 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/28l/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

δ^{'}

be any decision rule. For every

j

sup_{θ \in Θ} R (θ, δ^{'}) \geq \int_{Θ} R (θ, δ^{'}) π_{j} (d θ) \geq r_{j},

because

r_{j}

is the infimum of the integrated risk under

π_{j}

. Taking the lower limit gives

sup_{θ} R (θ, δ^{'}) \geq lim_{j \to \infty} r_{j} = r .

The given rule

δ

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.

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