= Solution
The <risk of a decision rule> is its expected loss at a fixed parameter value:
$$
R(\theta,\delta)
=\mathbb E_\theta\!\left[L(\delta(X),\theta)\right].
$$
For a prior $\pi$, a <Bayes estimator> minimizes the integrated risk
$$
r(\pi,\delta)
=\int_\Theta R(\theta,\delta)\,\pi(d\theta),
$$
or equivalently minimizes posterior expected loss for almost every observation. A <minimax decision rule> minimizes
$$
\sup_{\theta\in\Theta}R(\theta,\delta)
$$
over all decision rules.
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