Complete Causal Explanations in DAG Models

• Complete causal explanations are formal constructs that exactly match a DAG's conditional independencies with a dependency model, ensuring no surplus or omissions.
• The approach uses a four-phase algorithm to reconstruct and orient the DAG from observed independencies and background knowledge, guaranteeing consistency.
• Incorporating background knowledge, the method distinguishes robust causal directions from ambiguous ones using maximally oriented graphs.

Complete causal explanations are formal objects that aim to capture, without surplus or omission, the causal content warranted by a representation of data and prior knowledge. In the graph-theoretic framework of dependency models and directed acyclic graphs, the term denotes a DAG whose entailed conditional independencies are exactly the observed independencies, optionally subject to required and forbidden edge orientations from background knowledge (Meek, 2013). In later work, the same phrase is extended to symbolic inference systems, structural causal models, concept-based explainers, drift analysis, and image classification, but the common theme remains the same: completeness is always relative to a specified causal representation, query class, and admissible interventions, rather than a claim of unrestricted access to all causal facts about the world (Besnard et al., 2010).

1. Formal definition in dependency models and DAGs

The canonical formalization starts from a finite set of variables $V$ and a dependency model $M$, where statistical information from a sample is represented as a list of conditional independence statements $A \perp B \mid S$. A DAG $G$ links this representation to probability through the local directed Markov property, and entailment is understood equivalently by $d$-separation. Within this setting, a DAG $G$ is a complete causal explanation of $M$ iff the set of independencies entailed by $G$ is exactly $M$: $G$ neither misses any independence in $M$0 nor implies any independence not in $M$1 (Meek, 2013).

This definition is tightly connected to Markov equivalence. Two DAGs are Markov equivalent iff they have the same adjacencies and the same unshielded colliders, equivalently iff they have the same pattern. The pattern of a DAG records all adjacencies and all oriented unshielded colliders, but need not orient every remaining edge. Consequently, completeness at the level of statistical explanation does not imply uniqueness of the underlying DAG. A complete causal explanation may therefore be one member of a whole Markov equivalence class.

The framework also incorporates background knowledge as $M$2, where $M$3 is a set of forbidden directed edges and $M$4 is a set of required directed edges. A complete causal explanation consistent with $M$5 is then a DAG whose implied independencies match $M$6 exactly and whose orientations admit a consistent DAG extension respecting all requirements and forbiddances. This makes completeness explicitly relational: it is defined jointly by the dependency model and the admissible orientation constraints.

2. Existence of complete causal explanations

The existence problem asks whether there is a complete causal explanation for $M$7, either without background knowledge or consistent with $M$8. The algorithmic solution proceeds in four phases. Phase I reconstructs the pattern from independencies by first building the undirected skeleton through separator sets and then orienting unshielded colliders. In particular, $M$9 is retained exactly when there is no separating set $A \perp B \mid S$0 such that $A \perp B \mid S$1, and an unshielded triple $A \perp B \mid S$2 is oriented as $A \perp B \mid S$3 when $A \perp B \mid S$4 (Meek, 2013).

Subsequent phases resolve orientation and verification. Phase II′ closes the pattern under sound orientation rules to obtain all orientations compelled by the equivalence class when no background knowledge is used. Phase II″ incorporates $A \perp B \mid S$5: it fails immediately on hard conflicts, orients required edges, and repeatedly closes under orientation rules $A \perp B \mid S$6–$A \perp B \mid S$7. Phase III then extends the partially directed graph to a DAG. A key correction to earlier work is that no backtracking is needed: if a consistent extension exists at all, any choice of orientation for an undirected edge can be extended to a DAG consistent with the pattern and the background knowledge. Phase IV finally checks acyclicity and exact agreement with $A \perp B \mid S$8, either by verifying all independencies or, when $A \perp B \mid S$9 is a graphoid, by verifying the local Markov independencies induced by a topological order.

The resulting existence criterion is exact. There exists a complete causal explanation $G$0 consistent with background knowledge $G$1 iff Phase I yields a valid pattern, Phase II″ does not fail, Phase III fully orients the graph into a DAG, and Phase IV verifies that $G$2 entails exactly the independencies in $G$3. In this sense, completeness is not merely an informal ideal; it is algorithmically decidable under the stated assumptions.

3. Common causal relations across all complete explanations

A second question concerns what is invariant across all complete causal explanations. If at least one complete causal explanation exists, then all such DAGs are Markov equivalent, and the pattern recovered in Phase I characterizes the entire class. The issue is therefore not only whether an explanation exists, but which edge directions are common to every explanation (Meek, 2013).

Without background knowledge, this is answered by closing the pattern under rules $G$4–$G$5. The resulting graph $G$6 is maximally oriented: every directed edge in $G$7 has the same direction in every DAG in the equivalence class, while every undirected edge can be oriented either way in some DAG in the class. Directed edges in $G$8 are therefore robust causal directions; undirected edges mark genuine ambiguity left unresolved by independence information alone.

With background knowledge, Phase II′ is followed by Phase II″ to obtain $G$9, the maximally oriented graph with respect to $d$0. In that graph, every directed edge is oriented identically in every DAG that both belongs to the equivalence class and respects the background knowledge, while every undirected edge remains reversible in some consistent DAG extension. This directly blocks a common misconception: a complete causal explanation of the dependency model does not imply complete orientation of all causal relations. What is complete here is the match between graphical independencies and $d$1, together with a complete account of which orientations are compelled and which remain underdetermined.

The same framework shows that $d$2 is a chain graph, and likewise $d$3 under background knowledge. Chain graphs thus serve as compact representations of “partial but complete” causal information: all compelled directions are explicit, and all unresolved directions remain explicit as undirected edges.

4. Background knowledge, maximality, and soundness

Background knowledge in this framework is purely orientational, but its effect is substantial. Independence relations alone determine a Markov equivalence class, not a unique DAG. Required and forbidden orientations can therefore either rule out entire members of that class or force new orientations that then propagate through the sound orientation rules. In the extreme case, $d$4 can eliminate existence altogether when it contradicts an orientation already forced by the pattern (Meek, 2013).

The formal notion of maximality is central. The graph $d$5 is maximally oriented with respect to $d$6: every edge that must be oriented under the pattern and $d$7 is oriented, while each remaining undirected edge is still reversible in some consistent DAG extension that respects all requirements and forbiddances. This yields an exact separation between identified and unidentified causal directions under the joint information in $d$8 and $d$9.

The combined algorithm is sound and complete for two tasks: deciding whether there exists a DAG that is a complete causal explanation of $G$0 consistent with $G$1, and inferring all and only those edge directions common to every such DAG. Its validity, however, depends on restrictive assumptions. The analysis is for DAGs over observed variables only; latent variables are excluded. The dependency model $G$2 is treated as the true independence model rather than as an estimate from finite data. The framework also presupposes faithfulness or stability, namely that the observed independencies are exactly the Markov independencies of some DAG. This suggests that the original notion of complete causal explanation is exact but deliberately idealized.

5. Related explanatory formalisms beyond dependency models

In other literatures, “complete causal explanations” retain the idea of exactness relative to a formal base, but the represented objects differ. In causal Bayesian networks, causal explanation trees construct explanations by recursively selecting variables with maximal causal information flow to the explanandum under interventions. Completeness is not defined as a theorem there, but is approximated by continuing the tree until residual causal information falls below a threshold, so that the tree captures the major causal contributors while remaining concise (Nielsen et al., 2012).

In symbolic causal reasoning with ontology, completeness is relative to a fixed causal base $G$3, an ontology $G$4, and a background theory $G$5. Explanation atoms of the form $G$6 explains $G$7 bec_poss $G$8 are generated by base rules, IS-A inheritance, transitivity, and simplification. The intended completeness notion is syntactic or pattern-wise: the system aims to provide all the eventually tentative explanations that can be obtained from the given causal and ontological information, rather than a theorem about all possible causal explanations in a broader semantic sense (Besnard et al., 2010).

In structural-causal-model work, completeness is often phrased as structural coverage. Structural Causal Explanation recursively traverses all parents of a queried variable and continues to roots, so every causal ancestor on directed paths to the target is included in the explanation tree; the authors describe this as structural completeness relative to the given SCM and the class of single-variable why-questions, while explicitly leaving open whether the proposed explanation rules are logically complete (Zečević et al., 2021). In the action-guiding framework for XAI, the closest analogue is a good sufficient explanation $G$9, together with actual causes and, when recourse is required, a good counterfactual explanation sharing witness and safeguarded network. There, completeness is tied to strong sufficiency, minimality, and action constraints, not to exhaustive upstream coverage (Beckers, 2022). In stochastic sequential decision-making, completeness is model-relative again: for layered MDPs with finite horizons and discretized variables, the framework can in principle enumerate all weak and actual causes of an action across state-factor, reward, transition, and value explanations (Nashed et al., 2022).

These usages are not identical, but they share a common structure. Completeness is always indexed by a representation—dependency model, ontology, SCM, or MDP—not by an unrestricted metaphysical notion of total explanation.

6. Contemporary extensions, limits, and recurring misunderstandings

More recent work extends the vocabulary of complete causal explanations to settings where the explanatory target is a model, a concept set, a data disparity, or a drift process. For visual models, causal attribution over latent factors is presented as a step toward complete causal explanations because interventions propagate through a structural causal model rather than through observational perturbations, yet the work is explicit that current image generators prevent fully complete explanations for real images (Parafita et al., 2019). In concept-based black-box distillation, DiConStruct is completeness-aware in the sense that, because the target is the black-box score $M$0, the completeness criterion reduces to how well a concept-only model $M$1 predicts that score; the result is a local SCM over concepts and the black-box output rather than a full explanation of the underlying world (Moreira et al., 2024).

Other contemporary usages are even more operational. ExDis treats a disparity explanation as a pair $M$2 consisting of a subpopulation pattern over immutable attributes and a treatment pattern over mutable attributes, and seeks a set of such explanations maximizing total disparity score under support and diversity constraints; a more complete explanation set is therefore one that covers more of the differential causal behavior across the data space (Blau et al., 9 Dec 2025). For concept drift, complete causal explanations are drift-reversing intervention sets: the children of time form the minimal core whose mechanisms must change, while the children of time together with their ancestors form a full drift-reversing set (Komnick et al., 31 Jul 2025). In image classification, a complete explanation is defined even more sharply as a 1-exact explanation, namely a subset of pixels that is classified with exactly the same confidence as the original image (Kelly et al., 31 Jul 2025).

A final adjacent development concerns not empirical explanation but logic. In the categorical setting of $M$3, completeness refers to a sound and complete characterization of causal consistency via graph types and a causal logic extending pomset logic. Here the relevant result is not a complete causal explanation of data, but a complete logic for when higher-order processes are causally consistent (Simmons et al., 2024).

These variations make one recurring misunderstanding especially clear. “Complete causal explanation” is not a single invariant technical term across the literature. In the DAG-and-independence tradition it means exact equality between the independence model of a DAG and $M$4, possibly with background knowledge. In later explanation literatures it usually means complete relative to a causal base, a graph, a concept set, a query class, or a confidence criterion. This suggests that the phrase is best read as a family resemblance term whose exact meaning is fixed by the representation, admissible interventions, and verification conditions adopted in each framework.