Causal directed acyclic graph

A causal directed acyclic graph represents variables by vertices and direct causal relations by arrows. Its graphical separation rules encode conditional independences implied by the causal model.

A probability distribution is causally minimal for a Directed acyclic graph when it satisfies the global Markov property for a directed acyclic graph for that graph and for no proper subgraph obtained by deleting edges on the same vertex set. It excludes redundant edges without excluding every extra conditional independence.

Let $E_1,E_2,E_3$ be independent random variables with the standard normal distribution and set $Z_1=E_1$, $Z_2=Z_1+E_2$, $Z_3=Z_2-Z_1+E_3$. The complete acyclic graph with edges $1\to 2$, $1\to 3$, $2\to 3$ is causally minimal: deleting each edge imposes a false marginal or conditional independence. Yet $Z_1\perp Z_3$ because $Z_3=E_2+E_3$, so faithfulness of a directed acyclic graph fails. Nonzero structural coefficients alone do not ensure faithfulness.

The PC algorithm starts with a candidate undirected graph, removes edges using conditional independence tests over increasing conditioning-set sizes, records separating sets, then orients unshielded colliders and propagates orientations. Under oracle tests and faithfulness of a directed acyclic graph, it estimates the graph's Markov equivalence class.

With an exact conditional independence oracle and faithfulness of a directed acyclic graph, the skeleton phase of the PC algorithm deletes precisely the nonedges. A true edge cannot be D-separated by a set excluding its endpoints. For nonadjacent vertices, choose the later vertex $v$ in a topological ordering; its parents separate it from the earlier nonparent. True parent edges survive throughout, so this separator remains available among the algorithm's candidate neighbour sets. For an unshielded triple $u-v-w$, any recorded separator of $u,w$ excludes $v$ exactly when the triple is an unshielded collider. The skeleton and collider characterization of Markov equivalence therefore identifies the correct equivalence class. Directions compelled throughout that class give its completed partially directed acyclic graph.

Initialize the PC skeleton search with an estimated conditional independence graph to reduce candidate pairs and conditioning sets. An exact graph contains the true Directed acyclic graph skeleton because it equals its moral graph under faithfulness. For initially absent pairs record the full complement as a separating set; an empty default can create false collider orientations. Estimation errors can affect this screening step.

A distribution factorizes according to a Directed acyclic graph when its joint density is the product of the conditional density of each variable given its graphical parents. This factorization implies the graph's global Markov property of a directed acyclic graph.

A distribution has the global Markov property for a Directed acyclic graph when every D-separation of vertex sets by a conditioning set implies the corresponding conditional independence of their random variables.

A distribution is faithful to a Directed acyclic graph when, for all disjoint vertex sets $A,B,S$, $Z_A\perp Z_B\mid Z_S$ holds if and only if $A$ and $B$ are D-separated by $S$. This includes the graph's global Markov property of a directed acyclic graph and excludes additional independences caused by parameter cancellations. If faithfulness is used to name only the reverse implication, the Markov property must be assumed separately.

A nonparametric structural equation model assigns each observed variable an arbitrary measurable function of its graphical parents and an exogenous variable. Independence or dependence among exogenous variables determines the graph's latent-confounding structure.

An acyclic directed mixed graph contains directed edges without directed cycles and bidirected edges representing latent dependence. It extends a directed acyclic graph to causal models with unobserved common causes.

A Verma constraint is an equality restriction on an observed distribution implied by a latent-variable causal graph that need not be a conditional independence. Such constraints can therefore distinguish latent causal models whose ordinary observed conditional-independence relations coincide.

A district is a maximal set of vertices connected by a path consisting entirely of bidirected edges.

A vertex is fixable when its district contains no proper directed descendant of that vertex. Equivalently, $\mathrm{dis}(v)\cap \mathrm{de}(v)=\{v\}$.

M-separation extends d-separation to mixed graphs: a path is open given a conditioning set when every noncollider is unconditioned and every collider has a conditioned descendant.

Front-door adjustment identifies an exposure effect through an observed mediator when the mediator intercepts every directed exposure-outcome path, the exposure-mediator relation has no unblocked backdoor path, and exposure blocks every backdoor path from mediator to outcome.

Conditional front-door adjustment applies the front-door argument within strata of observed covariates and then averages over those covariates. The covariates must remove exposure-mediator confounding, while the exposure together with the covariates must remove mediator-outcome confounding.

A backdoor path creates noncausal association by entering the exposure through an arrowhead rather than beginning with a causal arrow leaving it.

A backdoor path from exposure $X$ to outcome $Y$ begins with an arrow entering $X$. An adjustment set identifies the total effect when it blocks every such path without conditioning on descendants of $X$.

A backdoor adjustment set contains no descendant of the exposure and blocks every backdoor path from the exposure to the outcome. Averaging the conditional outcome distribution over such a set identifies the total causal effect.

A sufficient adjustment set is a collection of pretreatment variables for which treatment and every relevant potential outcome are conditionally independent. Adjustment for that set identifies the corresponding causal contrast under consistency and positivity.

A minimal sufficient adjustment set is a sufficient adjustment set none of whose proper subsets is sufficient. Minimality is by set inclusion and does not imply minimum cardinality or uniqueness.

In a linear structural equation model, each variable is a linear combination of its graphical parents and an exogenous error term. Edge coefficients quantify direct effects.

Wright's path tracing rule expresses a covariance in a standardized linear structural equation model as a sum of products of edge coefficients over admissible unblocked paths.