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A conditional probability distribution of a random
variable, to be contrasted with its unconditional or
[[A priori distribution|a priori distribution]].

Let
<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
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100302.png">,
let
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100303.png">
be a random result of observations and let
<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="images/a010/a010030/a0100309.png"></td>
</tr></table></html>


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