Derive the shrinkage weight

Topic: Hierarchical Bayes · more questions on this topic
‹ Pooling Exchangeability is a scientific claim ›

Starting from \overline Y_j\mid\theta_j\sim\operatorname{N}(\theta_j,\sigma^2/n_j) and \theta_j\mid\mu,\tau^2\sim\operatorname{N}(\mu,\tau^2), complete the square and derive the posterior mean as w_j\overline y_j+(1-w_j)\mu. Check its limits as n_j, \tau^2, and \sigma^2 vary.

This exercise has no worked answer.