Exploiting Contextual Independence In Probabilistic Inference -- Evidence

Evidence

Evidence

As in VE, evidence simplifies the knowledge base. Suppose E₁=o₁ &...&E_s=o_s is observed. There are three steps in absorbing evidence:

Remove any confactor whose context contains E_i=o_i', where o_i != o_i'.
Remove any term E_i=o_i in the context of a confactor.
Replace each table t with set(t,E₁=o₁ &...&E_s=o_s) (as in the tabular VE algorithm).

Again note that incorporating evidence only simplifies the confactor base.

Once evidence has been incorporated into the confactor-base, the program invariant becomes:

The probability of the evidence conjoined with a context c on the non-eliminated, non-observed variables is equal to the product of the probabilities of the confactors that are applicable in context c. For each context c on the non-eliminated, non-observed 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(E|A,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(B|Y,Z) is replaced by
<y,
B Value

true 0.17

false 0.83

>
The first confactor for P(D|Y,Z) is removed and the second is replaced by
<true,
Y Value

true 0.21

false 0.41

>
where true represents the empty context.

David Poole and Nevin Lianwen Zhang,Exploiting Contextual Independence In Probabilistic Inference, Journal of Artificial Intelligence Research, 18, 2003, 263-313.

Evidence