D-separation

d-separation (directional separation) is a graph-theoretic criterion for reading conditional independence relations from a directed acyclic graph. Two sets of nodes X and Y are d-separated by a conditioning set Z if Z blocks every undirected path between X and Y, where blocking depends on whether the path contains chains, forks, or colliders. Under the Causal Markov condition, d-separation in the graph implies probabilistic conditional independence in the data, making it the bridge between causal structure and statistical pattern. The criterion fails, however, when the graph contains cycles or when latent confounders create paths that the observed graph cannot represent — a limitation that exposes the acyclicity assumption as a modeling convenience rather than a metaphysical truth.

The concept extends naturally to m-separation in mixed graphs that include both directed and bidirected edges, though the algorithms for computing m-separation are substantially more complex.