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

The new prior density contributes the factor ${\lambda }_i^{15}{e}^{-16{\lambda }_i}$. Multiplication by the Poisson distribution likelihood givesIts posterior mean displays the resulting shrinkage:The observed risk ratio $y_i/E_i$ is pulled towards one, most strongly for small exposures. The posterior variance is $(y_i+16)/(E_i+16{)}^2$.