value and the notation to denote the joint event. For simplicity, a joint event is abbreviated as F=f where is a configuration of F, corresponding to a realization of the field. For a discrete label set , the probability that random variable takes the value is denoted , abbreviated unless there is a need to elaborate the expressions, and the joint probability is denoted and abbreviated . For a continuous , we have probability density functions (p.d.f.'s), and .
F is said to be a Markov random field on with respect to a neighborhood system if and only if the following two conditions are satisfied:
where is the set difference, denotes the set of labels at the sites in and
stands for the set of labels at the sites neighboring i. The positivity is assumed for some technical reasons and can usually be satisfied in practice. For example, when the positivity condition is satisfied, the joint probability of any random field is uniquely determined by its local conditional probabilities [Besag 1974]. The Markovianity depicts the local characteristics of F. A label interacts with only the neighboring labels. In other words, only neighboring labels have direct interactions with each other. It is always possible to select sufficiently large so that the Markovianity holds. The largest neighborhood consists of all other sites. Any F is an MRF with respect to such a neighborhood system.
An MRF can have other properties such as homogeneity and isotropy. It is said to be homogeneous if is regardless of the relative position of site i in . The isotropy will be illustrated in the next subsection with clique potentials.
It may need to define for some problem a few coupled MRFs, each defined on one of the spatially interwoven sets of sites. For example, in the related tasks of image restoration and edge detection, two MRFs, one for pixel values () and the other for edge values (), can be defined on the image lattice and its dual lattice, respectively. They are coupled to each other e.g. via conditional probability (see Section 2.3.1).
The concept of MRFs is a generalization of that of Markov processes (MPs) which are widely used in sequence analysis. An MP is defined on a domain of time rather than space. It is a sequence (chain) of random variables defined on the time indices . An n-th order unilateral MP satisfies
A bilateral or non-causal MP depends not only on the past but also on the future. An n-th order bilateral MP satisfies
It is generalized into MRFs when the time indices are considered as spatial indices.
There are two approaches for specifying an MRF, that in terms of the conditional probabilities and that in terms of the joint probability . Besag (1974) argues for the joint probability approach in view of the disadvantages of the conditional probability approach: Firstly, no obvious method is available for deducing the joint probability from the associated conditional probabilities. Secondly, the conditional probabilities themselves are subject to some non-obvious and highly restrictive consistency conditions. Thirdly, the natural specification of an equilibrium of statistical process is in terms of the joint probability rather than the conditional distribution of the variables. Fortunately, a theoretical result about the equivalence between Markov random fields and Gibbs distribution [Hammersley and Clifford 1971 ; Besag 1974] provides a mathematically tractable means of specifying the joint probability of an MRF.