Bayes Theorem
Warning
This post is more than a year old. Information may be outdated.
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
$P(H_j | E) = {{P(E|H_k) P(H_k)} \over {\sum\limits_{j} P(E|H_j) P(H_j)}}$