 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-44/2/g/solution

(y_{i} + 1) / (1 + ψ) = (y_{i} + 1) ϕ

E [θ_{i} ∣ y] = (y_{i} + 1) \frac{s}{s + n} .

\overline{y} = s / n

and

w = s / (s + n) = \overline{y} / (1 + \overline{y})

yields

E [θ_{i} ∣ y] = w y_{i} + (1 - w) \overline{y} .

Thus partial pooling pulls every local mean toward the overall sample mean. The common weight arises from the fixed shape-one mixing distribution; it is the Poisson–exponential posterior shrinkage formula.

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