We are considering a sample of size $n$ from an exponential distribution, with parameter $w >0$. We wish to produce an estimate for $d$, for $w$ , with loss function: $L(w, d)=w(w-d)^2$
The prior distribution for $w$ is an inverse-gamma distribution, with hyperparameters $\alpha> 0$ and $\beta>0$
Find the Baye's decision and Baye's risk for an immediate decision
The Baye's risk is defined as $p^*(P)={inf}$ $ p(P, d)$
Where:
$ d\in D $, the decision space
$ p(P, d) $ is the risk of decision $d$ given prior $P$
$ p(P, d) =E(L(w, d))$, the expected value of our loss
So we our given our loss function as: $L(w, d)=w(w-d)^2$
How could we use these facts to calculate Baye's risk?
Bayes decision is a decision $d^*$ for which $ p(P, d^*) =p^*(P)$
Thank you in advance