Move Information 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 $\mathbb{R}^m$, 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 $w=(w_1,\dots,w_n)\in\mathbb{R}^n$ (the weights), and let $\rho_{p,a,w}:\mathsf{S}\to\mathbb{R}$ be the move score. The move information of $p$ for $a$ and $w$ is

$$\mathsf{J}(p;a,w)=\sum_{x\in\mathsf{S}}p(x)\,\rho_{p,a,w}(x)^{2}\in[0,\infty],$$

the sum over $\mathsf{S}$ of the nonnegative function $x\mapsto p(x)\rho_{p,a,w}(x)^{2}$ on $\mathsf{S}$.