Finite Fisher Information, the Score and the Fisher Information of a Probability Measure

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, with $L^{2}(\mu;\mathbb{R}^{d})$, test functions, gradients and Laplacians and tangent space $T_{\mu}$ as fixed there. The symbol $\mathcal{I}$ always denotes the Fisher information, never the quadratic cost $I$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (iv).

1. (Finite Fisher information)(i) The measure $\mu$ has finite Fisher information if there is a real number $C\ge0$ such that

$$\Bigl|\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu\Bigr|\le C\,\lVert\nabla\psi\rVert_{\mu}\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}^{d}).$$

The set of all $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with finite Fisher information is denoted $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$.

2. (Score)(ii) Let $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$. The score of $\mu$ is the unique $\xi_{\mu}\in T_{\mu}$ such that

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

It exists and is unique 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 (iv), applied to $\ell(\psi)=-\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu$: this $\ell$ is linear in $\psi$ by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure (iii) and the linearity of the integral (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion), and, for any $C$ as in clause 1, $|\ell(\psi)|=\bigl|\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu\bigr|\le C\lVert\nabla\psi\rVert_{\mu}$ by claim 2 of Properties of the Absolute Value in an Ordered Field.

3. (Fisher information)(iii) For $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, the Fisher information of $\mu$ is the nonnegative real number

$$\mathcal{I}(\mu)=\lVert\xi_{\mu}\rVert_{\mu}^{2}.$$