(i) A public health official is seeking a rational policy of vaccination against a relatively mild ailment which causes absence from work. Surveys suggest that $60 \%$ of the population are already immune, but accurate tests to detect vulnerability in any individual are too costly for mass screening. A simple skin test has been developed, but is not completely reliable. A person who is immune to the ailment will have a negligible reaction to the skin test with probability $0.4$, a moderate reaction with probability $0.5$ and a strong reaction with probability 0.1. For a person who is vulnerable to the ailment the corresponding probabilities are $0.1,0.4$ and $0.5$. It is estimated that the money-equivalent of workhours lost from failing to vaccinate a vulnerable person is 20 , that the unnecessary cost of vaccinating an immune person is 8 , and that there is no cost associated with vaccinating a vulnerable person or failing to vaccinate an immune person. On the basis of the skin test, it must be decided whether to vaccinate or not. What is the Bayes decision rule that the health official should adopt?

(ii) A collection of $I$ students each sit $J$ exams. The ability of the $i$ th student is represented by $\theta_{i}$ and the performance of the $i$ th student on the $j$ th exam is measured by $X_{i j}$. Assume that, given $\boldsymbol{\theta}=\left(\theta_{1}, \ldots, \theta_{I}\right)$, an appropriate model is that the variables $\left\{X_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right\}$ are independent, and

$$X_{i j} \sim N\left(\theta_{i}, \tau^{-1}\right),$$

for a known positive constant $\tau$. It is reasonable to assume, a priori, that the $\theta_{i}$ are independent with

$$\theta_{i} \sim N\left(\mu, \zeta^{-1}\right),$$

where $\mu$ and $\zeta$ are population parameters, known from experience with previous cohorts of students.

Compute the posterior distribution of $\boldsymbol{\theta}$ given the observed exam marks vector $\mathbf{X}=\left\{X_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right\} .$

Suppose now that $\tau$ is also unknown, but assumed to have a $\operatorname{Gamma}\left(\alpha_{0}, \beta_{0}\right)$ distribution, for known $\alpha_{0}, \beta_{0}$. Compute the posterior distribution of $\tau$ given $\boldsymbol{\theta}$ and $\mathbf{X}$ Find, up to a normalisation constant, the form of the marginal density of $\boldsymbol{\theta}$ given $\mathbf{X}$.