Note on a Fundamental Relationship Between Admissible and Bayesian Decision Rules

Abstract

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 is a Bayes rule against some proper prior where π _i are proper priors of δ _i We will also apply these theorems in both the parametric and non-parametric settings.

Keywords:

• Admissibility
• Bayes rules
• fundamental theorem of decision theory