Bayes estimator under squared error loss

Under squared error loss, the Bayes estimator is the posterior mean.

For $X\mid \theta \sim N_p(\theta ,I_p)$ and $\theta \sim N_p(0,c^2{I}_p)$, the posterior isThe Bayes estimator under quadratic loss is the posterior mean and its Bayes risk is the trace of the posterior covariance, $p{c}^2/(1+c^2)$.