Contextual constraints on two labels are the lowest order constraints to convey contextual information. They are widely used because of their simple form and low computational cost. They are encoded in the Gibbs energy as pair-site clique potentials. With clique potentials of up to two sites, the energy takes the form
where ``'' is equivalent to ``'' and ``'' to ``''. The above is a special case of (1.27), which we call a second-order energy because it involves up to pair-site cliques. It is most frequently used form owing to the mentioned feature that it is the simplest in form but conveys contextual information. A specific GRF or MRF can be specified by proper selection of 's and 's. Some important such GRF models are described subsequently. Derin and Kelly (1989) present a systematic study and categorization of Markov random processes and fields in terms of what they call strict-sense Markov and wide-sense Markov properties.
When and , where are arbitrary functions and are constants reflecting the pair-site interaction between i and , the energy is
The above is called auto-models [Besag 1974]. The auto-models can be further classified according to assumptions made about individual 's.
An auto-model is said to be an auto-logistic model, if the 's take on values in the discrete label set (or ). The corresponding energy is of the following form
where can be viewed as the interaction coefficients. When is the nearest neighborhood system on a lattice (4 nearest neighbors on a 2D lattice or 2 nearest neighbors on a 1D lattice), the auto-logistic model is reduced to the Ising model. The conditional probability for the auto-logistic model with is
When the distribution is homogeneous, we have and regardless of i and .
An auto-model is said to be an auto-binomial model if the 's take on values in and every has a conditionally binomial distribution of M trials and probability of success q
The corresponding energy takes the following form
It reduces to the auto-logistic model when M=1.
An auto-model is said to be an auto-normal model , also called a Gaussian MRF [Chellappa 1985] , if the label set is the real line and the joint distribution is multivariate normal. Its conditional p.d.f. is
It is the normal distribution with conditional mean
and conditional variance
The joint probability is a Gibbs distribution
where f is viewed as a vector, is the vector of the conditional means, and is the interaction matrix whose elements are unity and off-diagonal element at () is , i.e. with . Therefore, the single-site and pair-site clique potential functions for the auto-normal model are
respectively. A field of independent Gaussian noise is a special MRF whose Gibbs energy consists of only single-site clique potentials. Because all higher order clique potentials are zero, there is no contextual interaction in the independent Gaussian noise. B is related to the covariance matrix by . The necessary and sufficient condition for (1.47) to be a valid p.d.f. is that B is symmetric and positive definite.
A related but different model is the simultaneous auto-regression (SAR) model. Unlike the auto-normal model which is defined by the m conditional p.d.f.'s, this model is defined by a set of m equations
where are independent Gaussian, . It also generates the class of all multivariate normal distributions but with joint p.d.f. as
where B is defined as before. Any SAR model is an auto-normal model with the B matrix in (1.47) being where . The reverse can also be done, though in a rather unnatural way via Cholesky decomposition [Ripley 1981]. Therefore, both models can have their p.d.f.'s in the form (1.47). However, for (1.51) to be a valid p.d.f., it requires only that be non-singular.