The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space

In the settings of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, with the coordinates $x_{k}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $c$ be a variance sequence and $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$. $\mathcal{F}C^{1}_{b}(X)$ is the set of bounded $C^{1}$ cylindrical functions and $\partial_{k}\varphi$ are their partial derivatives, which exist by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §gradient. Let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, and let $L^{2}(\mu)=L^{2}(X,\mathcal{B}(X),\mu)$ be the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with the inner product $\langle\cdot,\cdot\rangle_{L^{2}(\mu)}$ and norm $\lVert\cdot\rVert_{L^{2}(\mu)}$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, the classes of $\varphi$ and $\partial_{k}\varphi$ belong to $L^{2}(\mu)$ and are again written $\varphi$ and $\partial_{k}\varphi$, and $\partial_{k}\varphi$ is integrable with respect to $\mu$, by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable; the function $x\mapsto x_{k}\varphi(x)$ is integrable with respect to $\mu$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §coordinate-integrable; hence, $c_{k}$ being positive by Variance Sequences and Their Truncations §variances, the function $x\mapsto x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)$ is integrable with respect to $\mu$ by Linearity and Monotonicity of the Lebesgue Integral §integrable.

(Relative score)¶ The measure $\mu$ has a relative score with respect to $\gamma_{c}$ if for every $k\in\mathbb{N}$ there is $\zeta_{k}\in L^{2}(\mu)$ with

$$\langle\zeta_{k},\varphi\rangle_{L^{2}(\mu)}=\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\,\varphi(x)-\partial_{k}\varphi(x)\Bigr)\,\mu(dx)\qquad\text{for every }\varphi\in\mathcal{F}C^{1}_{b}(X).$$

For each $k$ there is at most one such $\zeta_{k}$: the difference $g$ of two of them satisfies $\langle g,\varphi\rangle_{L^{2}(\mu)}=0$ for every $\varphi\in\mathcal{F}C^{1}_{b}(X)$, by linearity of the inner product in its first argument, and so is the zero vector by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density, so that the two coincide. The relative score of $\mu$ with respect to $\gamma_{c}$ is then the sequence $\zeta^{c}_{\mu}=(\zeta_{k})_{k\in\mathbb{N}}$, and $\zeta_{k}$ is its $k$-th component.