Bayes Theorem

In Probability,

if $\{H_k\}$ partitions $\Omega$ then

$$P(H_j | E) = {P(H_j \cap E) \over P(E)} = {{P(H_j) P(E|H_j)} \over {\sum\limits_{k} P(H_k) P(E|H_k)}}$$

$H_j$ is posterior in this case.

The Odds form of Bayes Theorem is

$$O(H|E) = O(H) {P(E|H) \over P(E|H^C)}$$

If ${H_k}$ partitions $\Omega$ then

$P(H_j | E) = {{P(E|H_k) P(H_k)} \over {\sum\limits_{j} P(E|H_j) P(H_j)}}$