Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 4 / a / Solution

The common normal distribution for the child-specific random intercepts produces partial pooling. Children with little information borrow strength from the common population, while well-observed children retain more of their individual estimate. For fixed slopes and hyperparameters, let ${\overline{r}}_i$ average the three responses after subtracting their slope terms. Multiplying the normal likelihood by the normal intercept prior distribution giveswith conditional variance $(3/{\sigma }^2+1/{\tau }^2{)}^{-1}$. Thus small between-child variation produces stronger shrinkage towards $\delta$. The common mean and variation are themselves learned from all children in this hierarchical Bayesian model.