= Proper-prior obstruction to homogeneous variance Bayes rules
{title2=$\delta(X)=\alpha\sum_iX_i^2$}
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 $m(s)=Cs^{-1-1/(2\alpha)}$, 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.
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