 Extended Bayes rule

A decision rule $\delta$ is extended Bayes if, for every $\epsilon >0$, there is a proper prior $\pi$ with $r(\pi ,\delta )\le r^{\ast }(\pi )+\epsilon$, where $r^{\ast }(\pi )$ is the infimum Bayes risk. Equivalently a sequence of priors makes the rule's excess Bayes risk tend to zero. If its risk function is also the finite constant $c$, then $r^{\ast }(\pi )\ge c-\epsilon$, so every rule has maximum risk at least $c-\epsilon$. Letting $\epsilon \to 0$ proves that this equalizer rule is minimax.