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

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

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

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

Steven M LaValle 2012-04-20