Proof of The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency

Each result cited is universally quantified over the data in its own statement. The notation of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force. The space $L^{2}(\mu;X^{a})$ is that of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, that is, the space of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis on $(X,\mathcal{B}(X),\mu)$ with $E=X^{a}$ and the orthonormal basis $(f_{k})_{k\in\mathbb{N}}$, $f_{k}=a_{k}^{1/2}e_{k}$ (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space); its scalar space $L^{2}(\mu)=L^{2}(X,\mathcal{B}(X),\mu)$ is the space of The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space in which the $\zeta_{k}$ lie. For $v\in L^{2}(\mu;X^{a})$ and $k\in\mathbb{N}$, $v_{k}\in L^{2}(\mu)$ is the coordinate of $v$ along $f_{k}$ (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates), the class of $x\mapsto\langle v(x),f_{k}\rangle_{a}$ for a representative $v$. Every $a_{k}$ is positive (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §weights), and for $y\in X^{a}$ one has $\langle y,f_{k}\rangle_{a}=a_{k}^{-1/2}\langle y,e_{k}\rangle$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis.

Step 1 (Claim 1). Since $\mu$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$, the series $\sum_{k=1}^{\infty}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ converges, with sum $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ (Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information). For each $k$, $a_{k}^{1/2}\zeta_{k}\in L^{2}(\mu)$ and $\lVert a_{k}^{1/2}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ by the homogeneity of the norm, so the series $\sum_{k=1}^{\infty}\lVert a_{k}^{1/2}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ converges, and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis provides exactly one $Z^{a}_{\mu}\in L^{2}(\mu;X^{a})$ with $(Z^{a}_{\mu})_{k}=a_{k}^{1/2}\zeta_{k}$ for every $k\in\mathbb{N}$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates,

$$\lVert Z^{a}_{\mu}\rVert_{\mu}^{2}=\sum_{k=1}^{\infty}\lVert a_{k}^{1/2}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=\sum_{k=1}^{\infty}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c}).$$

This proves claim 1.

Step 2 (Coordinates of a noise gradient). Let $n\in\mathbb{N}$, $g\in C^{1}_{b}(\mathbb{R}^{n})$ and $\varphi=g\circ p_{n}$, a bounded $C^{1}$ cylindrical function with representation $(n,g)$ (Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation). Let $x\in X$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, $\nabla_{a}\varphi(x)\in X^{a}$ and $\langle\nabla_{a}\varphi(x),e_{k}\rangle$ equals $a_{k}\,\partial_{k}\varphi(x)$ for $k\le n$ and $0$ for $k>n$; and $\partial_{k}\varphi=(\partial_{k}g)\circ p_{n}$ for $k\le n$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial. Hence the coordinate of $\nabla_{a}\varphi(x)$ along $f_{k}$ is $a_{k}^{-1/2}a_{k}\,\partial_{k}g(p_{n}(x))=a_{k}^{1/2}\,\partial_{k}g(p_{n}(x))$ for $k\le n$, and $0$ for $k>n$. The map $\nabla_{a}\varphi$ is square-integrable with respect to $\mu$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient) and represents its own class, so $(\nabla_{a}\varphi)_{k}$ is the class of $a_{k}^{1/2}(\partial_{k}g)\circ p_{n}$ for $k\in[n]$ and the zero vector of $L^{2}(\mu)$ for $k>n$.

Step 3 (Claim 2). In the situation of Step 2, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates gives $\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=\sum_{k=1}^{\infty}\langle(Z^{a}_{\mu})_{k},(\nabla_{a}\varphi)_{k}\rangle_{L^{2}(\mu)}$. The terms with $k>n$ vanish, the second argument being the zero vector, so the partial sums (Series of Real Numbers §partial-sums) are constant from the index $n$ on and the sum (Series of Real Numbers §convergent) is the $n$th partial sum. By Steps 1 and 2 and the bilinearity of the inner product of $L^{2}(\mu)$,

$$\langle Z^{a}_{\mu},\nabla_{a}(g\circ p_{n})\rangle_{\mu}=\sum_{k=1}^{n}\bigl\langle a_{k}^{1/2}\zeta_{k},a_{k}^{1/2}(\partial_{k}g)\circ p_{n}\bigr\rangle_{L^{2}(\mu)}=\sum_{k=1}^{n}a_{k}\,\langle\zeta_{k},(\partial_{k}g)\circ p_{n}\rangle_{L^{2}(\mu)} .$$

This proves claim 2.

Step 4 (Claim 3). Let $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$; then $g\in C^{1}_{b}(\mathbb{R}^{n})$, as noted in the statement, so claim 2 applies to $g$. Let $k\in[n]$. As recorded in the preamble of The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates, $\partial_{k}g\in C^{1}_{b}(\mathbb{R}^{n})$, so $(\partial_{k}g)\circ p_{n}$ is a bounded $C^{1}$ cylindrical function with representation $(n,\partial_{k}g)$, and by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial its $k$th partial derivative is $(\partial_{k}\partial_{k}g)\circ p_{n}$. The defining identity of the relative score (The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score), applied with this cylindrical function, gives

$$\langle\zeta_{k},(\partial_{k}g)\circ p_{n}\rangle_{L^{2}(\mu)}=\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\,\partial_{k}g(p_{n}(x))-\partial_{k}\partial_{k}g(p_{n}(x))\Bigr)\mu(dx),$$

the numbers $c_{k}$ being those of the variance sequence $c$ of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian in both places. Multiplying by $a_{k}$ and summing over $k\in[n]$, claim 2 and The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional give $\langle Z^{a}_{\mu},\nabla_{a}(g\circ p_{n})\rangle_{\mu}=L^{a}_{\mu}(g)$. This proves claim 3.

Step 5 (Claim 4). Let $n\in\mathbb{N}$. By The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information §head-score, $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$; write $\tilde{\zeta}=\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}$, an element of $T_{\tilde{\mu}_{n}}\subseteq L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$ (Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §score), and let $\eta=(\eta_{1},\dots,\eta_{n})$ be a representative of $\tilde{\zeta}$, a Borel map with $\int\lVert\eta\rVert^{2}\,d\tilde{\mu}_{n}<\infty$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu). Each $\eta_{k}$ is a Borel representative of the component $\tilde{\zeta}_{k}$ of $\tilde{\zeta}$ (preamble of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions, as fixed in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §euclidean). By Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift, $\Lambda_{n}\eta$ takes values in $X^{a}$, is square-integrable with respect to $\mu$, and represents $\Lambda_{n}\tilde{\zeta}$. For $x\in X$, $\Lambda_{n}\eta(x)=\sum_{j=1}^{n}a_{j}^{1/2}\eta_{j}(r_{n}(x))\,e_{j}$, so by the linearity of the inner product of $X$ and the orthonormality of $(e_{j})$, $\langle\Lambda_{n}\eta(x),e_{k}\rangle$ is $a_{k}^{1/2}\eta_{k}(r_{n}(x))$ for $k\le n$ and $0$ for $k>n$; hence the coordinate of $\Lambda_{n}\eta(x)$ along $f_{k}$ is $\eta_{k}(r_{n}(x))$ for $k\le n$ and $0$ for $k>n$. By The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information §head-score, applied with the Borel representative $\eta_{k}$ of $\tilde{\zeta}_{k}$, $\Pi_{n}\zeta_{k}$ is the class of $a_{k}^{-1/2}(\eta_{k}\circ r_{n})$, so the class of $\eta_{k}\circ r_{n}$ is $a_{k}^{1/2}\Pi_{n}\zeta_{k}$. Therefore $(\Lambda_{n}\tilde{\zeta})_{k}=a_{k}^{1/2}\Pi_{n}\zeta_{k}$ for $k\in[n]$ and $(\Lambda_{n}\tilde{\zeta})_{k}$ is the zero vector for $k>n$, and by the linearity of coordinates (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates) the element $W=Z^{a}_{\mu}-\Lambda_{n}\tilde{\zeta}$ has

$$W_{k}=a_{k}^{1/2}(\zeta_{k}-\Pi_{n}\zeta_{k})\quad(k\in[n]),\qquad W_{k}=a_{k}^{1/2}\zeta_{k}\quad(k>n).$$

The subspace $V_{n}$ is a closed linear subspace of the real Hilbert space $L^{2}(\mu)$ (Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §subspace), so Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras gives $\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=\lVert\Pi_{n}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}+\lVert\zeta_{k}-\Pi_{n}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$. Hence $\lVert W_{k}\rVert_{L^{2}(\mu)}^{2}=a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}-a_{k}\lVert\Pi_{n}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ for $k\in[n]$ and $\lVert W_{k}\rVert_{L^{2}(\mu)}^{2}=a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ for $k>n$. Let $s_{N}=\sum_{k=1}^{N}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ and $t_{N}=\sum_{k=1}^{N}\lVert W_{k}\rVert_{L^{2}(\mu)}^{2}$ be the partial sums (Series of Real Numbers §partial-sums). For $N\ge n$, splitting the finite sums at $n$ and using The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information §head-fisher,

$$t_{N}=s_{N}-\sum_{k=1}^{n}a_{k}\lVert\Pi_{n}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=s_{N}-\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n}).$$

As $N\to\infty$, $s_{N}\to\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ by Step 1 and Series of Real Numbers §convergent, and $t_{N}\to\lVert W\rVert_{\mu}^{2}$ by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates; so

$$\bigl\lVert Z^{a}_{\mu}-\Lambda_{n}\tilde{\zeta}\bigr\rVert_{\mu}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c})-\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n}).$$

By The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information §head-fisher-convergence, $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\to\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ as $n\to\infty$, so the right-hand side tends to $0$. This proves claim 4.

Step 6 (Claim 5). For $n\in\mathbb{N}$, the relative score $\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}$ lies in $T_{\tilde{\mu}_{n}}$ (Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §score), so its lift $v_{n}=\Lambda_{n}\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}$ lies in $T^{a}_{\mu}$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §tangent. By claim 4, the distance $\lVert Z^{a}_{\mu}-v_{n}\rVert_{\mu}$ is the nonnegative square root of a sequence of nonnegative numbers tending to $0$, so it tends to $0$; that is, the sequence $(v_{n})_{n\in\mathbb{N}}$ in $T^{a}_{\mu}$ converges to $Z^{a}_{\mu}$ in the metric space $L^{2}(\mu;X^{a})$ (Real Hilbert Space §topology). By Sequential Characterization of the Closure in a Metric Space, $Z^{a}_{\mu}$ lies in the closure of $T^{a}_{\mu}$. Now $T^{a}_{\mu}$ is the closure of $G^{a}_{\mu}$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent), which is closed by claim 2 of The Closure is the Smallest Closed Superset, and so equals its own closure by claim 4 there. Hence $Z^{a}_{\mu}\in T^{a}_{\mu}$, which proves claim 5.