Bayesian networks place no restriction on how a node depends on its parents. Unfortunately this means that in the most general case we need to specify an exponential (in the number of parents) number of conditional probabilities for each node. There are many cases where there is structure in the probability tables that can be exploited for both acquisition and for inference. One such case that we investigate in this paper is known as `causal independence'.
In one interpretation, arcs in a BN represent causal
relationships; the parents 
of a variable
e are viewed as causes that jointly bear on the effect e.
Causal independence refers to the situation where
the causes 
contribute independently to
the effect e.
More precisely, 
are said to be causally
independent w.r.t. effect e if there exist random variables
that have the same frame, i.e., the
same set of possible values,
as e such that
probabilistically depends on 
and is
conditionally independent of all other 
's and all other 
's given 
, and
over the frame of e such that 
.
mean that X is independent of Y given Z, the first
condition is:
and similarly for the other variables. This entails 
and 
for each 
and 
where 
.
We refer to 
as the contribution of 
to e. In
less technical terms, causes are causally independent w.r.t. their
common effect if individual contributions from different causes are
independent and the total influence on the effect is a combination of
the individual contributions.
We call the variable e a convergent variable as it is where
independent contributions from different sources are collected and
combined (and for the lack of a better name). Non-convergent
variables will simply be called regular variables. We call
the base combination operator of e.
The definition of causal independence given here is slightly different than that given by Heckerman and Breese [16] and Srinivas [34]. However, it still covers common causal independence models such as noisy OR-gates [14,28], noisy MAX-gates [10], noisy AND-gates, and noisy adders [6] as special cases. One can see this in the following examples.
The following example is not an instance of any causal independence models that we know:
In the traditional formulation of a Bayesian network we need
to specify an exponential, in the number of parents, number of conditional probabilities for a variable.
With causal independence, the number of
conditional probabilities 
is linear in m.
This is why causal independence can
reduce complexity of knowledge acquisition
[17,28,26,25].
In the following sections we show how causal independence can also be
exploited for computational gain.