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Let
 
Let
−
<html><img align="absmiddle" border="0" src="images/a010/a010030/a0100301.png">
+
<html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100301.png">
 
be a random parameter with an a priori density
 
be a random parameter with an a priori density
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100302.png">,
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100302.png">,
 
let
 
let
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100303.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100303.png">
 
be a random result of observations and let
 
be a random result of observations and let
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100304.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100304.png">
 
be the conditional density of
 
be the conditional density of
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100305.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100305.png">
 
when
 
when
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100306.png">;
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100306.png">;
 
then the a posteriori distribution of
 
then the a posteriori distribution of
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100307.png">
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100307.png">
 
for a given
 
for a given
−
<img align="absmiddle" border="0" src="images/a010/a010030/a0100308.png"></html>,
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100308.png"></html>,
 
according to the
 
according to the
 
[[Bayes formula|Bayes formula]],
 
[[Bayes formula|Bayes formula]],

Revision as of 16:28, 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


todo

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Sidebar. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sidebar&oldid=2994
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