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

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 coordinate maps $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and push-forwards as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, let $c$ be a variance sequence with truncations $c^{(n)}$, and $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$. The relative score with respect to $\gamma_{c}$ is that of that definition; weight sequences, finite Fisher information relative to $\gamma_{c}$ with weights $a$ and $\mathcal{I}_{a}(\cdot\,|\,\gamma_{c})$ are those of Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information. 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 $L^{2}(\mu)$ be as in The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space, with its closed linear subspaces and the orthogonal projections onto them. For $n\in\mathbb{N}$ let $\mu_{n}=(p_{n})_{\#}\mu$, which lies in $\mathcal{P}_{2}(\mathbb{R}^{n})$ by claim 1; let $L^{2}(\mu_{n})=L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\mu_{n})$ be the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert; and let $V_{n}$ be the set of the classes in $L^{2}(\mu)$ of the functions $g\circ p_{n}$ with $g:\mathbb{R}^{n}\to\mathbb{R}$ Borel and $\int_{X}(g\circ p_{n})^{2}\,d\mu<\infty$. The items Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information, The Weighted Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space and Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions are read, in the settings named there, with $n$ in place of the dimension written $d$ there and with the variance vector $c^{(n)}$. Finite Fisher information relative to $\gamma_{c^{(n)}}$ is that of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §finite; when $\mu_{n}$ has it, $\zeta^{(n)}_{k}\in L^{2}(\mu_{n})$ ($k\in[n]$) are the components of its relative score, as in Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions §agreement, and, with $a^{(n)}=(a_{1},\dots,a_{n})$ for a weight sequence $a$, we write $\mathcal{I}^{(n)}_{a}(\mu_{n})=\mathcal{I}_{a^{(n)}}(\mu_{n}\,|\,\gamma_{c^{(n)}})$, the weighted Fisher information of $\mu_{n}$ relative to $\gamma_{c^{(n)}}$ with weights $a^{(n)}$.

1. (Projected measures)¶ For every $n\in\mathbb{N}$, $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ in the sense of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space.

2. (Cutoff isometry)¶ Let $n\in\mathbb{N}$ and let $g:\mathbb{R}^{n}\to\mathbb{R}$ be Borel. Then $g\circ p_{n}$ is $2$-integrable with respect to $\mu$ if and only if $g$ is $2$-integrable with respect to $\mu_{n}$, and in that case the $L^{2}(\mu)$-norm of the class of $g\circ p_{n}$ equals the $L^{2}(\mu_{n})$-norm of the class of $g$, and the former class depends only on the latter. For $h\in L^{2}(\mu_{n})$, $h\circ p_{n}\in V_{n}$ denotes the class of $g\circ p_{n}$ for any representative $g$ of $h$.

3. (Cutoff subspaces)¶ For every $n\in\mathbb{N}$, $V_{n}$ is a closed linear subspace of $L^{2}(\mu)$, and $V_{n}\subseteq V_{n+1}$.

4. (Projections of the score)¶ Let $\mu$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$, and let $n\in\mathbb{N}$. Then $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$, and for every $k\in[n]$ the orthogonal projection of $\zeta_{k}$ onto $V_{n}$ is $\zeta^{(n)}_{k}\circ p_{n}$.

5. (Monotonicity)¶ Let $a$ be a weight sequence and let $\mu$ have a relative score with respect to $\gamma_{c}$, so that $\mathcal{I}^{(n)}_{a}(\mu_{n})$ is defined for every $n\in\mathbb{N}$ by claims 1 and 4. Then the sequence $\bigl(\mathcal{I}^{(n)}_{a}(\mu_{n})\bigr)_{n\in\mathbb{N}}$ is nondecreasing.

6. (Convergence)¶ Let $a$ be a weight sequence and let $\mu$ have a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. Then $\mathcal{I}^{(n)}_{a}(\mu_{n})\le\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ for every $n\in\mathbb{N}$, and $\mathcal{I}^{(n)}_{a}(\mu_{n})\to\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ as $n\to\infty$.

7. (Converse)¶ Let $a$ be a weight sequence. Suppose that for every $n\in\mathbb{N}$ the measure $\mu_{n}$ has finite Fisher information relative to $\gamma_{c^{(n)}}$, and that there is $C\in\mathbb{R}$ with $\mathcal{I}^{(n)}_{a}(\mu_{n})\le C$ for every $n\in\mathbb{N}$. Then $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le C$.