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)
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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