What is meant by a convex decision problem? State and prove a theorem to the effect that, in a convex decision problem, there is no point in randomising. [You may use standard terms without defining them.]

The sample space, parameter space and action space are each the two-point set $\{1,2\}$. The observable $X$ takes value 1 with probability $2 / 3$ when the parameter $\Theta=1$, and with probability $3 / 4$ when $\Theta=2$. The loss function $L(\theta, a)$ is 0 if $a=\theta$, otherwise 1 . Describe all the non-randomised decision rules, compute their risk functions, and plot these as points in the unit square. Identify an inadmissible non-randomised decision rule, and a decision rule that dominates it.

Show that the minimax rule has risk function $(8 / 17,8 / 17)$, and is Bayes against a prior distribution that you should specify. What is its Bayes risk? Would a Bayesian with this prior distribution be bound to use the minimax rule?