Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering
Authors: Clément Yvernes
Emilie Devijver
Adèle H. Ribeiro
Marianne Clausel
Eric Gaussier
Venue: NeurIPS 2025 poster ()
Keywords: Causality
Causal Inference
Cluster-DAGs
Do-Calculus for Higher-Level abstractions
Identification
Primary area: probabilistic_methods
TL;DR: We extend C-DAGs to support arbitrary (possibly cyclic) clusterings and introduce a sound, atomically complete calculus for cluster-level causal reasoning.

Abstract: Cluster DAGs (C-DAGs) provide an abstraction of causal graphs in which nodes represent clusters of variables, and edges encode both cluster-level causal relationships and dependencies arisen from unobserved confounding. C-DAGs define an equivalence class of acyclic causal graphs that agree on cluster-level relationships, enabling causal reasoning at a higher level of abstraction. However, when the chosen clustering induces cycles in the resulting C-DAG, the partition is deemed inadmissible under conventional C-DAG semantics. In this work, we extend the C-DAG framework to support arbitrary variable clusterings by relaxing the partition admissibility constraint, thereby allowing cyclic C-DAG representations. We extend the notions of d-separation and causal calculus to this setting, significantly broadening the scope of causal reasoning across clusters and enabling the application of C-DAGs in previously intractable scenarios. Our calculus is both sound and atomically complete with respect to the do-calculus: all valid interventional queries at the cluster level can be derived using our rules, each corresponding to a primitive do-calculus step.

Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering
Clément Yvernes
Emilie Devijver
Univ Grenoble Alpes, CNRS, Grenoble INP, LIG
Univ Grenoble Alpes, CNRS, Grenoble INP, LIG
Adèle H. Ribeiro University of Münster, Institute of Medical Informatics
Marianne Clausel
Université de Lorraine, CNRS, CRAN, 54000 Nancy
Eric Gaussier
Univ Grenoble Alpes, CNRS, Grenoble INP, LIG
Abstract
Cluster DAGs (C-DAGs) provide an abstraction causal graphs in which nodes represent clusters of variables, and edges encode both cluster-level causal relation- ships and dependencies arisen from unobserved confounding. C-DAGs define an equivalence class of acyelic causal graphs that agree on cluster-level relationships, enabling causal reasoning at a higher level of abstraction. However, when the chosen clustering induces cycles in the resulting C-DAG, the partition is deemed inadmissible under conventional C-DAG semantics. this work, we extend the C-DAG framework to support arbitrary variable clusterings by relaxing the parti- tion admissibility constraint, thereby allowing cyclic C-DAG representations, We extend the notions of d-separation and causal calculus this setting, significantly broadening the scope of causal reasoning across clusters and enabling the appli- cation of C-DAGs previously intractable scenarios. Our calculus both sound and atomically complete with respect to the do cooccaluluu: all valid interventional queries at the cluster level can be derived using our rules, each corresponding to a primitive do-calculus step.
Knowing the effect of treatment X outcome Y, encoded by an intervention do(X) in the inter- ventional distribution P(Y)d(())) is crucial in many applications. However, performing interventions often impractical due ethical concerns, potential harm or prohibitive costs. In such cases, one can instead aim to identify do-free formulas that estimate the effects of interventions using only observational experrmmie))))) data and causal graph (Pearl 2009). Solving the identifiability problem typically involves establishing graphical criteria under which the total effect is identifiable, and providing do-free formula for estimating it from observational data. However, specifying causal diagram requires prior knowledge of the causal relationships between all observed variables, a requirement that is often unmet real-world applications. This challenge particularly acute in complex, nigh-dimensional settings, limiting the practical applicability of causal inference methods.
One way to circumvent this difficulty is rely on abstract representations which group several vari- ables, a mapping usually referred to as causal representation learning (Schölkopf et al. (202]). which are connected through causal relationships and dependencies arisen from unobserved confounding. Several studies have been devoted to causal discovery of and causal inference in specific abstract representations, both for static and dynamic (time series) variables, as Assaad et al. (2022): |eereiiaa
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and Assaad (2024) Anand et al. (2023); ) Wahl et al. (2023). Recent studies have also tackled the related problem of defining mappings from clusters to variables while preserving specific causal properties (Chalupka et al. 2015 2016 Rubenstein et al. 2017, Beckers and Halpern 2019)) and
have provided thorough theoretical analysis the relationship between micro- and macro-level causal models with view on causal discovery assumptions (Wahl et al. 2024).
The starting point of our study is the framework recently proposed Anand al. (2023). which relaxes the strict requirement of a fully specified causal diagram and provides a foundation for valid inference over clusters of variables. However, focuses on abstract graphs. called Cluster-DAGs, which do not contain cycles between clusters, a restriction known as partition admissibilityy We extend this framework in this paper by removing this restriction and consider arbitrary clusterings of variables, potentially resulting in abstract graphs with self-loops and cycles between clusters. Several real-world scenarios illustrate this concept as macroeconomies where sector-level relationships investment) but not the firm-to-firm or household-to- hooos-h---hh
are known (as consumption causal links, or in neuroscience where functional MRI region interactions can be established but not necessarily the causal links between neurons.
To tackle identification in causal abstractions, recent work has introduced separation criteria that generalize d-separation to specific abstraction types (Jaber et al. [0022] Perković et al. 20188 Ferreira and Assaad, 2025). While effective, this has led proliferation of abstraction sppeifie rules. Yet, all these methods operate over the same underlying object: the class of graphs compatible with the abstraction. An alternative line of work could seek construct transformed graph on which standard d-separation can be directly applied. This is feasible, for example, when the union of all compatible graphs is itself compatible cycles. Our approach adopts a hybrid strategy. Rather than relying solely path-based d-separation, we define a structure- criterion that captures all necessary information for assessing separation These structures avoid pitfalls such as reattaching colliders into conditioned paths and are simple to construct - typically by tracing backward along directed edges from root nodes. Crucially, our criterion remains tightly aligned with standard d-separation: every d-connecting path can define such a structure, and every connecting structure contains a d-connecting path. To support identification under abstraction, introduce two-step method grounded in a tractable search space. First, derive an extended graph from the abstraction that, while not necessarily compatible, conservatively includes all potentially connecting structures. This enables efficient exploration using standard graph-traversal techniques. Second, a lightweight compatibility test is applied to filter out invalid structures, ie., those which do not correspond any graph in the compatible class.
but such cases are rare, especially when the abstraction introduces
Specifically, we make the following contributions:
1. We extend the framework Anand al. (2023) by removing the assumption of partition admissibility, thereby broadening its applicability to a wider range causal abstractions.
2. We reformulate the d-separation criterion in an ADMG using a peructure-based separation criterion that remains faithful to classical d-separation.
3. We introduce calculus which is sound and atomically complete.
4. We further show that any cluster can be reduced to cluster of limited size, leading to efficient calculus rules.
The remainder of the paper is structured as follows: Section T introduces the main notions while Section presents our main result regarding sound and atomically complete calculus: Section B presents an efficient way to look at causal abstractions based on clusters; lastly, Section discusses some extensions of our work while Section concludes the paper. All proofs are provided in the Technical Appendices.
Preliminaries
We follow the notations of Pearl (009). single variable is denoted by an uppercase letter X and its realized value by a small letter x. A calligraphic uppercase letter X denotes a set.
Graphs. We denote by Ane(X. 9) and Desc(X. G) the sets of ancestors and descendants of X in the graph G. respectively. By convention, each node is regarded as its own ancestor and descendant.

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Figure 1: Left: a C-DAG G = Right: a graph Gmh (V _.") that is compatible with GC. For example, A 'V corresponds to A2) EV" In g structure of interest (Definition is highlighted in bold black, with all other nodes and edges shown gray.
We denote by Root(G) the set of roots of G. i.c., the vertices that have no child in G. A vertex V is a collider and V or any of its
said to be active on a path relative subset of variables Z if descendants are in Z or 2) V is non-collider and is not in Z. A path π is said to be active given (or conditioned on) every vertex on active relative to Z. Otherwise, is said be inactive given Z. Given graph G. the sets X and y are said to be d-separated by Z if every path between X and inactive given Z. We denote this by g Z. Otherwise, X and are d-connected given Z, which we denote by X Z. The mutilated graph Gxz is the result of removing from a graph G edges with an arrowhead into X (e.g., - X, A < X), and edges with tail from Z (e.g..A < Z). Let π be path in graph G and let A and B be two nodes of æT. We denote by æ[a.a] the subpath of π between and B. For two graphs G1 = (V,, E and G2 = (2), the union is Gi UG2 (VV, 2 U &2). We denote by Xn G the set of nodes in & that belong to X. i.c., the
intersection between X and the vertex set of G. We denote by G \ X the subgraph of G obtained by removing all vertices in X together with any edges incident to X.
Structural Causal Models. Formally, a Structural Causal Model (SCM) M is 4-tuple where U is set of exogenous (latent) mutually independent variables and V set of endogenous (measured) variables F is collection of functions (fil such that each endogenous variable V, E V is a function fi F of U, U Pa(Vi), where U, S U and Pa(V)) CV V,. The uncertainty is encoded through probability distribution over the exogenous variables, P(U)). Each SCM M induces a directed acyclic graph (DAG) with bidirected edges- or an acyclic directed causal diagram, that encodes the structural
(Ep. Eg)), known relations among V U U, where every V; E V is a vertex. We potentially distinguish edges into directed edges En, which connect each variable and bidirected edges, which appear as dashed edges (V, <-> Vi) between variables V,, V, V that share common exogenous parent, i.e., such that 44 U1 0. Performing intervention X=x is represented through the do-operator, do(X= =x), which represents the operation of fixing a set X to a constant x, and induces a submodel Mx, which is M with fx replaced for every X E X. The iost-interventional distribution induced by Mx is denoted by P(V Mld(X)).
mixed graph (ADMG) -CVMG)
E V to its parents V, Pa(V)) as (V, V)),
Cluster-DAGs In this study, we further develop the Cluster-DAC framework introduced in Anand Øt al. (2023). The individual variables, called micro- variabless and denoted by V"M, are grouped into clusters forming a partition QV, where each cluster contains one more micro-variables.
Definition (Custer-DAG), Let G"h (V""."") be an ADMG and let VCC be a partition vm We construct the mixed graph G -VC, possibly with self-loops and cycles, by defining g as follows. For all clusters V, W qv::
V - W is in E if and only if there exists V, in V and W., in W such that V, Ww in GT.
V e> W is in E if and only if there exists V, in V and W., in W such that V, 4-> W., in G"r
We say that GD and G are compatible.
The induced distribution P is Markovian with respect to the graph associated to the SCM (Proposition 6.31 Peters et al. (2117).
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A key novelty of our approach is the allowance of cycles at the cluster level, in contrast to Anand et al. (2023), which restricts the cluster graph to be acyclic. We nonetheless retain the term Blusttr-DAG" (C-DAG) to emphasize that the underlying graph on
remain acyclic. We denote a node in G by Ve, and its corresponding set of micro-variables in a compatible graph G"' by VM = [V,,-.. Vavi where the indices follow a topological ordering associated with G" (chosen arbitrarily if the ordering is not unique). We will use the same notations for any intersection or union of clusters. The cardinality of each cluster is displayed in the upper left corner of its corresponding node, as represented in Figure Ta
A C-DAG g is, by definition, derived from an ADMG over the micro-variabe set V however, in practical applications the true causal diagram on V typically unknown. Then, we are interested in all the ADMGs compatible with g
Definition (Class of Compatible Graphs), Lei Le be partition of and G be a mixed graph on Vv. We denote C(GC) == (g"T 9 compatible with G the equivalence class of graphs compatible with GO.
C-DAG is a valid causal abstraction of any underlying causal diagram on variiblas if and only if it contains no directed cycle composed entirely of singleton clusters, which ensure the existence of least one ADMG compatible with the mixed graph (see Proposition in Appendix).
2 A Causal Calculus for Cyclic C-DAGs
We now introduce atomically complete calculus for reasoning about cluster queries in C-DAGs. For each rule of Pearl's calculus, Theorem 2 gives a graphical criterion that is sound and complete (Theorem 3): if the separation criterion for given rule applies, then the rule is valid in all compatible graphs; if not, then there is at least one compatible graph in which the rule fails. Our work is constructive: if such graph exists, then our criterion enables its construction. To establish Theorems 2 and 3 we first introduce the concept of structure of interest, then describe the associated graphs needed to define efficiently our caleulus, and finally define the corresponding calculus in the third subsection.
2.1 Structure of interest
Demonstrating that rule of Pearl's calculus fails compatible graph (at least one) requires exhibiting a graph on micro-variables in which the corresponding d-separation fails. Concretely, this involves the three following steps on the micro-variables, for given C-DAG: (i) find a path connecting variables; (ii) for each collider on that path, provide a directed path to a conditioning variable; and (iii) ensure all these paths coexist single compatible graph. While (i)-ii) could be decided via a graphical test on the C-DAG, step (iii) is nontrivial since paths may conflict and form cycles. To avoid this, we directly look at structures of interest. which correspond we add for each collider directed path to a conditioning variable. Thus, we only need to test that this structure of interest connects two sets of variables. Definitions B and refine and formalize this intuition.
paths which
Definition 3. A structure of interest o is an ADMG, with a single connected component, in which each node satisfies the following property:
V has at most one outgoing arrow, or,
V has two outgoing arrows but no incoming arrow.
In an ADMG, executing breadth-firee search from the root set against the arrow orientation produces subgraph that, by construction, satisfies the first condition of Definition 3 Moreover, by enforcing the second condition at each exploration step, one obtains an efficient procedure for constructing the desired structures of interest within the ADMG.
Figure in Appendix shows how arrows look like around a vertex in structure of interest. We draw in Figure lb in bold an example of structure of interest in a graph on micro-variables.
"If we denote dG"" QV) the cluster-DAG obtained from GDd via Definition) | the equivalence relation is detined by G" moye G <> "G"": V dGG"; V.

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Figure 2: On the first row (Figures 2a 2b and 2cp. three examples of C-DAG are given. On the second row (respectively, Figures 2d 2e and 2f). we represent the corresponding unfolded and canonical compatible graphs. The plain and dashed arrows corresponds to G whereas the dotted arrows represent the "eligible" arrows. Lemma 2 and Figure 2e show that there no graph compatible with the C-DAG depicted in Figure such that A₁ and Ci are connected by directed path. Similarly, Proposition 3 and Figure sho that there is no graph G"d compatible with the C-DAG depicted in Figure 2c such that A E Anc(Z1,G")))
Definition 4 (Connecting structure of interest). Let G = (V,E) be a mixed graph. Let X.Y. Z be pairwise disjoint subsets V. We say that a structure of interest or S 9 connects and y under Z and we write X o Z if the following conditions hold:
X no # 0 and y no #
(a connects X and Y)
Root(or) s ZUXUU and
(all vertices or are ancestors of Z ZUXUM)
(o) (ooo(()) 0 0
(neither chains nor forks of o are in Z)
Example 1. Let us consider the graph GD and the structure of interest 6Th depicted in Figure Tb) The rools of rt are Root(o"") = (D1, C1, Yi). According Desinition 4 T connects Xı and Y1 under C U D = [C C2, D2). Indeed, T contains the path x²" (X,, Bi, C1, Yi) which d-connects X and under C U D in G
We remark that a path is always structure of interest, but it may not be connecting structure of interest. Theorem shows that there exists d-connecting path and only if there exists a connecting structure of interest.
Theorem 1 (D-connection with structures of interests). Let G be ADMG. Let X.M. Z be pairwise disjoint subsets of nodes of G. The following properties are equivalent:
1. X4g812 Z.
2. G contains structure of interest such that X X Z.
Theorem reduces the problem of determining whether rule from Pearl's calculus fails for given C-DAG the search for a compatible graph which contains a structure of interest that violates the corresponding d-separation.
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2.2 Associated graphs
Enumerating compatible graphs is generally infeasible due to their potentially large number. To address this challenge, define two mixed graphs. The canonical compatible graph (Definition 5 allows efficient verification whether given structures of interest exist in some compatible graph. The unfolded graph (Definition 6 aggregates all structures of interest present in at least one compatible graph. To look for a structure of interest, we first check the unfolded graph. However, since it may
include spurious structures that do not correspond to any actual compatible graph, we afterward filter out these spurious structures from the canonical compatible graph.
Definition 5 (Canonical Compatible Graph). Let G be C-DAG. Its corresponding canonical compatible graph is the ADMG G (VVa can Cam)) where the set of nodes is V == = and the set of edges is constructed by the following procedure:
For all dushed-bbidrrecte---rvwws Yc <-> W in G add the dusheed-bidirectted--rwss V, e> for all v. w E (1, #VV "I, #WC) such that V, * W
2. For all self-loop C VC, add the arrow V; Vj all i, (l,... #VC such that
3. For all arrows yC - WC, with yc WC, add the arrow V1 - Wawe.
Examples of canonical compatible graphs are given in Figure 2 As stated in Proposition [ the canonical compatible graph is itself compatible and canonical in the sense that can be added to any compatible graph without violating compatibility.
Proposition 1. Let G be C-DAG and Goan be its corresponding canonical compatible graph. Then, the following properties hold:
1. Ptmm CC(C).
2. For all G"I E C(G²), Poa U E C(G))
Consequently, structure of interest exists in some compatible graph, must also coexist with the canonical compatible graph, meaning that its addition to the canonical compatible graph does not create cycles
We now introduce the unfolded graph.
Definition 6 (Unfolded graph). Let G be a C-DAG and G.T.. (Vi E be its corresponding cam
canonical compatible graph. Its corresponding unfolded graph is GT, (V ns En), the mixed graph defined by the following procedure:
Va V
Let us consider the following set:
[vC wC GG, and, @m can (V, Ww. acyclic.
Eeliiibbe = V. Ww
Then E := 8 U Eeligible-
Examples of unfolded graphs are given in Figure 2 is supergraph of any compatible graph. Therefore, if there exists compatible graph containing structure of interest o",, follows that o must also appear in the unfolded graph. This implies that it is no longer necessary to enumerate all compatible graphs in the search for a structure of interest; instead, it suffices to search within the unfolded graph alone. qulurtrillsttrtee how the unfolded graph can be used to demonstrate the non-exisee non-existence of certain structures within the set of compatible graphs.
As shown in Proposition 2] the unfolded graph
Proposition 2. Let G be a C-DAG and @Th be its corresponding unfolded graph. Then, any compatible graph Gth is subgraph .T up permutation of indices in each cluster.
oL 5 Since Gmr G is acyclic, T² 6 also acyclic.

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The unfolded graph defines the search space for the structures of interest, while the canonical compatible graph, as stated in Proposition 3] ensures that these structures can indeed be realized within a compatible graph.
Proposition 3. Let G be a C-DAG and 9h be its corresponding unfolded graph. Let am be structure of interest in -TT If U an is acyclic, then P U a compatible with G
By keeping these two notions distinct, we are able to apply mutilations directly on the unfolded graph without restricting the overall class of compatible graphs.
2.3 Calculus
It is important to note that, in general, mutilating all graphs compatible with a C-DAG yields a strictly smaller set of graphs than the of graphs compatible with the mutilated C-DAG, as illustrated in Example 2 f This means that one cannot do the do-calculus on the class of graphs defined by the mutilated C-DAG.
Example 2. Let consider the C-DAG G
AA 2 B. We have the following identities:
A1 +
A
GTmH
g E ECGG))
=
Bi
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A1 >
A1
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c(G mr)
B1
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And then, (G B, 9 C(G C(( 8))
However, thanks to the unfolded graph and the canonical compatible graph, the rules of do-calculus for cluster queries can be encoded in a sound and complete manner. This principle is formally established in Theorems 2 and An illustrative application of Theorem 2 provided Example (a complementary example is provided in Figure 8 in Appendix).
Theorem 2 (Calculus). Lei Gi be a C-DAG and let G be its corresponding unfolded graph. Let x Z WW be pairwise distinct subsets of nodes. Then, for any density P. Pinduced by SCM compatible with G the following rules apply:
Rl. P(y" do(w")), ,xx" P(y
do(w 1nn) if 9T does not contain a structure of interest 6th such that X" y W" T and GtL. U ot is acyclic.
R2. P(y" do(w"), do(x"") z=)) P(y" do(w")), x,"", if gw does not contain structure of interest Tthh such that X ("' Tm T and Gmn U 0 acyelic.
R3. P(ym do(x") zm)) Pmym do(w"), zm) if 9"'p does not contain a structure of interest Tt such that X ym Wm Z"n, 9 U T is acyclic and Root(o")) C (W"M Z") y
The first two rules of Theorem arr very similar to the first two rules of Pearl's do-calculus. In contrast, the third rule of Pearl's do-caleulus requires verifying the d-separation condition y 1 कान zorze,
X
""" Z"", in all compatible graph GT", where (Z") >== = XM \ Anc( "o" 9 (.".).)..........)""... Since XTx is not, in general, union of clusters, the associated mutilation depends the particular graph G"". As result, it is not possible to directly apply this mutilation to the unfolded graph to derive an atomically complete criterion for Rule 3.
Nonetheless, if Rule does not hold in some compatible graph GTD then there exists a structure of interest between yd and X" in G" 'Wo ("(") If this structure includes root X, e X", then X, must
4A similar phenomenon was observed by Zhang 2008 in the context of ancestral graphs. 'In Pearl's framework, we would consider DAGS and any positive, compatible density.
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be an ancestor of some Z E Z in the mutilated graph. In such case, we can augment the structure of interest by explicitly adding the directed path from X, to ZTh resulting new structure whose roots lie outside X This constructive process of eliminating roots from X is behind the third rule
in Theorem 2
For all the rules, structures of interest are sought in the unfolded graph. The canonical compatible graph is then used ensure that the identified structure of interest actually exists in a compatible graph.
All the caleulus rules given in Theorem 2 are atomatically complete, stated by Theorem
Theorem 3 (Atomic completeness). The calculus in Theorem 2 is atomically complete i.e. if the rule does not hold given a C-DAG, then there exists a compatible graph in which the corresponding rule in Pearl's calculus fails.
Figure dissplass the correspond- ing unfolded graph and canonical compatible graphs. The plain and dashed arrows represent Pame while the dotted arrows denote the "eligible" edges. According the second rule of Theorem 2
Example 3. Let us consider gC, the C-DAG depicted in Figure 2c
have Phy | )) = P(ym )² Indeed any structure of interest o"h which connects Y and Z. under O contains the arrows A B2 ZA. Since Gean contains Zi - As, we know that goon U tth contains a cycle. Therefore, G" does not contain a structure of interest 6 that connects Y and Tm under 0 such that G.am V o is acyclic.
3 Computational efficiency: reducing clusters of large size to 3 nodes
While Theorem provides sound and atomically complete calculus, its direct application may be impractical for large clusters, as computing G"" becomes intractable due to a combinatorial explosion in the number of edges. To address this, associate with any C-DAG ge simplified C-DAG Le on the same set of nodes (but with different cardinals), where each cluster of size greater than 3 is reduced to size 3. The set of edges of G is the set of edges of G.. The key difference is that CGG) # C(G'), because the graphs compatible with GC, contain fewer nodes and different edges than the graphs compatible with GC. We illustrate this in Figure 3 Notably, Theorem 4 shows that applying the calculus on G or on GC leads to the same results.
Theorem 4 (Infinity is at most three). Let G be a C-DAG and GC be the corresponding C-DAG where all clusters of size greater than 3 are reduced to size 3. Let WC, Xe. y and Z be pairwise disjoint subsets of nodes. For E (1, let R,W N.W......... XX.Y. Z. be the th rule of Pearl's Calculus applied to (W.X. "WX.M,.), and say it "does not holds in G" whenever its associated d-separation condition in the associated mutilated graph is not satisfied. The following propositions are equivalent:
1. There exists Gt" E C(GC) in which R,WW", _"",,"""""""" , does not hold.
2. There exists G CIG in which W,W s3. X y Zma does not hold.
Where W
23 X you and Znd are the sets of nodes corresponding to WC, XC, yc and Z G
Figure 3 iillustrates Theorem 4 by showing how a graph G"r E C(G in which a given d-separation does not hold, is transformed into G E C(G which the corresponding d-separation does not hold as well. The figure highlights how this transformation impacts the structure of interest that violates the d-separation by omitting all irrelevant dependencies (see Figure 3d..
Theorem 4 shows that reducing cluster size to at most three preserves all relevant dependencies. This bound is tight: in some C-DAGs, any further reduction of the size of the clusters (by removing more nodes) would necessarily lose causal information. Example iillustaats such a case.
Example 4 (Infinity is at least three). Let G be the C-DAG defined by Figure 4a Let X = (XC), y = ( and Z c z² There exists a compatible graph G" in which 7 ym Z"r: we displayed it in Figure 4b
There is only one graph compatible with G CG = |G (the one displayed in Figure 4ch. Moreover, in gry, the corresponding dependence does not hold.
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Figure 3: Top: G (left) and G (right). Bottom: graph compatible with G (left) and graph compatible with G (right). In Figure 3c the arrows in bold black represent structure of interest that connects X and yd under C"" UA In Figure 3d the arrows in bold black represent the structure of interest that connects X and ym under C U A which is obtained by applying the strategy used in the proof of Theorem
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Figure 4: Figure 4a depicts cluster G Figure 4b illustrates graph compatible with GC, Figure 4c illustrates the unique graph compatible with g
Discussion
On Cluster d-Separation Theorem 2 provides sound and atomically complete calculus for causal identification. In addition, our results offer sound and atomically complete solution to the problem of cluster d-separation. Specifically, the criterion for cluster d-separation corresponds to the first rule of Theorem 2 when taking W = O. Furthermore, Theorem 6 in Appendix establishes cluster d-separation under cluster-level mutilations. Our results thus encompass both the first (association) and second (intervention) rungs of Pearl's ladder of causation.
Recovering the results on standard (acyclic) C-DAGs Theorem recovers the results of Anand al. 2023). When C-DAG G is acyclic, its corresponding unfolded graph G", is also acyclic. As a result, lt, 9 is a compatible graph and standard d-separation is both sound and complete in G
2-step strategy In our study, the unfolded graph defines the search space for the structures of interest, while the canonical compatible graph ensures that these structures can indeed be realized within a compatible graph. By keeping these two notions distinct, we are able to apply mutilations directly on the unfolded graph without restricting the overall class of compatible graphs. This approach resolves the non-commutativity between mutilation and enumeration compatible graphs, since performing mutilation before enumeration generally produces strictly larger set of graphs

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than enumerating first before mutilating, A similar phenomenon was observed in Zhang 2008 in the context of ancestral graphs.
Unknown size of clusters The typical use case for C-DAGs assumes access to individual ob- served variables (i.e., micro-variables Clusters are constructed by grouping variables based on interpretability needs and domain knowledge. In such settings, causal and confounding relationships between clusters are explicitly modeled, while dependencies within clusters are left unspecified. Crucially, the specification of C-DAG does not require the inclusion modeling of any unobserved variables within cluster. Unobserved variables are only explicitly modeled when they act as latent confounders between clusters, in which case bi-directed dashed edges introduced. This means that the cardinality each cluster corresponds to the number of observed variables contains, a quantity that is generally known. Therefore, the assumption of known cluster cardinality reflects realistic scenarios and does not compromise the validity of our thcoretical results.
Nonetheless, if the cardinality of a cluster is overestimated, then the calculus remains sound, though necessarily complete. Conversely, if the cardinality underestimated, the calculus is complete, but soundness is no longer guaranteed. In cases where the true cluster size is unknown, Theorem ensures that assuming a cardinality of yields sound calculus.
Conclusion
We have addressed in this study the problem of identification in causal abstractions based arbitrary clusterings of variables in ADMGs, extending the framework considered in Anand et al. (2023) to abstract graphs which potentially contain self-loops and cycles between clusters. This extension is important practice as the structure induced on clusters of variables in a given ADMG is likely contain cycles between clusters. In this framework, d-separation in an ADMG using structures of interest, a reformulation which remains faithful to the original formulation finding structure of interest is sufficient d-connect two sets, and then provided a causal calculus which is both sound and atomically complete. We further showed that any cluster can be reduced a cluster of limited size, leading efficient calculus rules.
have first reformulated the notion of
In the future, we aim to establish the global completeness of the calculus, as it is currently only atomically complete. We also plan to extend this work by considering micro-level interventions, iee. interventions on individual variables rather than on clusters of variables, when only the C-DAG is known.
Acknowledgments and Disclosure of Funding
This work was partly supported by fundings from the French government, managed by the National Research Agency (ANR), under the France 2030 program, reference ANR-23-IACL-0006 and ANR-23-PEIA-0007.
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References
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Assaad, C. K., Devijver, E., and Gaussier, E. (2022). Discovery of extended summary graphs in time series. In Cussens, J. and Zhang, K., editors, Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, volume 180 of Proceedings of Machine Learning Research, pages 96- 106. PMLR.
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Chalupka, K., Eberhardt, F., and Perona, P. (2016). Multi-level cause-effect systems. In Gretton, A. and Robert, C. C., editors, Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine Learning Research, pages 361-369 Cadiz, Spain. PMLR.
Chalupka, K., Perona, P., and Eberhardt, F. (2015). Visual causal feature learning. In Proceedings of the Thirty-First Conference on Uncertainty Artificial Intelligence, page 181-190, Arlington, Virginia, USA. AUAI Press.
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Zhang, J., and Bareinboim, E. (2022). Causal identification under markov equivalence: Calculus, algorithm, and completeness. In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., and Oh, A., editors, Advances in Neural Information Processing Systems, volume 35, pages 3679-3690. Curran Associates, Inc.
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Contents
Preliminaries
2
A Causal Calculus for Cyclic C-DAGS
Structure of interest
2.2
Associated graphs
Calculus
3
Computational efficiency: reducing clusters of large size to 3 nodes
4
Discussion
Conclusion
10
Glossary of Notations
13
Technical Appendices and Supplementary Material
14
B. Basic Properties of C-DAGs
14
B.2
Proof of Theorem
1
15
B.3
Proofs of the Properties of the canonical compatible Graph and the Unfolded Graph
18
B.3.1
Canonical Compatible Graph
18
B.3.2
Unfolded Graph
19
B.3.3
Proof of Proposition
20
Proofs of the Calculus
20
B.4.1
Cluster D-separation with Cluster Mutilations
20
B 3,4.2 Proofs of the Three Rules of the Calculus
21
B.5
Proof of Theorem
23
NeurIPS Paper Checklist
28
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A Glossary of Notations
X: A variable.
X: A realized value of X.
X: A set.
Anc(X.g): The set ancestors of X in G (including X itself).
Desc(X. The descendants
in & (including X itself).
Root(g):: The roots of G, i... vertices with no child.
Active vertex: A vertex V is active path relative to Z if (i) Vss collider and V or one of its descendants is in Z, or (ii) V is non-collider and e Z.
Active path: A path x is active given if all vertices on π are active relative Z.
D-separation: Sets X and y are d-separated by if every path between them is inactive given Z, denoted g Z.
Gxz The mutilated graph obtained by removing edges with arrowheads into X and tails from Z.
T[A.B]: The subpath of π between vertices A and
G1 V U2: The union of graphs Gi &V) and G2 (V2,&2), defined as (V, UV U2)).
X Xng: The set of nodes in & that belong to X.
G \ X: The subgraph of G obtained by removing all vertices in X and their incident edges.
ymh Micro-variables.
qc: Partition of y
Gt: An ADMG
micro-variable
G:: A C-DAG defined in Definition] 1
C(G ): The class of compatible graphs defined in Definition 2
σ: A structure interest.
can The canonical compatible graph defined in Definition
9Th The unfolded graph defined in Definition 6
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B Technical Appendices and Supplementary Material
Notation 1. Let Gf be a C-DAG, and let V be cluster in G When seen as node of V will be written as ;e; but when V is seen as a set of variables of a graph Gmh on vicro-variables, V will be written as VL = IV...... Vav] where the indices follow a topological ordering induced by G (chosen arbitrarily if the ordering is not unique). We will use the same notations for any intersection or union of cluster.
Terminolgy on paths. Let π be path and X subset of variables. We say that intersects or encounters X if they share least one common vertex. We treat paths as ordered lists of variables, which allows us to define the first and last encounter of π with X. The first encounter is the first vertex in the order of π that also belongs to X. Similarly, the last encounter is the last such vertex in the order of π. Let be path from X to y. Let A be vertex on œ. We denote by X|xAi the subpath of from its first vertex to A and TAM from A to its last vertex.
B.1 Basic Properties of C-DAGs
In this section, following the notations of Perković et al. (2018), an arrow (++) represents either a directed arrow (-) fashed-bidirecter arrows (++))
Proposition 4. Let QVc be partition of VMM. Let G be mixed graph over VVC and let C(g²)
denote the class of graphs compatible with G.. Then the following propositions are equivalent:
C C(GC) +
G does not contain any cycle on clusters of size 1.
Proof. Let prove the two implications:
=: If G contains the cycle 'A ... IA, then any compatible graph would contain the A,, which is not allowed because compatible graphs have to be acyclic.
cycle A
c: If ge does not contain any cycle on cluster of size 1. Let us construct a compatible graph. For all cluster V of size we put incoming and outgoing edges at V1- This does not create cycle because, otherwise, G would contain cycle on cluster of size 1. For all other clusters V, as they are at least of size 2, we can deal with V1 and Vz- We put all outgoing edges at Vi and all incoming edges at Vz. This construction cannot introduce directed cycle, since no vertex in Td ever has both an incoming and an outgoing edge.
Proposition 5. Let G be a C-DAG, 9" be a compatible graph with G and vc and WC be nodes of G.. G contains two similar (same type) arrows between 4 and Wm then removing of these arrows create another compatible graph.
Proof. By definition, only arrow between /" and Wm is necessary.
Let G be a C-DAG, GTd be a compatible graph with G and VC be node in GC, i.e. cluster. If there exists (i. j) with i > j such that G" contains the arrow - Ww. then there exists compatible graph G" that contains the arrow V, + Wi, and not V; W., if desired.
Proposition 6.
Proof. If j. thus by the convention Notation I we know that V, is before V, in a topological order of GTh Thus, V, is not descendant of V, in Gmd Thus, V, is not descendant of W,, in G"h Therefore, adding the arrow V, - W,, into G"I does not create a cycle, because otherwise V, would be a descendant of W,. in G"
By Proposition 5. Vi W,, can be thereafter removed if desired.
Corollary 1. Let G be a C-DAG, G" be compatible graph and vc be cluster. G contains path which contains a fork on V, with j. then there exists compatible graph G which contains the same path except that the fork is on Vz.
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Proof. Apply Proposition 6 twice.
Proposition 7. Let G be C-DAG, 9mT be compatible graph and vc be a cluster. If there exists i < j such that GTH contains the arrow V,t Ww. where to, then there exists a compatible graph Gmd that contains the arrow jt W and not VieeW, w if desired.
Proof. i < j. thus by the convention of Notation 1 Therefore, adding the arrow W, does not create cycle.
we know that Wi is not a descendant of Vj.
By Proposition
5]
W can be removed desired.
B.2 Proof of Theorem
We introduce Figure 5 which helps the reader understanding structures of interest.
We represent the three forms that arrows can take around vertex with multiple arrow in structure of interest. Some vertices have incoming arrows (left), some have incoming arrows and single outgoing arrow (middle), and some have exactly two outgoing arrows and no incoming arrows (right).
Figure 5:
During the proofs, target set. However, the graph containing such a structure also includes a directed path from this problematic root (outside the target set) to vertex within the target set. Lemma I shows how to exploit this directed path to construct a new structure of interest where the problematic root has been removed. Figure 6 depicts this idea.
often led construct structures of interest in which root lies outside the
A
B
Ri
R2
R
R
(b)
Figure 6: Figure 6a depicts a graph containing structure of interest shown in bold black. Let us assume that the target set of roots (R, R2). In this graph, is root outside the target set. However, the graph contains a directed path (B, C, Rí) from to vertex in the target set. Figure 6b] illustrates how this path can be used construct a structure of interest in which no longer root, without introducing new root outside the target set.
Lemma 1 (Add Path Remove Problematic Root). Let G be ADMG. Let X, y and Z be pairwise distinct subsets of nodes G. Let or be structure of interest such that:
a C G
X o # and o # 0.
(or) Root((r)) = 0.
there exists R E (Root(or) ZZ) Anc(Z. then contains structure of interest o' such that:
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(a) o' CG
(b) Xno # 0 and y no's * 0.
(c) (o') RRoo((r")) n Z = 0.
(d) Root(or') \ Z (Root(o)) \ Z) \ (R).
Sketch of proof. R E Anc(Z. G), thus & contains directed path π from R to Z. We add this path o to remove R from the root set. The end of the path is in Z. thus we do not add a problematic root.
Proof. R E Anc(Z. thus & contains a directed path from R to Z. Without loss of generality, we assume that π meets only its last vertex. We construct o' with the following procedure:
If o a \ [R] = 0, then or U π is structure of interest. In this case, we set o' - o U.. We have the following properties:
(a) o' & because S and π G.
(b) Xn SXno and y nor y no thus ooo # 0 and y no' # #0.
(c) By construction, Root(o")) = Root(or) U Root(T) \ (R). Thus, o' \ Root(or)) = (R) U (or UR) (Root(or) Root(T)) _ (R) U (ar Root(or) U (π \ Root(x)) Since R & Z. (or \ Root(cr)) Z 0, and (π Root(T)) n 0 (because π meets Z only at its last vertex), we can conclude that (a' Root(o"))
0.
(d) Root(o")= Root(() U Root(z) \ (R). Since Root(z) S Z, we conclude that Root(a"))
Z S (Root(or) Z) (R).
Otherwise, Z \ (R) 0. Let W be the first encounter of and \ (R) and let r' be the subpath of π from R to W 00 In this case, we set o' + U x'. We have (a), (b), but also
(c) By construction, Root(o")) = Root(or) (R). Thus, o' Root(o")) = UOT Root(a)) (x' Root(or)). Since meets Z only at its last vertex, we know that ' n Z (W) Thus, (r'' \ Root(rr)) = (x' n Z) Root(or) _ (W) Root(or) S or RRoot(r).. Since Z and (or Root(or)) Z = 0, we can conclude that (a' Root(o")) Z 0. (d) Root(o")) = Root((or) (R). Therefore, Root(o")) Z (Root(or) Z) \ (R).
However, o' not necessarily a structure of interest. By construction, is the only node of o U xt which does not necessarily satisfy the conditions of Definition 3 Since or is structure of interest, the arrows around W in or are necessarily in one the three cases described by F The right hand case is the only case where adding an incoming arrow prevents o' from being a structure interest. Thus, if o' is not structure of interest, means that W has one incoming arrow and two outgoing arrows in o'. Let A and B be the two children of in o'. Moreover, since or π # O, we know that of has a single connected component. Let x² be a path between X E X and Y y in o'. Let us show that we can assume that x² does not use A < W B:
nT uses A + W - B. Without loss of generality, we assume that A is before B in Λ². Let x be path from X to R in We distinguish two cases:
* XR does not encounter W (cf Figure 7a We consider x U U - is a subgraph of o' in which X and y are connected By construction, 0 does not contain W. Thus 0 o' contains path between X and y that does not use
A - W - B.
* Otherwise, x encounters W. Since XR S o, XM uses A - W + B. We
distinguish
cases:
UT 0 is subgraph of o' in which X and y are connected. By construction, 0 does
If A is before B in in (cf Figure 7bp. we consider 0 == [xw] Un' U XR
"/ and may be equal.
'x 0 Z is empty if x' * π, otherwise, it is equal to (W).
*0 is not necessarily a path.
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not contain W - B. Thus 0 S a² contains path between X and y that does not W B.
XR x |B.r Otherwise, after in x8 (cf Figure 7cb consider x [x,0| a subgraph of o' in which X and are connected. By construction, does contain A W - B. Thus 0 S o contains path between X and y that does use W B.
Thus, without loss of generality, we assume that x* does use A W We remove from o' one outgoing arrow from W that is not used by x². By doing so, W satisfies the conditions of Definition Thus, all the vertices in o' now satisfy the conditions of Definition 3 and satisfies (a) and (b), and
(c) (o')) Root(or)) = because removing the arrow does not change the vertices nor the roots of o'.
(d) Root((r") Z (Root(or) Z) (R) because removing the arrow does not change the roots of o'
However, o' does not contain necessarily single connected component. Thus, we only keep the connected component of X, which contains via x². By doing so, o' now structure of interest and we have:
(c) (o') \ Root(o")) (d) Root(or') Z (Root(or) Z) (R) because consider subgraph.
0 because we consider subgraph.
Y
W
(a)
X
R
Y
XR
A
B
(b)
Figure 7: Helping figure for the Proof of Lemma x² and XR are represented by the dotted paths following the arrows. A squiggly arrow represents arbitrary path.
Corollary 2. Let G be ADMG. Let X, y, Z and R be pairwise distinct subsets of nodes of G. Let G contains a structure of interest or such that:
X o # 0 and y no #
Root(or) ZuXuuuu
(or Root((r)) 0 === 0
for all R R, R Anc(Z.G), then G contains structure of interest that connects X and y under
Z.
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Proof. We apply Lemma
iteratively for each R E R. we get structure of interest o² such that:
3 g
X o' # and y no' # 0
Root(or") Z _ (Root(or) \ (Z) \ R. Thus, Root(or) S ZUXUM.
(or)) Root(or")) Z 0
Therefore, o' structure of interest that connects and y under Z.
Theorem 1 (D-connection with structures of interests). Let & be ADMG. Let X,y Z be pairwise disjoint subsets of nodes of G. The following properties are equivalent:
1. X
2. G contains structure of interest such that X y Z.
Proof. Let prove the two implications:
2 X Z. then, by definition, the ADMG contains path d-connecting X and D. Without loss of generality, we assume that л encounters X only at its first vertex and y only its last vertex. Since is d-connecting path, it is structure of interest which satisfies the following properties:
π SG.
Xn # 0 and y n # 0.
(rr Root(z)) n Z 0.
However, some roots of π may not be in ZUXUO y, preventing л from being connecting structure of interest. Necessarily, theses roots are colliders. Define R = Root(r)) (Z U J) y) to be the set of these colliders. Since is d-connecting. for all R E R. R E Anc( (. By Corollary 2 G contains structure of interest which connects and y under Z.
D By definition, has single connected component. Thus contains path from X V. Without loss of generality, we assume that π encounters only at its first vertex and y only at its last vertex. Let us first prove that without loss of generality, we can assume that all colliders on π are ancestors of Z. If it's not the case, since Root(or) C Z U X U V, all colliders that not ancestors of Z are ancestors of X U y. Without loss of generality, assume that a collider C on π is an ancestor of X. Thus, O contains a directed path π² from C to X. Let x be the subpath of between C and y. Since C belongs to both x¹ and x² these two paths intersect at C, and possibly at other vertices. Let be the last vertex x! that is in x! nx? Let == x U [ n is a path. Moreover, between X and T. x' is directed path, and, after to V. is a subpath of πτ. Therefore, N' contains least Z. Repeating this procedure iteratively from X to in which all colliders are ancestors of Z.
one fewer collider than that is not an ancestor of
allows us to construct path
Finally, note that all forks and chains on π are not roots by definition, and hence do not belong to Z. Therefore, the resulting path is d-connecting between X and y given Z.
B.3 Proofs the Properties of the canonical compatible Graph and the Unfolded Graph
B.3.1 Canonical Compatible Graph
Proposition 1. Let G be C-DAG and Gmh be its corresponding canonical compatible graph. Then, the following properties hold:
1. GH C(G
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2. For all G" E ECG Pm. Ug E C(G)).
We break the proof of Proposition Din Lemma Lemma 3
Lemma 2. Let G be a C-DAG. Then, Feam its canonical compatible graph, is compatible with G
Proof. By construction, all arrows in G are represented in G Moreover, all arrows that are added correspond an arrow in GC. Therefore, we only need to check for acyclicity to prove that G is compatible graph with G By contradiction, let us assume that G contains a cycle æ. First, we know that not within single cluster. Indeed, in each cluster Gany is a Vav-rooted tree. Thus, encounters at least two clusters and has an arrow between two clusters. Let A - BBR be such an arrow. We prove that necessarily, #A = Indeed, #A 2 2, then no arrow in G pointing on Ai. Similarly, we can show that #B = 1. Thus, the next arrow cannot be pointing into B thus the next arrow is also an arrow between two clusters. By induction, we show that all the clusters encountered by have a cardinal of 1. This contradicts Proposition 4 Therefore, G is acyclic.
Therefore. P is a compatible graph with g
Lemma 3. Let G be C-DAG, Iram be its corresponding canonical compatible graph, and Gmh be compatible graph. Then, Gean U is a compatible graph.
Proof. Since G"r and Gemn are compatible, then we only need check for acyclicity. Let label indices according to Notation Let be an arrow in G that is not in Gm", We distinguish three cases:
If is
arrow, then adding does not create cycle.
If is a directed arrow inside a cluster Ve, since GTT follows Notation | then the indices in ym follow a topological ordering associated with GTD. Therefore, a can be added without creating cycle.
Otherwise, corresponds to arrow between two clusters V and U G" is compatible thus it also contains an arrow from U to V"t By applying Propositions 6 and 7 we can add a without creating cycle.
we see that
Therefore, Ptm U g"r compatible graph.
B.3.2 Unfolded Graph
Proposition 2. Let G be a C-DAG and 9T be its corresponding unfolded graph. Then, any compatible graph G"h is subgruph of TTh up to a permutation of indices in euch cluster.
Proof. Let G (1"", 8th be compatible graph. By definition, we already know that V = V.. G" is a DAG, thus, in each cluster, we can permute the indices of the vertices so that a topological order of G"" agrees with the order of the indices. Let be an arrow of G"". We distinguish three cases:
If a a seahhed-bidirected--ee which contains all possible dashed-bidirected bidirected-arrowss
then a is also in G because G is a super graph of 9mh
If a corresponds to self-loop CVC in GC, Necessarily, in G",, a - Vy with i < j. Thus, a corresponds to an arrow added during step 2 of Definition 5 Therefore, a is also an arrow in 9tT
Otherwise, corresponds an arrow U - VC, with UC yc, We distinguish two cases:
If is added step 3 of Definition 5] then is also arrow in GTh
Otteewisee, by Lemma we know that Gmh U U is compatible. Thus 9 G" is acyclic. Since 9TH can. U a subgraph of GTH U g", we can conclude that does not create a cycle in G Thus, a E Eeligible Therefore is an arrow G
Hence, 8d C &g. Therefore, G"I is a subgraph of Gu-
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B.3.3 Proof of Proposition 3
Proposition 3. Let G be C-DAG and Gi be its corresponding unfolded graph. Let o be structure of interest in Ga- I om acyclic, then g U nmh compatible with g
Proof. By Proposition G is compatible with G Therefore, G n contains all necessary arrows to be compatible with G Moreover, by definition of G all arrow V, - W, in o S G correspond an arrow - W G", Therefore, @ 0 does contain any arrow preventing from being compatible with G.T G U o² is acyclic, thus it is an ADMG. Therefore, G V o is compatible with G
B.4 Proofs of the Calculus
First of all let recall the rules of Pearl's calculus for cluster queries.
Theorem 5 (Do-Calculus Rules for Cluster Queries (Pearl 2009).. Let G be C-DAG and Xe .cy .ze, rules hold:
Ie, WC be pairwise distinct subsets of nodes. Let G"I be compatible graph. The following
1. Insertion/deletion of observations:
P(y" | x, z ) P(y do( ym 1 X" 1m 2T
2. Action/observattin exchange:
P(y" (do(w")),o(x"" 2m P(ym ),(w")), 2m) y ll Gaper yo
X" 1m m
3. Insertion/deletion of actions:
P(y" do(w do(x"" P(y" do(w")). zm Y 1l Ggn X WM 2
where (T"(Z")) X" Anc(Z"" gGw)).
The first two rules hinge graphical conditions expressed d-separations under cluster-level mutilations, We begin by rigorously characterizing this form dependency; Theorem provides precisely this characterization.
B.4.1 Cluster D-separation with Cluster Mutilations
Theorem 6. Let Le = =Vve, (1vve88))r be a C-DAG. Let Gtan be its corresponding canonical compatible graph. Let 9th be the corresponding unfolded graph. Let X .yc and Zc be pairwise distinct subsets of nodes of G. Let A and B be subsets of nodes of G Then the following properties are equivalent:
g C(G) X 4 ym Tm
2. 9"h oz contains a structure of interest T" such that X -r. 2 and G am is acyclic.
Proof. Let us prove the two implications:
Let G" be compatible graph such that G" Aeigg contains structure of interest " which connects X" and y" under T By Lemma
C Ag.. _ S - Moreover, G"h 3rgg. we know that Tt does not contain any incoming arrow in AM and no outgoing arrow from 8d Therefore, " C G"" gr and T connects n and y under "
since TM
By Lemma 9m UG" is compatible graph, thus acyclic. Moreover, Gmh U C G UG" Therefore, g at!! is acyclic.
Cee t is acyclic. By Lemma 2 G is compatible. Moreover, all arrows from come from G Therefore, GHh U T compatible graph.
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Moreover, since o" S Fh ge we know that oth does not contain any incoming arrow in R" and outgoing arrow from 8Th Thus, ṭ C (G and y" under Z",, we can conclude that A HG" yc -
U gm.m Since o² connects X
Figure illustrates Theorem
y
Yi
Y2
B
3
Z
A
B
2x
X1
X2
B3
(a) g
(b) G
Figure 8: (a) a C-DAG G (b) its corresponding unfolded graphs. The plain and dashed arrows represent G while the dotted arrows denote the "eligible" edges. According to Theorem 6 we have V G"h E (g ". Iz", all structures of interest connecting X"" and Y"" given Z" in 9 include directed path from A₁ Z1. Since Gtan already contains the edge Z1 A1, such paths would necessarily form a cycle.
B.4.2 Proofs of the Three Rules of the Calculus
As soon as Theorem 6 has been established, Rules and 2 of the calculus follow almost immedi- ately. In contrast, the third rule of Pearl's do-calculus requires verifying the d-separation condition y"d L" qpo parzen
Xm WT Z"", in all compatible graph G"", where (Z") X" Anc(Z" . ww".. Since (²" is not, in general, a union of clusters, the associated mutilation depends on the particular graph GmD As result, this rule does not fall under the scope of Theorem
Nonetheless, if Rule does not hold in some compatible graph GTD then there exists structure of interest between yd and X"" in g" w='x"(Z"> If this structure includes root X, E X"", then X, must be an ancestor of some 7- E = in the mutilated graph. In such case, we can augment the structure of interest by explicitly adding the directed path from X, to Z"", resulting new structure whose roots lie outside X"
Theorem (Calculus). Lei G be a C-DAG and let GTh be its corresponding unfolded graph. Let X ,, Z -W be pairwise distinct subsets of nodes. Then, for any density P induced by SCM compatible with G the following rules apply:
Rl. P(y" | do(w")), "" P(y do(w")), x²) if 9 waa does not contain a structure of interest ot such that X ym W"" 2 and 9Th U T is acyelic.
R2. P(y" do(w"), do(x")) )m²) structure of interest Tt such that X" """ 1 " 7 and G U U acyelic.
P(ym do(w")) xm z! if Gh does contain
R3. P(y do(w"), do(x")), doo(x" 2m) = Pmy" do(w")), ), z if G does not contain a structure of interest o such that X ym m Wm Zm, G U t is acyclic and Root(o"") C (W" Z") Y
Proof. The first two rules are proven by Theorem The third one is proved by the following reasoning.
We will show that if the third rule applies, then in all compatible graph g"", the third rule of Pearl's calculus applies. More precisely, we prove the contrapositive. Let Gm be compatible graph which
PIn Pearl's framework, would consider DAGs and any positive, compatible density.
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the third rule does not apply. Then, 9 "h xx() contains a structure of interest o" that connects y and n under the conditioning set T ZTh where (""(Z")) = x"" Anc(Z"". """ By definition, we already know that follows the following properties:
n E _
Wh ("")
W
R n o 0 and yd aF 0.
Root(o=") (W T U ydm U x
(o" Rooot(o")) n (W Z 0
Let us remark that (ooot(=")nn*" Ç A (Z" Anc(Z". G GG"T,.). Indeed, let X; be an element of Root(o"") O X Since X, is a root and 6" has a single connected component, X, must have an incoming edge within o²" Thus, X₂ must have an incoming edge within g wh (2")) Thus X, (""). For element Root(or") n Am", we iteratively update using Lemma the end of this process, we obtain a structure of interest o!"! which satisfies the following identities:
Ord!! C G '
S n o 0 and yd n n * 0.
(a"" "Roo(o"")) n (1 I")
Root(o"")) (W" U = S (Root(o"") (W" U Z"") \ (Root(o"") x")) y
Thus, Root(o""") (T" SM Z U ydd and a² is structure of interest which connects X"" and y under 4 U Z" Since GTh W S F Guwwo it follows that G contains ot", Moreover, since Pean U Jom S R U it follows that Gram U acyclic.
Therefore, 5 Gim contains structure interest o" which X"" and y under " T U Zmh with Root(o"") S (W" U Z"") y"", and such that G U o² is acyclic.
Theorem 3 (Atomic completeness). The calculus in Theorem is atomically complete i.e. the rule does not hold given a C-DAG, then there exists compatible graph in which the corresponding rule in Pearl's 's calculus fails.
Proof. The first two rules are proven by Theorem The third one is proved by the following reasoning.
If the rule does not hold, then GT contains structure of interest ot that connects X and y under " -m such that Root(or") S (W" Z y and Gmm U " is acyclic. Gmm o- is acyclic, thus GTT == G " is compatible graph. Moreover, since dt C G" then 8 C G" Ww will show that we can assume that o"" - C """""" where X""(Z") X" Anc(Z". . "."h""")) "pm)):
By definition, X n T
Let X, be element of X n o", We distinguish the cases:
If T contains an outgoing arrow from X, i.c. X, - o Then this arrow is not deleted by the mutilation (Z"" exists Gmh "('"""""""
If T contains an incoming arrow X₂ i.c. X, Tth Since Root(o"")) S (W" we know that X, is not root. Let R, be root corresponding to X, i.e. an element of Desc(X,, o Rooo((r" Since T GtI "pe we know that R, & W Thus, E y"" We distinguish two cases:
If R, E Z Then X, (Z")... Therefore, - X, is not deleted by the mutilation X""("") and it exists in G"" ""
Otherwise, R, E n Thus, . contains a proper causal path from X to y We update n to be this path. Now 6 have no incoming arrows on X thus it exists in G W "Z")"
22

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Therefore, we can assume that o " xm Therefore, by Theorem y L " Zm i.e. the third rule of Pearl's do-calculus does not hold in g'".
6 ape pe12a))
X
B.5 Proof of Theorem 4
Theorem 4 presents three equivalences, one for each rule of do-calculus, To streamline the proofs, we introduce Corollary uniform manner.
3]
which restates Theorem 2 in a form better suited to treating all three rules in
Corollary 3. Let Gc be C-DAG. Let we, ,x yy and Z be pairwise disjoint subsets of nodes. Let we, X yc and Z be pairwise disjoint subsets of nodes. For i E 2,,,,3,, R, X. Z) be the th rule of Pearl's Calculus applied to (W N. Z). and say it "does not holds in G whenever its associated d-separation condition in the associated mutilated graph is not satisfied. The following propositions are equivalent:
There exists GmD E C(GC) in which R,CW" .,",,""" , I does not hold.
Sww contains structure of interest 6 such that x + ym ( Wm T and G U o is acyclic and Root(o"") S R.T
where MT (XmI
if and R.T.
(W"" U L (1"" T U u y otherwise.
fi
otherwise.
Proof. Directly follows from Theorem 2
In order to prove Theorem we need to prove Lemma 4 on structures of interest.
Lemma 4. Let G (V, E) a mixed graph. Let X.M, Z be pairwise disjoint subsets of Let GSG be structure of interest such that X Z. Then there exists structure of interest of Co such that y Z and such that #o' n = and #o n.y=
Proof. X Z, thus a contains a d-connecting path from X to V. For all collider C on x, or contains a directed path ae from to Z. We can assume, without loss of generality, that for all collider C, πe does not encounter X. Indeed, let C* denote the last collider AC- encounters X. Let be the first encounter of X and πс.. We just need to consider the path x' [ V Kict,yy .i. Similarly, we assume, without loss of generality that for all collider C, Te does not encounter y. We apply the construction of Lemma
such that
By doing so, we obtain a structure of interest o' C π Uc collider on Tc S a such that X M Z. Therefore, #o' n and #o ny=1. 1.
Theorem 4 (Infinity is at most three). Let G be a C-DAG and G be the corresponding C-DAG where all clusters of size greater than 3 are reduced size Let WC Xc. ye and Z be pairwise disjoint subsets of nodes. For E (1,2,3), let R,W. X. Z) be the th rule of Pearl's Calculus applied to (W.X.J) XXM.Z) Z), and say it "does not holds in G" whenever its associated d-separation condition in the associated mutilated graph is not satisfied. The following propositions are equivalent:
1. There exists GDd E CCGC) in which R,W"", ,"" ZZ does not hold.
2. There exists G E C(G in which R,CW" X"
S X y Zmm does not hold.
Where m
X y and Zm are the sets of nodes corresponding to WC XC y and Z in GC
23
Proof. We prove the two implications.
Proof of
We add many vertices without arrow as necessary construct graph compatible with gC in which RWW"" _ y Zz") does not hold.
Proof of (Figure illlustatts the key steps of this implication with concrete example.)
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By Corollary 3 G0 Sunwiss contains a structure of interest 6 such that XnM +m ym
(m -"
{
if
and Fean o is acyclic and Root(o")) s Rg where MT
and R.
0
otherwise.
(WW" U u U
3,
I(W" U =T n u y
otherwise.
Let oth be such structure of interest. By Lemma 4 we can assume that #o" X and #o ny = Let ye be cluster. We split o n V in two subsets as follows:
T = (V, E0 n pm V, has no incoming arrows in )m))
NT == o 0 \T
We will show that we can assume that #NF without loss of generality. Consider the case where #NF 2 2. Necessarily, since #o n X = 1 and #o n y = 1, we know that ve is different from any cluster in X y.C. Let Vy denote the element of NT with maximal index. We distinguish two cases:
If NF contains a root of oth Since X"" Ln , we know that NF S (W"" U Z"" X" U ymr Since yC is different from any cluster in X yC, we can conclude that NF C Mm u Zmr Since, X +m" Wm Zm we know that (or" RRoo((o")) n (W" U 2m) = 0. Therefore, all elements in NT are roots in ṭth By Proposition 7 and since mutilations are done at cluster level, we know that for all arrows W,, - V, with V, E NF, Feh Fu also contains the arrow W., VT. Therefore, F"r "m also contains
structure of interest o-d which is equal to o" except that all arrows W. - V, with V, E NF are pointing toward Vmax NT. Therefore, Vmax NT is the only element of ot!!! n V that has incoming arrows. Therefore, in this case, we can assume that #NF = 1.
Otherwise, every element in NF has an outgoing arrow in ot". Since o" is a structure of interest, we know that o" contains path nr from X"" to ym We will construct graph oT satisfying the following conditions:
Td C a C _ "M"
Root(o""') Root(o"
n differs from 8" only inside VM
and such that "T contains a path x=d from X to y which encounters NF at most once with a chain collider and that this intersection occurs Vmax NT. If o and x" does not satisfy these conditions, let us consider a and az be respectively the first and last arrows of I" in NT. We distinguish the cases:
1. If both and da are incoming arrows in NF, i.e n" .... W W VL ... V-UU, By Proposition and since mutilations are done at cluster level, we know that GT WM""
contains the arrows W W. Vmax NY and Vmax U We consider o" == O² U(W,, Viax N9 J (V max (VmmxNTT No and x" == x w.,1 WW N Un) (U...a Note that n encounters NF only at V and that Root(o""") ")ot(o"").
If both aj and a2 are outgoing arrows in NT, i. nr = ... W V2 U, By Proposition/oland since mutilations done at cluster level, we know that G"hh """M"
contains the arrows W Wr, V, and Vi Un. We consider 8t" O' VIV U.! and n² := [xx w. UWW CoV V Unl Un "UU,."" Note th does not encounter NF and that Root(o-"t)) ")
3. Otherwise, without loss of generality, towards y"d i.c. x" = W ++ Vl. V..V.. o+ U, U..... Indeed, otherwise, we just need to consider the path from y" to X Note that Vi, or V3 could be equal to Viax NY but not both. Since all elements in NT have outgoing arrow, let consider C., the child
assume that aj and a₂ pointing
24

---

of V Venax NY in o By Propositions 6 7 and since mutilations are done at cluster level,
we know that Ge Pun contains the arrows Ww, +- V Vmax NF and C. V2
We consider
arl = a U (W VIna (C, ( and x²' == I [x,",,,] U ( Vimx NF
C. V, 1V3 Note x"" x" encounters NT only once with collider in V VF and once with a fork in V2 P.
and that Root(o""') Root(o"
Note that in all cases, we have used Propositions o and 7 thus 6nd U Gean is acyclic. Moreover, in all cases, Root(o²m) Root(or"), Therefore, Root(or"")) & RT In addition, ot contains x²', a path from T to y which encounters NF at most once with a chain or collider and that this intersection occurs a V Vmax NT.
However, in Cases and 3, we have added outgoing arrows to some vertices different from NF. NY. This could prevent o!" from being structure of interest. We apply the following transformation to construct structure of interest from o""::
1. Move all incoming arrows to V Maxx NY By Proposition 7 and since mutilations are done at cluster level, we know that for all arrows W,, V, with V, NF, NFF.TWm.. Ww
also contains the arrow W. - Vax Therefore, G Fu also contains on which
is equal to pt except that all arrows W - V, with V, E NF are now pointing toward Vmma NY. Since we are using Proposition 7 we know that aumer U G is acyclic, Note that n² still exists in "mh and that all vertices in o=!!'' n V* except V Vinux NY have no incoming arrows in ot" Moreover, note that Root(a = Root(o""")
2. Remove problematic outgoing arrows: To keep the notations simple, we update 6T - Some vertices, different from Vina NY, may have more than two outgoing arrows, preventing o"" from being a structure of interest. We remove from o"", all outgoing arrow from V that not used by Λ"' Since, x²" uses most two arrows around vertex, we know that all vertices have now at most two outgoing arrows. Since we have just removed some arrows, we know that yed U G remains acyclic. Moreover, since r² is preserved, we know that X and y are still connected.
Remove the problematic vertices: At the end of the previous steps, some vertices are not connected to the others in o preventing o from being a structure of interest. More precisely, these vertices are NT (VV NY except V1 in case and except V2 in case 3. Indeed, at step 1, they have lost their incoming arrows and at step 2, they have lost their outgoing arrows. We remove these vertices from o"h, By doing so, 8-"! U remains acyclic and o is now structure of interest, Root(o"")) = Root(o""") S Ry
To summarize, we have constructed structure of interest T!! in ""T such that
X y | Wmm Z" GTh U o is acyclic and Root(o""') R.m. Moreover, V Vinx NYF is the only element of td n n with incoming arrows. Therefore, in this case, we can assume that #NT 1.
Therefore, in all cases, we can assume that #NF S 1 without loss generality.
We will now show that we can assume that F C (Vi) without loss of generality. If it is not the case we apply the following transformations:
1.
Move all arrows to Vi: By Proposition aand since mutilations are done at cluster level, for all arrow V, - W,, F n o"", we know that GT ""'"""""" contains the arrow Vi - W,, Thus, we consider "h the subgraph of T" obtained by moving all the arrows V, - W with V. in F Fn" to V1 - W.. Since, o is a structure of interest, contains a path x" from X to y"" Note that the above transformation yields a path nh in " which connects x and y
2. Remove problematic outgoing arrows: In o Vi may have more than two outgoing arrows, preventing m from being a structure of interest. Yet " uses at most two of these arrows. Thus, only keep two of them without altering r"
25

---

3. Remove the problematic vertices: At the end of the previous steps, some vertices are not connected the others in o preventing o from being structure of interest. More precisely, these vertices are F [V1). Indeed, they have lost their outgoing arrow at step and had no incoming arrows are they are F.
To summarize, we have constructed a structure of interest 6-' in F "M..e such that X +l.
W Zz 9 U "" is acyclic and Root(o"")) - R. Moreover, V1 is the only element of V n o with no incoming arrows. Therefore, we can assume that S (V1).
End of the proof I 2 Without loss of generality, we assume that #NT S and T S (V1). Let VNy be the only element of NF (if exists). We consider the graph GTT g U o.". By Corollary 3 R&C "M y -Z"" does not hold in G and by construction Gm is compatible with
G For every vertex in VM (F U NF), we apply Proposition to move all outgoing arrows to Vi and Proposition 7 to move all incoming arrows to Vave. This yields graph GTD E CCG in which R, x ymm ") does not hold, and where vertex outside NF U (V1) U ( is incident to an arrow.
We repeat this construction for each cluster, ultimately obtaining G where R,(1"".. X y"" ZT") still fails and every cluster has arrows on at most three vertices. Finally, by removing all vertices that are not incident any arrow in G"* we obtain GTh R;( $3. X y $3* (x does not hold.
53 graph compatible with G s3 in which
Therefore, there exists 9% E C(G in which W 23
Am D
Z 5 23
does not hold.
We come back to the example presented in Figure to illustrate the successive steps of this proof in Figure 9 Particularly, Figure 9a is the initial graph, then in Figure 9b we consider an analogous structure of interest such that # #NF I I. Then, in Figure 9c we consider analogous structure of interest such that F S [V1). Finally, in Figure 9d we move every unused gray arrows such that only three vertices of 8 incident any arrows. From this graph, we deduce G in Figure 9e on which d-connection with structures of interests are equivalent. We apply equivalent rules to gel which corresponds Figure
26

---

Yi
B1
Y1
X1
A
D1
X1
A1
D1
B2
Y2
B2
Y1
X2
D2
X2
D2
B3
Y
B3
Y3
X3
X3
B4
C1
Y4
B4
Y4
(a) 9" C(G)
(b) Application of Proposition to the arrow B3
Cr. thereby adding B₁ Cj. Note that now
#NTF
B
Yi
B
Yi
X
A
D
X1
A
D
B2
Y2
Bz
Y
X2
D2
X₂
D2
B3
Y3
B3
Y:3
X3
X3
B4
C1
Y4
B4
C1
Y4
(c) Application of Proposition to shift Bg
Ci to
(d) Application of Propositions 6 and to reposition all unused (gray) arrows. Note that only three vertices of B" are incident any arrows.
Bi Ct. Note that now F C (V
X1
B1
D
Y
X1
B
D
Y1
A
A
X2
B2
D2
Y2
X2
B2
D2
Y2
X3
B3
C
Y3
X3
B₁
C₁
Y3
(e) Remove some vertices that are not incident to any arrow to get GTH Note that G E C(GS).
(f) Application Propositions G 7 graph of Figure 3d
get the same
Figure 9: Figure shows the graph GTD containing the structure of interest T (in bold black), which connects X and Td under C U A". Figures 9b] 9c and 9d illustrate the successive transformations of G"d and o (as carried out in the proof of Theorem 4) for the cluster BC Figure 9e shows the last step of the proof. Figure 9 shows how to transform the graph in Figure 9e to get the graph in Figure 3d
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Reviews:
[{"review_id": "tZRtH1oW9R", "reviewer": "Reviewer_PDWb", "summary": "This paper generalizes Anand et al.'s (2023) result on do-calculus for Cluster-DAGs (C-DAGs), a kind of graphical structure derived from partitioning the vertex set of an underlying acyclic directed mixed graph and adding appropriate types of edge between cells of the partition according to the edges in the underlying graph. While Anand et al. (2023) only considers C-DAGs that do not contain any directed cycle and so effectively rules out certain partitions of vertices, the current work allows C-DAGs that contains cycles and as a result takes all partitions as admissible. In addition to presenting a do-calculus for the less constrained C-DAGs and demonstrating its soundness and atomic completeness, the paper also establishes that for the purpose of checking the applicability of a do-calculus rule to a given C-DAG, it suffices to check it in a simplified C-DAG in which all the clusters of size greater than three (i.e., contains more than three micro-vertices) are treated as if they contain only three micro-vertices.", "questions": "Minor questions:\n\n1. I have not tracked recent usage of the term \"root\" in graph theory, but what are called \"roots\" in this paper seem to me more appropriately called \"leaves\".\n\n2. On p. 3, line 6 from the bottom, what does \"*the* topological order\" mean? Topological ordering isn't unique in general, is it?\n\n3. The informal remark about a \"structure of interest\" reminds me of the notion of a \"d-connecting walk\", which is also a kind of combination of a d-connecting path and paths from colliders to conditioning vertices. Is there any connection worth noting here between the two notions?\n\n4. On p. 5, last line, why is the qualifier \"V_v != W_w\" needed? \n\n5. In the statement of Theorem 2, the notion of \"density compatible with [a] compatible graph\" is not explained previously. Moreover, I have a vague impression that Pearl's do-calculus requires positive density. Do I remember it wrong?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Despite being a highly theoretical paper with dense details on graphical definitions and manipulations, it is clearly written and fairly readable.\n\n2. The central results are elegant. \n\n3. Although it builds on Anand et al.'s (2023), there is at least one non-trivial innovation (the introduction of the canonical compatible graph), and one interesting finding on the possibility of reducing the computational cost, so the improvement is not merely incremental.  \n\nWeaknesses:\n\n1. No real examples are given to illustrate the utility of the theoretical constructs. As a result, it is hard to estimate how much impact it will make.\n\n2. The authors often use the terms \"micro-variables\" and \"macro-variables\" to describe the contrast between variables represented by the vertices in the underlying graph and those represented by clusters of vertices. However, it is not entirely clear to me that this terminology is appropriate. If the state space of a cluster-variable is merely the Cartesian product of the state spaces of its constituent variables (i.e., there is no coarse-graining of the state space), then referring to it as a macro-variable seems misleading, at least from my perspective.", "was_revised": true, "final_justification": "My evaluation remains positive after taking into account other reviews and feedback.", "rebuttal": "We thank you for your valuable feedback.\nWe agree with your remarks and provide below clarifications to your questions and comments.\nTo better motivate our result, we will include an illustrative example in the final version.\nRegarding the terminology around micro- and macro-variables, we agree it may be confusing and will revise the paper to consistently refer to variables and clusters of variables instead.\n\nAnswers to your questions: \n- Q1: you are absolutely right that, in graph theory, nodes without descendants are typically referred to as “leaves”. We have adopted here the terminology commonly used in causal inference, where such nodes are often called “roots” (see \"Identification of Conditional Interventional Distributions\", Ilya Shpitser and Judea Pearl, UAI 2006). We chose to align with this convention to remain consistent with the causal framework.\n- Q2: Thank you for pointing this out. You are correct that the topological ordering of a DAG is not unique in general. In our context, it suffices to fix an arbitrary topological order of \\( G^m \\). The only property we rely on is that if \\( i < j \\), then \\( V_i \\) is not a descendant of \\( V_j \\) in \\( G^m \\). We will clarify this point explicitly in the final version of the paper.\n- Q3: We are not aware of a formal notion of a d-connecting walk. If this refers to a walk that satisfies the usual d-connection criterion but allows repeated vertices, then it differs from our structure of interest, which explicitly accounts for additional paths from colliders to conditioning variables.\nIf the reviewer had a different definition in mind or could point us to a relevant reference, we would be very interested in learning more.\n- Q4: It is needed because the canonical graph is intended to be a compatible graph, and therefore must be a valid ADMG. Without this qualifier, the construction could introduce a self-loop (e.g., on \\( V_1 \\)), which is not permitted in an ADMG.\n- Q5: You are right, this notion was not clearly defined before Theorem 2. What we meant is a density that admits a Markovian factorization over the observed nodes with respect to the C-DAG.\nYou are also correct that the positivity assumption is needed, similarly to the standard assumptions in Pearl’s framework. We will revise the paper to clarify both points in the final version."}, {"review_id": "MGNZgcu20c", "reviewer": "Reviewer_bfWX", "summary": "This paper extends cluster DAGs (Anand2023) by providing a do-calculus in the presence of cycles, which one would expect to be present between clusters in many applications. The calculus is based on the assumption of an underlying acyclic graph, and the interventions considered act on all members of a cluster. On the basis of a novel d-separation criterion, they show that it is possible to determine whether a causal query would fail in any underlying graph that is compatible with a given cluster DAG in a sound and atomically complete fashion. Moreover, they find that that the result does not change if all clusters of size greater than 3 are reduced to size 3, which greatly improves the computational efficiency of the calculus.", "questions": "Main points:\n- line 117: what do you do if the topological ordering is not unique? (also, what is $V$ without index here?)\n- Definition 4: should it have a name? $\\sigma$ is a tuple of sets but treated as a single set in some observations; please clarify. Line 163: should it say \"ancestors of or in\"?\n- Theorem 2: In R2, shouldn't it be $\\bar{X}$? In R3, why the brackets in the union?\n- line 257: What is $X^m(Z^m)$?\n- Suggestion: add main-text pointers to corresponding proofs in appendix\n- Suggestion: rename _structure of interest_ to something less generic\n\nDetails:\n- line 28: representations don't have to be learnt\n- line 80: $Z$ not yet defined\n- Line 166: what do you mean by \"under\"?\n- Line 223: what do you mean by \"generally\"?\n- Footnote 1: what is the $\\sim$ relation?", "limitations": "A more extensive discussion on limitations and implications of your work would be welcome.", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "# Strengths\n\n- Their results provide a do-calculus for what's arguably the most practically relevant setting: cyclic clusters.\n- Exceptionally clear, despite the great volume of theoretical results.\n- Strong result on how to improve the computational efficiency of applying their calculus.\n\n**Weaknesses**\n\n- Some notational issues (see \"Questions\").\n- Rather short discussion of limitations and implications of their results.\n\n---\n\nThis is the best paper I have read in a while.", "was_revised": true, "final_justification": "This is an excellent paper of high technical quality and remarkable clarity.", "rebuttal": "We thank you for your valuable feedback — in particular, we are proud and pleased that you found this to be “the best paper [you] have read in a while.”\nWe agree with your remarks and provide below clarifications to your questions and comments.\nIn the final version, we will develop the discussion of limitations and implications of our results in the conclusion, by discussing the points raised by the reviewers. \n\nAnswers to your questions: \n- Regarding the topological ordering: Thank you for pointing this out. You are correct that the topological ordering of a DAG is not unique in general. In our context, it suffices to fix an arbitrary topological order of \\( G^m \\). The only property we rely on is that if \\( i < j \\), then \\( V_i \\) is not a descendant of \\( V_j \\) in \\( G^m \\). We will clarify this point explicitly in the final version of the paper.\n- Regarding Definition 4: we thank the reviewer for this helpful suggestion. We agree that it would be clearer to assign it a name to Definition 4, and we propose 'connecting structure of interest'. For the second point of \\sigma being a tuple of sets treated as a single set, we will proofread carefully for the final version.\n- Regarding the notion of ancestors: we assume that every node is considered as an ancestor of itself, so the term “ancestors” already subsumes “ancestors of or in,”. We will clarify this convention in the final version of the paper.\n- Regarding Rule 2 and Rule 3: we have carefully checked Rule 2 and believe it is correct as stated. In this step, we apply do-calculus to remove the do-operator on $X^m$, which results in the mutilation \\underline{X^m}. As for Rule 3, you are right, brackets are not necessary since union is an associative operator.\n- Regarding X^m(Z^m), we defined it in line 251:\n\\(\\X^m(\\Z^m) = X^m \\setminus \\Anc(\\Z^m,\\Gm_{\\overline{\\W^m}})\\)\n- Regarding the CRL reference: you are right representations don't have to be learnt. We will improve this paragraph in the final version.\n- Regarding $\\mathcal{Z}$: thanks for catching this up, we will introduce $\\mathcal{Z}$.\n- Regarding L166: by \"under,\" we mean \"under the conditioning of.\" We will revise the sentence to clarify this and avoid ambiguity.\n- Regarding L223:in this context, \"generally\" is meant to convey that one cannot expect the two operations to commute in general. That is, while the inclusion between the two sets always holds, equality does not. We have updated the sentence to clarify this point.\nRevised sentence:\n“It is important to note that, in general, mutilating all graphs compatible with a C-DAG yields a strictly smaller set of graphs than those compatible with the mutilated C-DAG, as illustrated in Example 2”\n- Regarding Footnote 1: we are defining an equivalence relation in this footnote, denote by tilde. This will be clarified."}, {"review_id": "v2R3efQCI5", "reviewer": "Reviewer_MpsW", "summary": "In a lot of complex real-world scenarios, access to a fine-grained causal graph is difficult, and often we have to deal with a coarser representation of these graphs. This paper studies one such abstraction of fine-grained causal graph called Cluster-DAG (CDAG) and extends the limitation of previous work, which needed an acyclicity constraint on the C-DAG. They also extend the notion of d-separation and do-calculus for the CDAG.", "questions": "- In Definition 5 (Line 193), when adding the arrows for V^{C} -> W^{C}, why do we only add one arrow between these two clusters of variables? Why not add as many as possible so that the compatible graph remains acyclic? I understand that this might lead to a question of which edges to choose, and we are adding all those edges anyway in the unfolded graph. Please clarify / give intuition for this.\n \n- Theorem 4 is interesting since it shows that using a C-DAG with cluster size at most 3 leads to the same do-calculus results. Please add some intuition on why a cluster size of 3 was necessary and what information is theoretically lost (if any) when we reduce the size of the cluster up to 3.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- The paper is well written, and the presentation is clear with supporting examples throughout to give intuition.\n\n- The authors provide atomically complete do-calculus for the C-DAG (Theorem 2), which can then be used to answer both Level 1 (observational) and Level 2 (interventional) queries in Pearl’s hierarchy.\n\n- They provide a compact representation of the search space for the microlevel DAGs compatible with the C-DAG in the form of an unfolded graph (Definition 6) instead of enumerating over the whole search space. Their definition of a compatible graph (Definition 5) is also constructive, which helps create instances to test for a particular d-separation (Proposition 3).\n\n\n**Weaknesses**\n\n- (Major) This work assumes that the cardinality of the cluster variable is known, which is then used to create the compatible or unfolded graph. There is a discussion to limit the number of variables to 3 in Theorem 4, showing that this preserves all the relevant dependencies. But, I am not sure if the result will hold if the cardinality is misspecified, which is very likely in the real-world scenarios where we only have access to the coarse C-DAG. \n\n- (Minor) This work assumes that when an intervention is performed on a cluster node all the micro-level nodes are also intervened on (the author also mentions this in Line 355). This assumption might be difficult to achieve in practice, where an intervention on such a large number of nodes simultaneously might be infeasible.", "was_revised": true, "final_justification": "I retain my very positive rating of 5. I'm not further increasing it to 6, since it isn't clear what happens when the cardinality of the cluster variable is mis-specified, which is very likely in real-world scenarios.", "rebuttal": "We thank you for your valuable feedback which we will discuss in the final version of our paper, if accepted, especially regarding clusters of unknown or misspecified cardinality.\n\nAnswers to your questions: \n- Q1: The canonical graph is deliberately constructed to impose the minimal number of topological constraints. For instance, suppose a compatible graph contains an edge \\(V_{2} \\rightarrow W_{w}\\). Acyclicity then forbids \\(W_{w}\\) from being an ancestor of \\(V_{2}\\) in \\(G^{m}\\), and, by our indexing convention, it also forbids \\(W_{w}\\) from being an ancestor of \\(V_{1}\\). Instead, by including only the edge \\(V_{1} \\rightarrow W_{w}\\), we eliminate the second constraint.\n- Q2: First of all, example 4 shows that Theorem 4 may fail if we restrict cluster size to 2. In particular, for rule R1 with \\(W^{m} = \\emptyset\\). When we reduce the size of the cluster up to 3, the resulting C‑DAG no longer corresponds exactly to the original micro‑system. However, from the standpoint of causal reasoning (i.e., the application of do‑calculus rules), no information is lost: the reduced C‑DAG and the original C‑DAG yield the same identification conclusions."}, {"review_id": "ji7LE3iSnE", "reviewer": "Reviewer_vAM9", "summary": "The paper addresses causal effect identification problem in abstract causal graphs, in which each macro-level node is a cluster of the micro-level variables of some underlying causal mixed DAG. The authors extend the existing results in Anand et al. (2023) to allow cycles in the abstract graphs with a reformulation of d-separation and do-calculus rules that are provable sound and complete.", "questions": "Although I appreciate the theoretical contributions, I remain uncertain about their practical implications. While I am fully aware of the scope of the work, my questions relate primarily to causal discovery, which may help clarify the potential application of the theory.\n1. The results suggest that if the d-separation rules do not hold in a C-DAG, there exists a compatible micro-level graph where they do not hold. What is the implication of this? Does it mean that the current clustering is invalid, and we should instead seek one where do-calculus consistently holds across all compatible graphs?\n2. The paper assumes known abstract structure. In practice, in what scenarios is the macro-level structure known while the micro-level structure remains unknown?\n3. Given a clustering of macro-level variables, there may be multiple compatible micro-level DAGs. Does the true underlying DAG among micro-level variables always belong to this set of compatible graphs?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "### **Strengths:**\n\nThe paper is well-written and easy to follow, with clear illustrative examples. The authors present solid theoretical results that address the problem at hand. The authors should it is sufficient to search for a compatible graph of size-3 clusters to efficiently apply do-calculus in practice. The method does not require knowledge of the micro-level causal structure but instead assumes known structure at the macro-level. Moreover, the proposed rules are agnostic to specific clustering types.\n\n### **Weaknesses:**\n* The paper supports causal reasoning up to L2-inference, whereas Anand et al. (2023) additionally support counterfactual inference. \n\n* A major limitation is the absence of numerical experiments to validate the theoretical findings. It would significantly strengthen the paper if the authors included a numerical demonstration showing that, given a valid clustering, one can derive the same  conclusions of interventional inference from the abstract DAGs as those drawn from the micro-level DAGs. \n\n* Additionally, it would be beneficial for the authors to further characterize the algorithm’s computational complexity.", "was_revised": true, "final_justification": "It is clear the paper provides a solid theoretical work. I retain my rating where the borderline evaluation pertains to the lack of numerical experiments as empirical evidence that the theory can be reliably applied in practice.", "rebuttal": "We thank you for your valuable feedback which we will consider for the final version of our paper, if accepted, especially regarding numerical experiments and computational complexity.\n\nAnswers to your questions:\n- Q1: When the clustering fixed, the implication is that the available causal information is insufficient to guarantee d‑separation (and identifiability) across all compatible micro‑graphs. In such cases, our do‑calculus procedure will construct a a micro‑level graph in which the targeted d‑separation condition fails and which can help users understand what causal information is missing. Furthermore, if one is free to choose a different clustering, it may be worthwhile to search for one under which d‑separation holds uniformly across compatible graphs. That said, if d‑separation is genuinely violated in the true underlying micro‑graph, no choice of clustering can restore it. \n- Q2: There are several real‑world scenarios in which the macro‑level C‑DAG is known but the micro‑level structure remains unknown, as in macroeconomics where sector‑level relationships are known (as consumption→investment) but not the firm‑to‑firm or household‑to‑household causal links, or in neuroscience where functional MRI region interactions can be established but not necessarily the causal links between neurons.   \n- Q3: By Definition 1 (see also lines 119-121), a cluster DAG is assumed to be obtained from an ADMG which corresponds to the true underlying DAG. By construction, this DAG belongs to the set of compatible graphs."}]

Author rebuttal:

Meta-review: This paper vastly extends the applicability of Cluster-DAG models, which cluster variables together and model causality at the higher level of clusters, by relaxing the constraint of acyclicity. The key is to develop a proper do-calculus that makes sense in this context and to characterize what graphs are compatible. The paper also shows that it is always possible to work with small compatible graphs to reason at the level of the clusters.

Reviewers appreciated the paper. The presentation is clear, but it could benefit from better guidance about notation and acronyms, e.g., glossary, making sure no acronym is used before it is defined. An illustrative example for the model could also go a long way in establishing its applicability. We urge the authors to address these points and any other clarification made during the discussions.
Decision: Accept (poster)