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| − | A conditional probability distribution of a random
| + | * navigation |
| − | variable, to be contrasted with its unconditional or
| + | ** mainpage|mainpage-description |
| − | [[A priori distribution|a priori distribution]].
| + | ** Special:Allpages|Pages A-Z |
| − | | + | ** :Category:Statprob|StatProb Collection |
| − | Let
| + | ** recentchanges-url|recentchanges |
| − | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100301.png">
| + | ** currentevents-url|currentevents |
| − | be a random parameter with an a priori density
| + | ** randompage-url|randompage |
| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100302.png">,
| + | ** Help:Contents|help |
| − | let
| + | ** Talk:EoM:This_project|Project talk |
| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100303.png">
| + | ** Special:RequestAccount|Request account |
| − | be a random result of observations and let
| + | * SEARCH |
| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100304.png">
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| − | be the conditional density of
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100305.png">
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| − | when
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100306.png">;
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| − | then the a posteriori distribution of
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100307.png">
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| − | for a given
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100308.png"></html>,
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| − | according to the
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| − | [[Bayes formula|Bayes formula]],
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| − | has the density
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| − | <table class="eq" style="width:100%;">
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| − | <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>
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| − | </tr></table>
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| − | If
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| − | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003010.png"></html>
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| − | is a
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| − | [[Sufficient statistic|sufficient statistic]]
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| − | for the family of distributions with densities
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| − | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003011.png">,
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| − | then the a posteriori distribution depends not on
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003012.png">
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| − | itself, but on
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003013.png">.
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| − | The asymptotic behaviour of the a posteriori distribution
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003014.png">
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| − | as
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003015.png">,
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| − | where
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003016.png">
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| − | are the results of independent observations with density
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003017.png">,</html>
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| − | is
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| − |  "almost independent" 
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| − | of the a priori distribution of
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| − | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003018.png"></html>.
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| − | For the role played by a posteriori distributions
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| − | in the theory of statistical decisions, see
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| − | [[Bayesian approach|Bayesian approach]].
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| − | | |
| − | ====References====
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| − | <html><table><TR><TD valign="top">[1]</TD>
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| − | <TD valign="top">
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| − |  S.N. Bernshtein, 
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| − |  "Probability theory"
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| − | , Moscow-Leningrad
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| − |  (1946)
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| − |  (In Russian)</TD></TR></table></html>
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| − | ''Yu.V. Prokhorov''
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| − | ====Comments====
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| − | ====References====
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| − | <html><table><TR><TD valign="top">[a1]</TD>
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| − | <TD valign="top">
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| − |  E. Sverdrup, 
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| − |  "Laws and chance variations"
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| − | , <b>1</b>
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| − | , North-Holland
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| − |  (1967)
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| − |  pp. 214ff</TD></TR></table></html>
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| − | This text originally appeared in Encyclopaedia of Mathematics
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| − | - ISBN 1402006098
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