# Difference between revisions of "MediaWiki:Sidebar"

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</tr></table></html> | </tr></table></html> | ||

If | If | ||

− | <html><img align="absmiddle" border="0" src=" | + | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003010.png"></html> |

is a | is a | ||

[[Sufficient statistic|sufficient statistic]] | [[Sufficient statistic|sufficient statistic]] | ||

for the family of distributions with densities | for the family of distributions with densities | ||

− | <html><img align="absmiddle" border="0" src=" | + | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003011.png">, |

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

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itself, but on | itself, but on | ||

− | <img align="absmiddle" border="0" src=" | + | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003013.png">. |

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

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as | as | ||

− | <img align="absmiddle" border="0" src=" | + | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003015.png">, |

where | where | ||

− | <img align="absmiddle" border="0" src=" | + | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003016.png"> |

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

− | <img align="absmiddle" border="0" src=" | + | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003017.png">,</html> |

is | is | ||

 "almost independent"  |  "almost independent"  | ||

of the a priori distribution of | of the a priori distribution of | ||

− | <html><img align="absmiddle" border="0" src=" | + | <html><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003018.png"></html>. |

## Revision as of 16:38, 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=2999