The previous numerical approaches produced algorithms that converge to optimizers of the posterior distribution, without guarantees about global optimality. In this subsection we sketch an approach to obtain convergence to the global minimum of the posterior distribution. Empirical tests are due to study the practical applicability of this approach.

Lavine's bracketing algorithm is a method to obtain the posterior
quantity __p__(x_{q} = a) =
_{q} = a, e)/p(e) ).
The idea is to settle for deciding whether or not
__p__(x_{q} = a) is
larger than a given value k. When we obtain this result, we can construct
an algorithm by bracketing the interval [0,1] with k. This
algorithm is convergent and improves monotonically.

Notice that __p__(x_{q} = a) = _{q} = a, e)/p(e) )
is larger than k if and only if
_{q} = a, e) - k p(e) ) is larger than zero.
The point of Lavine's algorithm is that minimization of the latter
quantity may be simpler than minimization of the former, since
there are no ratios involved. This is in fact true for type-1 combinations
in Quasi-Bayesian networks. Consider the expression that must be minimized:

sum_{x is in (xq, e)}
prod_{i} p_{i}^{e} -
k sum_{x is in e} prod_{i} p_{i}^{e}.

© Fabio Cozman[Send Mail?]

Fri May 30 15:55:18 EDT 1997