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Causal discovery seeks information about causal structure from observations or interventions, rather than assuming the complete structure in advance. In a causal directed acyclic graph, observed conditional independences can restrict possible arrows, but generally do not identify a unique graph without further assumptions.

For three variables, the chains $X\to Y\to Z$ and $X\leftarrow Y\leftarrow Z$, and the fork $X\leftarrow Y\to Z$, all imply $X$ and $Z$ are independent conditional on $Y$ in the corresponding faithful models. These independences alone cannot distinguish those structures. By contrast, the collider $X\to Y\leftarrow Z$ generically makes $X$ and $Z$ marginally independent but dependent after conditioning on $Y$.

This example explains both the appeal and the limitation of searching for graphical structure. Conditional independence tests offer clues, while hidden common causes, selection and unfaithful parameter cancellations can undermine the inference. An experiment that changes $X$ can distinguish directions that observational data alone leave ambiguous.^[ref]

The fixing operation in a nested Markov model turns a random vertex of a conditional acyclic directed mixed graph into a fixed input and transforms its probability kernel. A vertex $v$ is fixable if no other vertex in its bidirected-connected district of an acyclic directed mixed graph is a descendant of $v$. Graphically, fixing removes edges with an arrowhead at $v$ and moves $v$ from the random to the fixed vertex set.^[ref]

For a positive Markov kernel $q$, the transformation divides $q$ by the conditional factor for $v$ given its district's other vertices and the parents of that district. Fixed arguments are retained as kernel inputs. Fixability is what permits the resulting object to be a normalized kernel over the remaining random variables. Valid fixing sequences must check this condition again after each step.

For the directed acyclic graph $A\to B\to C$, write $p(a,b,c)=p(a)p(b\mid a)p(c\mid b)$. Fixing $B$ givesOrdinary conditioning instead gives $p(a\mid b)p(c\mid b)$. If $A$ is a fair binary variable and $p(B=1\mid A=1)=0.8$, $p(B=1\mid A=0)=0.2$, then conditioning on $B=1$ makes $p(A=1)=0.8$, whereas fixing $B=1$ leaves it $0.5$. The operation is algebraically defined even without interpreting a graph causally.

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]

A nested Markov model is a statistical model associated with an acyclic directed mixed graph. Besides ordinary conditional independence, it imposes Markov restrictions on probability kernels obtained through valid sequences of a fixing operation. Fixing changes a random vertex into a fixed input and transforms the distribution by an appropriate conditional-density ratio. It is not simply conditioning. Formally, every kernel reached by a valid sequence must satisfy the global Markov property of its reached graph.^[ref]

A concrete equality can arise from the positive joint distributionwith $U$ unobserved. DefineSubstitution and Bayes' theorem reduce this to ${\sum }_{u}p(d\mid c,u)p(u)$, independent of $a$. This is a Verma constraint; the graph can impose it without implying any ordinary conditional independence among the observed variables.^[ref]

For a numerical instance, make $U,A$ independent fair binary variables, and choose success probabilities $p(B=1\mid a,u)=0.1+0.2a+0.4u$, $p(C=1\mid a,b)=0.15+0.2a+0.3b$, and $p(D=1\mid c,u)=0.1+0.3u+0.4c$. Then $q(D=1\mid a,c)=0.25+0.4c$ for either $a$. Yet $p(D=1\mid A=0,C=0)=181/760$ differs from $p(D=1\mid A=1,C=0)=29/125$.

A latent-variable model can impose additional inequalities, so its observed margin may be strictly smaller than the associated nested Markov model.^[ref]

Robin J. Evans studies causal inference, graphical models and the behavior of statistical models after hidden variables are marginalized. His work on margins of Bayesian networks characterizes algebraic restrictions that are not captured by a list of conditional independences alone.^[ref]

With Thomas S. Richardson and other collaborators, he coauthored Nested Markov Properties for Acyclic Directed Mixed Graphs by Thomas S. Richardson and others. A central idea is to describe constraints on the observed distribution without committing to an identifiable parameter for every unobserved mechanism.^[ref]

This matters when two models agree on ordinary independence tests yet differ in other restrictions on probabilities. Separating these cases can improve both model interpretation and estimation: a convenient parameterization of the observed model need not identify the underlying hidden causal story. His institutional personal-page redirect leads to his research-group site.^[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.