# 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.

#### References

<html>
 [1] S.N. Bernshtein,   "Probability theory" , Moscow-Leningrad  (1946)  (In Russian)
</html>

Yu.V. Prokhorov

#### References

<html>
 [a1] E. Sverdrup,   "Laws and chance variations" , 1 , North-Holland  (1967)  pp. 214ff
</html>

This text originally appeared in Encyclopaedia of Mathematics

                      - ISBN 1402006098

How to Cite This Entry:
Sidebar. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sidebar&oldid=3040