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

The gamma full conditional distribution has mean $(y_i+1)/(1+\psi )=(y_i+1)\varphi$. The tower property and the beta distribution mean giveWriting $\overline{y}=s/n$ and $w=s/(s+n)=\overline{y}/(1+\overline{y})$ yieldsThus 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.