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A conditional probability distribution of a random
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* navigation
variable, to be contrasted with its unconditional or
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** mainpage|mainpage-description
[[A priori distribution|a priori distribution]].
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** Special:Allpages|Pages A-Z
 
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** :Category:Statprob|StatProb Collection
Let
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** recentchanges-url|recentchanges
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100301.png">
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** currentevents-url|currentevents
be a random parameter with an a priori density
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** randompage-url|randompage
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100302.png">,
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** Help:Contents|help
let
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** Talk:EoM:This_project|Project talk
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100303.png">
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* SEARCH
be a random result of observations and let
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* TOOLBOX
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100304.png">
 
be the conditional density of
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100305.png">
 
when
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100306.png">;
 
then the a posteriori distribution of
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100307.png">
 
for a given
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100308.png"></html>,
 
according to the
 
[[Bayes formula|Bayes formula]],
 
has the density
 
<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></table></html>
 
If
 
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/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="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003011.png">,
 
then the a posteriori distribution depends not on
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003012.png">
 
itself, but on
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003013.png">.
 
The asymptotic behaviour of the a posteriori distribution
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003014.png">
 
as
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003015.png">,
 
where
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003016.png">
 
are the results of independent observations with density
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003017.png">,</html>
 
is
 
&#160;"almost independent"&#160;
 
of the a priori distribution of
 
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003018.png"></html>.
 
 
 
 
 
For the role played by a posteriori distributions
 
in the theory of statistical decisions, see
 
[[Bayesian approach|Bayesian approach]].
 
 
 
====References====
 
<html><table><TR><TD valign="top">[1]</TD>
 
<TD valign="top">
 
&#160;S.N. Bernshtein,&#160;
 
&#160;"Probability theory"
 
, Moscow-Leningrad
 
&#160;(1946)
 
&#160;(In Russian)</TD></TR></table></html>
 
 
 
 
 
 
 
''Yu.V. Prokhorov''
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
====Comments====
 
 
 
 
 
 
 
 
 
 
 
====References====
 
<html><table><TR><TD valign="top">[a1]</TD>
 
<TD valign="top">
 
&#160;E. Sverdrup,&#160;
 
&#160;"Laws and chance variations"
 
, <b>1</b>
 
, North-Holland
 
&#160;(1967)
 
&#160;pp. 214ff</TD></TR></table></html>
 
 
 
 
 
This text originally appeared in Encyclopaedia of Mathematics
 
                      - ISBN 1402006098
 

Latest revision as of 09:51, 29 March 2016

  • navigation
    • mainpage|mainpage-description
    • Special:Allpages|Pages A-Z
    •  :Category:Statprob|StatProb Collection
    • recentchanges-url|recentchanges
    • currentevents-url|currentevents
    • randompage-url|randompage
    • Help:Contents|help
    • Talk:EoM:This_project|Project talk
  • SEARCH
  • TOOLBOX
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Sidebar. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sidebar&oldid=3006