# Difference between revisions of "MediaWiki:Sidebar"

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[[Bayes formula|Bayes formula]], | [[Bayes formula|Bayes formula]], | ||

has the density | has the density | ||

− | |||

<html><table class="eq" style="width:100%;"> | <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><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> | </tr></table></html> | ||

+ | If | ||

+ | <html><img align="absmiddle" border="0" src="images/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="images/a010/a010030/a01003011.png">, | ||

+ | then the a posteriori distribution depends not on | ||

+ | <img align="absmiddle" border="0" src="images/a010/a010030/a01003012.png"> | ||

+ | itself, but on | ||

+ | <img align="absmiddle" border="0" src="images/a010/a010030/a01003013.png">. | ||

+ | The asymptotic behaviour of the a posteriori distribution | ||

+ | <img align="absmiddle" border="0" src="images/a010/a010030/a01003014.png"> | ||

+ | as | ||

+ | <img align="absmiddle" border="0" src="images/a010/a010030/a01003015.png">, | ||

+ | where | ||

+ | <img align="absmiddle" border="0" src="images/a010/a010030/a01003016.png"> | ||

+ | are the results of independent observations with density | ||

+ | <img align="absmiddle" border="0" src="images/a010/a010030/a01003017.png">,</html> | ||

+ | is | ||

+ |  "almost independent"  | ||

+ | of the a priori distribution of | ||

+ | <html><img align="absmiddle" border="0" src="images/a010/a010030/a01003018.png"></html>. | ||

+ | |||

+ | For the role played by a posteriori distributions | ||

+ | in the theory of statistical decisions, see | ||

+ | [[Bayesian approach|Bayesian approach]]. | ||

todo | todo |

## Revision as of 16:37, 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

<html>

</html>

If <html></html> is a sufficient statistic for the family of distributions with densities <html>, then the a posteriori distribution depends not on itself, but on . The asymptotic behaviour of the a posteriori distribution as , where are the results of independent observations with density ,</html> is "almost independent" of the a priori distribution of <html></html>.

For the role played by a posteriori distributions
in the theory of statistical decisions, see
Bayesian approach.

todo

**How to Cite This Entry:**

Sidebar.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Sidebar&oldid=2998