A group of $n$ hospitals is to be 'appraised'; the 'performance' $\theta_{i}$ of hospital $i$ has a $N(0,1 / \tau)$ prior distribution, different hospitals being independent. The 'performance' cannot be measured directly, so an expensive firm of management consultants has been hired to arrive at each hospital's Standardised Index of Quality [SIQ], this being a number $X_{i}$ for hospital $i$ related to $\theta_{i}$ by the commercially-sensitive formula

$$X_{i}=\theta_{i}+\varepsilon_{i}$$

where the $\varepsilon_{i}$ are independent with common $N\left(0,1 / \tau_{\varepsilon}\right)$ distribution.

(i) Assume that $\tau$ and $\tau_{\varepsilon}$ are known. What is the posterior distribution of $\theta$ given $X$ ? Suppose that hospital $j$ was the hospital with the lowest SIQ, with a value $X_{j}=x$; conditional on $X$, what is the distribution of $\theta_{j}$ ?

(ii) Now, instead of assuming $\tau$ and $\tau_{\varepsilon}$ known, suppose that $\tau$ has a Gamma prior with parameters $(\alpha, \beta)$, density

$$f(t)=(\beta t)^{\alpha-1} \beta e^{-\beta t} / \Gamma(\alpha)$$

for known $\alpha$ and $\beta$, and that $\tau_{\varepsilon}=\kappa \tau$, where $\kappa$ is a known constant. Find the posterior distribution of $(\theta, \tau)$ given $X$. Comment briefly on the form of the distribution.