Identifiability in Causal Abstractions: A Hierarchy of Criteria

Abstract: Identifying the effect of a treatment from observational data typically requires assuming a fully specified causal diagram. However, such diagrams are rarely known in practice, especially in complex or high-dimensional settings. To overcome this limitation, recent works have explored the use of causal abstractions-simplified representations that retain partial causal information. In this paper, we consider causal abstractions formalized as collections of causal diagrams, and focus on the identifiability of causal queries within such collections. We introduce and formalize several identifiability criteria under this setting. Our main contribution is to organize these criteria into a structured hierarchy, highlighting their relationships. This hierarchical view enables a clearer understanding of what can be identified under varying levels of causal knowledge. We illustrate our framework through examples from the literature and provide tools to reason about identifiability when full causal knowledge is unavailable.

• The paper rigorously formalizes a hierarchy of identifiability notions for causal queries, extending classical do-calculus frameworks to collections of causal diagrams.
• It demonstrates that identifiability is determined solely by maximal elements, reducing computational complexity in settings with partial structural knowledge.
• The work highlights practical implications for scalable causal inference, improved model uncertainty resolution, and the design of automated discovery pipelines.

Identifiability in Causal Abstractions: A Structured Hierarchy

This work provides a comprehensive formalization of identifiability notions for causal queries in settings where the full causal structure is only partly specified—an increasingly common occurrence in empirical research, high-dimensional applications, and real-world deployments of causal inference. By modeling causal abstractions as collections of causal diagrams rather than single, fixed graphs, the framework acknowledges and systematizes reasoning under partial knowledge, elucidating both the theoretical and practical limits of causal identification.

Formalization and Hierarchy of Identifiability Notions

The paper introduces and rigorously defines several identifiability notions for causal queries under abstraction:

1. Identifiability in a Single Graph (IG): The classic setting where identifiability requires a unique estimand to exist for all structural causal models (SCMs) compatible with a single fixed graph. The proof relies on do-calculus, which is both sound and complete.
2. Identifiability in a Class of Graphs (IG): Here, a causal query is identifiable if a single estimand is valid for every graph in a given class (collection) of possible causal diagrams. This reflects reasoning under structure uncertainty, as in equivalence classes due to limited or ambiguous data.
3. Identifiability with Knowledge of Observational Distribution (IGP): This further restricts attention to SCMs with observational distributions matching the true underlying data distribution. It assumes, in principle, infinite data or perfect knowledge of the observational distribution, thus imposing strong but often impractical assumptions.
4. Identifiability by Common Do-Calculus (ICD): A tractable, practically motivated notion. It requires the existence of a single do-calculus proof, valid for all graphs in the collection. Crucially, only those conditional independence statements that are preserved across all considered diagrams can enter the proof, and the resulting estimand must be universally valid under these constraints.
5. Identifiability by Common Graphical Criterion (ICGC): This generalizes to any common graphical criterion (e.g., backdoor, frontdoor, or adjustment set conditions) that can be simultaneously satisfied across all graphs in the collection. It includes ICD as a special case but is flexible enough to encompass heuristic or domain-specific graphical criteria.

The authors organize these definitions in a clear logical hierarchy, establishing implications and (lack thereof) between notions. For instance, ICGC and ICD are shown to be equivalent, with both strictly stronger than IG, which in turn is stronger than IGP. Non-equivalences are established with explicit counter-examples, and one implication—whether IG and ICD are strictly distinct—remains open as a conjecture.

The logical relations are succinctly represented:

• Every ICGC (or ICD) implies IG, which implies IGP.
• There exist settings where IGP holds but not IG, reflecting cases where only observational equivalence is assumed.
• ICGC and ICD are equivalent, and strictly stronger than ICB (by backdoor) or ICF (by frontdoor).
• Ultimately, the conjectured (but unproven) separation between IG and ICD is posed as an important open question.

Properties and Computational Gains

A fundamental property is established: identifiability in a collection of graphs is determined solely by the maximal elements under graph inclusion. The paper proves that identifiability in the full collection is equivalent to identifiability in the subclass of (possibly much fewer) maximal graphs, greatly reducing computational burden. In particular, if a unique maximal graph exists, the problem reduces fully to the classic single-graph case.

This insight has significant implications for designing and analyzing causal abstractions. Practically, it encourages seeking abstractions with small sets of maximal diagrams, as in cluster DAGs or summary graphs, making verification tractable.

Examples and Links to Existing Literature

The framework is instantiated with canonical examples, such as identifiability in partial ancestral graphs (PAGs), summary causal graphs for time series, and clusters or groupings relevant in reinforcement learning and neural network interpretability. These connections demonstrate both conceptual generality and practical relevance.

Notably, the paper identifies that identifiability by common adjustment, already characterized in summary graphs and Markov equivalence classes, is a specific instance of the ICGC/ICD approach.

Implications for Practice and Theory

For Applied Research and Systems Design:

• Model Uncertainty: The explicit hierarchy clarifies what can and cannot be concluded when only partial causal knowledge is available. It guides practitioners regarding which identification criteria are defensible given their domain assumptions.
• Algorithmic Efficiency: The focus on maximal subgraph reduction provides a pathway for scalable implementations, as only a limited set of candidate graphs must be analyzed for identifiability.
• Automated Causal Discovery Pipelines: The stratified criteria offer blueprints for automated identification routines, where systems can fall back to weaker but still valid identification schemes (e.g., move from ICD to IG to IGP) as knowledge is reduced.

For Theoretical Development:

• Completeness and Separations: The IG-vs-ICD conjecture—whether identification via disparate do-calculus proofs with the same estimand, in different graphs of a collection, can always be unified into a common proof—is posed as a central open problem, potentially yielding insights into the limits of graph-based inference with partial information.
• Compositionality and Abstraction: By situating causal abstractions within this framework, the paper provides bridges to developments in causal representation learning, group-level reasoning, and continual abstraction.

Limitations and Future Directions

• Verification Complexity: Although maximal subgraph reduction can ease computational burdens, the number of maximals can still be exponential in the worst case; further work on efficient enumeration is warranted.
• Expressivity of Graphical Criteria: No known complete graphical criterion exists for ADMGs, limiting the reach of ICGC/ICD without further advances in causal graphical theory.
• Empirical Risk and Approximate Identification: The framework is predicated on exact identification; extensions to settings where only approximate or probabilistic identification is acceptable would be valuable in large-scale and noisy systems.

Prospects for Future AI Developments

Building on this work, future systems may feature adaptive identification pipelines able to migrate between abstraction levels and identifiability notions based on available evidence and computational limits. This has promise for robust AI in uncertain environments, principled explainability via mechanistic interpretability, and scientifically-grounded deployment in healthcare, policy, and industrial control where full structural knowledge is unattainable.

Ultimately, the formal hierarchy and machinery developed in this paper lay the groundwork for sound, scalable, and defensible causal inference in the presence of partial knowledge—an essential component for mature, reliable AI systems in complex domains.