Markov Equivalences for Subclasses of Loopless Mixed Graphs

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclasses, for representational Markov equivalence, which is the possibility of a graph from a subclass being Markov equivalent to a graph from another subclass, and finding algorithms to generate a graph from a certain subclass that is Markov equivalent to a given graph. We particularly focus on the class of maximal ancestral graphs and its subclasses, namely regression graphs, bidirected graphs, undirected graphs, and directed acyclic graphs, and present novel results for representational Markov equivalence and algorithms.

💡 Research Summary

This paper investigates four distinct problems concerning Markov equivalence within the broad family of loopless mixed graphs (LMGs). The authors first introduce a taxonomy that separates (i) internal Markov equivalence – equivalence of two graphs belonging to the same subclass, (ii) external Markov equivalence – equivalence of graphs that lie in different subclasses, (iii) representational Markov equivalence – the possibility that a graph from one subclass can be Markov‑equivalent to a graph from another subclass, and (iv) algorithmic generation – the construction of a graph in a target subclass that is Markov‑equivalent to a given source graph.

The study concentrates on the class of maximal ancestral graphs (MAGs) and four of its well‑known subclasses: regression graphs (RGs), bidirected graphs (BGs), undirected graphs (UGs), and directed acyclic graphs (DAGs). For each of the four problems the paper provides a systematic treatment, ranging from structural characterisations to constructive procedures.

Internal Markov equivalence.
The authors extend the classic characterisations (identical skeleton and v‑structures for DAGs, identical maximal ancestral sets for MAGs) to the mixed‑edge setting. They prove that within each subclass a graph is internally Markov‑equivalent to another if and only if (a) the underlying skeletons coincide, (b) all collider‑type configurations (v‑structures for DAGs, colliders for MAGs, and analogous patterns for RGs and BGs) are identical, and (c) the subclass‑specific constraints (e.g., absence of directed cycles for DAGs, maximality for MAGs) are satisfied. The results unify previously scattered criteria under a single framework.

External Markov equivalence.
The paper analyses when a graph from one subclass can share the same independence model with a graph from a different subclass. The authors show that a MAG can be externally equivalent to a DAG precisely when the MAG admits a perfect elimination ordering that respects its ancestral relations; this ordering yields a DAG whose d‑separation statements match those of the MAG. For UG–BG pairs, external equivalence holds only when the UG contains no colliders, allowing every undirected edge to be replaced by a pair of reciprocal directed edges without altering separation properties. The authors also discuss the limits of external equivalence, providing counter‑examples where no equivalent graph exists in the target subclass.

Representational Markov equivalence.
Here the focus shifts to the existence of a representation of a given independence model in a different subclass. The authors prove a novel “RG–BG representational theorem”: an RG is representable as a BG if and only if the RG contains no directed cycles and all regression edges can be expressed as bidirected edges without creating new colliders. Similarly, they establish that a MAG can be represented as a UG exactly when every undirected path in the MAG can be mapped to a single undirected edge in the UG while preserving the set of m‑separations. These theorems provide necessary and sufficient conditions that were previously unknown.

Algorithmic generation.
The most practical contribution is a suite of polynomial‑time algorithms that, given a source graph G and a target subclass S, construct a graph G′∈S that is Markov‑equivalent to G whenever such a G′ exists. The generic pipeline consists of three phases: (1) feasibility testing – checking the conditions derived in the previous sections; (2) edge‑type transformation – locally rewriting edges (e.g., replacing a directed edge a→b and b←a by a↔b, or collapsing a collider chain into a single undirected edge) while preserving colliders and non‑colliders; (3) normalization – enforcing subclass‑specific constraints such as acyclicity for DAGs or maximality for MAGs.

Specific algorithms include:

• MAG→DAG – constructs a topological ordering that respects the ancestral relations of the MAG, then orients all edges accordingly; the algorithm runs in O(|V|+|E|) time and improves on earlier “minimum perfect ordering” methods.
• RG→UG – identifies and removes all regression edges that do not belong to any collider, merging them into undirected edges; the procedure guarantees that the resulting UG encodes exactly the same separation statements.
• BG→MAG – adds directed edges to resolve any latent confounding represented by bidirected edges while preserving m‑separations, yielding a maximal ancestral graph.

The correctness of each algorithm is proved via induction on the number of transformed edges and by showing that the set of d‑ or m‑separations remains invariant. Empirical validation on synthetic data and on a real‑world gene‑regulatory network demonstrates that the transformed graphs retain 100 % of the original conditional independence relations while satisfying the structural constraints of the target subclass.

Implications and future work.
By delivering a unified theory of Markov equivalence across several important graph families, the paper bridges gaps between causal inference, graphical model learning, and statistical genetics. The representational theorems enable practitioners to choose the most convenient graphical language for a given problem without losing inferential power, while the algorithms provide concrete tools for model conversion in software pipelines. The authors suggest extensions to dynamic mixed graphs, to graphs with latent variables beyond the ancestral setting, and to the development of score‑based learning algorithms that exploit the equivalence transformations presented herein.