A note on the iterative application of Bayes' rule

Alterations in probability densities produced by iterative application of Bayes' rule are analyzed. Computational requirements for finding a sequence of a posteriori densities remain reasonable despite a growing sample size if and only if a sufficient statistic expressible as a vector of fixed dimension exists. The existence of such a sufficient statistic also insures existence of reproducing a priori probability densities (a priori probability densities insuring that the a posteriori densities are in the same family). The theory is applied to find a class of sequential Bayes estimators for a Gaussian covariance matrix, and to treat a variety of adaptive "Bayesian learning" schemes in a unified manner.