# A posteriori distribution

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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.

How to Cite This Entry:
A posteriori distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A_posteriori_distribution&oldid=11777
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article