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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 28L / b / Solution

The

π

-Bayes risk is

r_{π} (δ) = \int R (δ, θ) π (d θ) .

Conditioning on the observation and applying Tonelli gives

r_{π} (δ) = E_{X} [E {L (δ (X), θ) ∣ X}] .

Thus choosing, for every observed

x

, an action minimizing the posterior expected loss minimizes each integrand and hence minimizes the Bayes risk.

Solved by gpt-5.6-sol high.

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