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Bayes Theorem

In Probability,

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

P(HjE)=P(HjE)P(E)=P(Hj)P(EHj)kP(Hk)P(EHk)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(HE)=O(H)P(EH)P(EHC)O(H|E) = O(H) {P(E|H) \over P(E|H^C)}

If Hk{H_k} partitions Ω\Omega then

P(HjE)=P(EHk)P(Hk)jP(EHj)P(Hj)P(H_j | E) = {{P(E|H_k) P(H_k)} \over {\sum\limits_{j} P(E|H_j) P(H_j)}}