Exploiting Contextual Independence In Probabilistic Inference -- Examples of Eliminating Variables

Example. When we eliminate B from the confactors of Figure *, we only need to consider the four confactors that contain B. The preconditions for summing out B or for multiplying are not satisfied, so we need to split. If we split the first confactor for P(E|A,B,C,D) on Y (as in Example *) and split the first confactor for P(B|Y,Z) on A (as in Example *), we produce two confactors, (*) and (*), that can be multiplied producing:

a&y,
B E Z Value

true true true 0.4235

true true false 0.0935

true false true 0.3465

true false false 0.0765

false true true 0.069

false true false 0.249

false false true 0.161

false false false 0.581

This is the only confactor that contains B and is applicable in the context a&y, so we can sum out B from the table, producing the confactor:
a&y,
E Z Value

true true 0.4925

true false 0.3425

false true 0.5075

false false 0.6575

The other nontrivial confactors produced when summing out B are:
a&~y,
E Value

true 0.3675

false 0.6325

~a&~c&d&y,
E Z Value

true true 0.21475

true false 0.70975

false true 0.78525

false false 0.29025

~a&~c&d&~y,
E Value

true 0.62725

false 0.37275

See Example * 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. Suppose that instead we were to eliminate D from the confactors of Figure *. 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 (*)), and <~a &~c &~d ,t₄[E]> (confactor (*)) 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 (*) on Z producing:

<~a &~c &d &z ,t₃[B,E]>

<~a &~c &d &~z ,t₃[B,E]>

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
<a &z,t₇[D]>

<~a &z,t₇[D]>

There are no other confactors containing D with context compatible with confactor (*). 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 *).
The confactor (*) can be split on C, producing
<~a &c &z,t₇[D]>

<~a &~c &z,t₇[D]>

We can sum out D from confactor (*), producing <~a &c &z,1>, as in the previous case.
We can split confactor (*) on D producing:
<~a &~c &d &z,0.29>

<~a &~c &~d &z,0.71>

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 (*) and (*), producing:
<~a &~c &d &z ,0.29 t₃[B,E]>

where 0.29 t₃[B,E] is the table obtained by multiplying each element of t₃[B,E] by 0.29.
We can also split confactor (*) on Z, producing:
<~a &~c &~d &z ,t₄[E]>

<~a &~c &~d &~z ,t₄[E]>

We can multiply confactors (*) and (*), producing:
<~a &~c &~d &z ,0.71 t₄[E]>

We now have only complementary confactors for D in the context ~a &~c &z, namely confactors (*) and (*) so we can sum-out D in this context resulting in
<~a &~c &z ,t₉[B,E]>

where t₉[B,E] is 0.29 t₃[B,E]+_t0.71 t₄[E]. In full form this is:
~a&~c&z,
B E Value

true true 0.36225

true false 0.63775

false true 0.6015

false false 0.3985

The other confactor produced when summing out D is:
~a&~c&~z,
B E Y Value

true true true 0.12475

true true false 0.21975

true false true 0.87525

true false false 0.78025

false true true 0.7765

false true false 0.7065

false false true 0.2235

false false false 0.2935