Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 28L / b

The risk of a decision rule is its expected loss at a fixed parameter value:For a prior $\pi$, a Bayes estimator minimizes the integrated riskor equivalently minimizes posterior expected loss for almost every observation. A minimax decision rule minimizesover all decision rules.

Let ${\delta }^{\prime }$ be any decision rule. For every $j$,because $r_j$ is the infimum of the integrated risk under ${\pi }_j$. Taking the lower limit givesThe given rule $\delta$ has constant risk $r$, and hence worst-case risk exactly $r$. It attains this universal lower bound and is therefore minimax. This is the constant-risk limit-of-Bayes-risk criterion.

Given $X=x$, the posterior expected loss from reporting $d$ isThis is a strictly convex quadratic in $d$ whenever the conditional inverse moment is finite and positive. Differentiating with respect to $d$ givesso the Bayes estimator under reciprocal weighted quadratic loss is