Consider a decision problem with parameter space $\Theta$. Define the concepts of a Bayes decision rule $\delta_{\pi}$ and of a least favourable prior.

Suppose $\pi$ is a prior distribution on $\Theta$ such that the Bayes risk of the Bayes rule equals $\sup _{\theta \in \Theta} R\left(\delta_{\pi}, \theta\right)$, where $R(\delta, \theta)$ is the risk function associated to the decision problem. Prove that $\delta_{\pi}$ is least favourable.

Now consider a random variable $X$ arising from the binomial distribution $\operatorname{Bin}(n, \theta)$, where $\theta \in \Theta=[0,1]$. Construct a least favourable prior for the squared risk $R(\delta, \theta)=E_{\theta}(\delta(X)-\theta)^{2}$. [You may use without proof the fact that the Bayes rule for quadratic risk is given by the posterior mean.]