 Causal sufficiency

A collection of observed variables is causally sufficient when no unobserved variable is a common cause of two or more observed variables. In a causal directed acyclic graph model, a common sufficient condition is that the observed graph includes all such common causes; independent unobserved noise affecting only one observed variable is allowed.

For example, $U\to X$ and $U\to Y$ with $U$ unobserved violates causal sufficiency for $\{X,Y\}$. Observing $U$ removes that particular hidden common cause. In contrast, $U\to X\to Y$ alone makes $U$ an ancestor of $Y$ through $X$, without supplying an additional common-cause fork outside the observed graph.

Causal sufficiency supports the use of a causal directed acyclic graph on the observed variables with independent disturbances. It does not by itself imply randomization, faithfulness of a directed acyclic graph, or identification of every causal effect. When hidden common causes remain, a latent projection may require bidirected edges to represent the induced dependence.