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 δi is a Bayes rule against some proper prior π = ⊗i=1n πi where πi are proper priors of δi. We will also apply these theorems in both the parametric and non-parametric settings.
- Bayes rules
- Fundamental theorem of decision theory