Markov equivalence class

A Markov equivalence class is the set of all directed acyclic graphs (DAGs) that encode exactly the same set of conditional independence relationships. Two DAGs are Markov equivalent if and only if they share the same skeleton (the same undirected edges, regardless of direction) and the same v-structures (patterns of the form X → Y ← Z where X and Z are not adjacent). This equivalence means that no amount of observational data can distinguish between members of the class: they are statistically indistinguishable, and only intervention or additional assumptions can resolve which member is the true causal graph.

The Markov equivalence class is represented compactly as a CPDAG (completed partially directed acyclic graph), where directed edges indicate arrows whose direction is shared by all members of the class, and undirected edges indicate arrows whose direction varies. Algorithms like GES and the PC algorithm search in the space of equivalence classes rather than individual DAGs precisely because the class is the finest grain of causal structure learnable from observation alone. The existence of Markov equivalence classes is not a technical inconvenience; it is a fundamental limit on what passive observation can reveal about causation. The universe does not write its causal structure in the data. It writes its independence structure, and the rest is silence.