Define the normal and extensive form solutions of a Bayesian statistical decision problem involving parameter $\Theta$, random variable $X$, and loss function $L(\theta, a)$. How are they related? Let $R_{0}=R_{0}(\Pi)$ be the Bayes loss of the optimal act when $\Theta \sim \Pi$ and no data can be observed. Express the Bayes risk $R_{1}$ of the optimal statistical decision rule in terms of $R_{0}$ and the joint distribution of $(\Theta, X)$.

The real parameter $\Theta$ has distribution $\Pi$, having probability density function $\pi(\cdot)$. Consider the problem of specifying a set $S \subseteq \mathbb{R}$ such that the loss when $\Theta=\theta$ is $L(\theta, S)=c|S|-\mathbf{1}_{S}(\theta)$, where $\mathbf{1}_{S}$ is the indicator function of $S$, where $c>0$, and where $|S|=\int_{S} d x$. Show that the "highest density" region $S^{*}:=\{\theta: \pi(\theta) \geqslant c\}$ supplies a Bayes act for this decision problem, and explain why $R_{0}(\Pi) \leqslant 0$.

For the case $\Theta \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$, find an expression for $R_{0}$ in terms of the standard normal distribution function $\Phi$.

Suppose now that $c=0.5$, that $\Theta \sim \mathcal{N}(0,1)$ and that $X \mid \Theta \sim \mathcal{N}(\Theta, 1 / 9)$. Show that $R_{1}<R_{0}$.