In this section, we shall first introduce an operator for combining factors that contain convergent variables. The operator is a basic ingredient of the algorithm to be developed in the next three sections. Using the operator, we shall rewrite equation (2) in a form that is more convenient to use in inference and introduce the concept of heterogeneous factorization.
Consider two factors f and g. Let , ..., be the convergent variables that appear in both f and g, let A be the list of regular variables that appear in both f and g, let B be the list of variables that appear only in f, and let C be the list of variables that appear only in g. Both B and C can contain convergent variables, as well as regular variables. Suppose is the base combination operator of . Then, the combination of f and g is a function of variables , ..., and of the variables in A, B, and C. It is defined by:
for each value of . We shall sometimes write as to make explicit the arguments of f and g.
Note that base combination operators of different convergent variables can be different.
The following proposition exhibits some of the basic properties of the combination operator .
Proof: The first item is obvious. The commutativity of follows readily from the commutativity of multiplication and the base combination operators. We shall prove the associativity of in a special case. The general case can be proved by following the same line of reasoning.
Suppose f, g, and h are three factors that contain only one variable e and the variable is convergent. We need to show that . Let be the base combination operator of e. By the associativity of , we have, for any value of e, that
The proposition is hence proved.
The following propositions give some properties for that correspond to the operations that we exploited for the algorithm . The proofs are straight forward and are omitted.