Gaussian posterior law in an isonormal inverse experiment

Set $S=C_0^{1\mathrm{/}2}$. The positive trace-class operator $C_0$ is compact by [F4], so $S$ is compact and self-adjoint. Then $B=S{K}^{\ast }KS$ is compact, self-adjoint, and positive, because a composition containing a compact operator is compact. By [F5], applied to $KS$, $B+{ϵ}^2{I}_X$ is bijective and $R$ is bounded. Its inverse is self-adjoint: writing $x=(B+{ϵ}^{2}I)u$ and $z=(B+{ϵ}^{2}I)v$ transfers symmetry of $B+{ϵ}^{2}I$ to $R$. It is positive because $⟨Rz,z⟩=⟨v,(B+{ϵ}^{2}I)v⟩\ge {ϵ}^2\mathrm{\parallel }v{\mathrm{\parallel }}^2$ when $z=(B+{ϵ}^{2}I)v$. The inverse identities give $BR=RB=I-{ϵ}^{2}R$. For $z\in X$, put $v=Rz$, so $z=(B+{ϵ}^{2}I)v$. Expanding this equation and using self-adjointness of $B$ gives $\mathrm{\parallel }z{\mathrm{\parallel }}^2-{ϵ}^2⟨Rz,z⟩=\mathrm{\parallel }Bv{\mathrm{\parallel }}^2+{ϵ}^2⟨Bv,v⟩\ge 0$. Thus $0\le {ϵ}^{2}R\le I_X$, so $BR=I-{ϵ}^{2}R$ is positive and $I-BR={ϵ}^{2}R\ge 0$. Define $D=SBRS$; then $D$ is positive, $C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}={ϵ}^{2}SRS$ is positive, and $D+C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}=S(BR+{ϵ}^{2}R)S=C_0$. The products are trace class by [F8] because $S$ is Hilbert–Schmidt and the intervening operators are bounded. Consequently $0\le D\le C_0$ and $\mathrm{tr}D\le \mathrm{tr}{C}_0$.

For $g\in H$, write $Y_0(g)=Y(g)-⟨K{m}_0,g{⟩}_H$. The signal and noise are independent, so covariance expansion gives $\mathbb{E}[Y_0(g)Y_0(k)]=⟨K{C}_0{K}^{\ast }g,k{⟩}_H+{ϵ}^2⟨g,k{⟩}_H$. For $h\in X$, $Y_0({\mathrm{\Phi }}_h)=y({\mathrm{\Phi }}_h)-⟨K{m}_0,{\mathrm{\Phi }}_h{⟩}_H$ when the observed field is $y$. Using $S{K}^{\ast }{\mathrm{\Phi }}_h=BRSh$ and ${\mathrm{\Phi }}_h=KSRSh$, the covariance identity yields $\mathbb{E}[Y_0({\mathrm{\Phi }}_h)Y_0({\mathrm{\Phi }}_k)]=⟨SBRSh,k{⟩}_X=⟨Dh,k{⟩}_X$. Likewise $\mathbb{E}[⟨U-m_0,h{⟩}_X{Y}_0({\mathrm{\Phi }}_k)]=⟨Dh,k{⟩}_X$. These equalities follow by expanding the two covariance terms, using $BR=RB$ and $(B+{ϵ}^{2}I)R=I$.

For the supplied finite or countable basis $(v_j)$ of $X$, [F8] and step 1.1 give $\mathbb{E}{\sum }_j\mathrm{\mid }{Y}_0({\mathrm{\Phi }}_{v_j}){\mathrm{\mid }}^2={\sum }_j⟨D{v}_j,v_j{⟩}_X=\mathrm{tr}D\le \mathrm{tr}{C}_0<\mathrm{\infty }$. Therefore the nonnegative sum is finite almost surely: if it were infinite on a set of positive probability, its expectation would exceed every finite bound. Parseval gives ${\sum }_j\mathrm{\mid }⟨m_0,v_j{⟩}_X{\mathrm{\mid }}^2=\mathrm{\parallel }{m}_0{\mathrm{\parallel }}_X^2<\mathrm{\infty }$, and $\mathrm{\mid }a+b{\mathrm{\mid }}^2\le 2\mathrm{\mid }a{\mathrm{\mid }}^2+2\mathrm{\mid }b{\mathrm{\mid }}^2$ shows ${\sum }_j{b}_j(Y{)}^2<\mathrm{\infty }$ on that probability-one set. Hence the first branch of the Statement's definition applies and $M_Y=m_0+{\sum }_j{Y}_0({\mathrm{\Phi }}_{v_j})v_j$ there. The same argument includes finite and empty bases. The series converges in $X$ by [F7]. Its partial sums are measurable; their Cauchy set is a countable intersection and union of norm-comparison events, hence measurable, and completeness of $X$ gives a limit there. For each nonempty closed set $F\subseteq X$, distance to $F$ is continuous, so the distance of the limit to $F$ is the limit of the measurable distances of the partial sums; [F11] makes this limit measurable. Because each coordinate map $y\mapsto y({\mathrm{\Phi }}_{v_j})$ is measurable for the cylinder sigma-algebra, the convergence construction and the default value $m_0$ show that $y\mapsto M_y$ is measurable from $({\mathbb{R}}^H,{\mathcal{C}}_H)$ to $X$ with its Borel sigma-algebra. For each fixed $h$, expand $h$ in the supplied basis. The corresponding finite linear combinations of $Y_0({\mathrm{\Phi }}_{v_j})$ converge in $L^2$ to $Y_0({\mathrm{\Phi }}_h)$, since [F7] makes the basis projections converge to $h$ and [step 2.1] bounds the squared error by $\mathrm{\parallel }D\mathrm{\parallel }$ times the squared projection error. The same finite combinations converge almost surely to the inner product of the coordinate-series limit with $h$. Uniqueness of limits in probability gives $⟨M_Y-m_0,h{⟩}_X=Y_0({\mathrm{\Phi }}_h)$ almost surely for this fixed $h$. This proves the displayed posterior-mean formula, with an exceptional set allowed to depend on $h$.

For fixed $h\in X$ and $g\in H$, covariance expansion gives $\mathrm{Cov}(⟨U-m_0,h{⟩}_X,Y_0(g))=⟨K{C}_{0}h,g{⟩}_H$. The covariance of $Y_0({\mathrm{\Phi }}_h)$ with $Y_0(g)$ is $⟨K{C}_0{K}^{\ast }{\mathrm{\Phi }}_h+{ϵ}^2{\mathrm{\Phi }}_h,g{⟩}_H$. Since $C_0=S^2$, $S{K}^{\ast }KS=B$, and $(B+{ϵ}^{2}I)R=I$, one has $K{C}_0{K}^{\ast }{\mathrm{\Phi }}_h+{ϵ}^2{\mathrm{\Phi }}_h=KSBRSh+{ϵ}^{2}KSRSh=KS(B+{ϵ}^{2}I)RSh=KSSh=K{C}_{0}h$. By step 3.1, $⟨M_Y-m_0,h⟩=Y_0({\mathrm{\Phi }}_h)$ almost surely, so the residual $W=U-M_Y$ has zero covariance with every observation $Y(g)$. Both $⟨U-m_0,h⟩$ and $Y_0({\mathrm{\Phi }}_h)$ are centered, so the same identity shows $\mathbb{E}⟨W,h⟩=0$ for each $h$; thus $W$ is centered. Its covariance is $C_0-D=C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$ by the two covariance identities in step 2.1 and the covariance of $U$.

For each fixed basis vector $v_j$, step 3.1 identifies $⟨M_Y-m_0,v_j⟩$ almost surely with $Y_0({\mathrm{\Phi }}_{v_j})$. For any finite list of basis vectors, intersect the corresponding probability-one events; the associated coordinates of $W$ are then linear images of finite Gaussian projections of $U-m_0$ and $Z$. The same holds jointly with any finite list of observations. The underlying finite $U-m_0$ and $Z$ blocks are independent by [F12], so their joint linear images are Gaussian by [F3] and [F6]. Step 4.1 makes the residual-coordinate and observation blocks have zero cross-covariance, so [F6] makes those blocks independent. The rational finite linear combinations of the supplied basis $(v_j)$ form a countable dense subset of $X$. For each center in this set, Parseval expresses the squared distance from $W$ as the limit of finite sums of squared basis-coordinate differences, so open balls with rational radii about these centers are measurable in the basis-coordinate sigma-algebra; they form a countable base, proving that these coordinates generate the Borel sigma-algebra of $W$. For a fixed finite list of observations, finite basis-coordinate cylinder rectangles and observation rectangles have the product of their marginal probabilities by the independence just proved. These rectangles form a generating pi-system for the product of the residual Borel sigma-algebra and the finite observation Borel sigma-algebra, so [F9] extends the product identity and shows that $W$ is independent of this finite observation block. Now fix any $A\in \mathcal{B}(X)$. The finite measures $C\mapsto \mathbb{P}(W\in A,C)$ and $C\mapsto \mathbb{P}(W\in A)\mathbb{P}(C)$ agree on the pi-system of finite observation cylinder events, including the whole space. By [F9] they agree on $\sigma (Y(g):g\in H)$. Since this holds for every $A$, $W$ is independent of the full field. This argument includes finite and empty supplied bases and requires no inclusion in the ordinary completion of a basis-observation sigma-algebra.

The map $y\mapsto M_y$ is measurable by step 3.1. Let $\nu$ be the Borel law of the residual random element $W$, whose mean is zero and covariance is $C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$ by step 4.1. Define ${\mathrm{\Pi }}_y(A)={\int }_X{\mathbf{1}}_A(M_y+w)\nu (dw)$ for $A\in \mathcal{B}(X)$. The map $(y,w)\mapsto M_y+w$ is measurable, and [F10] makes $y\mapsto {\mathrm{\Pi }}_y(A)$ measurable; for each $y$, translation of $\nu$ is a probability measure. Since $W$ is independent of the full data field by step 5.1 and $U=M_Y+W$, integrating first with respect to the fixed law $\nu$ gives $\mathbb{P}(U\in A,Y\in C)={\int }_C{\mathrm{\Pi }}_y(A)P_Y(dy)$ for every Borel $A\subseteq X$ and $C\in {\mathcal{C}}_H$. This is the regular conditional distribution identity from [F2].

For every finite list $h_1,\dots ,h_q\in X$, step 3.1 gives almost surely, simultaneously for this finite list, $⟨W,h_a⟩=⟨U-m_0,h_a⟩-Y_0({\mathrm{\Phi }}_{h_a})$. The vector of these residual functionals is therefore a linear image of finite Gaussian projections of $U-m_0$ and $Z$, whose blocks are independent by [F12] and jointly Gaussian by [F3] and [F6]; hence it is jointly Gaussian. Step 4.1 gives that $W$ is centered with covariance $C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$, so $W$ is a centered Borel Gaussian random element with that covariance. Translating by $M_y$ shows that every section ${\mathrm{\Pi }}_y$ is Gaussian with mean $M_y$ and covariance $C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$, including the zero-dimensional and finite-dimensional cases. For fixed $h$, step 3.1 identifies the posterior mean with $⟨m_0,h⟩+y({\mathrm{\Phi }}_h)-⟨K{m}_0,{\mathrm{\Phi }}_h⟩$ almost surely, and the scalar marginal under a translated Gaussian is normal with variance $⟨C_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}h,h⟩$. This proves all asserted posterior-law and feature-marginal formulas; the exceptional set remains feature-dependent.