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[[Bayes formula|Bayes formula]],
 
[[Bayes formula|Bayes formula]],
 
has the density
 
has the density
 
 
<html><table class="eq" style="width:100%;">
 
<html><table class="eq" style="width:100%;">
 
<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></html>
 
</tr></table></html>
 +
If
 +
<html><img align="absmiddle" border="0" src="images/a010/a010030/a01003010.png"></html>
 +
is a
 +
[[Sufficient statistic|sufficient statistic]]
 +
for the family of distributions with densities
 +
<html><img align="absmiddle" border="0" src="images/a010/a010030/a01003011.png">,
 +
then the a posteriori distribution depends not on
 +
<img align="absmiddle" border="0" src="images/a010/a010030/a01003012.png">
 +
itself, but on
 +
<img align="absmiddle" border="0" src="images/a010/a010030/a01003013.png">.
 +
The asymptotic behaviour of the a posteriori distribution
 +
<img align="absmiddle" border="0" src="images/a010/a010030/a01003014.png">
 +
as
 +
<img align="absmiddle" border="0" src="images/a010/a010030/a01003015.png">,
 +
where
 +
<img align="absmiddle" border="0" src="images/a010/a010030/a01003016.png">
 +
are the results of independent observations with density
 +
<img align="absmiddle" border="0" src="images/a010/a010030/a01003017.png">,</html>
 +
is
 +
&#160;"almost independent"&#160;
 +
of the a priori distribution of
 +
<html><img align="absmiddle" border="0" src="images/a010/a010030/a01003018.png"></html>.
 +
  
 +
For the role played by a posteriori distributions
 +
in the theory of statistical decisions, see
 +
[[Bayesian approach|Bayesian approach]].
  
 
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Revision as of 17:37, 16 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.

todo

How to Cite This Entry:
Sidebar. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sidebar&oldid=2998