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 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 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.

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 $X$.

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.