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 under suitable conditions any admissible rule δ = ⊗_i=1ⁿ δ_i is a Bayes rule against some proper prior π = ⊗_i=1ⁿ π_i where π_i are proper priors of δ_i. We will also apply these theorems in both the parametric and non-parametric settings.