Directional Form of the Multivariate van Trees Inequality

Adopt the setting, notation and hypotheses (i)--(iv) of the multivariate van Trees inequality: the probability space $(\Omega,\mathcal{F},P)$, the natural number $l\ge1$, the measurable space $(Y,\mathcal{G})$ with its $\sigma$-finite measure $\mu$, the square-integrable random variables $\Theta_{1},\dots,\Theta_{l}$ with $\Theta=(\Theta_{1},\dots,\Theta_{l})$, the measurable map $D:\Omega\to Y$, and the van Trees information matrix $J$, which is positive definite by hypothesis (iv). Write $\mathbb{E}$ for the expectation, $x\cdot y$ for the dot product on the Euclidean space $\mathbb{R}^{l}$ and $Jz$ for the matrix-vector product. Let $a,z\in\mathbb{R}^{l}$ with $z$ nonzero, so that $z\cdot(Jz)>0$ by claim 1 of Rank-One Lower Bound for the Inverse of a Positive Definite Matrix, and write $a\cdot\Theta=\sum_{j=1}^{l}a_{j}\Theta_{j}$, a square-integrable random variable by the closure properties of square-integrability.

Then for every $g:Y\to\mathbb{R}$ measurable with respect to $\mathcal{G}$ and the Borel $\sigma$-algebra such that $g(D)=g\circ D$ is square-integrable,

$$\mathbb{E}\bigl[(g(D)-a\cdot\Theta)^{2}\bigr]\ \ge\ \frac{(a\cdot z)^{2}}{z\cdot(Jz)} .$$