Consider the following model: $X_1,...,X_n$ are iid $N(\theta,1)$ and $\theta \sim N(0, \tau^2)$ for some known $\tau^2$. Determine the Bayes estimator of $\theta^2$ under the squared error loss.
I know the posterior distribution of $\theta$ given the observed sample $X_1,...,X_n$ is $N(\frac{\tau^2}{\frac 1n + \tau^2} \bar{X}, (\frac{1}{\tau^2} +n)^{-1})$.
So if the question asks me the Bayes estimator of $\theta$ under the squared error loss, I know that is $\frac{\tau^2}{\frac 1n + \tau^2} \bar{X}$. But now the question is about the Bayes estimator of $\theta^2$ .