Theorem: Bayes' Theorem

Given mutually exclusive and collectively exhaustive events \(A_1,A_2,\ldots,A_n\) with the probabilities \(p(A_i) > 0\) for \(i=1,2,\ldots,n\), and any event \(B\) with the probability \(p(B) > 0\), the probabilities of the events \(A_i\) given \(B\) can be calculated by the formula

\[p(A_i|B)=\frac{p(B|A_i)p(A_i)}{p(B)}=\frac{p(B|A_i)p(A_i)}{\sum_{i=1}^np(B|A_i)p(A_i)},\quad\quad i=1,2,\ldots,n.\]

Proofs: 1

Epochs: 1


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References

Bibliography

  1. Bosch, Karl: "Elementare Einf├╝hrung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition