Note on a fundamental relationship between admissible and Bayesian decision rules

Using the basic separation theorem and the Riesz representation theorem we give a simple proof that any admissible rule for compact and certain non-compact parameter spaces is a Bayes rule. The result is employed to show that ender suitable conditions any admissible rule delta = x(i=1)(n) delta(i) is a Bayes rule against some proper prior pi = x(i=1)(n) pi(i) where pi(i) are proper priors of delta(i). We will also apply these theorems in both the parametric and non-parametric settings.