Consider this Bayesian model:

\[ \begin{aligned} \theta&\sim\text{N}(m_0,\,\tau_0^2)\\ X_{1:n}\mid\theta&\overset{\text{iid}}{\sim}\text{N}(\theta,\,1). \end{aligned} \]

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 \(m_0\) and the MLE.