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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.

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]

Thomas S. Richardson develops graphical approaches to causal inference and statistical models, including ways to reason when some variables are hidden. His institutional research description lists graphical models, model selection, Bayesian inference and causal models.^[ref]

He is a coauthor of Nested Markov Properties for Acyclic Directed Mixed Graphs by Thomas S. Richardson and others. The work represents observable constraints of hidden-variable directed graphs using a nested Markov model. Its motivation is that eliminating a hidden cause can leave more than ordinary conditional independence restrictions among the measured variables.^[ref]

For example, a hidden common cause of two observed variables can be represented by a bidirected edge in a latent projection. The graph records a possible dependence source without pretending that the common cause itself was measured. Such representations separate what observations constrain from what remains unidentifiable.