VE allows us to exploit structure in a Bayesian network by providing
a factorization of the joint probability distribution. In this section
we show how causal independence can be used to factorize the joint
distribution even further. The initial factors in the VE algorithm
are of the form 
. We want to break this
down into simpler factors so that we do not need a table
exponential in m. The following proposition shows how causal
independence can be used to do this:
Proof:
The definition of causal independence entails the independence assertions
By the axiom of weak union [28, p. 84,], we have
. Thus all of the 
mutually
independent given 
.
Also we have, by the definition of causal independence
, so
Thus we have:
The next four sections develop an algorithm for exploiting causal independence in inference.
