
$$(16)^{\frac{1}{2}}$$
$$\left({\overline{{a}}}\wedge{\overline{{c}}}\wedge d\wedge y,\right.$$  $$\left.\left({\overline{{a}}}\wedge{\overline{{c}}}\wedge d\wedge{\overline{{y}}},\right.\right.$$
E Z *Value*

![0_image_0.png](0_image_0.png)

true true 0.21475
true false 0.70975
false true 0.78525
false false 0.29025
$$(14)$$
$\left(\begin{array}{c}5\\ 5\\ 5\end{array}\right)$
E *Value*
true 0.62725
false 0.37275

$$(15)$$

See Example 19 below for some trivial confactors produced and how to avoid them.

These confactors should be contrasted with the factor on A, C, D, E, Y, Z (of size 32) that is produced by eliminating B in VE.

Example 15 Suppose that instead we were to eliminate D from the confactors of Figure 4. This example differs from the previous example as D appear in the bodies as well as in the tables.

The two confactors for P(E|A, B,C, D) that contain D, namely -a ∧ c ∧ d, t3[B, E] (confactor
(5)), and -a ∧ c ∧ d, t4[E] (confactor (6)) are both compatible with both confactors for P(D|Y, Z).

So we cannot sum out the variable or multiply any confactors.

In order to be able to multiply confactors, we can split confactor (5) on Z producing:

$$\begin{array}{l}{{\langle\overline{{{a}}}\wedge\overline{{{c}}}\wedge d\wedge z,t_{3}[B,E]\rangle}}\\ {{\langle\overline{{{a}}}\wedge\overline{{{c}}}\wedge d\wedge\overline{{{z}}},t_{3}[B,E]\rangle}}\end{array}$$
-a ∧ c ∧ d ∧ z, t3[B, E] (16)
$\eqref{eq:walpha}$
-a ∧ c ∧ d ∧ z, t3[B, E] (17)
The confactors for P(D|Y, Z) are -z, t7[D] and -z, t8[D, Y]. We can split the first of these on A
producing

$$\langle a\wedge z,t_{7}[D]\rangle$$
-a ∧ z, t7[D] (18)
$$\langle{\overline{{a}}}\wedge z,t_{7}[D]\rangle$$
-a ∧ z, t7[D] (19)
There are no other confactors containing D with context compatible with confactor (18). The prerequisite required to sum out D in the context a ∧ z is satisfied. This results in the confactor
-a ∧ z, 1 where 1 is the factor of no variables that has value 1. This can be removed as the product of 1 doesn't change anything. Intuitively this can be justified because in the context when A is true D has no children. We can detect this case to improve efficiency (see Section 4.10).

The confactor (19) can be split on C, producing

$$\langle{\overline{{a}}}\wedge c\wedge z,t_{7}[D]\rangle$$
-a ∧ c ∧ z, t7[D] (20)
$$\langle\overline{{{a}}}\wedge\overline{{{c}}}\wedge z,t_{7}[D]\rangle$$
-a ∧ c ∧ z, t7[D] (21)
We can sum out D from confactor (20), producing -a ∧ c ∧ z, 1, as in the previous case.

We can split confactor (21) on D producing:

$$\langle\overline{{{a}}}\wedge\overline{{{c}}}\wedge d\wedge z,0.29\rangle$$ $$\langle\overline{{{a}}}\wedge\overline{{{c}}}\wedge\overline{{{d}}}\wedge z,0.71\rangle$$
-a ∧ c ∧ d ∧ z, 0.29 (22)
-a ∧ c ∧ d ∧ z, 0.71 (23)
where 0.29 and 0.71 are the corresponding values from t7[D]. These are functions of no variables, and so are just numbers.

We can now multiply confactor (22) and (16), producing:
-a ∧ c ∧ d ∧ z, 0.29t3[B, E] (24)
where 0.29t3[B, E] is the table obtained by multiplying each element of t3[B, E] by 0.29.

$$(18)$$ $$(19)$$
$$\begin{array}{l}{(20)}\\ {(21)}\end{array}$$

$$\begin{array}{l}{(22)}\\ {(23)}\end{array}$$
