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Causal Effect Identifiability in Causal Abstractions: A Strict Hierarchy of Criteria
Clément Yvernes, Marianne Clausel, Emilie Devijver, Eric Gaussier
TL;DR
Fully specified causal graphs are often unavailable, motivating causal abstractions that retain partial causal information and define collections of compatible diagrams. This paper formalizes and hierarchically compares identifiability notions over those collections, showing that graph-based identifiability can hold without a single derivation valid across all graphs.
Problem
Method
The paper formalizes several identifiability notions over graph collections and compares them using a hierarchy and derivation-graph and step-graph tools.
Source: §6 · p. 15Results
The hierarchy establishes strict separations, including that ICA does not imply ICD and IG strictly subsumes ICD.
Source: §1 · p. 2, §6 · p. 15Takeaways & Limitations
The hierarchy provides a basis for understanding and designing causal abstractions with controlled identifiability properties.
Source: §6 · p. 15IG and IGP are difficult to verify, and IGP depends on the usually unavailable true observational distribution.
Source: §3.1 · p. 7
Abstract
from arXiv · showhide
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.
1 Introduction
Fully specified causal graphs are often unavailable, while causal abstractions leave multiple compatible diagrams and make identifiability depend on how claims are required to hold across them. The paper unifies these notions and establishes strict relationships among them.
1 Introduction
- Identifying causal effects from observational data typically requires a fully specified graph, but that knowledge is often unavailable in complex or high-dimensional settings.Interventions may also be impractical because of ethical concerns, potential harm, or prohibitive costs.Source: §1 · p. 1
- Causal abstractions retain partial causal information and induce classes of compatible diagrams, where different graphs may yield different answers to the same query.Several identifiability notions have been proposed, but they are usually developed independently.Source: §1 · pp. 1–2
- The paper unifies several notions of identifiability over collections of causal graphs and establishes a strict hierarchy for comparing their relative strength.The hierarchy can assess whether existing theory misses identifiable queries for a given abstraction or fully captures them.Source: §1 · p. 2
- The hierarchy shows that adjustment-based identifiability (ICA) does not imply uniform do-calculus identifiability (ICD), while the known converse also fails.Graphical identifiability (IG) strictly subsumes ICD.Source: §1 · p. 2
2 Preliminaries
The preliminaries establish graph and structural causal model terminology, then review how causal effects are defined and identified in a single graph. They distinguish semantic identifiability from derivation-based procedures and graphical criteria.
2.1 Foundations of Causal Modeling
- An SCM specifies measured variables through structural functions and exogenous variables; its causal diagram is an ADMG, and interventions replace functions and remove incoming edges.A DAG is an ADMG without bidirected arrows.Source: §2.1 · pp. 2–3
- A causal query is an interventional probability, and an identification functional maps observational distributions to values that identify that query.The paper also defines compatibility through the existence of an SCM inducing a graph and observational distribution.Source: §2.1 · p. 3
2.2 Identifiability in a Single Graph
- In a single graph, identifiability is the existence of a valid functional, while a causal derivation is a sequence of do-calculus and probability transformations valid under the graph’s required conditions.Do-calculus is sound and complete for identification in causal diagrams, and the ID algorithm constructs a derivation when a query is identifiable.Source: §2.2 · pp. 3–5
- Do-calculus transforms interventional expressions through three rules whose applications are licensed by graphical conditions, together with standard probability manipulations.The rules include insertion or deletion of observations, action-observation exchange, and insertion or deletion of actions.Source: §2.2 · p. 4
- Graphical criteria can provide direct identification strategies: backdoor adjustment uses a set that is not a descendant of X and blocks paths into X, while frontdoor adjustment uses mediators under its stated path conditions.The frontdoor result also requires P(x, z) > 0.Source: §2.2 · p. 5
3 Identifiability in a Collection of Graphs.
For a collection of causal diagrams, identifiability has multiple meanings because graphs can support different answers and proof strategies. The paper distinguishes settings based on available information and on whether one identification strategy must work uniformly across the collection.
3 Identifiability in a Collection of Graphs.
- Causal abstractions induce collections of diagrams, and identifiability over a collection can be ambiguous because different compatible graphs may yield different answers to the same query.The broad formulation asks whether the same identification functional can be derived for every graph in the class.Source: §3 · p. 6
3.1 Identifiability Under Different Knowledge Conditions
- IG requires a functional valid across all SCMs whose graphs belong to the class, whereas IGP additionally conditions on knowledge of the true observational distribution.IGP retains a collection of admissible SCMs constrained by both graph and distribution.Source: §3.1 · pp. 6–7
- IG and IGP are difficult to verify: IG requires validity over all graphs in the class, while IGP depends on the usually unavailable true observational distribution.The paper therefore focuses on graphical characterizations and sufficient conditions for identifiability.Source: §3.1 · p. 7
3.2 Identifiability Under Different Methodological Conditions
- IG permits different graphical identification procedures for different graphs if they yield the same functional, whereas ICD requires a single derivation valid in every graph.Thus graph-based identifiability can hold without one common causal derivation.Source: §3.2 · pp. 7–8
- Method-based notions require one graphical criterion or causal derivation to hold uniformly across the collection, including common backdoor, frontdoor, adjustment, and do-calculus strategies.An atomically complete calculus characterizes which do-calculus transformations hold uniformly and can establish ICD.Source: §3.2 · pp. 7–8
- Common backdoor identifiability holds in the example because {Z} is a valid backdoor set in both graphs, yielding the same adjustment formula.The example identifies the effect using P(y | do(x)) = Σ_z P(y | x, z)P(z).Source: §3.2 · p. 8
4 The Strict Hierarchy of Identifiability in Collections of Graphs
The paper establishes a strict hierarchy among identifiability notions for collections of causal diagrams, with counterexamples showing that several converse implications fail. In particular, identifiability across graphs need not have a single do-calculus derivation that works in every graph.
4.1 The Hierarchy of Identifiability
- IG implies IGP, ICD implies IG, ICB and ICF imply ICD, and ICA implies IG.Theorem 5 summarizes the valid implication relations among the identifiability notions.Source: §4.1 · pp. 8–9
4.2 Strictness of the Hierarchy
- None of the converse implications in the hierarchy holds in general, and ICA and ICD are incomparable.Figure 1 marks implications that hold universally and counterexamples to converses.Source: §4.2 · p. 9, Figure 1 · p. 9
4.2.1 Straightforward Separations
- Known single-graph incompleteness results for adjustment criteria yield converse failures when a collection contains just one diagram.The paper cites the back-door, front-door, and generalized adjustment criteria as incomplete, unlike do-calculus (Perković et al., 2015).Source: §4.2.1 · p. 10
- Knowing the observational distribution can identify an effect over a graph collection even when the graph structure alone does not.For the class containing X → Y and X ← Y, independence of X and Y makes P(y | do(x)) identifiable given P⋆, but not from the class alone.Source: §4.2.1 · p. 10
4.2.2 ICA, ICD and IG
- ICA can hold when ICD fails: a common adjustment criterion may identify a query without a derivation shared across all graphs.The paper extends this separation to IG, which does not imply ICD; hence identifiability through graphs need not admit a common derivation.Source: §4.2.2 · p. 12
- Atomic completeness is stronger than necessary for IG, because IG permits graph-specific derivations where a uniform derivation may not exist.The paper argues that characterizing IG may require identification tools more flexible than atomically complete calculi.Source: §4.2.2 · p. 13
5 Discussion
The discussion treats IG as a completeness target for purely graphical abstractions and explains what its failure means. It also shows that identifiability can be assessed using only maximal graphs, under the stated existence assumption.
5.1 Interpretation of the Hierarchy
- Completeness claims are relative to the hierarchy: completeness for ICD does not establish completeness for IG.The paper notes that IG remains open for the abstractions and queries considered, although an atomically complete calculus solves ICD.Source: §5.1 · p. 13
- For a purely graphical abstraction, IG is the least restrictive notion and guarantees that no effect identifiable from that information is missed.If IG fails, identifying the effect requires additional information, such as a refined abstraction or non-graphical assumptions.Source: §5.1 · p. 13
5.2 Maximal Elements Is All You Need
- Under Assumption 1, each identifiability notion considered is equivalent over a graph class and its maximal elements.Assumption 1 requires every graph in the class to be contained in a maximal element.Source: §5.2 · pp. 13–14
- Identifiability is easier to assess when the class has few maximal graphs, and a greatest graph reduces the analysis to one graph.The reduction follows from monotonicity under edge deletion.Source: §5.2 · pp. 13, 15
- If the class has a greatest graph, IG and ICD are equivalent over that class.Theorem 7 reduces both notions to the single greatest graph, where do-calculus is sound and complete.Source: §5.2 · p. 15
6 Conclusion
The paper unifies identifiability notions for causal abstractions and establishes a strict hierarchy whose counterexamples distinguish their relative strength. Derivation graphs and step graphs help explain why existing approaches may not capture every identifiable query.
6 Conclusion
- The framework formalizes IG, ICD, IGP, ICA, ICB, and ICF as a common language for comparing previously disparate notions of identifiability.These notions apply to causal abstractions represented as collections of causal graphs.Source: §6 · p. 15
- Counterexamples establish a strict hierarchy, show that natural implications fail, and leave ICA and ICD incomparable.The hierarchy also motivates open questions about efficient procedures for deciding IG and ICD and their computational complexity.Source: §6 · p. 15
- Derivation graphs and step graphs provide a fine-grained way to analyze causal derivations across graph collections and explain why existing approaches may miss identifiable queries.The paper specifically notes that atomically complete calculi do not necessarily capture all such queries.Source: §6 · p. 15
A.1 Proofs of Section 4.1
The appendix verifies the hierarchy's stated implications by proving them separately. Its arguments use common adjustment, backdoor, and frontdoor criteria to obtain derivations valid across the graph collection.
A.1 Proofs of Section 4.1
- Theorem 5 states that the implications between identifiability notions summarized in Figure 1 hold for a class of causal diagrams and disjoint query variables.The proof treats each implication separately.Source: §A.1 · p. 18
- A common backdoor set yields common adjustment and common derivation, because the backdoor criterion supplies a valid derivation in every graph.The proof applies marginalization, the chain rule, and do-calculus rules to obtain the observational adjustment expression.Source: §A.1 · p. 18
- A common frontdoor set yields a common derivation, using the frontdoor criterion and do-calculus transformations in each graph.The resulting derivation is stated to hold for every graph in the collection.Source: §A.1 · p. 19
- The appendix establishes ICA ⇒ IG, ICD ⇒ IG, and IG ⇒ IGP through the cited theorems and propositions.The proof of ICA ⇒ IG begins with a common adjustment set; the other two implications are attributed to Propositions 2 and 1.Source: §A.1 · p. 19
A.2 Proofs of Section 4.2
The appendix proves that the hierarchy is strict and that ICA and ICD are incomparable. Counterexamples show failures of converse implications, including cases where identification notions differ over a graph collection.
A.2 Proofs of Section 4.2
- None of the converse implications in the hierarchy holds in general, and ICA and ICD are incomparable in both directions.The text identifies failures of ICA ⇒ ICD and IGP ⇒ IG, among others.Source: §A.2 · p. 20
- A single graph also witnesses ICA without ICB: a descendant variable can satisfy the adjustment criterion without satisfying the backdoor criterion.The example uses the graph Z ← X → Y with Z as the adjustment set.Source: §A.2 · p. 20
- The paper uses the Napkin Graph as a single-graph counterexample for failures including ICD ⇒ ICA, IG ⇒ ICA, ICD ⇒ ICB, and ICD ⇒ ICF.The query P(y | do(x)) is identifiable in the Napkin Graph.Source: §A.2 · p. 20
- In one collection, ICD and IG hold while ICB, ICA, and ICF fail because no candidate adjustment set works across the graphs.Conditioning on W or Z opens one backdoor path, while conditioning on neither leaves another open.Source: §A.2 · p. 20
A.2.1 Proofs of Section 4.2.1
A two-graph example separates identifiability from the graph collection alone from identifiability when the observed distribution is also known. For independent X and Y in the true data distribution, IGP holds although IG does not.
A.2.1 Proofs of Section 4.2.1
- For the collection {X → Y, X ← Y}, the query P(Y = 1 | do(X = 1)) is not identifiable through graphs alone.Two compatible structural causal models yield different interventional probabilities.Source: §A.2.1 · p. 21
- The two models give P(Y = 1 | do(X = 1)) values of 9/10 and 1/2, respectively.These differing values establish non-identifiability through the graph collection.Source: §A.2.1 · p. 21
- When X and Y are independent in the true data distribution, P(Y = 1) identifies the query across compatible models, so IGP holds but IG does not.The paper gives the functional F: P(X, Y) ↦ P(Y = 1) for this data-dependent identification.Source: §A.2.1 · p. 21
A.3 Proofs of Section 4.2.2
This proof gives a counterexample in which common adjustment identifies a causal effect across a graph class, but common derivation does not. Its argument uses two compatible graphs and a third graph to contradict the existence of a derivation uniform across the class.
A.3 Proofs of Section 4.2.2
- Common adjustment can identify P(y | do(x), z) across a graph class even when common derivation cannot.Proposition 4 states this failure of ICA ⇒ ICD.Source: §A.3 · p. 21
- In G1 and G2, {Z} satisfies the adjustment criterion, so common adjustment holds for P(y | do(x)).The proof takes C = {G1, G2}, with G1 := Z ← X → Y and G2 := X ← Z → Y.Source: §A.3 · p. 21
- The proof adds G3 := X → Z → Y and argues that a derivation common to G1 and G2 would also hold in G3.It uses an enumerated inclusion of derivation graphs, with the complete graphs shown in Figures 4 and 5.Source: §A.3 · pp. 21–22, Figure 4 · p. 22, Figure 5 · p. 23
- Because {Z} fails the adjustment criterion in G3, the constructed distribution yields a contradiction, ruling out common derivation for the class.The construction uses a linear Gaussian SCM whose distribution is Markovian with respect to G3.Source: §A.3 · p. 22
B Proofs of Section 5
This section proves monotonicity results when one causal graph is a subgraph of another. It constructs a model for the larger graph that preserves the smaller graph's distributions, supporting transfers of identification validity, derivations, and graphical criteria.
B Proofs of Section 5
- When G2 is a subgraph of G1, identification functionals valid across the specified G1 models are also valid across corresponding G2 models, and derivations and graphical criteria holding in G1 also hold in G2.The functional statement is conditioned on the same observational distribution P*(V).Source: B Proofs of Section 5 · pp. 23–24
- Consequently, an identification functional valid for all G1 models matching P*(V) is valid for all G2 models matching that distribution.This is the distribution-conditioned transfer established in the lemma's second claim.Source: B Proofs of Section 5 · pp. 23–24
- The proof builds an SCM inducing G1 from one inducing G2 by adding independent exogenous variables for missing bidirected edges and ignoring added arguments.The construction preserves observational and interventional distributions.Source: B Proofs of Section 5 · pp. 23–24