Marginalization

Let the events $F_1,F_2,\ldots,F_n$ be any partition of $S$. The probability of an event $E$ can be obtained through marginalization as

$$P(E) = \sum_{i=1}^n P(E\vert F_i) P(F_i) .$$ (9.8)

One of the most useful applications of marginalization is in the denominator of Bayes' rule. A substitution of (9.8) into the denominator of (9.6) yields

$$P(F\vert E) = {P(E\vert F)P(F) \over \displaystyle\strut \sum_{i=1}^n P(E\vert F_i) P(F_i)} .$$ (9.9)

This form is sometimes easier to work with because $P(E)$ appears to be eliminated.