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) |
Bayes formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bayes_formula&oldid=45997