Markov Equivalence Class in Causal Models

Updated 30 December 2025

Markov Equivalence Class is the set of all DAGs that share identical conditional-independence patterns, uniquely represented by a CPDAG.
It plays a vital role in causal inference by serving as the basis for algorithms in structure learning, enumeration, and intervention design.
Efficient algorithms leverage chordal decompositions and sampling techniques to enumerate and analyze MECs even in large, sparse graphs.

A Markov Equivalence Class (MEC) is the set of all Directed Acyclic Graphs (DAGs) that encode identical conditional-independence structures via d-separation. In observational or partially specified causal inference, a data-generating DAG cannot usually be determined uniquely from observational data; instead, only the implied MEC—i.e., the set of DAGs Markov-equivalent to the true structure—can be recovered. Each MEC can be represented by a unique completed partially directed acyclic graph (CPDAG), in which compelled (invariant) edge orientations appear as directed edges and ambiguously oriented edges remain undirected. MECs are central both as a statistical abstraction of identifiable features and as algorithmic primitives in structure learning, enumeration, intervention design, and causal effect identification. This article presents a comprehensive technical account of the definition, graphical representations, combinatorics, enumeration, algorithms, properties under interventions, and relevance of MECs across causal inference and graphical model research.

1. Formal Definition and Characterizations

Let $G$ be a DAG over node set $V$ . Two DAGs $G, G'$ are Markov equivalent if they encode the same set of conditional-independence (CI) statements; that is,

$\forall A,B,S \subset V: (A \perp B \mid S)_G \Longleftrightarrow (A \perp B \mid S)_{G'}.$

The Markov equivalence class (MEC) of $G$ is denoted $[G] = \{ G' : G' \text{ is a DAG on } V,\, G' \text{ Markov equivalent to } G \}$ (Guo et al., 2020, Desjardins, 2013, Zhang et al., 2024, Wienöbst et al., 2023).

Verma–Pearl Criterion:

Two DAGs are in the same MEC if and only if they have:

The same skeleton (underlying undirected graph),
The same set of v-structures (immoralities): induced subgraphs of the form $A \to B \leftarrow C$ with $A,C$ not adjacent (Desjardins, 2013, Guo et al., 2020, Zhang et al., 2024, Wienöbst et al., 2023).

The unique graphical representation of an MEC among DAGs is the completed partially directed acyclic graph (CPDAG):

Edges directed identically across all members remain directed.
Edges oriented differently across members are left undirected (Guo et al., 2020, Desjardins, 2013, Sharma, 2023).

2. Graphical Representations: CPDAGs, MPDAGs, PAGs

CPDAG (Completed Partially DAG):

Encodes all invariant orientations as directed, and ambiguous edges as undirected (Guo et al., 2020, Wienöbst et al., 2023). The mapping from MEC to CPDAG is unique.

MPDAG (Maximally Oriented Partially DAG):

Incorporates both the equivalence structure (from the CPDAG) and any available background knowledge (additional required edge directions). Undirected edges encode orientation ambiguity consistent with both observed CI constraints and background knowledge; invariant edges (directed in all conforming DAGs) are oriented (Guo et al., 2020, Teh et al., 18 Jun 2025, LaPlante et al., 2023, Wienöbst et al., 2023).

PAG (Partial Ancestral Graph):

Encodes equivalence classes of maximal ancestral graphs (MAGs), which generalize DAGs for latent-variable settings. Pag edges may have circles (ambiguity), tails, and arrowheads, indicating fine-grained invariance/ambiguity under both observed and latent variable structures (Jaber et al., 2018, Zhang, 2012).

3. Combinatorics and Enumeration of MECs

Size and Structure:

The size of an MEC is determined by the undirected components of the essential graph (CPDAG). Each undirected chain component is a chordal graph whose acyclic, v-structure–free orientations correspond one-to-one to members of the MEC (He et al., 2016, Wienöbst et al., 2023, AhmadiTeshnizi et al., 2020).

Counting and Algorithms:

The size $|[G]|$ of the MEC associated to CPDAG $C$ factors as $|[G]| = \prod_i |[C_i]|$ over its undirected chain components (He et al., 2016, AhmadiTeshnizi et al., 2020).
For trees, explicit generating functions for MECs by number of immoralities and class size are related to Fibonacci and Lucas polynomials (Radhakrishnan et al., 2017).
For arbitrary skeletons, there are recursive and closed-form formulas using core graph decompositions, which permit efficient symbolic computation of MEC size as a polynomial in the number of dominating vertices (He et al., 2016).
Fixed-parameter tractable algorithms count MECs with a fixed skeleton in time $O(n \cdot 2^{O(k^5 \delta)} + n^2)$ where $k$ is treewidth, $\delta$ is maximum degree (Sharma, 2023).
For logical constraints (e.g., degree bounds), enumeration of essential DAGs (singleton MECs) under $C^2$ first-order constraints is polynomial in $n$ (Bizzaro et al., 2024).

Enumeration:

Efficient, linear-delay algorithms enumerate all DAGs in an MEC (or MPDAG) using maximum cardinality search on chordal components, achieving $O(n+m)$ output delay (Wienöbst et al., 2023).
All DAGs in a MEC can be ordered so that consecutive members differ by structural Hamming distance at most $3$; $1$ or $2$ is not always achievable (Wienöbst et al., 2023).

4. Sampling, Search, and MCMC on MECs

Uniform Sampling and Chain Dynamics:

Markov chains with a perfect set of operators perform uniform sampling inside the space of CPDAGs (MECs) or inside all DAGs within an MEC. Valid moves include directed/undirected edge insertions and deletions, and v-structure manipulations constrained to stay within the CPDAG space (He et al., 2012, Bernstein et al., 2017).
On sparse graphs, most CPDAGs have most edges directed, and the undirected subgraph fragments into small components—favorable for fast mixing and sampling (He et al., 2012, Bernstein et al., 2017).

Model Search:

Essential graph (CPDAG) representations allow score-based and constraint-based algorithms (e.g., GES, PC) to traverse equivalence classes rather than individual DAGs, improving algorithmic tractability (Ali et al., 2012, Guo et al., 2020).
Integer-programming algorithms (e.g., MEC-IP) combine clique-based data-driven pruning and IP optimization to efficiently recover CPDAGs/MECs from observational data (Elrefaey et al., 2024).

5. Interventions and Extension to I-MECs

Interventional Markov Equivalence Classes (I-MECs):

Once interventions are performed, more edge orientations become compelled, so the corresponding I-essential graph has fewer undirected edges, and the I-MEC is smaller.
For random DAGs of bounded density, the expected log-size of the MEC is constant as $n\to\infty$ (for ρ=0.5, $E[\log_2 |MEC|] \leq 3.497$ ), and the number of interventions required to fully determine the DAG is also constant, typically 1–2 (Katz et al., 2019).
Lazy enumeration and counting algorithms exploit the incremental effect of interventions for efficient experiment design (AhmadiTeshnizi et al., 2020).

6. Causal Effect Identification and Adjustment under MECs

Identification Theory:

The identifiability of causal effects from an MEC (or a more refined MPDAG) depends on graphical path conditions and, for adjustment, the avoidance of undirected “backdoor” first edges (Guo et al., 2020, LaPlante et al., 2023, Jaber et al., 2018).
Minimal enumeration algorithms partition an MEC into subclasses corresponding to distinct total-effect functionals, yielding all possible distinct causal estimands compatible with the equivalence structure (Guo et al., 2020).
For robust identification under multiple candidate MECs with background knowledge, simultaneous identifiability criteria generalize adjustment to families of MPDAGs, provided certain graphical and marginalization equivalences hold for all members (Teh et al., 18 Jun 2025).
In settings with latent confounding, the equivalence class is represented as a PAG, and completeness of adjustment, identification, and partial identification are characterized via invariant edge marks (tails/arrowheads) and discriminating path properties (Jaber et al., 2018, Zhang, 2012, LaPlante et al., 2023).

7. Broader Generalizations, Latents, and Logical/Combinatorial Constraints

Latent Structure and Marginal Models:

In the presence of latent variables, the statistical object of interest is often a partial ancestral graph (PAG) or a joined graph summarizing a MAG (maximal ancestral graph) equivalence class. Such classes may be infinite when allowing arbitrary latent expansions, but minimal canonical models can be isolated using graphical operations (Ali et al., 2012, Desjardins, 2013).
Markov equivalence classes also extend to more general independence models (e.g., directed mixed graphs under μ-separation) and allow unique maximal graphical representatives (DMEG) to be constructed directly from independence statements (Mogensen et al., 2018).

Algorithmic and Logical Perspectives:

Fast algorithms for MEC discovery, enumeration, and experiment design leverage graphical properties, chordal decompositions, logical constraints (C $^2$ ), and integer programming, enabling practical application up to hundreds or thousands of nodes in sparse regimes (Wienöbst et al., 2023, AhmadiTeshnizi et al., 2020, Elrefaey et al., 2024, Bizzaro et al., 2024).
The complexity of membership testing (given an MEC and data) depends exponentially on the maximum undirected clique size $s$ but not on the maximum indegree (which typically governs learning complexity), allowing efficient verification in high-degree, low-clique-size scenarios (Zhang et al., 2024).

In summary, a Markov Equivalence Class (MEC) succinctly describes the set of DAG models statistically indistinguishable from observational data, with graphical characterizations (skeleton, v-structures), efficient representations (CPDAG, MPDAG, PAG), and a well-developed toolchain for enumeration, counting, experiment design, and inference that underpins modern algorithmic and theoretical research in causal discovery and graphical models (Guo et al., 2020, Wienöbst et al., 2023, Ali et al., 2012, Zhang, 2012, Jaber et al., 2018, Teh et al., 18 Jun 2025, LaPlante et al., 2023, AhmadiTeshnizi et al., 2020, Elrefaey et al., 2024, He et al., 2016, Sharma, 2023, Radhakrishnan et al., 2017, Bernstein et al., 2017, Katz et al., 2019, Bizzaro et al., 2024, Zhang et al., 2024).