Proof of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity

Each result cited is universally quantified over the data in its own statement. Throughout, $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ is the probability space of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures, random vectors on it are the Borel maps $\mathbb{R}^{d}\to\mathbb{R}^{d}$, and the results Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form are applied both on $(\Omega,\mathcal{F},P)$ and on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$, the expectation on the latter being the integral against $\mu$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §probability-space). Classes and the operations on them are those of The Space of Square-Integrable Random Vectors §classes, and the inner product of two classes is $\mathbb{E}[X\cdot Y]$ for any representatives, by The Space of Square-Integrable Random Vectors §inner-product.

Step 1: claim 1. Let $\xi\in L^{2}(\mu;\mathbb{R}^{d})$ and fix representatives, again written $\xi$ (a Borel map with $\int\lVert\xi\rVert^{2}\,d\mu<\infty$) and $X$ (a random vector with $\mathcal{L}(X)=\mu$). By Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition, $\xi\circ X$ is a random vector in $\mathbb{R}^{d}$. The random variable $\lVert\xi\circ X\rVert^{2}$ is $\varphi\circ X$ for the nonnegative Borel function $\varphi=\lVert\xi\rVert^{2}$, the composition of the Borel map $\xi$ with the Borel map $y\mapsto\lVert y\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), so Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation gives $\mathbb{E}[\lVert\xi\circ X\rVert^{2}]=\int\lVert\xi\rVert^{2}\,d\mu<\infty$: $\xi\circ X$ is square-integrable.

Independence of the representatives. Let $\xi'$ be another representative of the class of $\xi$, so that $\mu(\{\xi=\xi'\})=1$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §almost-sure (the set $E=\{x:\xi(x)=\xi'(x)\}$ is Borel by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §almost-sure applied on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$), and let $X'$ be another representative of $X$, so that $P(X=X')=1$ and $\mathcal{L}(X')=\mathcal{L}(X)=\mu$ by The Space of Square-Integrable Random Vectors §law; the first paragraph therefore applies to the pair $(\xi',X')$ as well, so $\xi'\circ X'$ is a square-integrable random vector. The event $\{X\in E\}$ has probability $\mathcal{L}(X)(E)=\mu(E)=1$ by Random Vector and Its Law §law. On $\{X=X'\}\cap\{X\in E\}$ we have $\xi(X(\omega))=\xi'(X(\omega))=\xi'(X'(\omega))$, so $\{X=X'\}\cap\{X\in E\}\subseteq\{\xi\circ X=\xi'\circ X'\}$. Put $A=\{X=X'\}$ and $B=\{X\in E\}$. By claim 3 of Basic Properties of a Measure ($P$ is finite), $P(\Omega\setminus A)=1-1=0$ and $P(\Omega\setminus B)=0$; the event $\Omega\setminus(A\cap B)=(\Omega\setminus A)\cup(\Omega\setminus B)$ then has probability at most $0+0=0$ by claim 4 there (applied to the sequence whose first two terms are these two events and whose remaining terms are empty), so $P(A\cap B)=1-0=1$ by claim 3 again, and $1=P(A\cap B)\le P(\xi\circ X=\xi'\circ X')\le1$ by claim 2 there (the set $\{\xi\circ X=\xi'\circ X'\}$ is an event by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §almost-sure); hence $P(\xi\circ X=\xi'\circ X')=1$. Hence the two compositions have the same class (Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §almost-sure).

Linearity and inner products. For $\xi,\eta\in L^{2}(\mu;\mathbb{R}^{d})$ and $a,b\in\mathbb{R}$, $(a\xi+b\eta)\circ X=a(\xi\circ X)+b(\eta\circ X)$ pointwise on $\Omega$, since sums and scalar multiples of maps are formed pointwise; passing to classes, the map $\xi\mapsto\xi\circ X$ is linear. The random variable $(\xi\circ X)\cdot(\eta\circ X)$ equals $\varphi\circ X$ with $\varphi=\xi\cdot\eta$, a Borel function (Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$) that is $\mu$-integrable by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations; so by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation, $\langle\xi\circ X,\eta\circ X\rangle_{L^{2}}=\mathbb{E}[(\xi\cdot\eta)\circ X]=\int\xi\cdot\eta\,d\mu=\langle\xi,\eta\rangle_{\mu}$. Taking $\eta=\xi$ and using $\lVert Z\rVert_{L^{2}}=\sqrt{\langle Z,Z\rangle_{L^{2}}}$ (The Space of Square-Integrable Random Vectors §inner-product with Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations) on both spaces gives $\lVert\xi\circ X\rVert_{L^{2}}=\lVert\xi\rVert_{\mu}$. Finally $\mathrm{id}\circ X=X$ pointwise.

Step 2: claim 2. By Step 1, $\lVert\xi_{\mu}\circ X\rVert_{L^{2}}^{2}=\lVert\xi_{\mu}\rVert_{\mu}^{2}=\mathcal{I}(\mu)$ (Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §information), and for $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, $\langle\xi_{\mu}\circ X,\nabla\psi\circ X\rangle_{L^{2}}=\langle\xi_{\mu},\nabla\psi\rangle_{\mu}=-\int\Delta\psi\,d\mu$ by Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score. Since $\Delta\psi$ is Borel and $\mu$-integrable (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §laplacian), Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation gives $\int\Delta\psi\,d\mu=\mathbb{E}[\Delta\psi\circ X]$.

Step 3: claim 3. Let $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, and let $\psi_{n}$ be as in Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (applied with $\nu$ in place of $\mu$); each $\psi_{n}$ is a test function, as recorded there. By Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score applied to $\psi_{n}$, the sequence $a_{n}=\langle\xi_{\nu},\nabla\psi_{n}\rangle_{\nu}=-\int\Delta\psi_{n}\,d\nu$ converges to $-d$ 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 §limits and claim 3 of Arithmetic of Limits of Real Sequences; and since $\nabla\psi_{n}\to\mathrm{id}$ in $L^{2}(\nu;\mathbb{R}^{d})$ 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 §identity, the same sequence converges to $\langle\xi_{\nu},\mathrm{id}\rangle_{\nu}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, applied with the constant sequence $\xi_{\nu}$ in the first argument and $\nabla\psi_{n}\to\mathrm{id}$ in the second. By claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, $\langle\xi_{\nu},\mathrm{id}\rangle_{\nu}=-d$. If $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, this applies to $\nu=\mu$, and by Step 1, $\langle\xi_{\mu}\circ X,X\rangle_{L^{2}}=\langle\xi_{\mu}\circ X,\mathrm{id}\circ X\rangle_{L^{2}}=\langle\xi_{\mu},\mathrm{id}\rangle_{\mu}=-d$.