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Invariance of Clustering Operations in Causal Effect Identification
Jani Nykänen, Otto Tabell, Santtu Tikka, Juha Karvanen
TL;DR
Arbitrary clustering can change causal relations, and identifiability in the clustered graph does not by itself settle nonidentifiability in the original graph. The paper develops coherent-cluster and c-component conditions for invariant clustering, establishing classes that preserve identifiability status.
Problem
Method
The paper studies coherent clusters and conditions related to c-components to determine when clustering preserves identifiability status.
Source: §1 · p. 2, §4 · p. 12Results
Plain clusters and c-coherent clusters yield universally identification-invariant clustering operations.
Source: §4 · pp. 10, 12Takeaways & Limitations
For acyclic clustering, identifying effects in the clustered graph also identifies them in the original graph, with the functional obtainable by substitution.
Source: §6 · p. 17The results require clustering to preserve acyclicity; handling cycles in the clustered graph remains future work and requires a modified characterization of nonidentifiability.
Source: §6 · p. 18
Abstract
from arXiv · showhide
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.
1 Introduction
Clustering lets causal models represent high-level relationships without specifying every lower-level mechanism, but it can distort causal structure. This paper seeks conditions for preserving identifiability in both the original and clustered graphs.
1 Introduction
- Clustering can simplify high-level causal modeling and graph-based inference, but arbitrary clustering can make an identifiable effect nonidentifiable in the clustered graph.Clustering also reduces graph size and may simplify identifying functionals or speed identification algorithms.Source: §1 · pp. 1–2
- Identifiability in the clustered graph transfers to the original graph under mild conditions, but the converse is not guaranteed for the graph at hand.Some prior results establish converse nonidentifiability only for some graph compatible with the clustered graph.Source: §1 · p. 2
- Transit-cluster conditions can establish identification invariance, but require stricter constraints than acyclicity alone.The paper notes that these conditions constrain how clusters connect to parents and children.Source: §1 · p. 2
- The paper introduces coherent clusters and c-component-based conditions for identification-invariant clustering operations, generalizing earlier results by Tikka et al. (2023).Its approach asks when clustering cannot create a hedge absent from the original graph.Source: §1 · p. 2
2 Notation and Definitions
The section establishes the graph and causal-effect vocabulary used to study whether clustering preserves identifiability. It also introduces c-components and hedges as structures relevant to that question.
2 Notation and Definitions
- The paper studies DAGs induced by recursive structural causal models and restricts attention to semi-Markovian models, where each unobserved variable appears in at most two structural equations.Observed variables have structural equations determined by their parents and associated unobserved variables.Source: §2 · pp. 3–4
- A causal effect is identifiable when it is uniquely computable from the joint observed distribution in every causal model inducing the graph.The paper considers positive distributions so causal effects and identifying functionals are well-defined.Source: §2 · p. 4
- An operation is identification invariant for an effect when it preserves that effect’s identifiable or nonidentifiable status between the original and transformed graphs.For a family of effects, the property must hold for each effect in the family.Source: §2 · pp. 4–5
- A c-component connects every pair of observed vertices by bidirected paths, while a c-forest generalizes a c-tree by allowing a root set.C-components are associated with identifiability; hedges formed from related c-forests characterize nonidentifiability of general causal effects.Source: §2 · p. 5
- A causal effect is identifiable if and only if no hedge for that effect exists in the graph.The ID algorithm is sound and complete and fails only when it detects such a hedge.Source: §2 · p. 5
3 Clustering in Causal Graphs
Clustering replaces a set of observed vertices with one representative vertex, but unrestricted clustering can create cycles or spurious directed connections. The section defines coherent clusters to constrain these effects.
3 Clustering in Causal Graphs
- Clustering replaces a vertex set with one representative vertex, and the operation excludes unobserved vertices from the cluster.Unobserved variables whose two children lie inside the cluster are omitted from the clustered graph.Source: §3 · p. 6
- Universal identification invariance requires preserving identifiability status for every causal effect whose treatment and outcome variables lie outside the clustered set.The paper focuses on this property for clustering operations.Source: §3 · p. 7
- Arbitrary clustering can create directed cycles or new directed connections, potentially turning an originally identifiable effect into a nonidentifiable one.The paper defines clustering as an operation that outputs a causal graph, or DAG.Source: §3 · p. 7
- A cluster is coherent when every parent-to-child connection through it has a directed path that passes through at least one cluster vertex.This path condition is intended to remedy the side effects of arbitrary clustering.Source: §3 · p. 7
- Coherent clustering does not create directed cycles or new directed connections, and any directed path intersects the cluster in at most one contiguous subpath.Directed reachability between vertices outside the cluster is preserved by clustering.Source: §3 · pp. 7–8
- Transit clusters are coherent, but coherence is broader: it can hold with fewer edges and allows bidirected arcs prohibited by transit-cluster conditions.The paper gives an example of a coherent cluster that is not a transit cluster.Source: §3 · p. 8
4 Identification Invariance of Clustering Operations
For effects outside the clustered set, identification invariance is characterized by whether hedges in the clustered graph also occur in the original graph. The section gives sufficient graph-based conditions for this property and extends it to iterated clustering.
Hedge criterion
- Identification invariance for an effect is equivalent to requiring that any hedge for it in the clustered graph also exists in the original graph.When clustering does not create directed cycles, identifiability in the clustered graph also transfers to the original graph.Source: §4 · pp. 9–10
Plain clusters
- Clustering a plain cluster is universally identification invariant for effects that do not involve its variables.Plainness keeps c-components involving cluster vertices strictly within the cluster.Source: §4 · p. 10
C-coherence
- A congested transit cluster may fail to preserve invariance; c-coherence instead supplies a sufficient condition for coherent clusters.The c-coherent condition requires a vertex t* and specified directed paths within the cluster.Source: §4 · pp. 10–12
C-isolation
- C-isolation is another sufficient condition, allowing bidirected connections to the cluster while excluding bidirected paths from t* to cluster descendants.Unlike c-coherence, c-isolation does not require assumptions about paths inside the cluster.Source: §4 · pp. 12–13
Scope of the conditions
- Plainness, c-coherence, and c-isolation are sufficient but not complete tests; invariant clusters can satisfy none of them.Completeness would require graph-specific verification that clustered-graph hedges imply hedges in the original graph.Source: §4 · p. 13, Figure 4 · p. 14
Applying invariance
- Theorem 24 transfers nonidentifiability from a clustered subgraph to the original graph when clustering is invariant in that subgraph.The examples apply this by removing selected edges, clustering, and using the resulting nonidentifiability to infer nonidentifiability in the original graph.Source: §4 · pp. 13–14
Iterated clustering
- A sequence of identification-invariant clustering steps remains invariant for the same effect when each intermediate graph is a DAG.For universal invariance, each step must be universally invariant; effects must not involve variables from any clustered set.Source: §4 · pp. 14–15
5 Demonstration
The demonstration applies the paper’s clustering conditions to a sleep-medication graph and shows how identifiability conclusions transfer between clustered and original graphs.
5 Demonstration
- The example clusters T1 and T2, then further clusters T3, to study effects of demography, socioeconomic status, and population density on medication use.Figure 5 marks T1 in green and T2 in blue; the graph is adapted from Evandt et al. (2017) with additional bidirected arcs.Source: §5 · p. 15, Figure 5 · p. 16
- The iterated clustering is universally identification invariant, so conclusions about effects outside the clusters transfer between the clustered and original graphs.The example applies c-isolation and c-coherence results to successive clusterings.Source: §5 · p. 15
- Effects on sleep medication use with interventions including demography or socioeconomic status are nonidentifiable in both the final clustered graph and the original graph.The passage states this for p(xm | do(xB)) when B intersects {d, e}.Source: §5 · pp. 15–16
- The effect of population density on sleep medication use is identifiable in the final clustered graph and remains identifiable in the original graph.The original-graph functional is obtained by substituting the clustered variables in sequence.Source: §5 · p. 17
6 Discussion
The paper gives structural conditions for identification-invariant clustering and discusses their scope, limitations, and potential use in algorithms.
6 Discussion
- Plainness, c-coherence, and c-isolation each suffice for universal identification invariance of a coherent cluster when clustering creates no directed cycle.Identifiable effects transfer from the clustered graph to the original graph by simple substitution; the conditions prevent creation of a new hedge.Source: §6 · p. 17
- The three conditions are sufficient but not necessary, and a necessary-and-sufficient structural characterization remains open.Some invariant clustering operations satisfy none of the three conditions.Source: §6 · p. 17
- For a smaller target family of effects, the conditions may be relaxed; coherence may be unnecessary if clustering forms no new directed paths between the effect variables.Source: §6 · pp. 17–18
- The results require clustering to preserve acyclicity; handling cycles is harder because ordinary hedges do not detect cycle-induced nonidentifiability in the clustered graph.Source: §6 · p. 18
- The structural results could support algorithms that detect coherent clusters or verify plainness, c-coherence, and c-isolation.Source: §6 · p. 18
Appendix A. Proofs for Section 3 (Clustering in Causal Graphs)
The appendix proves that clustering a coherent set preserves acyclicity and key directed-path and ancestry relationships.
Appendix A. Proofs for Section 3 (Clustering in Causal Graphs)
- Clustering a coherent set does not create directed cycles, so the clustered graph remains a DAG.The proof argues that a cycle involving the cluster vertex would imply a cycle in the original DAG.Source: Appendix A · p. 18
- Any directed path intersects a coherent vertex set at most once, with the cluster members on that path forming one contiguous subpath.A path crossing the cluster twice would become a repeated visit to its representative vertex after clustering.Source: Appendix A · pp. 18–19
- For vertices outside the cluster, ancestry in the original graph outside the cluster’s descendants matches ancestry in the clustered graph outside its representative.Directed connectivity between vertices outside the expanded cluster is likewise preserved.Source: Appendix A · p. 18
- The proof recovers original directed paths through cluster members by replacing each clustered-vertex segment with a path through the coherent set.Source: Appendix A · p. 19
Appendix B. Proofs for Section 4 (Identification Invariance of Clustering Operations)
The appendix establishes universally identification-invariant clustering for plain, c-coherent, and c-isolated clusters, and extends invariance to certain simplified graphs and iterative clustering. Its proofs connect identifiability to transferring hedges between clustered and original graphs.
Appendix B. Proofs for Section 4 (Identification Invariance of Clustering Operations)
- A hedge for an effect in the clustered graph implies a corresponding hedge for that effect in the original graph.The proof uses this transfer to establish nonidentifiability in the original graph.Source: Appendix B · p. 19
- The appendix also establishes identifiability transfer from the clustered graph to the original graph and uses both directions to characterize invariance.For the stated effects, identifiability in the clustered graph implies identifiability in the original graph; the proof also rules out a clustered hedge when the original effect is identifiable.Source: Appendix B · pp. 19–20
- Clustering plain, c-coherent, or c-isolated clusters is universally identification invariant.The results apply to causal effects outside the clustered variables.Source: Appendix B · pp. 20, 22
- If clustering is universally identification invariant in a simplified graph, nonidentifiability there transfers to the original graph.The simplified graph is formed by removing latent variables and their edges, plus selected observed-variable edges.Source: Appendix B · p. 22
- Iterated clustering preserves identification invariance when each intermediate graph is a DAG and each step is invariant for the effect.The corollary extends this to effects involving none of the clustered variables when every step is universally invariant.Source: Appendix B · pp. 22–23
Appendix C. Transit Cluster
A transit cluster is a coherent cluster characterized by its receivers and emitters, subject to shared-boundary and internal-connectivity conditions.
Appendix C. Transit Cluster
- A transit cluster is a special coherent cluster uniquely determined by its receivers and emitters; this paper restricts the cluster to observed vertices.The paper notes that adding unobserved vertices whose two children lie in the cluster yields the corresponding transit cluster used in earlier work.Source: Appendix C · pp. 23–24
- Receivers are cluster vertices with parents outside the cluster, while emitters have children outside it.Receiver status also accounts for unobserved parents whose other child lies outside the cluster.Source: Appendix C · p. 23
- All receivers must share their external parents, and all emitters must share their external children.The conditions compare parent sets outside T+ for receivers and child sets outside T for emitters.Source: Appendix C · p. 24
- Every cluster vertex must connect internally to a receiver or emitter, with additional reachability conditions linking the two boundary sets when present.The internal paths are taken in the subgraph formed by removing incoming edges to receivers and outgoing edges from emitters.Source: Appendix C · p. 24
Appendix D. Auxiliary Results
Lemma 30 states that clustering a coherent cluster preserves the ancestors of A within the graph restricted by B, outside the cluster.
Appendix D. Auxiliary Results
- Outside a coherent cluster disjoint from A and B, the ancestors of A in G[B] are exactly the ancestors of A in the clustered graph G′[B], excluding its cluster vertex.The proof replaces any path segment through the cluster with the representative vertex, or expands that vertex back into a path through the cluster.Source: Appendix D · p. 24
- For the forward inclusion, a path through the cluster can be replaced by its representative because the cluster forms a subpath.The condition T ∩ B = ∅ keeps the replacement path in G′[B].Source: Appendix D · p. 24
- For the reverse inclusion, a path through the representative can be expanded to a path through T, which remains in G[B].Acyclicity ensures the representative occurs at most once on the path.Source: Appendix D · p. 24
Appendix E. Explanation of Variables in the Demonstration Example
Table 1 explains the variables appearing in Figure 5.
Appendix E. Explanation of Variables in the Demonstration Example
- Table 1 provides explanations for the variables in Figure 5.Source: Table 1 · p. 25, Appendix E · p. 25