# A posteriori distribution

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.

#### References

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

#### Comments

#### References

[a1] | E. Sverdrup, "Laws and chance variations" , 1 , North-Holland (1967) pp. 214ff |

**How to Cite This Entry:**

A posteriori distribution. Yu.V. Prokhorov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=A_posteriori_distribution&oldid=11777