# Bayes formula

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) |

#### Comments

#### 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