1.2.2
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.
