6.1.2
A simple approximate scheme is
pseudo-likelihood [Besag 1975,Besag 1977]. In this scheme, each
in (6.11) is treated
as if 
are given and the pseudo-likelihood is defined as the
simple product of the conditional likelihood
where 
denotes the set of points at the boundaries of
under the neighborhood system 
(The boundary sites are excluded from the product to reduce artificiality). Substituting
the conditional probability (6.8)
into (6.14), we obtain the pseudo-likelihood for the
the homogeneous and isotropic auto-logistic model as
The pseudo-likelihood does not involve the partition function Z.
Because 
and 
are not independent, the pseudo-likelihood
is not the true likelihood function, except in the trivial case of nil
neighborhood (contextual independence). In the large lattice limit, the
consistency of the maximum pseudo-likelihood (MPL) estimate is proven by
[], that is, it converges to the truth with
probability one.
The following example illustrates the MPL estimation. Consider the
homogeneous and isotropic auto-logistic MRF model
described by (6.7) and
(6.8). Express its
pseudo-likelihood (6.15) as 
.
The logarithm is
The MPL estimates 
can be obtained by
solving
