The Multivariate van Trees Inequality

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $l\ge1$ be a natural number, let $(Y,\mathcal{G})$ be a measurable space, and let $\mu$ be a $\sigma$-finite measure on it. Let $\mathcal{B}_l$ and $\lambda_l$ be the $l$-fold product $\sigma$-algebra and product of Lebesgue measure on $\mathbb{R}^{l}$ from Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, and let $\lambda_l\otimes\mu$ be the product measure on the product $\sigma$-algebra $\mathcal{B}_l\otimes\mathcal{G}$ ($\lambda_l$ being $\sigma$-finite by claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$).

Let $\Theta_1,\dots,\Theta_l$ be square-integrable random variables on $(\Omega,\mathcal{F},P)$, write $\Theta=(\Theta_1,\dots,\Theta_l):\Omega\to\mathbb{R}^{l}$, and let $D:\Omega\to Y$ be measurable with respect to $\mathcal{F}$ and $\mathcal{G}$. The pair $(\Theta,D):\Omega\to\mathbb{R}^{l}\times Y$ is then measurable with respect to $\mathcal{F}$ and $\mathcal{B}_l\otimes\mathcal{G}$, by the generator criterion of Measurable Function and Real-Valued Measurable Function: the preimage of a set $A_1\times\dots\times A_l\times G$ is the intersection of the events $\{\Theta_1\in A_1\},\dots,\{\Theta_l\in A_l\}$ and $D^{-1}(G)$, and such sets generate $\mathcal{B}_l\otimes\mathcal{G}$ by claim 2 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$. Write $Q$ for the image measure of $P$ under $(\Theta,D)$ on $\mathcal{B}_l\otimes\mathcal{G}$, the joint law of $(\Theta,D)$. Assume:

(i) Density. There is a measurable $p:\mathbb{R}^{l}\times Y\to[0,\infty)$, with respect to $\mathcal{B}_l\otimes\mathcal{G}$ and the Borel $\sigma$-algebra, such that $Q$ is the measure with density $p$ with respect to $\lambda_l\otimes\mu$.

(ii) Smoothness. For every $y\in Y$ the function $\theta\mapsto p(\theta,y)$ is strictly positive on $\mathbb{R}^{l}$ and is a $C^1$ map on the open set $\mathbb{R}^{l}$; its $i$th partial derivative at $\theta$ is denoted $\partial_i p(\theta,y)$.

(iii) Integrability. For each $i\in\{1,\dots,l\}$ the function $(\theta,y)\mapsto\partial_i p(\theta,y)$ is measurable with respect to $\mathcal{B}_l\otimes\mathcal{G}$ and, with the integral of nonnegative measurable functions and the absolute value,

(the integrand is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, applied to the continuous map $(u_0,u_1,\dots,u_l)\mapsto(1+\sum_{j}|u_j|)\,|u_0|$ composed with the measurable map with components $\partial_ip$ and the coordinate maps $(\theta,y)\mapsto\theta_j$, whose preimages are Borel rectangles crossed with $Y$).

(iv) Score. For each $i$, the score

is a square-integrable random variable (it is a random variable: $\partial_i p/p$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, applied to the continuous map $(u,v)\mapsto u/v$ on $\mathbb{R}\times(0,\infty)$ composed with $(\partial_ip,p)$, and preimages compose), and the van Trees information matrix $J$ — the $l\times l$ matrix with entries $J_{ij}=\mathbb{E}[S_iS_j]$, finite since products of square-integrable random variables are integrable by Square-Integrable Random Variables and the Mean-Square Inner Product, with $\mathbb{E}$ the expectation, and symmetric, as $J_{ij}=J_{ji}$ — is positive definite. ($J$ is the information matrix of the joint density $p$ in the $\theta$-variable, aggregating prior and observation information; it is not the Fisher information matrix of the conditional model at a fixed parameter value.)

Then for all measurable $m_1,\dots,m_l:Y\to\mathbb{R}$ (with respect to $\mathcal{G}$ and the Borel $\sigma$-algebra) such that every $m_j\circ D$ is square-integrable, the error matrix $R$ with entries

where $m_j(D)=m_j\circ D$, is a well-defined symmetric matrix; the inverse $J^{-1}$ exists and is symmetric positive definite by Invertibility of Symmetric Positive Definite Matrices; and, in the semidefinite order,