# Bayes formula

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A formula with which it is possible to compute a posteriori probabilities of events (or of hypotheses) from a priori probabilities. Let be a complete group of incompatible events: , if . Then the a posteriori probability of event if given that event with has already occurred may be found by Bayes' formula:

 (*)

where is the a priori probability of , is the conditional probability of event occurring given event (with ) has taken place. The formula was demonstrated by T. Bayes in 1763.

Formula (*) is a special case of the following abstract variant of Bayes' formula. Let and be random elements with values in measurable spaces and and let . Put, for any set ,

where and is the indicator of the set . Then the measure is absolutely continuous with respect to the measure () and , where is the Radon–Nikodým derivative of with respect to .

#### References

 [1] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)

#### References

 [a1] R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1 , Springer (1977) pp. Section 7.9 (Translated from Russian)
How to Cite This Entry:
Bayes formula. A.N. Shiryaev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bayes_formula&oldid=16075
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098