Score Identities and the Mixture-Weight Directional van Trees Inequality

Adopt the setting and notation 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 $\sigma$-algebra $\mathcal{B}_l$ and the measure $\lambda_l$ on $\mathbb{R}^{l}$ of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ (which is Lebesgue measure on $\mathbb{R}^{l}$, $\mathcal{B}_l$ being the Borel $\sigma$-algebra by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), the product measure $\kappa=\lambda_l\otimes\mu$ on $\mathcal{B}_l\otimes\mathcal{G}$, the square-integrable random variables $\Theta_1,\dots,\Theta_l$ with $\Theta=(\Theta_1,\dots,\Theta_l)$, the measurable map $D:\Omega\to Y$, the joint law $Q$ of $(\Theta,D)$, and the density $p$ with its partial derivatives $\partial_ip$. Assume hypotheses (i), (ii) and (iii) of that theorem, and in place of its hypothesis (iv) assume only

(iv$'$) Square-integrable scores. For each $i\in\{1,\dots,l\}$ the score $S_i=\partial_ip(\Theta,D)/p(\Theta,D)$, a random variable exactly as noted in hypothesis (iv) there, is square-integrable.

Write $\mathbb{E}$ for the expectation, $\lVert U\rVert_2$ for the mean-square norm, $x\cdot y$ for the dot product and $x-y$ for the difference on the Euclidean space $\mathbb{R}^{l}$, $|\cdot|$ for the absolute value, and $\sqrt{t}$ for the nonnegative square root of a real $t\ge0$. For $u=(u_1,\dots,u_l)\in\mathbb{R}^{l}$ put $\partial_up=\sum_{i=1}^{l}u_i\,\partial_ip$ and $S_u=\sum_{i=1}^{l}u_iS_i$, and for $\alpha\in\mathbb{R}^{l}$ put $\alpha\cdot\Theta=\sum_{j=1}^{l}\alpha_j\Theta_j$; sums over finite index sets are the finite sums of the real numbers. Throughout, $m:Y\to\mathbb{R}$ denotes a map measurable with respect to $\mathcal{G}$ and the Borel $\sigma$-algebra such that $m(D)=m\circ D$ is square-integrable.

1. (Score identities) For every such $m$ and all $i,j\in\{1,\dots,l\}$,

$$\mathbb{E}\bigl[m(D)\,S_i\bigr]=0,\qquad \mathbb{E}\bigl[\Theta_j\,S_i\bigr]=-\delta_{ij},$$

where $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ otherwise. Consequently, for all $u,\alpha\in\mathbb{R}^{l}$,

$$\mathbb{E}\bigl[\bigl(m(D)-\alpha\cdot\Theta\bigr)\,S_u\bigr]=\alpha\cdot u .$$

2. (Mixture-weight directional bound) Let $n\ge1$ be a natural number, let $a_1,\dots,a_n\in\mathbb{R}^{l}$, and define the mixture weight $\bar p:\mathbb{R}^{l}\times Y\to\mathbb{R}$ by

$$\bar p(\theta,y)=\frac12\,p(\theta,y)+\frac{1}{2n}\sum_{j=1}^{n}p(\theta-a_j,y).$$

Then $\bar p$ is strictly positive and measurable with respect to $\mathcal{B}_l\otimes\mathcal{G}$ and the Borel $\sigma$-algebra, $\int\bar p\,d\kappa=1$, and for every $u\in\mathbb{R}^{l}$ the function $(\partial_up)^{2}/\bar p$ is measurable with

$$\mathcal{I}_u=\int_{\mathbb{R}^{l}\times Y}\frac{(\partial_up)^{2}}{\bar p}\,d\kappa\ \le\ 2\,\mathbb{E}\bigl[S_u^{2}\bigr]<\infty ,$$

called the mixture-weight information in the direction $u$ for the shifts $a_1,\dots,a_n$. Moreover, for every $m$ as above and every $\alpha\in\mathbb{R}^{l}$,

$$|\alpha\cdot u|\ \le\ \sqrt{\mathcal{I}_u}\,\Bigl(\bigl\lVert m(D)-\alpha\cdot\Theta\bigr\rVert_2+\max_{1\le j\le n}|\alpha\cdot a_j|\Bigr).$$