= 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> gives
$$
\mathbb E(\alpha_i\mid\text{others},y)=w\overline r_i+(1-w)\delta,\qquad
w=\frac{3\tau^2}{\sigma^2+3\tau^2},
$$
with 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>.
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