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.
