Minimax decision rule

A minimax rule minimizes the worst-case risk ${\sup }_{\theta \in \Theta }R(\theta ,\delta )$.

An equalizer rule has the same risk at every parameter value. A Bayes equalizer rule is minimax when its constant risk bounds from below the worst-case risk of every competing rule.

For two distinct nonsingular Gaussian class distributions, vary the class-zero prior continuously. The class-zero and class-one error probabilities of the corresponding Bayes classifier cross. At a crossing they are equal, making the classifier an equalizer rule, a minimax decision rule, and the associated prior a least favorable prior.

If a rule has constant risk $r$ and a sequence of priors has Bayes risks tending to $r$, then the rule is minimax. Every competing rule has worst-case risk at least every one of those Bayes risks.

For independent $X_i\sim N(\theta ,1)$ with $\theta \ge 0$, the sample mean has constant squared-error risk $1/n$ and is minimax. Smooth priors supported on $[0,L]$ whose prior Fisher information is $O(L^{-2})$ give, through the Van Trees inequality, a lower bound tending to $1/n$ for every rule's worst-case risk.