In comparing VE and
VE1 , we notice that
when summing out a variable, they both combine only
those factors that contain the variable.
However, the factorization that the latter works with
is of finer grain than the factorization used by the former.
In our running example, the latter works
with a factorization which initially consists of
factors that contain only two variables;
while the factorization the former uses
initially include factors
that contain five variables.
On the other hand, the latter uses the operator 
which
is more expensive than multiplication. Consider, for instance,
calculating 
. Suppose e is a
convergent variable and all variables are binary.
Then the operation requires 
numerical
multiplications and 
numerical summations.
On the other hand, multiplying 
and 
only requires
numerical multiplications.
Despite the expensiveness of the operator 
,
VE1 is more efficient
than VE .
We shall provide empirical evidence in support of
this claim in Section 10.
To see a simple example where this is true, consider
the BN in Figure 3(1), where
e is a convergent variable. Suppose all variables are binary.
Then, computing 
by VE using
the elimination ordering 
, 
, 
, and 
requires 
numerical multiplications
and 
numerical additions.
On the other hand,
computing 
in the deputation BN shown in Figure 3(2)
by VE1 using the elimination ordering
, 
, 
, 
, and 
requires only
numerical multiplications
and 
numerical additions.
Note that summing out 
requires 
numerical multiplications
because after summing out 
's, there are four heterogeneous factors,
each containing the only argument 
. Combining them pairwise
requires 
multiplications. The resultant factor needs to be
multiplied with the deputing factor 
, which
requires 
numerical multiplications.
Figure 3: A BN, its deputation and transformations.