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[[Bayes formula|Bayes formula]], | [[Bayes formula|Bayes formula]], | ||
has the density | has the density | ||
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<tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100309.png"></td> | <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100309.png"></td> | ||
| − | </tr></table> | + | </tr></table></html> |
If | If | ||
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003010.png"></html> | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003010.png"></html> | ||
Revision as of 07:57, 24 June 2010
A conditional probability distribution of a random variable, to be contrasted with its unconditional or a priori distribution.
Let
<html>
be a random parameter with an a priori density
,
let
be a random result of observations and let
be the conditional density of
when
;
then the a posteriori distribution of
for a given
</html>,
according to the
Bayes formula,
has the density
<html>
 |
</html>
If
<html>
</html>
is a
sufficient statistic
for the family of distributions with densities
<html>
,
then the a posteriori distribution depends not on
itself, but on
.
The asymptotic behaviour of the a posteriori distribution
as
,
where
are the results of independent observations with density
,</html>
is
"almost independent"
of the a priori distribution of
<html>
</html>.
For the role played by a posteriori distributions
in the theory of statistical decisions, see
Bayesian approach.
References
<html>
| [1] |
S.N. Bernshtein, "Probability theory" , Moscow-Leningrad (1946) (In Russian) |
</html>
Yu.V. Prokhorov
Comments
References
<html>
| [a1] |
E. Sverdrup, "Laws and chance variations" , 1 , North-Holland (1967) pp. 214ff |
</html>
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
Sidebar. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sidebar&oldid=3040