6.1.4
Mean field approximation from
statistical physics [Chandler 1987,Parisi 1988] provides other
possibilities. It originally aims to approximate the behavior of
interacting spin systems in thermal equilibrium. We use it to
approximate the behavior of MRFs in equilibrium. Generally, the mean of
a random variable X is given by 
.
So we define the mean field 
by the mean values
The mean field approximation makes the following assumption
for
calculating 
: The actual influence of 
(
) can be approximated by the influence of 
. This is reasonable when the field is in equilibrium. It leads
to a number of consequences. Equation (6.12)
is approximated by the mean field local energy expressed as
and the conditional probability is approximated by the mean field local probability
where
is called the mean field local partition function. Note in the above
mean field approximations, the values 
are
assumed given. The mean field values are approximated by
The mean field local probability (6.25) is an approximation to the marginal distribution
Because the mean field approximation ``decouples'' the interactions in equilibrium, the mean field approximation of the joint probability is the product of the mean field local probabilities
and the mean field partition function is the product of the mean field local partition functions
When 
is a continuous, such as the real line, all the above
probability functions are replaced by the corresponding p.d.f.'s and the
sums by integrals.
The mean field approximation (6.29) bears the form of
the pseudo-likelihood function.
Indeed, if 
(
) in the mean field computation is replaced by the current
, the two approximations become the same. However, unlike the
pseudo-likelihood, the mean values 
and the mean
field conditional probabilities 
are computed iteratively using (6.25),
(6.26) and (6.27) given an
initial 
.
The above is one of various approaches for mean field approximation. In Section 9.2, we will present an approximation for labeling with a discrete label set using saddle point approximation [Peterson and Soderberg 1989]. The reader is also referred to [Bilbro and Snyder 1989,Yuille 1990,Geiger and Girosi 1991] for other approaches. A study of mean field approximations of various GRFs and their network implementations is presented in a recent Ph.D thesis [Elfadel 1993]. Its application in MRF parameter estimation is proposed by [Zhang 1992,Zhang 1993].
