A clarification on the "conditioning" rule

[Subbarao Kambhampati <rao@asu.edu>]

In explaining how we can use "conditioning" to answer the query

P(Mary-calls|Earthquake), I used the following formula


P(M|E) = P(M|A) P(A|E) + P(M|~A) P(~A|E)                --(1)


It turns out that I was combining the conditioning rule and the bayes 
network semantics together in writing this formula.

The general rule for conditioning on any boolean random variable A is

P(M|E) = P(M|E,A) P(A|E) + P(M|E,~A) P(~A|E)            --(2)

Now we can simplify
P(M|E,A) to P(M|A) since M is independent of E (its non-descendant node) 
give A, its parent in the bayes network

Similarly

P(M|E,~A) to P(M|~A)

Once we apply these simplifications to 2, we get 1.

Rao