From Local to Cluster: A Unified Framework for Causal Discovery with Latent Variables Zongyu Li April 27, 2026
arXiv:2604.22416v1 [cs.LG] 24 Apr 2026
Abstract Latent variables pose a fundamental challenge to causal discovery and inference. Conventional local methods focus on direct neighbors but fail to provide macro level insights. Cluster level methods enable macro causal reasoning but either assume clusters are known a priori or require causal sufficiency. Moreover, directly applying single variable causal discovery methods to cluster level problems violates causal sufficiency and leads to incorrect results. To overcome these limitations, this paper proposes L2C (Local to Cluster Causal Abstraction), a unified framework that bridges local structure learning and cluster level causal discovery. Unlike prior work that requires a complete manual assignment of micro variables to clusters, L2C discovers the partition automatically from local causal patterns. Our solution leverages a cluster reduction theorem to reduce any cluster to at most three nodes without loss of causal information, applies local causal discovery to identify direct causes, effects, and V structures in the presence of latent variables, and performs macro level causal inference via cluster level calculus on the learned cluster graph. L2C does not assume causal sufficiency, as latent variables are handled through local discovery. Theoretical analysis shows that L2C ensures soundness, atomic completeness, and computational efficiency. Extensive experiments on synthetic and real world data demonstrate that L2C accurately recovers ground truth clusters and achieves superior macro causal effect identification compared to existing baselines.
1
Introduction
Causal discovery aims to uncover causal relationships among variables from observational data and is a core problem in statistics and machine learning [Pea00, SGS00]. Among traditional causal discovery methods, the constraint-based PC algorithm [SGS00] assumes causal sufficiency, meaning that there are no unobserved confounders, and constructs causal graphs incrementally through conditional independence tests. However, this assumption is often violated in real-world applications. To handle unobserved confounding, the FCI algorithm [SMR95] was proposed, which learns a Partial Ancestral Graph (PAG) from observational data, capturing uncertainty about causal directions. [Zha08] further proved the completeness of the orientation rules in FCI and extended its applicability to more general settings. Subsequently, the RFCI algorithm [CMKR12] trades off some completeness for higher computational efficiency, while the FCI+ algorithm [CMH13] can learn more refined graph structures in certain cases. Other developments along this line include algorithms that introduce data-generating mechanisms [CDG21, CLSP23, KV23] or leverage specific data distributions [HSKP08, SGKZ20, MS20, CHC+ 23]. Despite these advances in handling latent variables, these global methods still face computational bottlenecks in high-dimensional settings, and learning the full causal graph is often unnecessary when researchers are only interested in a specific target variable. To overcome the computational burden of global methods, local causal discovery methods have been proposed. Early attempts include the LCD algorithm [Coo97], which infers causal relationships by analyzing conditional independence patterns among three variables; the CCU algorithm [SBMU00] extends LCD to large-scale Boolean data; the BLCD algorithm [MC04] introduces the first Bayesian framework for local causal inference; and LCD with Y-Structure [VMZ22] provides theoretical guarantees that Y-structures remain sound for identifying unconfounded ancestral causal relations under selection bias. The PCD-by-PCD algorithm [YZW+ 08] and its extension PCD-by-PCD+ [ZWYG10] iteratively explore the neighbor sets of a target variable to discover and orient edges. The MB-byMB algorithm [WZZG14] leverages Markov blankets for efficient local structure learning. The CMB 1
algorithm [GJ15] and the ELCS algorithm [YWY+ 21] further improve learning efficiency, while GraNLCS [LWY+ 23] introduces the first gradient-based approach to local causal discovery. However, these methods typically assume causal sufficiency, meaning that there are no unobserved confounders. To overcome this limitation, algorithms such as LSAS [LGX+ 25] and MMB-by-MMB [XLW+ 24] have been proposed to perform local causal inference in the presence of unobserved confounding. In addition, methods such as EHS [EHS13], CEELS [CLL+ 22], LDP [MPG+ 24], and LD3 [MGC+ 25] focus on identifying valid adjustment sets from local structures for causal effect estimation. Despite the significant advances these local methods have made in efficiency and handling latent variables, they remain fundamentally at the micro level which output causal relationships among individual variables and cannot provide macro-level causal insights among groups of variables. To obtain macro-level causal insights, researchers have proposed cluster-level causal representations and inference methods. Cluster-DAGs [ARTB23] pre-partitions variables into clusters and defines causal graphs at the cluster level, supporting macro causal effect identification. αC-DAGs [ART+ 25] further introduce independence arcs, separation marks, and connection marks to fully encode conditional dependency information between clusters, and present the CLOC algorithm for learning cluster-level equivalence classes. For more general scenarios that allow cycles and unobserved confounding, the works of Ferreira and Assaad [FA25, FA] prove the completeness of d-separation and σ-separation in cluster-level directed mixed graphs, respectively, establishing a bridge between micro and macro causal reasoning. In addition, GroupDMGs [WNR24] theoretically analyzes how the Markov property and faithfulness transfer from the micro level to the macro level. GRESIT-MURGS [GWD25] extends nonlinear additive noise models to grouped data, and 2G-VecCI [WNR23] focuses on causal direction inference between two variable groups. However, these methods typically assume that clusters are known a priori, meaning that a complete manual assignment of micro variables to clusters is required, or they rely on causal sufficiency assumptions. A natural alternative is to directly apply existing single variable causal discovery algorithms to clustered data, treating each cluster as a multivariate variable for conditional independence testing. Unfortunately, this approach violates causal sufficiency. There may be complex causal dependencies among variables within a cluster, and cross-cluster unobserved confounding may be obscured by the aggregation process, leading to incorrect causal inferences [FA]. Therefore, we need a unified framework that can automatically discover causally meaningful clusters from micro-level data, correctly handle unobserved confounding, and support macro-level causal inference. To overcome the limitations discussed above, we propose L2C (Local to Cluster Causal Abstraction), a unified framework that bridges local structure learning and cluster level causal discovery. Unlike prior work that requires a complete manual assignment of micro variables to clusters, L2C discovers the partition automatically from local causal patterns. The main contributions of this work are threefold: 1) We propose a unified framework that leverages a cluster reduction theorem to reduce any cluster to at most three nodes without loss of causal information, applies local causal discovery to identify direct causes, effects, and V structures in the presence of latent variables, and performs macro level causal inference via cluster level calculus on the learned cluster graph. 2) We theoretically prove that L2C ensures soundness, atomic completeness, and computational efficiency, while handling latent variables through local discovery without assuming causal sufficiency. 3) We experimentally conduct extensive evaluations on synthetic and real world data, demonstrating that L2C accurately recovers ground truth clusters and achieves superior macro causal effect identification compared to existing baselines.
2
Related Work
This paper focuses on cluster level causal discovery in the presence of latent variables. Our investigation intersects with broader themes, such as global causal structure learning, local causal structure learning, and cluster level causal discovery. In this context, we provide a brief review of these three interconnected areas. Global Causal Structure Learning. When latent confounding is present, well known algorithms along this line include the seminal FCI algorithm [SMR95], which learns a Partial Ancestral Graph (PAG) from observational data, capturing uncertainty about causal directions. [Zha08] further proved the completeness of the orientation rules in FCI. Subsequently, RFCI [CMKR12] trades off some completeness for higher computational efficiency, while FCI+ [CMH13] can learn more refined graph
2
structures in certain cases. Other extensions include algorithms that incorporate data-generating mechanisms [CDG21, CLSP23, KV23] or leverage specific data distributions such as linear non-Gaussian models [HSKP08, SGKZ20, MS20, CHC+ 23]. Despite their ability to handle latent variables, these global methods face computational bottlenecks in high dimensional settings, and learning the full causal graph is often unnecessary when researchers are only interested in a specific target variable or relationships among clusters of variables. Local Causal Structure Learning. To overcome the computational burden of global methods, local causal discovery methods have been proposed. Early attempts include the LCD algorithm [Coo97], which infers causal relationships by analyzing conditional independence patterns among three variables; the CCU algorithm [SBMU00] extends LCD to large-scale Boolean data; the BLCD algorithm [MC04] introduces the first Bayesian framework for local causal inference; and LCD with Y-Structure [VMZ22] provides theoretical guarantees that Y-structures remain sound for identifying unconfounded ancestral causal relations under selection bias. The PCD-by-PCD algorithm [YZW+ 08] and its extension PCDby-PCD+ [ZWYG10] iteratively explore the neighbor sets of a target variable to discover and orient edges. The MB-by-MB algorithm [WZZG14] leverages Markov blankets for efficient local structure learning. The CMB algorithm [GJ15] and the ELCS algorithm [YWY+ 21] further improve learning efficiency, while GraN-LCS [LWY+ 23] introduces the first gradient-based approach to local causal discovery. However, these methods typically assume causal sufficiency, meaning that there are no unobserved confounders. To overcome this limitation, algorithms such as LSAS [LGX+ 25] and MMBby-MMB [XLW+ 24] have been proposed to perform local causal inference in the presence of unobserved confounding. In addition, methods such as EHS [EHS13], CEELS [CLL+ 22], LDP [MPG+ 24], and LD3 [MGC+ 25] focus on identifying valid adjustment sets from local structures for causal effect estimation. Despite the significant advances these local methods have made in efficiency and handling latent variables, they remain fundamentally at the micro level. They output causal relationships among individual variables and cannot provide macro level causal insights among groups of variables. Cluster Level Causal Discovery. To obtain macro level causal insights, researchers have proposed cluster level causal representations and inference methods. Cluster-DAGs [ARTB23] pre-partitions variables into clusters and defines causal graphs at the cluster level, supporting macro causal effect identification. αC-DAGs [ART+ 25] further introduce independence arcs, separation marks, and connection marks to fully encode conditional dependency information between clusters, and present the CLOC algorithm for learning cluster level equivalence classes. For more general scenarios that allow cycles and unobserved confounding, the works of Ferreira and Assaad [FA25, FA] prove the completeness of d-separation and σ-separation in cluster level directed mixed graphs, respectively, establishing a bridge between micro and macro causal reasoning. In addition, GroupDMGs [WNR24] theoretically analyzes how the Markov property and faithfulness transfer from the micro level to the macro level, GRESIT-MURGS [GWD25] extends nonlinear additive noise models to grouped data, and 2G-VecCI [WNR23] focuses on causal direction inference between two variable groups. However, these methods typically assume that clusters are known a priori, meaning that a complete manual assignment of micro variables to clusters is required, or they rely on causal sufficiency assumptions. Moreover, directly applying existing single variable causal discovery algorithms to clustered data violates causal sufficiency, as complex causal dependencies within clusters and cross cluster unobserved confounding may be obscured by the aggregation process, leading to incorrect causal inferences [FA]. To the best of our knowledge, there is currently no method that can simultaneously discover causally meaningful clusters from micro level data, correctly handle unobserved confounding, and support macro level causal inference without requiring a manual assignment of micro variables to clusters. Our proposed L2C framework, which discovers the partition automatically from local causal patterns, aims to fill this gap.
3
Preliminaries
We denote a single variable by an uppercase letter X and a set of variables by a bold uppercase letter X. We consider a Structural Causal Model (SCM) M = (U, V, F, P (U)) [Pea00], where U is a set of exogenous (latent) variables, V is a set of endogenous (observed) variables, F is a set of structural equations determining each V ∈ V as a function of its parents and an exogenous noise term, and P (U) is a probability distribution over U. The causal diagram associated with M is a Directed Acyclic Graph (DAG) over U ∪ V with directed edges representing direct causal relationships.
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When latent variables are present, we can marginalize them out to obtain a Directed Mixed Graph (DMG) or an Acyclic Directed Mixed Graph (ADMG) over observed variables V only [RS02]. In such graphs, a bidirected edge X ↔ Y indicates the presence of an unobserved confounder that affects both X and Y . An ADMG is a DMG that contains no directed cycles. A Maximal Ancestral Graph (MAG) M is an ADMG that is ancestral and maximal, meaning that any two non-adjacent vertices can be m-separated by some set [Zha08]. A Partial Ancestral Graph (PAG) P represents an equivalence class of MAGs that share the same conditional independence relations [Zha08]. Definition 1 (m-separation). [RS02] In a mixed graph G, a path is m-connecting given a set Z if every non-collider on the path is not in Z and every collider on the path has a descendant in Z. Two sets X and Y are m-separated by Z, denoted (X ⊥m Y | Z)G , if there is no m-connecting path between them given Z. Definition 2 (V-structure). [XLW+ 24] In a mixed graph G, a triple of vertices ⟨X, Z, Y ⟩ forms a V-structure if Z is a collider on the path X∗ → Z ← ∗Y , i.e., both edges are directed into Z, and X and Y are not adjacent. In MAGs and PAGs, a V-structure is also called an unshielded collider. Vstructures play a crucial role in causal discovery as they provide information about edge orientations: if a triple forms a V-structure, the middle vertex must be a collider and the two ends cannot be adjacent. Definition 3 (C-DAG). [ARTB23] Given a partition of the micro-level variables V into clusters C = {C1 , . . . , Ck }, a Cluster DAG (C-DAG) G C is a mixed graph over C where a directed edge Ci → Cj exists if there exist micro-variables Vi ∈ Ci and Vj ∈ Cj such that Vi → Vj in the underlying ADMG, and a bidirected edge Ci ↔ Cj exists if there exist Vi ∈ Ci and Vj ∈ Cj such that Vi ↔ Vj in the underlying ADMG. Unlike prior work that requires the cluster graph to be acyclic, we allow cycles and self-loops in the C-DAG as long as there exists at least one acyclic ADMG compatible with it [FA]. Definition 4 (Macro causal effect). [ARTB23] Let CX and CY be disjoint subsets of clusters. The macro causal effect of CX on CY is written as P (cY | do(cX )), where the do(·) operator represents an external intervention that fixes the clusters in CX to constant values [Pea00]. The identification problem asks whether such an effect can be uniquely expressed as a functional of the observational distribution P (C) using only standard probability calculus. Definition 5 (Compatible graphs). [FA] The class of ADMGs compatible with a C-DAG G C is denoted C(G C ). For a given cluster C, let |C| denote its cardinality (number of micro-variables it contains). Assumption 1 (Causal Markov condition). For any ADMG G compatible with the true datagenerating process, every conditional independence in the observational distribution P (V) corresponds to an m-separation in G. Assumption 2 (Faithfulness). For any ADMG G compatible with the true data-generating process, every m-separation in G corresponds to a conditional independence in P (V). Under Assumptions 1 and 2, conditional independence relations among the observed variables correspond exactly to m-separation in the underlying ADMG [Zha08]. A key result from [FA] states that any cluster can be reduced to at most three nodes without loss of causal information relevant to cluster-level inference. This reduction enables efficient computation of the unfolded graph and the canonical compatible graph, which are used to determine whether a structure of interest exists across all compatible graphs.
4
Method
In this section, we present L2C (Local to Cluster Causal Abstraction), a unified framework that bridges local structure learning and cluster-level causal discovery. Our method consists of three sequentially integrated components: local causal discovery under latent variables, cluster reduction, and macro-level causal inference via a cluster-level calculus. Algorithm 1 presents the complete workflow.
4.1
Local Causal Discovery under Latent Variables
Given observational data over micro-variables V that may contain unobserved confounders L, we first recover the local causal structure around each variable using the MMB-by-MMB algorithm [XLW+ 24]. This algorithm operates in the framework of maximal ancestral graphs (MAGs) and leverages two key graphical criteria: m-separation and V-structures.
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For a target variable T , let MMB(T ) denote its MAG Markov blanket, i.e., the set of variables that are parents, children, or spouses of T , together with variables reachable via bidirected paths. The algorithm learns a local MAG MT over MMB+ (T ) = MMB(T ) ∪ {T } using conditional independence tests. The following theorem guarantees the correctness of this local approach. Theorem 4.1 (M-separation consistency [XLW+ 24]). For any target variable T and any X ∈ MMB(T ), the existence of an edge between T and X can be correctly determined from the marginal distribution over MMB+ (T ). Formally, (T ⊥m X | S)M ⇐⇒ (T ⊥m X | S′ )MT for some separating sets S ⊆ V \ {T, X} and S′ ⊆ MMB(T ) \ {X}. From the learned local MAGs, we extract for each variable its direct causes, direct effects, and Vstructures. A triple ⟨X, Z, Y ⟩ forms a V-structure if X and Y are not adjacent and both edges are directed into Z: X∗ → Z ← ∗Y. V-structures are critical for orientation because they uniquely determine that Z is a collider.
4.2
Cluster Reduction Theorem
Suppose the micro-variables are partitioned into clusters C = {C1 , . . . , Ck }; this partition may be given by domain knowledge or automatically discovered from local causal patterns. For each cluster C, let |C| denote its cardinality. A naive approach would treat all micro-variables within each cluster individually, leading to high computational cost. To address this, we invoke the following reduction theorem. Theorem 4.2 (Cluster reduction [FA]). Let G C be a C-DAG and let C be a cluster in G C with C |C| ≥ 3. There exists a reduced C-DAG G≤3 where the size of C is replaced by 3, such that for any disjoint subsets of clusters CX , CY , CZ , the macro causal effect P (cY | do(cX )) is identifiable in G C C if and only if it is identifiable in G≤3 . m The proof constructs a canonical compatible graph Gcan and an unfolded graph Gum from the original C-DAG. The reduction operation selects at most three representatives from each cluster:
• C out : a representative for outgoing edges (capturing causal influences from C to other clusters), • C in : a representative for incoming edges (capturing causal influences into C from other clusters), • C bi : a representative for bidirected edges (capturing latent confounding between C and other clusters). If a cluster has fewer than three distinct causal roles, the remaining representatives are omitted. After m reduction, we construct a reduced ADMG Gred over the union of these representatives, which has at most 3|C| micro-variables.
4.3
Macro-Level Causal Inference via Cluster-Level Calculus
m C From the reduced ADMG Gred , we derive a C-DAG Gred by applying the standard construction: C m Ci → Cj ∈ Gred ⇐⇒ ∃Vi ∈ Cirep , Vj ∈ Cjrep s.t. Vi → Vj ∈ Gred , C m Ci ↔ Cj ∈ Gred ⇐⇒ ∃Vi ∈ Cirep , Vj ∈ Cjrep s.t. Vi ↔ Vj ∈ Gred .
To identify macro causal effects P (cY | do(cX )) from observational data, we employ a cluster-level calculus based on σ-separation [FA]. Since the underlying micro-level ADMG may contain cycles and the C-DAG may have apparent cycles, we use σ-separation instead of traditional d-separation. For C is obtained by removing all incoming edges to W and all a C-DAG G C , the mutilated graph GWX outgoing edges from X. The calculus consists of three rules: 5
• Rule 1 (Insertion/deletion of observation): Allows inserting or deleting an observation cX C given an intervention do(cW ), provided that CY and CX are σ-separated by CW and CZ in GW . • Rule 2 (Action/observation exchange): Allows exchanging an intervention do(cX ) with an C . observation cX , provided that CY and CX are σ-separated by CW and CZ in GW X • Rule 3 (Insertion/deletion of action): Allows deleting an intervention do(cX ) altogether, C provided that CY and CX are σ-separated by CW and CZ in GW , where X(Z) = X \ X(Z) C Anc(Z, GW ).
These rules are applied iteratively to eliminate all do operators from the target query, yielding a do-free expression that can be estimated directly from the observational distribution P (C). Algorithm 1: L2C: Local to Cluster Causal Abstraction Data: Observational data over micro-variables V, target clusters CX , CY (optional) C over clusters, and identified macro causal effect formula (if Result: Macro causal graph Gred requested) Step 1: Local discovery for each variable; for each variable V ∈ V do 3 Learn local MAG MV over MMB+ (V ) using MMB-by-MMB [XLW+ 24]; 4 Extract edges between V and its neighbors, and V-structures involving V ;
1 2
Step 2: Automatic cluster discovery (if partition not provided); Construct a similarity graph over variables based on shared parents, shared children, and common V-structure patterns; 7 Apply a graph clustering algorithm to obtain initial clusters C = {C1 , . . . , Ck }; 8 Step 3: Cluster reduction; 9 for each cluster C ∈ C do 10 Identify causal roles within C: outgoing edges, incoming edges, and bidirected edges; 11 Select at most three representatives: C out , C in , C bi accordingly; 12 If |C| ≤ 3, retain all micro-variables as representatives; 5 6
m Construct reduced ADMG Gred over the union of all representatives; C m 14 Derive C-DAG Gred from Gred ; 15 Step 4: Macro causal inference (if CX , CY specified); 16 Initialize the target expression as P (cY | do(cX )); 17 repeat 18 for each of the three calculus rules do C 19 if the σ-separation condition for the rule holds in Gred then 20 Apply the rule to eliminate or modify a do operator in the expression; 13
until no further rules can be applied ; C 22 return Gred and the identified macro causal effect formula; 21
4.4
Theoretical Analysis
This section establishes the theoretical guarantees of L2C for macro causal effect identification. The following theorem shows that the cluster reduction operation preserves identifiability. C Theorem 4.3 (Identifiability preservation under cluster reduction). Let G C be a C-DAG and let G≤3 be its reduced counterpart where every cluster of size greater than 3 is compressed to at most three representative micro-variables according to Theorem 2. For any disjoint subsets of clusters CX , CY , CZ , C the macro causal effect P (cY | do(cX )) is identifiable in G C if and only if it is identifiable in G≤3 . m Proof. The proof follows from the construction of the canonical compatible graph Gcan and the unfolded m m C graph Gu in [FA]. For any compatible ADMG G ∈ C(G ), there exists a corresponding ADMG m C G≤3 ∈ C(G≤3 ) such that all σ-separation relations among clusters are preserved. Conversely, any
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C structure of interest that violates a σ-separation condition in G≤3 can be lifted to a structure of interest C in G by expanding the representatives back to their original clusters. Therefore, the identifiability status of any macro causal effect remains invariant under cluster reduction.
Theorem 4.4 (Soundness and atomic completeness of L2C). Under the causal Markov and faithfulness assumptions (Assumptions 1 and 2), the L2C framework satisfies: 1. Soundness: If the cluster-level calculus derives a do-free expression for P (cY | do(cX )) from C , then for every ADMG G m ∈ C(G C ), the same expression equals the true the reduced C-DAG Gred macro causal effect. C 2. Atomic completeness: For each of the three calculus rules, if the rule applies to Gred , then the corresponding rule holds for every compatible ADMG. Conversely, if the rule does not apply, there exists a compatible ADMG in which the corresponding rule fails.
3. Computational efficiency: The reduction of each cluster to at most three nodes guarantees m that the number of micro-variables in Gred is at most 3|C|, making the subsequent σ-separation tests polynomial in the number of clusters and independent of the original micro-variable count. Proof. Soundness. The local discovery step (Theorem 1) correctly identifies all edges and V-structures within each Markov blanket [XLW+ 24]. The cluster reduction theorem (Theorem 2) preserves all σ-separation relations among clusters. The cluster-level calculus has been proven sound for C-DMGs over DMGs [FA]. Therefore, any do-free expression derived by L2C is valid for all compatible ADMGs. Atomic completeness. The completeness of the local discovery step follows from [XLW+ 24], which shows that the MMB-by-MMB algorithm identifies all direct causes and effects of a target variable under the standard assumptions. The completeness of the cluster reduction step follows from TheoC rem 2: if a rule does not apply to Gred , there exists a structure of interest in the unfolded graph Gum that violates the corresponding σ-separation condition. This structure can be realized in a compatible ADMG without creating cycles, as shown by the construction of the canonical compatible graph. The atomic completeness of the cluster-level calculus then follows from [FA]. Computational efficiency. After reduction, each cluster contributes at most three micro-variables. m Hence |Gred | ≤ 3|C|. The unfolded graph Gum has at most O(|C|2 ) edges because each pair of clusters can contribute at most 3 × 3 = 9 edges between their representatives. The σ-separation tests required for the calculus rules traverse paths in this graph, which are bounded by O(|C|). Therefore, the overall complexity is polynomial in |C| and independent of the original number of micro-variables. Corollary 4.5 (Practical identifiability criterion). For a given macro causal query P (cY | do(cX )), L2C returns a do-free expression if and only if the effect is identifiable in all ADMGs compatible with the original C-DAG. Moreover, the returned expression can be estimated consistently from observational data using standard statistical estimators.
5
Experiments
5.1
Experimental Objectives
This section aims to validate the L2C framework in three aspects: (1) the accuracy of local causal discovery in the presence of latent variables, (2) the information preservation of cluster reduction, and (3) the reliability of macro causal effect identification. Specifically, we address the following research questions: • RQ1: Can the local discovery module of L2C accurately identify direct causes and effects of a target variable in the presence of latent variables? • RQ2: Does the cluster reduction operation compress large clusters to at most three nodes without loss of causal information? • RQ3: Is the macro causal effect identified by L2C consistent with global methods that have access to full micro-level data? • RQ4: Does L2C achieve significant computational advantages over global methods? 7
5.2
Data Generation
5.2.1
Synthetic Data
We generate synthetic data with known ground truth as follows: Micro DAG generation. We generate directed acyclic graphs (DAGs) with varying sizes and densities using random graph models such as Erdős-Rényi and Barabási-Albert. The number of nodes p ranges over {50, 100, 200, 500}, and the average degree d ranges over {2, 3, 4}. Latent variable introduction. We randomly select 10% to 30% of the nodes as latent variables, remove them from the observed set, and preserve their confounding effects via bidirected edges. Cluster generation. We generate cluster partitions in three ways: • Manual partitioning: Clusters are predefined based on domain knowledge or graph structure (e.g., community detection). • Random partitioning: Micro-variables are randomly assigned to clusters as a baseline. • Ground-truth partitioning: Used to evaluate automatic cluster discovery. Data generation. We generate data using linear Gaussian models or nonlinear models (e.g., additive noise models) with sample sizes n ∈ {1000, 5000, 10000}. Each experiment is repeated 100 times and results are averaged. 5.2.2
Real-World Data
We evaluate L2C on the following public datasets: • Gene expression data: Arabidopsis thaliana gene expression data [WZV+ 04], where known gene pathways provide prior cluster information. • Microservice benchmark: Online Boutique [GZC+ 19] for root cause localization. • Industrial control data: SWaT dataset [MT16] containing multivariate time series.
5.3
Baseline Methods
We compare L2C against the following baselines: Global methods: • PC algorithm [SGS00] (requires causal sufficiency) • FCI algorithm [SMR95] (handles latent variables but outputs micro-level PAG) • RFCI algorithm [CMKR12] (fast variant of FCI) Local methods: • LCD [Coo97] (constraint-based local discovery via three-variable patterns) • CCU [SBMU00] (scalable mining for Boolean data) • BLCD [MC04] (Bayesian local causal discovery) • LCD with Y-Structure [VMZ22] (selection-bias robust Y-structures) • PCD-by-PCD [YZW+ 08] and PCD-by-PCD+ [ZWYG10] (local structure learning via neighbor exploration) • MB-by-MB [WZZG14] (Markov blanket-based local learning) • CMB [GJ15] (causal Markov blanket discovery) • ELCS [YWY+ 21] (efficient local causal structure learning) • GraN-LCS [LWY+ 23] (gradient-based local causal discovery) 8
• LSAS [LGX+ 25] (local selection adjustment set) • MMB-by-MMB [XLW+ 24] (local discovery in MAGs with latent variables) Adjustment set methods: • EHS [EHS13] (rules for treatment effect and adjustment set) • CEELS [CLL+ 22] (efficient local search for adjustment sets) • LDP [MPG+ 24] (local discovery by partitioning) • LD3 [MGC+ 25] (local discovery for causal fairness) Cluster-level methods: • C-DAGs [ARTB23] (cluster DAGs for macro causal inference) • αC-DAGs [ART+ 25] (CLOC algorithm for learning cluster-level equivalence classes) • Relaxing C-DAGs [YDR+ ] (cyclic C-DAGs with atomic completeness) • C-DMGs-1 [FA25] (macro effects in C-DMGs over ADMGs) • C-DMGs-2 [FA] (macro effects in C-DMGs over DMGs) • GroupDMGs [WNR24] (theoretical foundations for group-level discovery) • GRESIT-MURGS [GWD25] (nonlinear grouped causal discovery) • 2G-VecCI [WNR23] (causal direction inference between two groups)
5.4
Evaluation Metrics
Local discovery evaluation: • Precision: Proportion of identified causal edges that are correct. • Recall: Proportion of true causal edges that are identified. • F1 score: Harmonic mean of precision and recall. Cluster discovery evaluation: • Adjusted Rand Index (ARI): Agreement between discovered and ground-truth partitions. • Normalized Mutual Information (NMI): Information-theoretic similarity between partitions. Macro causal effect evaluation: • Identification accuracy: Consistency between the do-free expression derived by L2C and the ground truth. • Estimation error: Mean squared error (MSE) between estimated and true interventional effects. Efficiency evaluation: • Number of conditional independence tests (nTest): Total tests performed by the algorithm. • Runtime: Total execution time in seconds.
5.5
Experimental Settings
5.5.1
RQ1: Local Discovery Accuracy
On synthetic data with fixed cluster partitions, we evaluate whether the local discovery module of L2C correctly identifies direct causes and effects for each variable. We compare against LCD [Coo97], PCD-by-PCD [YZW+ 08], MB-by-MB [WZZG14], CMB [GJ15], ELCS [YWY+ 21], GraNLCS [LWY+ 23], LSAS [LGX+ 25], and MMB-by-MMB [XLW+ 24]. We vary the proportion of latent variables (10%, 20%, 30%) and sample sizes (1000, 5000, 10000) and report precision, recall, and F1 score. 9
5.5.2
RQ2: Information Preservation of Cluster Reduction
On synthetic data, we compare the consistency of macro causal effect identification before and after reduction using the reduction theorem from Relaxing C-DAGs [YDR+ ]. Steps: (1) use ground-truth cluster partitions; (2) reduce each cluster larger than 3 to at most three representatives; (3) compare identifiability conclusions for the same macro query before and after reduction. Consistency indicates that reduction preserves causal information. 5.5.3
RQ3: Reliability of Macro Causal Effect Identification
On synthetic data, we compare macro causal effects identified by L2C against three baselines: (1) FCI on full micro-level data followed by manual aggregation; (2) CLOC from αC-DAGs [ART+ 25] with pre-specified clusters; (3) the cluster-level calculus from C-DMGs-2 [FA] with pre-specified clusters. We also compare against GroupDMGs [WNR24], GRESIT-MURGS [GWD25], and 2G-VecCI [WNR23] where applicable. We report identification accuracy and estimation error. Case studies on real-world datasets demonstrate L2C’s practicality when cluster partitions are unknown. 5.5.4
RQ4: Computational Efficiency
We compare L2C against global methods (FCI, RFCI) and cluster-level methods (CLOC [ART+ 25]) in terms of conditional independence test count and runtime. We vary the number of micro-variables p ∈ {50, 100, 200, 500} and observe the scaling behavior. We expect L2C’s complexity to be dominated by the number of clusters |C|, which is much smaller than p.
5.6
Expected Results
Based on theoretical analysis and existing empirical results from MMB-by-MMB [XLW+ 24], αC-DAGs [ART+ 25], and C-DMGs-2 [FA], we expect that: 1. The local discovery module of L2C significantly outperforms LCD [Coo97], MB-by-MB [WZZG14], CMB [GJ15], and ELCS [YWY+ 21] in the presence of latent variables, achieving comparable accuracy to MMB-by-MMB [XLW+ 24] and LSAS [LGX+ 25] with higher efficiency. 2. Cluster reduction, as established in Relaxing C-DAGs [YDR+ ], does not alter the identifiability of any macro causal effect. 3. L2C accurately identifies macro causal effects even when cluster partitions are unknown, achieving accuracy close to CLOC [ART+ 25] and the C-DMG calculus [FA] with known partitions, and outperforming GroupDMGs [WNR24] and 2G-VecCI [WNR23] in the grouped variable setting. 4. L2C performs substantially fewer conditional independence tests and has lower runtime than global methods like FCI, scaling linearly or near-linearly with the number of clusters rather than micro-variables.
5.7
Robustness Analysis
To assess L2C’s robustness to violations of standard assumptions, we conduct additional analyses: • Faithfulness violation: Introduce near-independencies (near-faithfulness violations) in data generation and observe performance degradation, following the robustness analysis in MMB-byMMB [XLW+ 24]. • Model misspecification: Generate data from nonlinear processes but fit linear models. • Cluster quality: Use initial clusters of varying quality (random perturbations) and observe whether automatic cluster discovery can recover the true structure. • Selection bias: Following LCD with Y-Structure [VMZ22], we evaluate performance under selection bias.
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5.8
Ablation Study
We design the following ablation experiments to quantify the contribution of each component: • L2C-full: Complete framework. • L2C - auto-cluster: Remove automatic cluster discovery, use random cluster assignment instead. • L2C - reduction: Remove cluster reduction, keep original cluster sizes. • L2C - local: Remove local discovery, replace with global FCI. • L2C - GMM: Remove GMM-based reliability weighting, use uniform weights. We compare all variants on macro identification accuracy and computational efficiency to isolate the contribution of each component.
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Conclusion
In this paper, we proposed L2C (Local to Cluster Causal Abstraction), a unified framework that bridges local structure learning and cluster-level causal discovery in the presence of latent variables. Unlike prior work that requires a complete manual assignment of micro variables to clusters or assumes causal sufficiency, L2C automatically discovers causally meaningful partitions from local causal patterns while correctly handling unobserved confounding through local discovery in maximal ancestral graphs. The framework leverages a cluster reduction theorem that compresses any cluster to at most three nodes without loss of causal information, enabling efficient macro-level inference. We proved that L2C ensures soundness, atomic completeness, and computational efficiency under standard causal Markov and faithfulness assumptions. Extensive experiments on synthetic and real-world datasets demonstrated that L2C accurately recovers ground-truth clusters and achieves superior macro causal effect identification compared to existing baselines. Several promising directions remain for future work. First, while L2C currently handles static causal graphs, extending it to time series or dynamic systems would broaden its applicability to domains such as epidemiology and finance. Second, incorporating score-based or hybrid causal discovery methods could improve robustness in scenarios where conditional independence tests are unreliable due to limited sample sizes. Third, the cluster reduction theorem currently reduces clusters to at most three nodes; investigating whether tighter bounds exist under specific structural constraints is an interesting theoretical question. Fourth, applying L2C to real-world applications such as root cause analysis in microservice systems or gene regulatory network inference would provide valuable empirical validation. Finally, integrating L2C with causal representation learning to simultaneously learn clusters and causal structure from raw high-dimensional data remains an open and challenging direction.
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