Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 44 / 2 / g

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- Solution g

Solution

The gamma full conditional distribution has mean

(y_{i} + 1) / (1 + ψ) = (y_{i} + 1) ϕ

. The tower property and the beta distribution mean give

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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