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 Incoming links: Posterior expected loss

Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 19H / ii / Solution Created 2026-09-24 Updated 2026-09-24

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Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 19H / i / Solution Created 2026-09-24 Updated 2026-09-24

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Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 27I / Solution Created 2026-09-24 Updated 2026-09-24

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 28L / b / iii / Solution Created 2026-09-24 Updated 2026-09-25

Given

X = x

, the posterior expected loss from reporting

d

is

E [θ^{- 1} (θ - d)^{2} ∣ X = x] = E (θ ∣ X = x) - 2 d + d^{2} E (θ^{- 1} ∣ X = x) .

This is a strictly convex quadratic in

d

whenever the conditional inverse moment is finite and positive. Differentiating with respect to

d

gives

- 2 + 2 d E (θ^{- 1} ∣ X = x) = 0,

so the Bayes estimator under reciprocal weighted quadratic loss is

δ_{π} (x) = (E (θ^{- 1} ∣ X = x))^{- 1} .

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