Exact scalar MMSE denoiser for a two-point prior

By [F1]–[F2], $\mu =\frac{1}{2}{\delta }_{-a}+\frac{1}{2}{\delta }_a$ is a probability measure: for pairwise-disjoint Borel sets each of the two atoms occurs in at most one set, and $\mu (\mathbb{R})=1$. Since $a>0$, its two atoms are distinct and each has positive mass; every affine hyperplane in $\mathbb{R}$ is a singleton, so $\mu$ is not concentrated on any affine hyperplane. The posterior weights at $a$ and $-a$ are proportional to ${\rho }_{\sigma }(z-a)$ and ${\rho }_{\sigma }(z+a)$, hence $D_{\sigma }(z)=a\frac{{\rho }_{\sigma }(z-a)-{\rho }_{\sigma }(z+a)}{{\rho }_{\sigma }(z-a)+{\rho }_{\sigma }(z+a)}$.

Factoring $\exp (-(z^2+a^2)\mathrm{/}(2{\sigma }^2))$ from numerator and denominator and using [F3] yields $D_{\sigma }(z)=a(\exp (az\mathrm{/}{\sigma }^2)-\exp (-az\mathrm{/}{\sigma }^2))\mathrm{/}(\exp (az\mathrm{/}{\sigma }^2)+\exp (-az\mathrm{/}{\sigma }^2))$, which is the stated hyperbolic-tangent formula.

Fact [F4] applies to this nondegenerate two-point prior, so the posterior mean is the unique squared-error minimizer and the lemma's $h_{\sigma ,\gamma }$ makes $D_{\sigma }(z)$ the unique global minimizer of the displayed proximal objective for every $z$.