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&#160;S.N. Bernshtein,&#160;
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&#160;"Probability theory"
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, Moscow-Leningrad
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&#160;(1946)
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&#160;(In Russian)</TD></TR></table></html>
  
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''Yu.V. Prokhorov''
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Revision as of 10:30, 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

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How to Cite This Entry:
Sidebar. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sidebar&oldid=3003