A posteriori distribution

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A conditional probability distribution of a random variable, to be contrasted with its unconditional or a priori distribution.

Let 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 , according to the Bayes formula, has the density

If is a sufficient statistic for the family of distributions with densities , 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 , is "almost independent" of the a priori distribution of .

For the role played by a posteriori distributions in the theory of statistical decisions, see Bayesian approach.


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



[a1] E. Sverdrup, "Laws and chance variations" , 1 , North-Holland (1967) pp. 214ff
How to Cite This Entry:
A posteriori distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article