Identifying causal effects from observational data typically requires a fully specified causal graph, an assumption that is often unrealistic in complex or high-dimensional settings. Causal abstractions address this limitation by retaining only partial causal information, thereby defining equivalence classes of compatible causal graphs. In this paper, we study causal effect identifiability directly at the level of these graph collections, rather than at the level of individual causal diagrams. We formalize several notions of identifiability in this setting and organize them into a hierarchy that clarifies their relationships and differences. In particular, we establish that a causal effect may be identified over a collection of graphs without admitting a single causal derivation that holds uniformly across all compatible graphs. These results help reveal gaps and clarify ambiguities in the existing literature.
Figures & tables
Figure 1: Hierarchy between the notions of identifiability. An arrow indicates that the implication holds in every case. A red cross on an arrow indicates that the implication fails in at least one case.
Figure 2: Derivation graph of the DAG from Example 2 . Blue nodes denote interventional queries, while orange nodes denote do-free queries. Edges of the derivation graph represent valid applications of the do-calculus rules in the DAG: gray solid lines for Rule R1 , orange dashed lines for Rule R2 , and blue dotted lines for Rule R3 . Only vertices that are non-isolated are displayed.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: The Napkin Graph ( Pearl and Mackenzie, 2018 ; Guo et al., 2025 )
Figure 4: Intersection of the derivations graphs of G1 and G2 . Blue nodes denote interventional queries, while orange nodes denote do-free queries. Edges represent applications of the do-calculus rules: gray solid lines for Rule R1 , orange dashed lines for Rule R2 , and blue dotted lines for Rule R3 . Only non-isolated queries are displayed.
Figure 5: Derivation graph of G3 . Blue nodes denote interventional queries, while orange nodes denote do-free queries. Edges represent applications of the do-calculus rules: gray solid lines for Rule R1 , orange dashed lines for Rule R2 , and blue dotted lines for Rule R3 . Only non isolated queries are displayed.
Clustering variables in causal graphs reduces the size of the graph and simplifies causal inference. However, arbitrary clustering can alter crucial causal relations among variables and lead to erroneous conclusions. While the identifiability of a causal effect in the clustered graph implies the identifiability in the original graph under mild conditions, nonidentifiability in clustered graph does not imply nonidentifiability in the original graph without further assumptions. When both identifiability and nonidentifiability are preserved, the clustering operation is called identification invariant. We present a broad class of clustering operations that are identification invariant based on conditions related to the c-components of the original graph. Finally, we demonstrate use of the results in practical settings.
Jani Nykänen, Otto Tabell, Santtu Tikka +1
Department of Mathematics and Statistics FI-40014 University of Jyvaskyla, Finland
Determining identifiability of causal effects from observational data under latent confounding is a central challenge in causal inference. For linear structural causal models, identifiability of causal effects is decidable through symbolic computation. However, standard approaches based on Gröbner bases become computationally infeasible beyond small settings due to their doubly exponential complexity. In this work, we study how to practically use symbolic computation for deciding rational identifiability. In particular, we present an efficient algorithm that provably finds the lowest degree identifying formulas. For a causal effect of interest, if there exists an identification formula of a prespecified maximal degree, our algorithm returns such a formula in quasi-polynomial time.
Benjamin Hollering, Pratik Misra, Nils Sturma
Max Planck Institute for Mathematics in the Sciences, Germany · Binghamton University, USA · École Polytechnique Fédérale de Lausanne, Switzerland
We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.