# Difference between revisions of "MediaWiki:Sidebar"

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+ | This text originally appeared in Encyclopaedia of Mathematics | ||

+ | - ISBN 1402006098 |

## Revision as of 08:47, 17 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.

#### References

<html>

[1] |
S.N. Bernshtein, "Probability theory" , Moscow-Leningrad (1946) (In Russian) |

</html>

*Yu.V. Prokhorov*

#### Comments

#### References

<html>

[a1] |
E. Sverdrup,
"Laws and chance variations"
, |

</html>

This text originally appeared in Encyclopaedia of Mathematics

- ISBN 1402006098

**How to Cite This Entry:**

Sidebar.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Sidebar&oldid=3004