Directed acyclic graphs encode qualitative causal structure through arrows, conditional independences, and intervention semantics. Their value is not decorative: they make identification assumptions explicit and support algorithms for deciding which variables should and should not be adjusted for.
Graphs and ancestral relations
Definition. A directed graph \(G=(V,E)\) is acyclic if it has no directed cycle. If \(u\to v\), then \(u\) is a parent of \(v\). Ancestors and descendants are defined by directed paths. A topological ordering lists vertices so that every parent precedes its children.
Proposition. Every finite DAG has a topological ordering.
Proof. A finite DAG has a vertex with no incoming edge; otherwise following parents indefinitely would repeat a vertex and create a directed cycle. Remove such a source, topologically order the remaining DAG by induction, and place the source first.
Markov factorization
Definition. A distribution \(P\) is Markov relative to a DAG \(G\) if \[ p(v_1,\ldots,v_d)= \prod_{j=1}^d p\{v_j\mid \operatorname{pa}_G(v_j)\} \] for a topological ordering, with densities or probability kernels interpreted relative to suitable dominating measures.
Theorem. The factorization implies the local Markov property: every node is conditionally independent of its nondescendants other than parents, given its parents.
Proof outline. Marginalize descendants in reverse topological order; their conditional kernels integrate to one. In the resulting joint density of a node, its parents, and nondescendants, the only factor involving the node is its own conditional density given parents. The conditional law of the node therefore does not depend on the remaining nondescendants.
Structural causal models
Definition. A recursive structural causal model assigns \[ V_j=f_j\{\operatorname{pa}_G(V_j),U_j\}, \] where the exogenous variables \(U_j\) are jointly independent in a Markovian model. Acyclicity makes the equations solvable in topological order.
Proposition. A Markovian recursive structural model induces a distribution that factorizes over its DAG.
Proof. Conditional on its parents, \(V_j\) is a measurable transform of the independent noise \(U_j\). The conditional law of \(V_j\) therefore depends only on its parents. Multiplying these conditional kernels in topological order gives the joint law.
Interventions and truncated factorization
Definition. The intervention \(\operatorname{do}(A=a)\) replaces the structural equation for \(A\) by the constant \(a\), leaving all other equations unchanged. Graphically, incoming arrows to \(A\) are deleted.
Theorem. In a Markovian causal DAG, \[ p(v\setminus a\mid\operatorname{do}(a)) = \prod_{V_j\ne A} p\{v_j\mid \operatorname{pa}_G(v_j)\}\big|_{A=a}. \]
Proof. Intervention removes the conditional kernel generating \(A\) and fixes its value. Every other structural equation and exogenous-noise law is unchanged, so their observational conditional kernels multiply as before with \(A=a\).
Forks, chains, and colliders
Definition. Along a path: \[ X\leftarrow Z\to Y\quad\text{is a fork},\qquad X\to Z\to Y\quad\text{is a chain},\qquad X\to Z\leftarrow Y\quad\text{is a collider}. \]
Conditioning on the middle node blocks a fork or chain but opens a collider path. Conditioning on a descendant of a collider can also open it because the descendant carries information about the collider.
Example. Let \(X,Y\) be independent \(N(0,1)\) and \(Z=X+Y\). Marginally \(X\perp Y\), but \[ \operatorname{Cov}(X,Y\mid Z)=-\frac12. \] Observing their common effect makes a large value of one variable evidence for a smaller value of the other.
Latent common causes
If an unobserved \(U\) causes both \(A\) and \(Y\), the latent projection onto observed variables contains a bidirected edge \(A\leftrightarrow Y\). A pure observed-variable DAG cannot represent this as an ordinary directed arrow without changing causal meaning. Acyclic directed mixed graphs and nested Markov models generalize the graphical language to latent confounding.
Markov equivalence
Theorem. Two DAGs entail the same set of d-separation relations if and only if they have the same undirected skeleton and the same unshielded colliders.
Proof outline. Different skeletons produce a conditional independence involving a missing adjacency. Different unshielded colliders reverse whether a length-two path is open without conditioning. Conversely, once skeleton and unshielded colliders agree, edge orientations differ only through covered-edge reversals, each of which preserves the set of active paths.
Observational conditional independences therefore identify at best a Markov-equivalence class without extra assumptions, interventions, temporal order, or functional restrictions.
Faithfulness and causal discovery
Definition. A distribution is faithful to \(G\) if every conditional independence in the distribution is implied by d-separation in \(G\).
The Markov condition says graphical separation implies probabilistic independence; faithfulness gives the converse. It excludes exact algebraic cancellations. Constraint-based causal-discovery algorithms require some form of faithfulness, often strengthened to quantitative conditions for uniform consistency.
Graphs as assumption ledgers
An arrow is not merely an association. A causal DAG should be read as a set of intervention and exclusion assumptions. Missing arrows can be more consequential than present arrows, because they assert absence of direct effects. Every analysis should explain the temporal and substantive basis for the graph and examine plausible alternatives rather than presenting one diagram as observed fact.