The Role of Local Background Knowledge in Maximally Oriented Partially Directed Acyclic Graphs

In observational causal inference, background knowledge about specific edge orientations can refine a Markov equivalence class (MEC), with the resulting restricted class represented by a maximally oriented partially directed acyclic graph (MPDAG). Focusing on the causal effect of a treatment $X$ on an outcome $Y$, this paper addresses the question of how and when local background knowledge about edges incident to $X$ improves the identifiability of this effect. To answer this, we establish a suite of local graphical criteria. First, we derive a necessary and sufficient condition for the validity of proposed orientations of edges incident to $X$. Second, we provide local orientation rules that determine whether an adjacent undirected edge of $X$ can be directed by the given background knowledge, without enumerating all DAGs in the MEC. Finally, via a notion of possible mediators, we give a necessary and sufficient, polynomial-time criterion for determining when background knowledge improves identifiability, validated on benchmark networks.