Move Score of a Discrete Probability Mass Function

Let $m\ge1$ and $n\ge1$ be natural numbers, let $p$ be a discrete probability mass function on Euclidean space $\mathbb{R}^m$, and put $\mathsf{S}=\{x\in\mathbb{R}^m:p(x)>0\}$. Let $a=(a_1,\dots,a_n)$ be a family of points of $\mathbb{R}^m$ (the moves) and let $w=(w_1,\dots,w_n)$ be a point of $\mathbb{R}^n$ (the weights). Differences of points of $\mathbb{R}^m$ are those of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, and sums over $1\le j\le n$ are the finite sums of $\mathbb{R}$.

The move score of $p$ for the moves $a$ and weights $w$ is the function $\rho_{p,a,w}:\mathsf{S}\to\mathbb{R}$,

$$\rho_{p,a,w}(x)=\sum_{j=1}^{n}w_j\Bigl(1-\frac{p(x-a_j)}{p(x)}\Bigr).$$