Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

It may be noted that some of the probability of joint occurrences [ P (Ei & Hj)]
may be zero. It follows from expression (9.5) and expression (9.1) that
P (Ei / Hj) x P (Hj)
P (Hj / Ei) =
Σ P ( Ei / Hk ) x P (Hk)
1≤ k ≤ n
In the real world, however, the hypothesis Hj depends on a number of Ei s.
Thus,
P (Hi / E1, E2, ..., Em)
P { (E1, E2,..., Em) / Hj} x P (Hj)
=
Σ P { ( E1,E2, ..., Em) / Hk} x P (Hk)
1≤ k ≤n
However, the conditional probability of the joint occurrences of E1,
E2,...., En when Hj has happened, in many real life problems, is unknown. So
Ei is considered to be statistically independent, which unfortunately is never
true in practice.
When Ej s are independent, we can write
P (E1, E2,..., Em / Hj)
= P (E1/ Hj ) x P (E2 / Hj ) x ... x P (Em / Hj )
m
= Π P (Ei / Hj ) (9.8)
i=1
Substituting the right-hand side of expression (9.8) in (9.7) we find,
m
{ Π P (Ej / Hi ) } x P (Hi)
i=1
P (Hj / E1, E2,..., Em) = (9.9)
n m
∑ { Π P ( Ei / Hk) } x P (Hk)
k=1 i=1