Bayes estimator under squared error loss

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

For independent $N(0,v)$ observations, the quadratic-loss risk of the displayed estimator is $v^2[n(n+2){\alpha }^2-2n\alpha +1]$. Within homogeneous rules its uniform minimum occurs at $\alpha =1/(n+2)$; within affine rules a nonzero optimal intercept improves the integrated risk of a homogeneous rule unless $\alpha =1/n$, when the prior moments are finite. More generally a proper prior cannot produce an exactly homogeneous posterior mean: writing its marginal kernel $m(s)$ for $s=|X{|}^2$, the identity $E[v\mid s]=\alpha s$ forces $m(s)=C{s}^{-1-1/(2\alpha )}$, whose induced density for $s$ is a pure power and is not integrable at both zero and infinity. Finite posterior quadratic losses, or finite Bayes risk for the integrated-risk argument, exclude vacuous infinite-loss minimization.

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, $p{c}^2/(1+c^2)$.