Solution
= Solution
The MLE is $\hat\mu=\bar X$ with MSE $1/n$. Combining likelihood and the $N(0,\tau^{-2})$ prior gives posterior $N(n\bar X/(n+\tau^2),1/(n+\tau^2))$. The frequentist MSE of its mean is $[n+\tau^4\mu^2]/(n+\tau^2)^2$, which is below $1/n$ exactly when $\mu^2\lt 2\tau^{-2}+1/n$.