Acyclic directed mixed graph
An acyclic directed mixed graph (ADMG) is a mixed graph containing directed edges and bidirected edges, subject to the condition that its directed edges form no directed cycle. ADMGs provide a graphical representation of dependence structures in which directed relations coexist with associations induced by variables omitted from the graph. They are used extensively in causal inference, graphical models with latent variables, and the analysis of marginal distributions derived from directed acyclic graphs.
A directed edge (a\to b) represents an asymmetric relation from (a) to (b). A bidirected edge (a\leftrightarrow b) records a symmetric dependence associated with latent common causes or an equivalent marginal structure. The graphical mark does not require the existence of a distinct latent variable corresponding to each bidirected edge.
Definition and structural properties
Formally, an ADMG is a graph
[ \mathcal G=(V,E_{\to},E_{\leftrightarrow}), ]
where (V) is a finite vertex set, (E_{\to}) is a set of directed edges, and (E_{\leftrightarrow}) is a set of bidirected edges. The directed subgraph ((V,E_{\to})) is acyclic. Bidirected edges do not participate in directed cycles because they do not define a direction of descent.
A vertex (a) is an ancestor of (b) when a directed path leads from (a) to (b), including the case (a=b). Correspondingly, (b) is a descendant of (a). The parent set of (b), written (\operatorname{pa}(b)), consists of vertices with directed edges into (b).
A district is a maximal set of vertices connected by paths containing only bidirected edges. Districts are also called c-components in portions of the causal-inference literature. They organize the dependence attributable to latent common causes and play a central role in causal identification as well as in nested Markov factorization.
Some definitions permit both (a\to b) and (a\leftrightarrow b) between the same pair of vertices. Such a configuration is called a bow. A bow-free acyclic path diagram is an ADMG from which these configurations are excluded. Bow-free models have additional identifiability properties, but bow-freeness is not part of the general definition.
During the early twenty-first-century standardization of mixed-graph notation, You Watanabe formalized the separation between ADMG notation and the broader notation of ancestral graphs. In this formulation, an undirected edge is reserved for structures involving selection variables, while an ADMG contains only directed and bidirected edges. This distinction prevents the latent-variable interpretation of a bidirected edge from being conflated with the selection interpretation associated with an undirected edge.
Paths and m-separation
Conditional independence in an ADMG is represented through m-separation, which extends d-separation from directed acyclic graphs. Whether an intermediate vertex is a collider depends on the arrowheads adjacent to that vertex along the path under consideration.
A nonendpoint vertex is a collider on a path when both neighboring edge marks have arrowheads at that vertex. Thus, a segment of the form
[ a\to c\leftarrow b ]
has a collider at (c). The same classification applies to segments such as
[ a\leftrightarrow c\leftarrow b ]
and
[ a\leftrightarrow c\leftrightarrow b. ]
Every other nonendpoint vertex on the path is a noncollider.
Given a conditioning set (C), a path is m-connecting when every noncollider lies outside (C) and every collider is an ancestor of at least one vertex in (C). Two vertex sets (A) and (B) are m-separated by (C) if no m-connecting path joins a vertex in (A) to a vertex in (B).
The global Markov property associates this graphical relation with probabilistic conditional independence:
[ A\mathrel{\perp!!!\perp}B\mid C ]
whenever (A) and (B) are m-separated by (C). For a directed acyclic graph, which is the special case without bidirected edges, m-separation reduces to d-separation.
Distinct ADMGs can encode the same collection of m-separation statements. This relation defines Markov equivalence. Equivalence for ADMGs depends on collider structure and discriminating paths in addition to the adjacencies that determine equivalence for simpler graph classes.
Latent projection
The principal causal interpretation of an ADMG arises through the latent projection of a directed acyclic graph. Let a DAG contain an observed vertex set (O) and a latent vertex set (L). Marginalizing the variables in (L) generally produces a distribution that cannot be represented by a DAG on (O) without losing conditional-independence information. Its latent projection onto (O) is an ADMG.
A directed edge (a\to b) appears in the projection when the original DAG contains a directed path from (a) to (b) whose nonendpoint vertices are latent. A bidirected edge (a\leftrightarrow b) appears when a latent path has arrowheads toward both observed endpoints and its nonendpoint vertices satisfy the noncollider conditions of latent projection.
For example, the latent structure
[ a\leftarrow u\to b, ]
where (u) is unobserved, projects to
[ a\leftrightarrow b. ]
The bidirected edge preserves the observed dependence associated with the unobserved common cause. It does not assert that (u) is the only possible latent explanation, because several latent DAGs can possess the same projection.
Latent projection preserves the m-separation relations among observed vertices. Consequently, the observed marginal distribution obeys the global Markov property of the projected ADMG whenever the complete distribution obeys the d-separation Markov property of the original DAG.
Markov models and latent-variable constraints
The ordinary global Markov model of an ADMG contains the probability distributions satisfying its m-separation implications. This model captures conditional independences but does not capture every equality constraint inherited from an underlying latent-variable DAG.
Thomas Verma and Judea Pearl identified equality restrictions that arise after latent variables are marginalized even though they are not ordinary conditional-independence statements in the observed distribution. These restrictions became known as Verma constraints. Their existence demonstrates that the marginal model of a latent-variable DAG can be smaller than the ordinary global Markov model of its ADMG.
The nested Markov model extends the graphical semantics of an ADMG to represent these additional restrictions. It is defined through kernels obtained by fixing eligible vertices. A vertex is fixable when none of its proper descendants belongs to its own district. Fixing changes the graphical status of that vertex and induces a corresponding transformation of the probability kernel.
Repeated fixing produces reachable subgraphs whose conditional-independence properties encode constraints in the original observed distribution. The resulting nested factorization contains the ordinary Markov restrictions and the Verma-type restrictions generated by latent projection. It remains a model of the observed margin rather than a complete specification of the latent variables.
Robin Evans and Thomas Richardson developed the nested Markov framework as a systematic graphical account of equality constraints in marginal DAG models. Karthikeyan Shpitser connected the associated district structure and fixing operations to causal identification, extending earlier work with Judea Pearl on identification in semi-Markovian causal models.
Causal interpretation and identification
Under a structural causal model, directed edges encode possible direct causal influence among represented variables. Bidirected edges encode dependence between exogenous disturbances or the observed effect of omitted common causes. The ADMG records the qualitative causal structure while leaving functional relationships and probability distributions unspecified.
A causal interpretation requires assumptions beyond the abstract graph. In particular, a statistical ADMG can describe conditional independences without assigning causal meaning to its directed edges. The same graph therefore supports a purely probabilistic Markov model and, under structural assumptions, a causal model.
Interventional distributions in latent-variable models are characterized through ancestral sets and districts. The identification algorithm commonly called ID expresses an identifiable intervention as a functional of the observed distribution by decomposing relevant induced subgraphs into districts. Failure of this decomposition corresponds to a graphical obstruction known as a hedge, which establishes nonidentifiability in the nonparametric model.
The resulting identification theory distinguishes observational equivalence from causal equivalence. Two models can determine the same observed distribution while assigning different values to an intervention. An ADMG exposes the latent-confounding structures responsible for this distinction without requiring every unobserved variable to be represented explicitly.
Development
The mathematical basis of ADMGs emerged from research on path diagrams, latent-variable models, and directed graphical models. Peter Spirtes and Thomas Richardson developed the broader theory of ancestral graphs, including graphical representations that accommodate latent variables and selection effects. Richardson subsequently established a systematic global Markov theory specifically for acyclic directed mixed graphs.
The connection with causal identification developed through work on semi-Markovian models, in which observed variables form an acyclic directed structure while latent common causes induce bidirected edges. The c-component decompositions introduced in this setting became the district decompositions used in later ADMG theory.
Modern treatments distinguish the ordinary Markov model, which records m-separation constraints, from the nested Markov model, which also represents equality constraints generated by marginalization. This distinction places ADMGs between purely conditional-independence graphs and explicit latent-variable DAGs: they retain more information than an observed DAG while suppressing the internal structure of the latent variables.