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A conditional probability distribution of a random
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variable, to be contrasted with its unconditional or
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[[A priori distribution|a priori distribution]].
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<html><img align="absmiddle" border="0" src="">
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be a random parameter with an a priori density
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<img align="absmiddle" border="0" src="">,
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be a random result of observations and let
<img align="absmiddle" border="0" src="">
be the conditional density of
<img align="absmiddle" border="0" src="">
<img align="absmiddle" border="0" src="">;
then the a posteriori distribution of
<img align="absmiddle" border="0" src="">
for a given
<img align="absmiddle" border="0" src=""></html>,
according to the
[[Bayes formula|Bayes formula]],
has the density
<table class="eq" style="width:100%;">
<tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src=""></td>
<html><img align="absmiddle" border="0" src=""></html>
is a
[[Sufficient statistic|sufficient statistic]]
for the family of distributions with densities
<html><img align="absmiddle" border="0" src="">,
then the a posteriori distribution depends not on
<img align="absmiddle" border="0" src="">
itself, but on
<img align="absmiddle" border="0" src="">.
The asymptotic behaviour of the a posteriori distribution
<img align="absmiddle" border="0" src="">
<img align="absmiddle" border="0" src="">,
<img align="absmiddle" border="0" src="">
are the results of independent observations with density
<img align="absmiddle" border="0" src="">,</html>
&#160;"almost independent"&#160;
of the a priori distribution of
<html><img align="absmiddle" border="0" src=""></html>.
For the role played by a posteriori distributions
in the theory of statistical decisions, see
[[Bayesian approach|Bayesian approach]].
<html><table><TR><TD valign="top">[1]</TD>
<TD valign="top">
&#160;S.N. Bernshtein,&#160;
&#160;"Probability theory"
, Moscow-Leningrad
&#160;(In Russian)</TD></TR></table></html>
''Yu.V. Prokhorov''
<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;pp. 214ff</TD></TR></table></html>
This text originally appeared in Encyclopaedia of Mathematics
                      - ISBN 1402006098

Latest revision as of 17:59, 28 February 2021

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    • mainpage|mainpage-description
    • Special:Allpages|Pages A-Z
    • :Category:Statprob|StatProb Collection
    • recentchanges-url|recentchanges
    • currentevents-url|currentevents
    • randompage-url|randompage
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