The Relative Score on a Hilbert Space: the Gaussian Measure Has Score Zero, and the Score as a Square-Integrable Field in the Weighted Sequence 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, let $c$ be a variance sequence and $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$; $\mathcal{P}_{2}(X)$ is the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space; the relative score with respect to $\gamma_{c}$ and its components are those of that definition, with $L^{2}(\mu)$ as there; weight sequences, finite Fisher information relative to $\gamma_{c}$ with weights $a$ and its value $\mathcal{I}_{a}(\cdot\,|\,\gamma_{c})$ are those of Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information.

1. (The Gaussian measure)¶ $\gamma_{c}\in\mathcal{P}_{2}(X)$, and $\gamma_{c}$ has a relative score with respect to $\gamma_{c}$, every component of which is the zero vector of $L^{2}(\gamma_{c})$. Consequently, for every weight sequence $a$, $\gamma_{c}$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mathcal{I}_{a}(\gamma_{c}\,|\,\gamma_{c})=0$.

2. (The score field)¶ Let $a$ be a weight sequence, let $X_{a}$ be the weighted sequence space with its standard orthonormal basis $(f^{a}_{k})_{k\in\mathbb{N}}$, a real Hilbert space by Weighted Square-Summable Sequence Spaces are Real Hilbert Spaces: the Standard Orthonormal Basis and the Embedding of a Hilbert Space under Bounded Weights §hilbert, and let $\mu\in\mathcal{P}_{2}(X)$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. Let $L^{2}(\mu;X_{a})$ be the space of square-integrable maps from the measure space $(X,\mathcal{B}(X),\mu)$ to $X_{a}$ with the basis $(f^{a}_{k})_{k\in\mathbb{N}}$, with the norm $\lVert\cdot\rVert_{L^{2}(\mu;X_{a})}$ of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations and the coordinates $Z_{k}\in L^{2}(\mu)$ of its elements $Z$ along $f^{a}_{k}$ given by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates. Then there is exactly one $Z\in L^{2}(\mu;X_{a})$ with $Z_{k}=a_{k}^{1/2}\,\zeta_{k}$ for every $k\in\mathbb{N}$, where $a_{k}^{1/2}$ is the power of $a_{k}$ with exponent $1/2$, and

$$\lVert Z\rVert_{L^{2}(\mu;X_{a})}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c}).$$