Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V$ be an admissible cylindrical potential with head dimension $d$, with noise gradient $\nabla_{a}V$ and functions $\partial_{k}V$, let $\beta\in\mathbb{R}$ be positive, and let $\gamma^{V}_{\beta}$ be the Gibbs measure of $V$ at temperature $\beta$, with the weight $w_{V,\beta}$ of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §weight and the normaliser $Z_{V,\beta}$ of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser. $\log:(0,\infty)\to\mathbb{R}$ is the natural logarithm, the one used in Relative Entropy of Probability Measures §relative-entropy through The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm. $\mathcal{P}_{2}(X)$ 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. Finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy. The relative score with respect to $\gamma_{c}$, finite Fisher information relative to $\gamma_{c}$ with weights $a$ and $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ are those of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian; the relative score with respect to $\gamma^{V}_{\beta}$ and its components are those of The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score; finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ and $\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})$ are those of The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information. For $\mu\in\mathcal{P}(X)$, $L^{2}(\mu;X^{a})$ is the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields with norm $\lVert\cdot\rVert_{\mu}$; the coordinate of an element of it along $f_{k}$ is that of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates; and for $\mu$ with a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, $Z^{a}_{\mu}$ is its noise score field.

1. (Change of measure)¶ The measures $\gamma^{V}_{\beta}$ and $\gamma_{c}$ have the same null sets, $w_{V,\beta}$ being positive and bounded by The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §weight and $Z_{V,\beta}$ positive by The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser. For every Borel $f:X\to[0,\infty)$,

$$\int_{X}f\,d\gamma^{V}_{\beta}=\frac{1}{Z_{V,\beta}}\int_{X}f\,w_{V,\beta}\,d\gamma_{c},$$

and a Borel $f:X\to\mathbb{R}$ is integrable with respect to $\gamma^{V}_{\beta}$ if and only if $f\,w_{V,\beta}$ is integrable with respect to $\gamma_{c}$, the displayed formula then holding.

2. (Entropy)¶ Let $\mu\in\mathcal{P}(X)$. Then $\mu$ has finite relative entropy with respect to $\gamma^{V}_{\beta}$ if and only if $\mu$ has finite relative entropy with respect to $\gamma_{c}$ and $V$ is integrable with respect to $\mu$; in that case

$$H(\mu\,|\,\gamma^{V}_{\beta})=H(\mu\,|\,\gamma_{c})+\frac{1}{\beta}\int_{X}V\,d\mu+\log Z_{V,\beta}.$$

3. (Domain)¶ If $\mu$ has finite relative entropy with respect to $\gamma^{V}_{\beta}$, then $\mu\in\mathcal{P}_{2}(X)$ and $|\nabla_{a}V|_{a}$ and each $\partial_{k}V$ are integrable with respect to $\mu$; in particular The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential applies to $\mu$.

4. (Splitting of the score)¶ Let $\mu\in\mathcal{P}_{2}(X)$ be such that $V$ is integrable with respect to $\mu$, so that $|\nabla_{a}V|_{a}$ and each $\partial_{k}V$ are integrable with respect to $\mu$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable and The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential applies to $\mu$. Then $\mu$ has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ if and only if $\mu$ has a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$, finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. In that case the components of the relative score with respect to $\gamma^{V}_{\beta}$ are $\zeta^{V}_{k}=\zeta_{k}+\beta^{-1}\partial_{k}V$ for $k\in\mathbb{N}$, the class of $\partial_{k}V$ lying in $L^{2}(\mu)$.

5. (The splitting field)¶ Let $\mu\in\mathcal{P}_{2}(X)$ be such that $V$ is integrable with respect to $\mu$, and let $\mu$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$, finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$, so that, by claim 4, $\mu$ has a relative score $(\zeta^{V}_{k})_{k\in\mathbb{N}}$ with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$. Then the element $\beta Z^{a}_{\mu}+\nabla_{a}V$ of $L^{2}(\mu;X^{a})$ has coordinate $\beta\,a_{k}^{1/2}\zeta^{V}_{k}$ along $f_{k}$ for every $k\in\mathbb{N}$, and

$$\lVert\beta Z^{a}_{\mu}+\nabla_{a}V\rVert_{\mu}^{2}=\beta^{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta}).$$

6. (Fisher information bound)¶ There is $C_{*}\in\mathbb{R}$, depending only on $V$, $\beta$, $a$ and $c$, such that every $\mu\in\mathcal{P}_{2}(X)$ such that $V$ is integrable with respect to $\mu$ and that has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ (so that, by claim 4, the left-hand side is defined and finite) satisfies

$$\beta^{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma_{c})+\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu\ \le\ C_{*}\Bigl(1+\beta^{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})+\int_{X}\sum_{k=1}^{d}x_{k}^{2}\,\mu(dx)\Bigr).$$