Local Markov property

Let $G = (V,E)$ be a directed acyclic graph and $X = \{X_v\}_{v \in V}$ a set of random variables. We say $(G,X)$ satisfies the local Markov property if for all $v \in V$ and $w \in V$ such that $(w,v) \not \in E$ where there is no path from $v$ to $w$ then $X_v$ is conditionally independent of $X_w$ given $\cup_{(u,v) \in E} X_u$.