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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

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.




 S.N. Bernshtein,   "Probability theory" , Moscow-Leningrad  (1946)

 (In Russian)


Yu.V. Prokhorov





 E. Sverdrup,   "Laws and chance variations" , 1 , North-Holland  (1967)

 pp. 214ff


This text originally appeared in Encyclopaedia of Mathematics

                      - ISBN 1402006098
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