(i) In the context of decision theory, what is a Bayes rule with respect to a given loss function and prior? What is an extended Bayes rule?

Characterise the Bayes rule with respect to a given prior in terms of the posterior distribution for the parameter given the observation. When $\Theta=\mathcal{A}=\mathbb{R}^{d}$ for some $d$, and the loss function is $L(\theta, a)=\|\theta-a\|^{2}$, what is the Bayes rule?

(ii) Suppose that $\mathcal{A}=\Theta=\mathbb{R}$, with loss function $L(\theta, a)=(\theta-a)^{2}$ and suppose further that under $P_{\theta}, X \sim N(\theta, 1)$.

Supposing that a $N\left(0, \tau^{-1}\right)$ prior is taken over $\theta$, compute the Bayes risk of the decision rule $d_{\lambda}(X)=\lambda X$. Find the posterior distribution of $\theta$ given $X$, and confirm that its mean is of the form $d_{\lambda}(X)$ for some value of $\lambda$ which you should identify. Hence show that the decision rule $d_{1}$ is an extended Bayes rule.