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