ABSTRACT
This chapter deals with one of the two basic technical tools required in implementing the Empirical Bayes (EB) approach. The first is based on an explicit estimation of the unknown prior distribution. The second is based on a method of expressing the Bayes estimate or decision rule in terms of functionals of prior distribution G and estimating the Bayes rule itself directly. In the EB context such a restriction is perhaps somewhat unrealistic, but it deserves consideration because the experimenter may have good reason for faith in a certain type of prior distribution. Finite mixtures arise in problems of deciding between a finite number of alternative hypotheses. They are also important as probability models to describe some heterogeneous populations which can be regarded as being composed of a finite number of more homogeneous subpopulations. Finite mixing distributions occur naturally in problems involving a finite number of simple hypotheses.
