The probability of the evidence conjoined with a context c on the
noneliminated, nonobserved variables is equal to the product
of the probabilities of the confactors that are applicable in context
c. For each context c on the noneliminated, nonobserved
variables and for each variable X there is at least one confactor containing
X that is applicable in context c.
For probabilistic inference, where we will normalise at the end, we
can remove any confactor that doesn't involve any variable (i.e., with an empty
context and single number as the table) as a result of the second
or third cases. That is, we remove any confactor that only has
observed variables. We then need to replace "equal" with
"proportional" in the program invariant.
Example.
Suppose ~
d&~
z is observed given the confactors of
Figure *. The first two confactors for P(EA,B,C,D)
don't involve D or Z and so are not affected by the
observation. The third confactor is removed as its body is incompatible
with the observation. The fourth confactor is replaced by:
<~
a&~
c,
E  Value 
true  0.5 
false  0.5 

>
The first confactor for P(BY,Z) is replaced by
<y,
B  Value 
true  0.17 
false  0.83 

>
The first confactor for P(DY,Z) is removed and the second is replaced by
<true,
Y  Value 
true  0.21 
false  0.41 

>
where true represents the empty context.