Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound

Let $d\ge1$ be a natural number, let $\eta$ be a real number with $0<\eta\le1$, and let $\varphi_\eta$, $c_\eta$, $Z_b$, $\kappa_b=\lVert b\rVert^{2}/\eta$ and $R_b$ (for $b\in\mathbb{R}^d$) be as in The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, whose conventions for Euclidean space $\mathbb{R}^d$ (sums, scalar multiples and differences of points, the Euclidean norm $\lVert\cdot\rVert$, Lebesgue measure $\lambda_d$ on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$, measurability, sequential continuity, integrals, integrability, partial derivatives on the open set $\mathbb{R}^d$, and $t^{1/2}$ for the nonnegative square root) are in force. Sums over a finite index set are those of Sum over a Finite Index Set, and $\exp$ is the exponential function.

The discrete family and its counting measure. Let $p$ be a discrete probability mass function on $\mathbb{R}^d$ whose support $\mathsf{S}=\{x\in\mathbb{R}^d:p(x)>0\}$ is countable in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions (finite or the set of values of a sequence). Let $(\mathsf{S},2^{\mathsf{S}},\mathsf{c})$ be the countable disjoint union of the one-point measure spaces with unit mass at the points of $\mathsf{S}$, called the counting measure on $\mathsf{S}$: by claims 3 and 4 of that lemma every subset of $\mathsf{S}$ is measurable, every function on $\mathsf{S}$ is measurable, and for every $h:\mathsf{S}\to[0,\infty]$ the integral $\int_{\mathsf{S}}h\,d\mathsf{c}$ equals $\sum_{x\in\mathsf{S}}h(x)$, the least upper bound of the finite partial sums in the sense of the preamble of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, which for $[0,\infty)$-valued $h$ is the sum of $h$ over $\mathsf{S}$. Throughout, $\sum_{x\in\mathsf{S}}h(x)$ denotes $\int_{\mathsf{S}}h\,d\mathsf{c}$ also for $\mathsf{c}$-integrable real-valued $h$. Assume the first-moment condition

$$\sum_{x\in\mathsf{S}}p(x)\,\lVert x\rVert<\infty .$$

The kernel. Let $(\mathsf{R},\mathcal{R},\rho)$ be a measure space whose measure $\rho$ is $\sigma$-finite, and let $f:\mathsf{S}\times\mathsf{R}\to[0,\infty)$ be a kernel satisfying: (K1) for every $x\in\mathsf{S}$ the map $r\mapsto f(x,r)$ is measurable with respect to $\mathcal{R}$ and the Borel $\sigma$-algebra, and $\int_{\mathsf{R}}f(x,r)\,\rho(dr)=1$; (K2) there is an $\mathcal{R}$-measurable $M:\mathsf{R}\to[0,\infty)$ with $f(x,r)\le M(r)$ for all $(x,r)\in\mathsf{S}\times\mathsf{R}$. Put $\bar f(r)=\sum_{x\in\mathsf{S}}p(x)f(x,r)\in[0,\infty]$ and $\mathsf{R}_{+}=\{r\in\mathsf{R}:\bar f(r)>0\}$. Define the smoothed joint density $g:\mathbb{R}^d\times\mathsf{R}\to[0,\infty]$ by

$$g(\theta,r)=\sum_{x\in\mathsf{S}}p(x)\,\varphi_\eta(\theta-x)\,f(x,r).$$

Let $\mathcal{B}(\mathbb{R}^d)\otimes\mathcal{R}$ be the product $\sigma$-algebra and $\lambda_d\otimes\rho$ the product measure on it.

Moves and weights. Let $n\ge1$ be a natural number, let $a=(a_1,\dots,a_n)$ be points of $\mathbb{R}^d$ (the moves) such that $x+a_j\in\mathsf{S}$ for every $x\in\mathsf{S}$ and every $j$ (the moves are translation symmetries of the support, as for lattice supports; no exit-mass term arises), and let $w=(w_1,\dots,w_n)\in\mathbb{R}^n$ (the weights). Put $u=\sum_{j=1}^{n}w_ja_j\in\mathbb{R}^d$ and $\kappa_j=\kappa_{a_j}$. For $j\in\{1,\dots,n\}$ and $x\in\mathsf{S}$ define the shifted family by $p_j(x)=p(x-a_j)$ (which is $0$ when $x-a_j\notin\mathsf{S}$, as $p$ vanishes off $\mathsf{S}$) and $f_j(x,r)=f(x-a_j,r)$ if $x-a_j\in\mathsf{S}$, $f_j(x,r)=0$ otherwise. Let $\Phi_w$ be the symmetrised score functional for the weights $w$, and define the symmetrised kernel-weighted move information

$$\mathsf{J}^{\mathrm{sym}}=\sum_{x\in\mathsf{S}}\int_{\mathsf{R}}\Phi_w\bigl(p(x)f(x,r),\ p_1(x)f_1(x,r),\dots,p_n(x)f_n(x,r)\bigr)\,\rho(dr)\ \in[0,\infty].$$

For $\theta\in\mathbb{R}^d$ and $r\in\mathsf{R}$ put $g_j(\theta,r)=g(\theta-a_j,r)$ and define the mixture weight

$$\bar g(\theta,r)=\frac12\,g(\theta,r)+\frac{1}{2n}\sum_{j=1}^{n}g_j(\theta,r).$$

1. (Joint density) $\mathsf{S}$ is nonempty; $\bar f$ is $\mathcal{R}$-measurable with $\bar f\le M$, and $\mathsf{R}_{+}\in\mathcal{R}$; $g$ is finite and measurable with respect to $\mathcal{B}(\mathbb{R}^d)\otimes\mathcal{R}$; $0\le g(\theta,r)\le c_\eta\bar f(r)$; for every $x\in\mathsf{S}$ the map $r\mapsto\Phi_w\bigl(p(x)f(x,r),p_1(x)f_1(x,r),\dots,p_n(x)f_n(x,r)\bigr)$ is $\mathcal{R}$-measurable, so that $\mathsf{J}^{\mathrm{sym}}$ is well defined; for every $r\in\mathsf{R}$, $\int_{\mathbb{R}^d}g(\theta,r)\,\lambda_d(d\theta)=\bar f(r)$, and $\int g\,d(\lambda_d\otimes\rho)=1$; and $g(\theta,r)>0$ if and only if $r\in \mathsf{R}_{+}$. The same statements hold for each $g_j$ and for $\bar g$ in place of $g$.

2. (Smoothness and van Trees regularity) For every $r\in\mathsf{R}$, the function $\theta\mapsto g(\theta,r)$ is continuous at every point of $\mathbb{R}^d$, and for every $i\in\{1,\dots,d\}$ its partial derivative with respect to the $i$th variable exists at every $\theta$ and equals

$$\partial_ig(\theta,r)=\sum_{x\in\mathsf{S}}p(x)\,\partial_i\varphi_\eta(\theta-x)\,f(x,r),$$

the summand being $\mathsf{c}$-integrable; $\theta\mapsto\partial_ig(\theta,r)$ is continuous at every point, with $|\partial_ig(\theta,r)|\le c_\eta(2\eta)^{-1/2}\bar f(r)$. In particular, for every $r\in \mathsf{R}_{+}$ the function $\theta\mapsto g(\theta,r)$ is strictly positive and of class $C^1$ on $\mathbb{R}^d$. Moreover, $\partial_ig$ is measurable with respect to $\mathcal{B}(\mathbb{R}^d)\otimes\mathcal{R}$,

$$\int\Bigl(1+\sum_{k=1}^{d}|\theta_k|\Bigr)\,|\partial_ig(\theta,r)|\,d(\lambda_d\otimes\rho)(\theta,r)<\infty ,\qquad\text{and}\qquad \int\mathbf{1}_{\mathbb{R}^d\times \mathsf{R}_{+}}\,\frac{(\partial_ig)^{2}}{g}\,d(\lambda_d\otimes\rho)\le\frac1\eta ,$$

where $\mathbf{1}_{\mathbb{R}^d\times \mathsf{R}_{+}}$ is the indicator of $\mathbb{R}^d\times \mathsf{R}_{+}$ and the integrand of the second integral is read as $0$ off that set.

3. (Symmetrised directional score bound) With $\partial_ug=\sum_{i=1}^{d}u_i\,\partial_ig$ for $u=(u_1,\dots,u_d)$ as above, the function $\mathbf{1}_{\mathbb{R}^d\times \mathsf{R}_{+}}(\partial_ug)^{2}/\bar g$ (read as $0$ off $\mathbb{R}^d\times \mathsf{R}_{+}$) is measurable, and

$$\Bigl(\int\mathbf{1}_{\mathbb{R}^d\times \mathsf{R}_{+}}\,\frac{(\partial_ug)^{2}}{\bar g}\,d(\lambda_d\otimes\rho)\Bigr)^{1/2}\ \le\ \bigl(\mathsf{J}^{\mathrm{sym}}\bigr)^{1/2}+2^{1/2}\sum_{j=1}^{n}|w_j|\,\bigl(\exp(\kappa_j)-1-\kappa_j\bigr)^{1/2},$$

the left side being finite whenever $\mathsf{J}^{\mathrm{sym}}<\infty$ (and the inequality holding trivially otherwise).