 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-36/4/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Within the positive component, normal-normal conjugacy gives

p (θ_{i} ∣ y_{i}, +) \propto exp {- \frac{1}{2} (y_{i} - θ_{i})^{2} - \frac{θ _{i}^{2}}{2 V}} .

θ_{i} ∣ y_{i}, + \sim N (\frac{V}{1 + V} y_{i}, \frac{V}{1 + V}), E [θ_{i} ∣ y_{i}, +] = \frac{V}{1 + V} y_{i} .

Here positive denotes membership in the variable-effect component, not the condition

θ_{i} > 0

; its normal prior permits effects of either sign.

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