Decision rule ($\delta :\mathcal{X}\to \mathcal{A}$)

A measurable choice of an action from observed data, with a randomized rule allowing a conditional distribution of actions. Its performance under a loss function is the risk function $R(\theta ,\delta )={\mathbb{E}}_{\theta }L(\theta ,\delta (X))$. Admissibility compares risks pointwise, Bayes optimality integrates risk under a prior, and minimax optimality minimizes the worst-case risk.