Theorem: Gaussian four-center Bayes reference

For $s>0$,

For $s=0$, $Q^{\ast }(X)=1$. At $m=0,s>0$, the common class law gives $Q^{\ast }=1/4$.

For the supplied ideal probe laws,

Proof. Equal-prior Gaussian likelihood comparison assigns each coordinate its observed sign. For each coordinate, the probability of the correct sign is ${\Phi }_{\mathrm{G}}(m/s)$. Coordinate signs and noises are independent, so the joint correctness probability is the square. The degenerate cases are immediate.