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Bayes Theorem

In Probability,

if {Hk}\{H_k\} partitions Ω\Omega then

P(Hj∣E)=P(Hj∩E)P(E)=P(Hj)P(E∣Hj)∑kP(Hk)P(E∣Hk)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)}}

HjH_j is posterior in this case.

The Odds form of Bayes Theorem is

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