# MediaWiki:Sidebar

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.

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**How to Cite This Entry:**

Sidebar.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Sidebar&oldid=2999