Consider this Bayesian model:

\[ \begin{aligned} \theta&\sim\text{IG}(a_0,\,b_0)\\ X_{1:n}\mid\theta&\overset{\text{iid}}{\sim}\text{N}(0,\,\theta). \end{aligned} \]

1. What is the posterior distribution for \(\theta\) conditional on the data?
2. Derive the posterior mean and show that it is a convex combination of the prior mean and the MLE.