Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 2 / f

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- Solution f

Solution

The new prior density contributes the factor

λ_{i}^{15} e^{- 16 λ_{i}}

. Multiplication by the Poisson distribution likelihood gives

λ_{i} ∣ y_{i} \sim Gamma (y_{i} + 16, E_{i} + 16) .

Its posterior mean displays the resulting shrinkage:

E (λ_{i} ∣ y_{i}) = \frac{y _{i} + 16}{E _{i} + 16} = \frac{E _{i}}{E _{i} + 16} \frac{y _{i}}{E _{i}} + \frac{16}{E _{i} + 16} \cdot 1.

The observed risk ratio

y_{i} / E_{i}

is pulled towards one, most strongly for small exposures. The posterior variance is

(y_{i} + 16) / (E_{i} + 16)^{2}

.

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