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[[Bayes formula|Bayes formula]],
 
[[Bayes formula|Bayes formula]],
 
has the density
 
has the density
<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>
+
</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 08:57, 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
How to Cite This Entry:
Sidebar. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sidebar&oldid=3039