Gaussian MAP feature test

Assume the Axiom of Choice. Let $X,H$ be separable real Hilbert spaces, let $K:X\to H$ be bounded and linear, and let $ϵ>0$. Let $U$ be a Borel Gaussian prior with mean $m_0$ and positive self-adjoint, injective, trace-class covariance $C_0$, independent of a jointly Gaussian centered family $Z$ indexed by $H$, linear into $L^2$ with covariance $\mathbb{E}[Z(g)Z(k)]=⟨g,k{⟩}_H$. For parameter $x\in X$, the observation field is $Y_x(g)=⟨Kx,g{⟩}_H+ϵZ(g)$. Fix a feature vector $h\in X$. Let $S=C_0^{1\mathrm{/}2},B=S{K}^{\ast }KS,R=(B+{ϵ}^2{I}_X{)}^{-1},{\mathrm{\Phi }}_h=KSRSh,$ and define the raw threshold $c_h=⟨K{m}_0,{\mathrm{\Phi }}_h{⟩}_H-⟨m_0,h{⟩}_X\mathrm{.}$ Input a white-noise observation field $y\in {\mathbb{R}}^H$. Compute $q=y({\mathrm{\Phi }}_h)-c_h$. Return $1$ when $q>0$ and return $0$ when $q\le 0$. This strict raw comparison defines the Gaussian MAP feature test for every field input $y$.

Gaussian posterior law in an isonormal inverse experiment

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.2 and Theorems 2.3–2.5

Stuart, Inverse problems: A Bayesian perspective, §6.3