Extended Bayes rule
= Extended Bayes rule
{c}
A decision rule $\delta$ is extended Bayes if, for every $\varepsilon>0$, there is a proper prior $\pi$ with $r(\pi,\delta)\le r^*(\pi)+\varepsilon$, where $r^*(\pi)$ 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 $r^*(\pi)\ge c-\varepsilon$, so every rule has maximum risk at least $c-\varepsilon$. Letting $\varepsilon\to0$ proves that this equalizer rule is <minimax>.