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</tr></table></html>
 
</tr></table></html>
 
If
 
If
<html><img align="absmiddle" border="0" src="images/a010/a010030/a01003010.png"></html>
+
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003010.png"></html>
 
is a
 
is a
 
[[Sufficient statistic|sufficient statistic]]
 
[[Sufficient statistic|sufficient statistic]]
 
for the family of distributions with densities
 
for the family of distributions with densities
<html><img align="absmiddle" border="0" src="images/a010/a010030/a01003011.png">,
+
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003011.png">,
 
then the a posteriori distribution depends not on
 
then the a posteriori distribution depends not on
<img align="absmiddle" border="0" src="images/a010/a010030/a01003012.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003012.png">
 
itself, but on
 
itself, but on
<img align="absmiddle" border="0" src="images/a010/a010030/a01003013.png">.
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003013.png">.
 
The asymptotic behaviour of the a posteriori distribution
 
The asymptotic behaviour of the a posteriori distribution
<img align="absmiddle" border="0" src="images/a010/a010030/a01003014.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003014.png">
 
as
 
as
<img align="absmiddle" border="0" src="images/a010/a010030/a01003015.png">,
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003015.png">,
 
where
 
where
<img align="absmiddle" border="0" src="images/a010/a010030/a01003016.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003016.png">
 
are the results of independent observations with density
 
are the results of independent observations with density
<img align="absmiddle" border="0" src="images/a010/a010030/a01003017.png">,</html>
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003017.png">,</html>
 
is
 
is
 
&#160;"almost independent"&#160;
 
&#160;"almost independent"&#160;
 
of the a priori distribution of
 
of the a priori distribution of
<html><img align="absmiddle" border="0" src="images/a010/a010030/a01003018.png"></html>.
+
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003018.png"></html>.
  
  

Revision as of 16:38, 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://encyclopediaofmath.org/index.php?title=Sidebar&oldid=2999