Necessary Restrictions for Distributions A Posteriori

SUMMARY The combining by the method of Bayes of two samples each yielding a fiducial distribution, does not often yield a unique result. Necessary restrictions are given which it seems reasonable to impose on a posteriori distributions. If, in combining two samples having sufficient statistics following the same continuous distribution, the a posteriori distributions obtained by Bayes’s method satisfy these conditions, they are identical with the fiducial distribution based on the combined sample. The same result seems to hold for samples having sufficient statistics following two different continuous distributions. The paper is confined to single parameter distributions in which the range is independent of the parameter.