Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, with least variance $c_{\min}$ and scaling map $S_{c}$, and let $\gamma_{c}$ be the diagonal Gaussian measure with variances $c$. Borel is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and $M_{2}$ is the second moment. Test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, their gradient maps $\nabla\psi$ and Laplacians $\Delta\psi$, the spaces $L^{2}(\mu;\mathbb{R}^{d})$ with inner product $\langle\cdot,\cdot\rangle_{\mu}$ and norm $\lVert\cdot\rVert_{\mu}$, and the tangent spaces $T_{\mu}$ are those of the setting.

Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The map $S_{c}$ is Borel: its $i$th component $x\mapsto x_{i}/c_{i}$ is $c_{i}^{-1}$ times the coordinate projection $\pi_{i}$, which is measurable by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, so the component is measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; hence $S_{c}$ is measurable with respect to $\mathcal{B}_{d}$ by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\mathcal{B}_{d}=\mathcal{B}(\mathbb{R}^{d})$ by claim 5 there. Consequently $x\mapsto\lVert S_{c}(x)\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied with $u=S_{c}$. Since $0<c_{\min}\le c_{i}$ for every $i\in[d]$, one has $\lVert S_{c}(x)\rVert^{2}=\sum_{i=1}^{d}x_{i}^{2}/c_{i}^{2}\le c_{\min}^{-2}\lVert x\rVert^{2}$ for every $x\in\mathbb{R}^{d}$, so by monotonicity and positive homogeneity of the integral of nonnegative measurable functions (Linearity and Monotonicity of the Lebesgue Integral §nonnegative)

$$\int_{\mathbb{R}^{d}}\lVert S_{c}\rVert^{2}\,d\mu\le c_{\min}^{-2}\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\,\mu(dx)=c_{\min}^{-2}M_{2}(\mu)<\infty,$$

the second moment being finite because $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space). The class of $S_{c}$ therefore lies in $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $S_{c}$.

0. (The Ornstein-Uhlenbeck functional)¶ For $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ the function $\Delta\psi$ is integrable with respect to $\mu$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test. The Ornstein-Uhlenbeck functional of $\mu$ is defined, for $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, by

$$\ell^{c}_{\mu}(\psi)=\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu-\langle S_{c},\nabla\psi\rangle_{\mu},$$

a real number. The map $\psi\mapsto\ell^{c}_{\mu}(\psi)$ is linear: for $\psi,\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $s,t\in\mathbb{R}$, The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §linear gives $\Delta(s\psi+t\phi)=s\,\Delta\psi+t\,\Delta\phi$ and $\nabla(s\psi+t\phi)=s\,\nabla\psi+t\,\nabla\phi$ pointwise, hence also for the classes in $L^{2}(\mu;\mathbb{R}^{d})$; the integral term is then linear by Linearity and Monotonicity of the Lebesgue Integral §integrable, and $\langle S_{c},s\,\nabla\psi+t\,\nabla\phi\rangle_{\mu}=s\,\langle S_{c},\nabla\psi\rangle_{\mu}+t\,\langle S_{c},\nabla\phi\rangle_{\mu}$ by the symmetry and the additivity and homogeneity in the first argument of the inner product $\langle\cdot,\cdot\rangle_{\mu}$ (Real Inner Product Space §inner-product).

1. (Finite relative Fisher information)¶ The measure $\mu$ has finite Fisher information relative to $\gamma_{c}$ if there is a real number $C\ge0$ with $|\ell^{c}_{\mu}(\psi)|\le C\lVert\nabla\psi\rVert_{\mu}$ for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$.

2. (Relative score)¶ If $\mu$ has finite Fisher information relative to $\gamma_{c}$, its relative score with respect to $\gamma_{c}$ is the unique $\zeta^{c}_{\mu}\in T_{\mu}$ with

$$\langle\zeta^{c}_{\mu},\nabla\psi\rangle_{\mu}=-\ell^{c}_{\mu}(\psi)\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}^{d});$$

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 §representation, applied to the linear functional $-\ell^{c}_{\mu}$, which satisfies the bound of clause 1 since $|-\ell^{c}_{\mu}(\psi)|=|\ell^{c}_{\mu}(\psi)|$ by claim 2 of Properties of the Absolute Value in an Ordered Field.

3. (Relative Fisher information)¶ For such $\mu$, the Fisher information of $\mu$ relative to $\gamma_{c}$ is the nonnegative real number $\mathcal{I}(\mu\,|\,\gamma_{c})=\lVert\zeta^{c}_{\mu}\rVert_{\mu}^{2}$.