The Score of a Measure with a Positive Continuously Differentiable Density

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, let $m:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$ (in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii); $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space Is Open in Itself) with $m(x)>0$ for every $x\in\mathbb{R}^{d}$, with gradient map $\nabla m:x\mapsto Dm(x)$. The function $m$ and its partial derivatives $\partial_{i}m$ are continuous (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii)), hence Borel; so is the map $\eta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ given by $\eta(x)=m(x)^{-1}\,\nabla m(x)$, whose components $m^{-1}\partial_{i}m$ are continuous by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space and claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space; and so is $\lVert\eta\rVert^{2}$, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ be the measure with density $m$ with respect to $\lambda_{d}$, that is,

$$\mu(B)=\int_{\mathbb{R}^{d}}\mathbf{1}_{B}\,m\,d\lambda_{d}\qquad(B\in\mathcal{B}(\mathbb{R}^{d})),$$

and assume that $\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu<\infty$. Let $T_{\mu}$ be the tangent space at $\mu$, a closed linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (i), with the orthogonal projection $P_{T_{\mu}}$ onto it, and let finite Fisher information, the score $\xi_{\mu}$ and the Fisher information $\mathcal{I}(\mu)$ be as defined there.

1. (Integration by parts against the density)(i) The class of $\eta$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$, and

$$\langle\eta,\nabla\psi\rangle_{\mu}=-\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}^{d}).$$

2. (Finite Fisher information and the score as a projection)(ii) The measure $\mu$ has finite Fisher information, $\xi_{\mu}=P_{T_{\mu}}\eta$, and

$$\mathcal{I}(\mu)\le\lVert\eta\rVert_{\mu}^{2}=\int_{\mathbb{R}^{d}}m^{-1}\lVert\nabla m\rVert^{2}\,d\lambda_{d}.$$

3. (Equality)(iii) If $\eta\in T_{\mu}$, then $\xi_{\mu}=\eta$ and $\mathcal{I}(\mu)=\lVert\eta\rVert_{\mu}^{2}$.