Latent projection

A latent projection represents the influence of unobserved vertices of a directed acyclic graph on its observed vertices using an acyclic directed mixed graph. It retains an arrow $a\to b$ when the original graph has a directed path from $a$ to $b$ whose internal vertices are all unobserved. It adds $a\leftrightarrow b$ when there is a path with arrowheads at both observed endpoints whose internal vertices are unobserved noncolliders.

For example, the hidden fork $A\leftarrow U\to B$ becomes $A\leftrightarrow B$, while $A\to U\to B$ becomes $A\to B$. The bidirected edge encodes latent common-cause structure, not two opposing causal effects. Both edge types may coexist between a pair. The construction preserves the appropriate observed separation relations, while retaining more useful structure than simply deleting hidden vertices and their edges.^[ref]