In this subsection we show how the original Expectation-Maximization
algorithm [13] can be extended to a Quasi-Bayesian
Expectation-Maximization (QEM) algorithm with the same
convergence properties. We must maximize the posterior
log-likelihood L(Theta) defined previously.
The algorithm begins by assuming that the transparent variables are actual
random quantities with distributions specified by Theta. An initial
estimate Theta^{0} is assumed for Theta.

Suppose we had i sets of complete data for the transformed network, i.e.,
we had observed i trials for all variables in the network, including the
transparent variables. The log-likelihood for this complete data would be
L(Theta) = sum_{ijk} l_{i}(j,k) _{ijk},
where l_{i}(j,k) indicates the number of data points when the variable
x_{i} is instantiated in its j value with its parents instantiated
in their k value.

The first step of the QEM algorithm is to obtain the expected value of
the log-likelihood given the evidence and assuming Theta^{0} is
correct [10]:

^{k}) _{q} = a, e) _{ijk} p(x_{i}, _{i}) | x_{q} = a, e) _{ijk} _{ijk} p(x_{i}, _{i}) | e) _{ijk}.

The second step of the QEM algorithm is to
maximize Q(Theta|Theta^{k}) for Theta. Only a few terms in the
expression for Q(Theta|Theta^{k}) will be free, since only the theta_{ij}
for z'_{i} are estimated.
Collecting these terms we obtain:

sum_{ij} p(z'_{i} = j | x_{q} = a, e) _{ij}
- sum_{ij} p(z'_{i} = j| e) _{ij},

Fri May 30 15:55:18 EDT 1997