An acyclic directed mixed graph contains directed edges without directed cycles and bidirected edges representing latent dependence. It extends a directed acyclic graph to causal models with unobserved common causes.
A Verma constraint is an equality restriction on an observed distribution implied by a latent-variable causal graph that need not be a conditional independence. Such constraints can therefore distinguish latent causal models whose ordinary observed conditional-independence relations coincide.
A district is a maximal set of vertices connected by a path consisting entirely of bidirected edges.
A vertex is fixable when its district contains no proper directed descendant of that vertex. Equivalently, .
M-separation extends d-separation to mixed graphs: a path is open given a conditioning set when every noncollider is unconditioned and every collider has a conditioned descendant.
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