Consider the hierarchical Bayesian model with $\mu\sim Uniform(-\infty,\infty)$ prior. This is an improper prior, but go through the calculations setting the density to 1. Derive the Bayes estimator of $\mu+\alpha_i$ under squared error loss.

$$\begin{split}X_{ij}|\mu,a_i\sim N(\mu +\alpha_i, \sigma^2), j=1,...,n,i=1,...,s\\ \alpha_i\sim N(0, \sigma_A^2),i=1,...,s\end{split}$$

I begin by calculating the posterior for a single observation $$\xi(\mu|x)=f(x|\mu)\underbrace{\xi(\mu)}_{=1}\\ \propto e^{-\frac 1 2\left(\frac{x-(\mu+\alpha_i)}{\sigma}\right)^2}$$

Please suggest how to deal with $\alpha_i$, which has its own distribution. So there is a normal random variable nested inside another normal random variable, and I don't know how to deal with it. Thank you.


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