A random sample $X_{1}, \ldots, X_{n}$ is taken from a normal distribution having unknown mean $\theta$ and variance 1. Find the maximum likelihood estimate $\hat{\theta}_{M}$ for $\theta$ based on $X_{1}, \ldots, X_{n}$.

Suppose that we now take a Bayesian point of view and regard $\theta$ itself as a normal random variable of known mean $\mu$ and variance $\tau^{-1}$. Find the Bayes' estimate $\hat{\theta}_{B}$ for $\theta$ based on $X_{1}, \ldots, X_{n}$, corresponding to the quadratic loss function $(\theta-a)^{2}$.