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 §derivatives; $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) 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 §derivatives), 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 §pairs. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ be the measure with density $m$ with respect to $\lambda_{d}$, that is,

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 §closed, 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)¶ The class of $\eta$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$, and

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

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