Identifying Macro Causal Effects in C-DMGs over DMGs
Authors: Simon Ferreira
Charles K. Assaad
Venue: NeurIPS 2025 poster ()
Keywords: Causality
Cluster graph
do-calculus
sigma-separation
Primary area: probabilistic_methods

Abstract: The do-calculus is a sound and complete tool for identifying causal effects in acyclic directed mixed graphs (ADMGs) induced by structural causal models (SCMs). However, in many real-world applications, especially in high-dimensional settings, constructing a fully specified ADMG is often infeasible. This limitation has led to growing interest in partially specified causal representations, particularly through cluster-directed mixed graphs (C-DMGs), which group variables into clusters and offer a more abstract yet practical view of causal dependencies. While these representations can include cycles, recent work has shown that the do-calculus remains sound and complete for identifying macro-level causal effects in C-DMGs over ADMGs under the assumption that all clusters sizes are greater than 1. Nevertheless, real-world systems often exhibit cyclic causal dynamics at the structural level. To account for this, input-output structural causal models (ioSCMs) have been introduced as a generalization of SCMs that allow for cycles. ioSCMs induce another type of graph structure known as a directed mixed graph (DMG). Analogous to the ADMG setting, one can define C-DMGs over DMGs as high-level representations of causal relations among clusters of variables. In this paper, we prove that, unlike in the ADMG setting, the do-calculus is unconditionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. Furthermore, we show that the graphical criteria for non-identifiability of macro causal effects previously established C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs.

Identifying Macro Causal Effects in C-DMGs over DMGs
Simon Ferreira
Charles K. Assaad
Sorbonne Université, INSERM, Institut Pierre Louis d Epidémiologie et de Santé Publique, F75012, Paris, France
Sorbonne Université, INSERM, Institut Pierre Louis d' dpiiémmioooogi et de Santé Publique, F75012, Paris, France charles assaaddissaadddraaaam fr
simon. www..errrerrrdssrrmmeeeeiivveeee .fr
Abstract
The do-calculus is sound and complete tool for identifying causal effects in acyclic directed mixed graphs (ADMGs) induced by structural causal models (SCMs). However, in many real-world applications, especially in high-dimensional settings, constructing a fully specified ADMG is often infeasible. This limitation has led to growing interest in partially specified causal representations, particu- larly through cluster-directed mixed graphs (C-DMGs), which group variables into clusters and offer more abstract yet practical view of causal dependencies. While these representations can include cycles, recent work has shown that the do-calculus remains sound and complete for identifying macro-level causal effects in C-DMGs over ADMGs under the assumption that all clusters sizes are greater than 1. Nevertheless, real-world systems often exhibit cyclic causal dynamics at the structural level. To account for this, input-output structural causal models (ioSCMs) have been introduced ioSCMs induce another type of graph structure known as directed mixed graph (DMG). Analogous to the ADMG setting, one can define C-DMGs over DMGs as high-level representations of causal relations among clusters of variables. In this paper, we prove that, unlike in the ADMG setting, the do-calculus is uncondi- tionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. Furthermore, of macro causal effects previously established C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs.
generalization of SCMs that allow for cycles.
show that the graphical criteria for ioniidentifiability
Introduction
Understanding and identifying causal effects is central goal in many scientific disciplines. In recent years, structural causal models (SCMs) have emerged as foundational framework for reasoning about causality. These models encode causal assumptions through structural equations and are typically represented by acyclic directed mixed graphs (ADMGs), which capture both causal and confounding relationships. Within this framework, the do-calculus (Pearl 1995] based on the notion of d-separation [Pear] -povvides complete and sound set inference rules for identifying causal effects from observational data, assuming the causal structure is fully specified. However, SCMs do not fully capture systems with cyclic causal dependencies at the structural level, which are common in public health, biology, economics, and engineering systems. For example, there can be a cyclic relation between poor mental health (e.g., depression or anxiety) and substance use (e.g., alcohol, drugs). The worsening of mental health and increase in substance fime-frames (daily or even hourly), especially in high-risk populations. Over time, they may reach cyclic equilibrium where both reinforce each other without clear causal ordering. To address
oceur in tight
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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this, the notion of input-output structural causal models (ioSCMs) has been proposed FForé and Mooij 2020]. These models generalize SCMs by allowing for eycles and induce a new class of graphs known as directed mixed graphs (DMGs) [Richardson] 1997 Forré and Mooij 2017 2018
Forré and Mooij, 2020, Booken and Mooij 2024], which provide a richer representation of causal structures. Furthermore. Forré and Mooij [020] introduced an extension of d-separation to DMGs, called o-separation, and showed that the do-caleulus, when replacing d-separation by o-separation becomes sound for identifying causal effects in DMGs [Forré and Mooij, 2020).
However, in many wal-w applications- paatiinuulaa those involving diggh-dmensional data or limited domain knowledge- it is often unrealistic to assume a complete specification of the underlying causal graph. This has motivated the development of partially specified graphical models (Maathuis
and Colombo| [2013] PPrrkoicc et al. [016 Perkovic 2020, Paber et al.) [022] WWng et al.) 2023
Assaad et al. 2023] Anand et al. 2023 Wahl et al.4 Reiter et al. [0044 Boeken and Mooij, 2024
Ferreira and Assaad [024, 2025a]. and in particular cluster graphs. Cluster graphs abstract away
some of the fine-gres grained details by grouping variables into clusters, thus offering a more flexible and
scalable representation of complex systems. Importantly, cluster graphs allow for cycles, which can arise naturally in feedback systems or time-dependent processes, complicating the analysis compared to traditional ADMGs. In these graphs, causal effects can be separated into two types: a micro causal effect where the interest is the effect of variable within a cluster on another variable in another cluster; and the macro causal effect where the interest in the effect of set of an entire cluster on another
entire cluster. In this work, we focus on the latter. [Anand et al. [203)] Tikka et al. 2023 have shown that do-calculus (the version using d-separation) remains both sound and complete for identifying macro causal effects when the cluster graph representing ADMGS is acyclic. [eereira and Assaad 2025a b) showed that the do-calculus (the version using d-separation) is also sound and complete for identifying macro-level causal effects when the cluster graph representing an ADMG is cyclic, denoted here as C-DMG over ADMGs, assuming either that the size of the clusters is unknown, or that each cluster contains more than one variable [Ferreira and Assaad 2025b).
Motivated by these developments, we consider the problem of identifying macro-level causal effects in cluster graphs representing DMGs, denoted as C-DMGs over DMGs, a natural generalization of previous work. Our contributions are threefold:
We prove that sepparation Forré and Mooij 2018 a fundamental tool in causal reasoning in DMGs-is sound and complete in C-DMGs over DMGs.
We prove that do-calculus (the version using o-separation) FFor and Mooij 2020 is sound and complete for identifying macro-level causal effects in C-DMGs over DMGs cnniss--nodiiiinnlllyy. and without the constraints needed in the case of C-DMGs over ADMGs (Ferreira and Assaad 20256 Yvernes 2025].
We show that the graphical characterization of non-identifiability previously developed
for C-DMGs over ADMGs [Ferreira and Assaad 2025a b) also applies for C-DMGs over
DMGs under an additional assumption.
The remainder of the paper is organized as follows: In Section 2 we formally presents C-DMGs over DMGs. In Section 3 we show that o-separation and the do-calculus is sound and complete for macro causal effects in C-DMGs over DMGs and present a graphical characterization for the non-identifiability of these effects. Finally in Section 4 we conclude the paper while showing its limitations. All proofs are deferred to the appendix.
2 Preliminaries
To streamline the presentation and avoid repetitive explanations, will adopt the unified notation G = (V", E*) to refer to any type of graph. This notation allows us to generalize results and discussions without redundancy across different graph types. In the remainder, for every vertex V* V" in a graph g" = "V*,E") we will refer to its parents by Pa An (V*,g*), and its descendants by De (V*,9*), We consider that vertex counts as its own descendant and as its own ancestor. In addition, the strongly connected component of a vertex is defined as See (V,9") An (V,9*) n De (V.9".
its ancestors by
In this section, we present the essential definitions and notations that will be used throughout the paper, ensuring clarity and consistency in the exposition of our results. In this work, we assume
2

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causal relations are modeled using an input/output structural causal model (ioSCM) (Foréé and Mooij 2020
which extends classical structural causal models (SCMs) [Pearl 2009 by allowing for the presence of cycles. Unlike classical SCMs, ioSCMs allow structural equations to mutually depend on each other. For example, in the cyclie system:
X = Jx(,
Y fx(,
by),
X functionally depends on Y, and Y functionally depends on X, forming cycle. When cycles exist, instead of computing variables solutions. That is, joint assignment to the variables that simultaneously satisfies all equations. This analogous to finding an equilibrium dynamic systems. Firstly, let us properly define the notion of loops as it will be useful to guarantee the compatibility of the causal mechanisms in ioSCMs.
top-down order as in SCMs, ioSCMs rely on point
Definition (Loops). In a directed graph G* (V", E*), loop is a set of vertices $ S such that there exists directed path between every pair of distinct vertices in the subgraph induced by S i.e. YU + V V,S,V S,V De (U)
Note that every singleton S E {[V) V E {V} and every strongly connected components S e (See (V."(V I e VV* are loops. write the set of all loops of the graph 9 as & (g*). We call cycles the loops that are not singletons.
Next, we recall the definition of ioSCMs from Forré and Mooij domains of the variables.
2020 with the omission of the
Definition 2 (input/output Structural Causal Model (ioSCM)). An input/output structural causal model is tuple M (L, J, (.,Pr())) (1)), where
LL is set of laten//xxgeeoou variables, which cannot be observed but affect the rest of the model.
V Vos is a set of observed/endogenous variables, which are observed and every V e Vobs is functionally dependent some subset of u Vare u J) (V).
J is set of input/intervention variables which are not functionallyy dependent of any other variable but rather are fixed to specific values.
G* -VV,E*) )) graphical structure where:
1/+ Vobs u ULUJ
Vots = Ch (L (LUgg9*)
Pa((LuJ,g*)== =
F is set of functions such that for all S E L (G f is a function taking as input the values of Pa (S, \S and ouiputting values for S and such that F satisfies the global compatibility condition:
VS' WEES v,.,),,,, values of Pa (S,
(1Pa 4ys = 1s (PPaSS:))))) -
. Pr (1) is joint probubility distribution over L.
An ioSCM induces a directed graph, where every variable in V+ corresponds to a vertex in the graph. In this directed graph, directed edge is drawn from one variable to another if the former serves as an input the function that determines the latter. For simplicity, instead of working directly with these directed graphs, we consider alternative representation known as a directed mixed graph (DMG). In a DMG, only the observed and intervened variables (i.e., Vots U J) correspond to vertices, while hidden variables in L. that share common outputs are represented by bidirected edges <-> between the corresponding observed variables, thereby implicitly accounting for the hidden confounding, Formally, DMGs are defined as follows:
Definition 3 (Directed mixed graph (DMG)). Consider an ioSCM M. The directed mixed graph (V, E. === U. .) induced by M is the DMG where:
the vertices V= Vohs U I are the endogenous variables and the intervention variables of the ioSCM; and
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Y1
X
Y2
X
X3
Wi
W2
W1
Ws
W
W4
(a) DMGI
(b) DMG
(c) C-DMG.
Figure 1: Two DMGs and their compatible C-DMG. Red vertices represent the exposures of interest in and blue vertices represent the outcome of interest.
the directed edges are E_ EE (Vo.vou and
the bidirected edges g are E (X e, V. such that
L X,X,++YEE
However, in many fields, constructing, analyzing. and validating DMG remains significant challenge for researchers due to the inherent difficulty in accurately determining causal relationships among individual variables. This complexity primarily stems from the uncertainty surrounding causal relations, making it challenging to specify the precise structure of the graph. Nevertheless, researchers often provide a partially specified version of the DMG, which offers and compact representation of the underlying causal structure. These simplified representations. which we call Caluster-Direetedd Mixed Graphs over DMGs (C-DMGs over DMGs), group several variables into clusters, allowing for the representation of causal relationships at higher level of abstraction while retaining essential structural properties of the system. In a C-DMG over DMGs, directed edges between clusters represent causal influences at the higher level, while bidirected edges capture hidden confounding effects that exist between clusters. Formally, C-DMGs over DMGs are defined as follows:
practical
Definition 4 (Cluster directed mixed graph over DMGs (C-DMG over DMGs)), Lerg (V,E) be DMG induced from an ioSCM M and C = (',, Cc) partition of V. A C-DMG over DMGs compatible with G according to a graph G (C, E9) where V VC,, Cj the edge C: Cj (resp. Ci t.>> C2) is in [ES if and only if there exists V, C, and V3 Cj such that V Vj (resp. V1 6 Vj) is in
Figure I presents a simple C-DMG over DMGs along with two of its compatible DMGs. Cycles in C-DMG over DMGs can arise for two distinct reasons. First, unlike C-DMG over ADMGs,
C-DMG over DMGs can contain a cycle if there is genuine cyclic relationship in the underlying DMG between nodes belonging to different clusters. For example, in Figure la the cycle between X₂ and W3 in the DMG induces cycle between clusters Cx and Cw in the corresponding C-DMG over DMGs in Figure Tc Secondly, even in the absence of an actual cycle in the underlying DMG, cycles appear in the C-DMG over DMGs due to its partial specification. This is illustrated in Figure [b] where the edges X₂ + W1 and W X3 together create a cycle between clusters Cx and Ow the C-DMG over DMGs in Figure Ic Lastly, cycles that are contained in single cluster do not appear the C-DMG over DMGs. This is illustrated in Figure Ta with the cycle X₂ 전े Xg that does not show in the C-DMG over DMGs in Figure Ic]
We distinguish between two types of causal effects in the context of C-DMGs, the macro causal effect [nnnd et al. 2023 Ferreira and Assaad b025a and the micro causal effect [Assaad et al. 2024 Assaad 2025 In this paper we focus on the former and we formally define it below:
Definition 5 (Macro causal effect). Consider DMG G over variables V induced from ioSCM and let g EC) EC) be a compatible C-DMG. macro causal effect a causal effect from a set of macro-variables Cx another set of macro-variables Cy where Cx and Ey are disjoint subsets

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of C. It is written Pr (Cy cy do (Cx cx)), where the do (-) operator represents an external intervention.
The identification problem in causal inference aims to establish whether causal effect of set of variables on another set of variables can be expressed exclusively in terms of observed variables and standard probabilistic notions, such conditional probabilities. Formally, the identification problem in the context of macro causal effects and C-DMGs over DMGs is defined as follows:
Definition 6 (Identifiability in C-DMGs over DMGs), Let Cx and Cv be disjoint sets of ver- tices in a C-DMGs over DMGs go, The macro causal effect of Cx on Cy is identifiable ge if (Cy =x ty do (Cx= = cx)) is uniquely computable from any observational positive distribution compatible with g
In the following, we will abuse the notation by writing Pr (cy do (cx)) instead of Pr Pr(Cy= = cv do (Cx = cx)) when the setting is clear. In addition, whenever the context is clear, we will refer to C-DMGs over DMGs simply as C-DMGs.
3 Identification of Macro Causal Effects in C-DMGs over DMGs
In this section, aim to establish that the do-calculus is both sound and complete for identifying macro-level causal effects a C-DMG over DMGs. We begin by showing, in the first subsection, that o-separation o-separrtionn-orriinallll developed for DMGs as a tool for identifying conditional independen- cies-remains sound and complete when extended to C-DMGs over DMGs for detecting macro-level conditional independencies. In the second subsection, this section: the soundness and completeness of do-calculus for macro causal effect identification in this setting. Finally, we provide graphical characterization of non-identifiability. shedding light on cases where causal effects cannot be inferred from observational data alone.
present the core theoretical contribution of
3.1 The o-separation in C-DMGs over DMGs
The standard notion of d-separation Pearl 1988 was originally introduced for acyclic directed mixed graphs (ADMGs). was later shown to remain valid when extended to C-ADMGs over ADMGs [Anand et al. 2023] and C-DMGs over ADMGs [Ferreira and Assaad 2025a b). However, d-separation does not apply to DMGs, which may contain cyclic causal relations in the SCM. To address this limitation, o-separation was introduced and Mooij. 2020). In this subsection, we demonstrate that o-separation can be naturally applied to C- DMGs over DMGs. We begin by formally defining a-blocked walks and the concept of o-separation in this generalized setting.
generalization suitable for DMGs (Forré
Definition 7 (a-blocked walk [For and Mooij 2020)). ). In a graph G (V",E"), walk म (Vi mm Vit) is said be o-blocked by a set of vertices W g V* if:
1. Vf e w V* W",
2. 31 < < n such that -,, - i and V & W or
3. 31 <i<ii n such that (V + V ã and V, e (V or
4. 31 <i<<< n such that (V + V/-" + Vi+) S ã and V; e W (V; or
5. 31 < such that (V V,- + Vi s and
V E W (V See (V)).
where + represents + or 6-,,, ** represents + or 6-),, or and ** represents any of the three arrow type - A walk which is not a-blocked is said to be o-active.
Definition 8 (o-separation FForéé and Mooij 2020]). In a graph G (V",E"), let X*, Y", W be distinct subsets Y". 'W" is said to a d-separate X and " if and only Wf o o-blocks every walk from vertex in X* vertex in V". It is written (X",,") "2' [W*) ).
The following theorem shows that o-separation is applicable is to C-DMGs over DMGs.
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(a)
(b)
(c)
(d)
Figure C-DMGs with identifiable macro causal effects. Each pair of red and blue vertices represents the causal effect we interested in.
Theorem (Soundness of respparation in C-DMGs over DMGs), Let Ge = (C, E9) be a C-DMG and Cx, Cy, Cw be disjoint subsets of C. If Cx and Cy are o-separated by Cw in G then, in any compatible DMG 9 (V, E), X UCetx C and Ucec, are a-separated by W UCeC C.
Theorem D establishes that o-separation in C-DMGs over DMGs ensures the existence of corre- sponding macro-level o-separation across all compatible DMGs. According to [Forré and Mooij] 2020, Theorem 5.2]. this implies that some conditional independencies in the underlying probability distribution can inferred directly from the C-DMG. By extending the applicability of -spearatiion to C-DMGs over DMGs, this result enables the identification of macro-level conditional independen- cies even when the underlying causal structure is only partially specified. To illustrate the practical value of this result, now present two examples demonstrating the application of o-separation in C-DMG over DMGs.
Example 1. Let G be the true unknown DMG and consider that its compatible C-DMG, denoted as g is one given Figure 2a Using Definition we can directly deduce (CulL,, Cy Cx)g.. Thus according to Theorem 7 andd (Forré and Mooij 2020 Theorem 5.2], Crp is conditionally independent of Cy given Ox in every distribution compatible with the true ADMG.
Example Let G be the true unknown DMG and consider that its compatible C-DMG, denoted as 9 is one given in Figure 2c] Using Definition & we can directly deduce that (Czu,Cy Cx2, Cw)ge-
Thus according Theorem [ and (Forré and Mooij, 2020 Theorem 5.21, Cz is conditionally independent of Cv given Cx and Ow in every distribution compatible with the true ADMG.
The following theorem shows that o-separation is also complete in C-DMGs over DMGs.
Theorem 2 (Completeness of a-separation in C-DMGs), Let G (C,E9) be a C-DMG, Cz Cy, Cw be disjoint subsets of C, X Ucecx C, Ucedy C and W Ucery C. If Cx and Cy are not o-separated by Ow in go, then there exists compatible DMG G = (V, E) such that & and Y are not a o-separated by
The findings of Theorems a establish that identifying o-separation in C-DMGs over DMGs ensures the recovery all common macro-level -separatiion across all compatible DMGs. This result is particularly valuable for constraint-bart goal is to infer the structure of C-DMG without needing to fully specify an underlying DMG. Most importantly, these insights lay the theoretical groundwork for the results developed in the next subsection.
causal discovery methods, especially when the
3.2 The do-calculus in C-DMGs over DMGs
Pearl 1995) introduced an important tool in causal reasoning referred to as the do-calculus. This do-calculus consists of three rules each relying on some d-separation in the ADMG to guarantee an equality between different probabilities. Every one of these three rules can be interpreted differently: the first one allows the insertion or deletion of observation, the second one allows for the exchange between actions and observations and the third one allows the insertion or deletion of actions. The do-caleulus in ADMGs is complete and thus allows, whenever it possible, to identify causal effects. In other words, it allows, whenever it is possible, to express a causal effect containing do (-) operator as probabilistic expression without any do (-) operator and thus allows to compute it from a positive observational distribution. Since the do-calculus was initially introduced for ADMGs, it is not easily

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(b)
(c)
Figure 3: C-DMGs with not identifiable macro causal effects. Each pair of red and blue vertices represents the total effect we are interested in.
extendable cyclic graphs. However, Forré and Mooij 2020 showed that replacing d-separation by a-separation the three rules allows the do-calculus to be applicable on DMGs induced by ioSCMs. In this subsection, we show that this version of the do-calculus is also readily applicable C-DMGs over DMGs.
There exists multiple equivalent ways of writing the do-calculus rules, for example Pearl 2009 uses the notion of mutilated graph to define the three rules do-calculus. In this paper, we will follow the notation used in Forré and Mooij 20200] To do we will extend every graph 9 = (V* E'). by adding an intervention vertex Ix and the edge Ix + X for every vertex of the graph X E V". Moreover, will use the o-separation notation with do (-) operator (e.g., (Au, B C, do (D)) to place ourselves in the intervened graph ie., where all arrows going in /) are deleted and which D is conditioned or More formal definitions of extended graphs and intervened graphs can be found in the appendix. Using these newly defined notations and o-separation we can now state the rules of do-caleulus and show their applicability C-DMGs over DMGs.
Theorem 3 (do-caleulus for C-DMGs over DMGs and macro causal effects). Let = (C,E9) be C-DMG over DMGs and Cx, Cr, Cz, Cw be disjoint subsets of C. The three following rules of the do-calculus are sound.
Rule 1:Pr (cy do (c=) Cx (w) Pr (cy/ do (w))
if (CyLoCx| Cw do (Cz))g=
Rule 2:Pr (cy do (c2) do (cæ) (w)) = Pr (cy do (cz) Czi c.)
if (Cyl Cx, Cw, do (C2))
Rule 3:Pr (c, do (cx) do (cx) cww) Pr Pr(cy) do (w)
if (Cylolox Ow, do (Cz))
Now that the rules of do-calculus have been stated for C-DMGs over DMGs and their soundness proven, one can use them sequentially to identify the causal effect Pr (cy do (cx)) in all C-DMGs over DMGs in Figure 2 That is say, to express the causal effect as probabilistic expression without any do (-) operator. This is done in the following examples.
Example 3. Both in Figure 2a and 2c one can verify that (Cymsoexx Cx) thus Rule 2 of the do-calculus is applicable and Pr (c, do (c)) Pr (c, c2).
Example 4. Notice that Figure 2b does not contain any cycle other than self-loops and is very similar to Figure 1(b) of Anand et al. (2023) which corresponds to the well-known front-door criterion [Pearl (2099]. [Forré and Mooij 2018 have shown that the acyclic case, o-separation coincides with d-separation. Thus, using the corresponding sequence of classical rules of probability and rules of do-calculus as the one given in (Pearl 2009 p.83], one obtains Pr (Gy do (ce))
Σ., Pr (r(cw ( Ee,, Pr(cy (w, Pr (cx).
Example 5. Consider the C-DMG over DMGs in Figure 2d containing cycle between Ox, Cr. and Cy and a hidden confounding between Cy and Ow. Let Pr (c,, do (2), be the causal effect of interest. Using the rule of total probability we can rewrite Pr (cy do (C2, as
Pr do (c2, Pr (cw, do (c2,
In other words write (A do (D)) mean, Pearl 2009 notation, (Au, B| 22,,, D)
where G is obtained from G by removing every edge going in D.
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We first focus on Pr (cy do (cx: (2) Notice that (CyHoIox Cz. Cw, do (C2)) ge and that (CyLaloz) Cz, Cx, Cw)ge which means using two consecutive applications of Rule 2 we can rewrite Pr (cyl do (C2,C2) (w) ) as: (cy C2s Czi Gw).
Now we focus Pr (cw do (c2, Notice that (Cwllolcy do (C which means by Rule 3 of the do-calculus can completely remove do (c2) from the expression. Furthermore, have (Culllllcz Cz) which means by Rule 2 we can replace do (C2) by Cz. So we can rewrite Pr (cw do (cx; as Pr (cwl(z) (z).
These three examples show how the rules of do-calculus in C-DMGs on DMGs can be used to write macro causal effects as an expression of observed probabilities. Therefore, the macro causal effects in these C-DMGs can be estimated from the observational data, provided there is no further issues in the data (e.g., positivity violations).
In the following theorem, we show that not only is the do-caleulus applicable in C-DMGs over DMGs, but it is also complete.
Theorem 4 (Completeness of do-caleulus for C-DMGs and macro causal effects). If one of the do-calculus rules does not apply for a given C-DMG over DMGs, then there exists a compatible DMG for which the corresponding rule does not apply.
The proof of Theorem 4 relies a specific compatible DMG called the maximal compatible DMG which is properly defined in the appendix, therefore it can be reformulated as: if one of the do-calculus rules does not apply for a given C-DMG over DMGs, then this same rule does not apply for its maximal compatible DMG.
Note that this completeness result links C-DMGs to the underlying compatible DMGs, however it does not guarantee the absence of other rules to identify causal effects. While the do-calculus based on d-separation in ADMGs introduced in (Pearl [1995] has been proven to be complete (Shpitser and Pearl 2006 Huang and Valtorta, 2006]. the do-calculus based on o-separation in DMGs is not yet proven be complete and thus our results suffer the same limitation. Using the completeness of the do-calculus in C-DMGs (Theorem 4), one can be convinced, by going through every possible sequence of rules of the do-caleulus, that the causal effects of interest in every C-DMGs depicted in 3iss is not identifiable. Unfortunately, going through every possible sequence of rules of the do-calculus time-consuming and can become impractical when considering larger graphs. In an effort to solve this issue, the following subsection introduces a sub-graphical structure which allows recognize more efficiently when causal effect is not identifiable using the do-calculus.
[ Figure and and
3.3 Non-Identiffability:: a graphical characterization
In ADMGs, there exists a sub-grappical structure, called a hedge (Shpitser and Pearl 2006), which employed to graphically characterize non-identifiabiiity as shown Shpitser and Pearl 2006 Theorem 4. This graphical criterion is complete when considering ADMGs, that is to say, if the total effect of on Y is not identifiable an ADMG then there exists hedge for (X, Y) (Shpitser and Pearl 2006). However, it has been shown that this characterization is too weak to characterize all non-identifiabiities in the case C-DMGs over ADMGs [Ferreira and Assaad, 2025b]. Thus, a modified version of the hedge structure called the SC-hedge (strongly connected hedge) has been introduced in Ferreira and Assaad [2025a]. The presence of such SC-hedge in C-DMGs over ADMGs guarantees non-identifiability [Ferriraa and Assaad 2025a bn.In this subsection, we will show that SC-hedge can also be used to characterize non- con-identifiabiliiy in C-DMGs over DMGs in specific conditions. First, let recall some useful concepts to define hedge.
Definition 9 (C-component, Tian and Pearl (2002 ). Let G vertices Vi S such that A VVT.V, V. E V 3 VV."" C-component.
be graph. A subset of Vi e V with visi< n, V eo Via is called
Definition 10 (C-forest, Shpitser and Pearl 2006)). Let G* = (V*, be graph. IfG is acyclic, g* is a forest (i.e., every of ils vertices has at most one child), and G* is a C-component then G is called a C-forest. The vertices which have children called roots and we say a C-forest is R rrootd if it has roots R* V
Definition 11 (Hedge, Shpitser and Pearl 2006
2008
Consider a graph G* = (V*,E*) E*) and two and F (Vzr, Ez.) be two RD -rooted
disjoint sets of vertices X', Y S V. Let F
(.
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(a)
SC-
(b)
SC-
projection
projection of Figure
(c) SC-projection of Figure 2c
(d) SC-projection of Figure
of Figure 2a
(c) SC- projectiin of Fig- SC- projection of Fig- ure
SC projection of Fig- ure
(h) SC- Propoeectioon of Fig- ure 3d
ure 3b
Figure 4: SC-projections of the C-DMGs in Figures and Figures Each pair of red and blue vertices represents the total effect we are interested in, and the red edges indicate those added through the SC-projection.
C-forests subgraphs G" such that X Vy + Ø, X nv B, F. and R" An (8", 9*) Then and F' form a hedge for the pair (x* Y*) in g".
As mentioned before, a hedge turned out to be too weak to cover non-identifiability in C-DMGs [Ferreira and Assaad 2025b). For example, the C-DMG in Figure Ba contains hedge but the macro causal effect is not identifiable due to the cycle between Cx and Cy. In the following, we formally define SC-hedges in the context of C-DMGs over DMGs and demonstrate that this substructure serves as sound criterion for detecting non-identifiable macro causal effects if every cluster in a cycle is of size strictly greater than
Definition 12 (Strongly connected projection projecccjcii)) Consider C-DMG g = E.) The SC-projection H. g is the graph that includes all vertices and edges from g plus dashed bidirected edge between each pair Cx, Oy e such that See (Cx,9*) See (Cr, go) and Cx Oy.
Definition 13 (Strongly Connected Hedge (SC-Hedge)). Consider an C-DMG G EC, its SC-projection H. and two disjoint sets of vertices Cx, Cy S C. A hedge for (Cx, Cy) in H. is an SC-hedge for (Cx, Cy) in G
The following theorem guarantees the soundness of the SC-hedge criterion in C-DMGs over DMGs when every cluster in cycle is of size strictly greater than 1. This assumption is useful it allows the existence of compatible ADMGs even when the C-DMG over DMGs contains cycles. Thus, one use Ferreira and Assaad 2025b Theorem 5 in the presence of SC-hedge to show that for every identifying sequence of do-calculus rules, there exists a compatible ADMG in which this sequence is applicable.
Theorem 5. Consider an C-DMG g
EF, such that every cluster which is in a cycle is of size least 2 and two disjoint sets of vertices Cx, Cy C. If there exists an SC-hedge for (Cx, Cx) in g then Pr (c, do (cx)) is not identifiable.
The SC-projections of the C-DMGs illustrated in Figures 2 and 3 are given in Figure One can notice that the SC-projections of the C-DMGs in Figure 3 contain a hedge, thus the C-DMGs in Figure 3 contain SC-hedge and the causal effect of interest is therefore not identifiable according to Theorem 5 In contrast, the projections of the C-DMGs Figure 2 do contain hedge, this highlights the usefulness of SC-hedges for causal effect identification.

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4 Conclusion
In this paper, established the soundness and completeness a o-separation and the do-calculus using a-separation for identifying macro causal effects in C-DMGs over DMGs. There are two main limitations this work. The first limitation that the completeness result in Theorem does not take into account that there might exist different sequences of rules of the do-calculus in different DMGs that can give the same final identification of the causal effect. Moreover, the completeness results only link the cluster graphs to the compatible underlying graphs, our results say nothing on the completeness of o-separation and do-calculus in DMGs. A second related limitation is that we provided graphical characterization for the non- con-identifiabiiity of macro causal effects, however this characterization not proven to be complete, even though we did not find any counter-example of its completeness. Proving it to be complete remains open problem.
Acknowledgments and Disclosure of Funding
This work was supported by the CIPHOD project (ANR-23-CPJ1-021--0))
References
Tara V. Anand, Adele H. Ribeiro, Jin Tian, and Elias Bareinboim. Causal effect identification in cluster dags. Proceedings of the AAAI Conference on Artificial Intelligence, 37120111172212179,
Jun. 2023. doi: 10.1609/aaai 33.10..2643.
Charles K. Assaad. Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion. Transactions on Machine Learning Research, 2025. ISSN 2835-8856.
Charles K. Assaad, Imad Ez-Zejjari, and Lei Zan. Root cause identification for collective anomalies in time series given an acyclic summary causal graph with loops. In Francisco Ruiz. Jennifer Dy, and Jan- Willem van de Meent, editors, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, volume 206 of Proceedings of Machine Learning Research, pages
PMLR, 25-27 Apr 2023.
Charles K. Assaad, Emilie Devijver, Eric Gaussier, Gregor Goessler, and Anouar Meynaoui. Iden- tifiability of total effects from abstractions of time series causal graphs. In Proceedings of the Fourtieth Conference on Uncertainty in Artificial Intelligence, Proceedings of Machine Learning Research. PMLR, 2024.
Philip Boeken and Joris M. Mooij. Dynamic structural causal models, 2024. URL https://arxiv org/ hrgg/bbs/4400 01161
Simon Ferreira and Charles K. Assaad. Identifiability direct effects from summary causal graphs. Proceedings of the AAAI Conference on Artificial Intelligence, 38(18):00387-2039, Mar. 2024.
doi: 10.1609/aaai,
Simon Ferreira and Charles K. Assaad, Identifying macro conditional independencies and macro total effects in summary causal graphs with latent confounding, Proceedings of the AAAI Conference on Artificial Intelligence, 39, 2025a.
Simon Ferreira and Charles K. Assaad. Identifying macro causal effects in C-DMGs over ADMGs. Transactions on Machine Learning Research, 2025b.
Patrick Forré and Joris M. Mooij. Causal calculus in the presence of cycles, latent confounders and
selection bias. In Ryan P. Adams and Vibhav Gogate, editors, Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, volume 115 of Proceedings of Machine Learning Research, pages 71-80. PMLR, 22-25 Jul 2020.
Patrick Forré and Joris M. Mooij. Markov properties for graphical models with cycles and latent variables. arXiv: Statistics Theory, 2017.
Patrick Forré and Joris M. Mooij. Constraint- causal discovery for non-linear structural causal models with cycles and latent confounders, 2018. URL httph:ttt//aaatt org/abs.//......oooo.oo ht//a/b/////0000/
10

---

Yimin Huang and Marco Valtorta. Pearl's calculus of intervention is complete. In Proceedings of the Twenty-Second Conference on Uncertainty Artificial Intelligence, UAI'06, page 217-224, Arlington, Virginia, USA, 2006. AUAT Press. ISBN 0974903922.
Amin Jaber, Adele Ribeiro, Jiji Zhang, and Elias Bareinboim. Causal identification under markov equivalence: Calculus, algorithm, and completeness. In S. Koyejo. S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems, volume 35, pages 3679-3690. Curran Associates, Inc., 2022.
Marloes Maathuis and Diego Colombo. A generalized backdoor criterion. The Annals of Statistics, 43, 07 2013. doi: 10. wee.11///44440 A0S11295
Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1988. ISBN 1558604790.
Judea Pearl. Causal diagrams for empirical research. Biometrika, 82(4))669 12 1995. ISSN 0006-3444, doi: (0.1093/hiomet//2.44
Judea Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, USA, 2nd edition, 2009. ISBN 052189560X.
Emilija Perkovic. Identifying causal effects in maximally oriented partially directed acyclic graphs In Jonas Peters and David Sontag, editors, Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), volume 124 of Proceedings of Machine Learning Research, pages 530-539. PMLR, 03-06 Aug 2020.
Emilija Perkovic, Johannes Textor, Markus Kalisch, and Marloes H. Maathuis. Complete graphical characterization and construction of adjustment sets in markov equivalence classes of ancestral graphs. J. Mach. Learn. Res., 1:222011-200:6: 2016.
Nicolas-Domenic Reiter, Jonas Wahl, Andreas Gerhardus, and Jakob Runge. Causal inference process graphs, part ii: Causal structure and effect identification, 2024.
Thomas Richardson. characterization of markov equivalence for directed cyclic graphs. In- ternational Journal of Approximate Reasoning, 17(2):07--622, 1997. ISSN 0888-613X. doi: https://doi.org/I1 10. 11011//0888611133777000000000000
Uncertainty in AI (UAI'96) Conference.
Ilya Shpitser and Judea Pearl. Identification of joint interventional distributions in recursive semi- markovian causal models. In Proceedings of the 21 National Conference on Artificial Intelligence Volume 2, AAAI'06, page 1219-1226. AAAI Press, 2006. ISBN 9781577352815.
Iya Shpitser and Judea Pearl. Complete identification methods for the causal hierarchy. Journal of Machine Learning Research, 9:1941-1979, 2008
Jin Tian and Judea Pearl. On the testable implications of causal models with hidden variables. In Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence, UAI 02. page 119-527,6 San Francisco, CA, USA, 2002. Morgan Kaufmann Publishers Inc. ISBN 1558608974.
Santtu Tikka, Jouni Helske, and Juha Karvanen. Clustering and structural robustness in causal diagrams. Mach. Learn. Res., 24(1), January 2023. ISSN 1532-4435.
Jonas Wahl, Urmi Ninad, and Jakob Runge. Foundations of causal discovery on groups of variables. Journal of Causal Inference, 12(1)22230041, 2024. doi: 1oi: 15.1 15:1111115//ii22230001111
Tian-Zuo Wang. Tian Qin, and Hua Zhou. Estimating possible causal effects with latent variables via adjustment. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors, Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 36308-36335. PMLR, 23-29 Jul 2023.
Clement Yvernes. Note the identification of total effect in cluster-dags with cycles. Unpublished note, 2025.

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A Appendix
A.1 Additional Notations and Properties
Firstly, the proofs require the definitions of intervened graphs and extended graphs which are the graphs induced respectively by intervened ioSCMs and extended ioSCMs Forré and Mooij] [020]. Definition 14 (Intervened graph). Let G = (V,F) be a DMG induced from ioSCM M and A = V set of variables. The intervened graph (Glao()) is obtained by removing all edges going to A in g.
The intervened graph (Glan()) is also written as 9x in Pearl 2009
Definition 15 (Extended graph). Les G = =V, E) be a DMG induced from an ioSCM M. The extended graph [9] = ([V), [E] is obtained by adding for each vertex V e V an extended vertex Tv e [V] and the edge Tv ~ V e [E]. ie., [V] = Yu (Iv V e V) and [F] E u (Tv - V Ve V).
In order to map the vertices in a C-DMG Ge with the vertices in compatible DMG g, we will use the notion of corresponding cluster: VC e C, VV eC, CV C. Additionally, we will write for every set of clusters C2 S C, A UCEC C.
Definition 16 (Maximal compatible DMG). Let g (C, (9) be a C-DMG. Let us define the following sets:
== UC CcC
E" (V- VV V' E C' such that C + C' E E (V= 6- V' V' C' such that C 6-+ C³ [E}}
E=
The graph 9 = EV,E) is compatible with G and for every compatible DMG G, we have G S 9 thus 9 is called the maximal compatible DMG g
Property (Compatibility of extended graphs). Let G be a C-DMG and be a compatible DMG. Let us consider [9-] and [9] the corresponding extended graphs. Take [V]=Vu[le V e V) the extended micro variables and [C] Cu {[Tv V C} Ce C} the extended partition. [9] is compatible with [G9] according to partition [C].
Proof. Firstly, VV,V" E V, V V' [re -...) Cl (V. + (VV,99) é (ressp. ()) = CI(V.g" + CI(V", g9) E [ e-)). Secondly, VV V, Iv Ve [[]] and [C(()))) - CIV. e [[ [9'] and 3V C,, Iv [C] because C partition so &C' C, C+
V" V
(resp, (..)
Lastly, VC C, Ic Ce
Property 2 (Extended maximal compatible graphs). Let G be C-DMG, G be the maximal compatible graph of go, [ be the extended graph of G and [ Jm be the maximal compatible graph of [ These graphs verify
*[9%] S [ I. and
vV [V] See (V, (9%.) = See Seceee (V.
Therefore, any macro-level a-connection that holds in [ge], also holds in [
Proof. Firstly, YV, V V, - V E [ (resp. c) = V + V E go (resp. () = (V. CI'V, (V',g))*) (resp. c) = CV,
Cl (V' .G) E [[ (resp. ()) =
V 1/ E [go] Jm (resp. (..).. Secondly, VV e eV, Iv + V [ and Iv e (gv, V
(G))) Ci (G), CI (V, [99] thus Iv ~ Ve [ m
Regarding the strongly connected components, VV v, See (V. [] See (V,9%) (V) (Uessa((oiV:3))00 C (V} (Uosssaconvii)).too)) C) See (V, [g]].. Moreover, VV e
WWhile the usual strongly connected component, (V" ,g"). always contains the vertex V itself, this tweaked version, See ( (V*,9") ).",. contains the vertex V if and only if there exists self-loop on it iee. V* V e
12

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V. the set of edges including ly in [ is {Iy + } and the set of edges including Iv in [g"],, is 1) (V thus See (Iv.[9m)) {Iv} Seo (Iv.[9))
In conclusion, using the Definition 8 any macro-level a-connection that holds in [g"),.. also holds in [-
Property 3 (Compatibility of intervened graphs). Let G be C-DMG and g be comparible DMG. Let Ca S C be a set of clusters. Let us consider (9 and (Aan(ar) the graphs obtained respectively by intervening on C. in G and by intervening on & in G. (Alaote) compatible with (9%a)))))-
Moreover, let 9 be the maximal compatible graph of g (Imldo(a)) its intervened graph and (99a)c))) be the maximal compatible gruph of (dlc)))) (Amlo(()) and (9%dott))), are the same graph.
= V V" e c..) and V & (V, (V .o) e (resp.. (.)) and CI(VI, Co = (V. - CI (V', go) e
Proof. VV, VV, V -V' (glao(e)) (resp. (.)))
(9%a)(c)) (resp. 2).
Property 4 (Intervened maximal compatible graphs). Let G be a C-DMG, I be the maximal compatible graph g", (Amlat(a)) ) be the intervened graph of Im and (aa(tw))) be the maximal compatible graph of (G))do(w)) (Sillo(as) and (99)ax(a))) are the same graph.
Proof. VV.V' E V + V'' (gmlao(A)) (resp. <) = V V (resp. <) and V $ = (V(V ~ (V"V" E (resp. (-)) and Cl (V'V' & Ca = (V, ~ ((VV,9) (9%o)(.)) (resp. c.>) = V + +V* (99 (resp. c.)).
U
A.2 Proof of Theorem
Proof. Suppose Cx and Cy are o-separated by Cw in G and there exists a compatible DMG G (V, E) and walk = (Vi,",, Vv) in G from Vi E X to Vn E which is not o-blocked by W. Consider the walk = (C1, (C.,.., C.) with v1 visis n, C, = CI (Vi,9*) and V1 s < n, (C, + C+a) S (resp. ).)) (V: Vi+a) S (resp. +, -.). # is walk from Cx to Cy in G.. Since Cx and Cy are s-sparateed by Cws we know that Cw o b-blocks म.
If C1 E W or Cn e Cw then Vi W or Vn e W and thus a is a-blocked by W which contradicts the initial assumption.
Otherwise, take 1 1< i < n such that (Ci-1, Cita) is Cl-ooblocked.
g and C, Cw then, VV, s and V, é W. Thus is a-blocked by W which contradicts the initial assumption.
If
If (Ci-1 - Ci+CCan1) S ग and C₁ CwNSSee (C--,, then, (Vi-1 < VieVial s m. Moreover, Sce (V-1,9) UCesee(Ci.1.g9 go) C and thus V;; W/,cc (V-.,, Therefore, is o-blocked by W which contradicts the initial assumption.
U
If (C.-1=C,- + Ca1) S ग and C₁ CwNSco (Ca1,9 then, (Vi-, + Va+1 s m. Moreover, See (V=1,9) 9) UCeSee(Cian .0) C and thus V;; W/,0c (V. Therefore, is a-blocked by W which contradicts the initial assumption.
If (Ct-1 C. Ca1) S $ and C. then, (Vi-1 Vi Vun) म. Moreover, (Sco(Vi-1, (V=1,9)) See (V+1,9)) UCe(Se((CL,1. (ec(Se((C.,,SennSsccC.. Ge)nScc(C..1.09) C' and thus V; W (Scc (Vi-1, n See (V)+1,9)). There- fore, π o-blocked by W which contradicts the initial assumption.
(C,,,, See (Cu1,,
In conclusion, the o-separation is sound in C-DMGs over DMGs.
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A.3 Proof of Theorem 2
Proof. Suppose Cx and Cy are not o-separated by Cw in g There exists an C path T (C,, C.) with Ci Cx and Cn e Cy. Take 9 EV the maximal compatible DMG of ge as in Definition 16 Take for every cluster C' e a representative of this cluster Vc e C. The maximal compatible graph 9 contains the path mm (Va,, Vc..) and for every cluster C e C and every variable in that cluster V & C, Sec (V, wi UC*eSe((CCC" C". Therefore, π being Cw-a-active in 9o clearly implies that Tm is W-o-active g In conclusion, the o-separation criterion in C-DMGs over DMGs is complete.
Notice, that not only did we prove Theorem 2 i.e..if if o-separation does not hold in a C-DMG then there exists a compatible DMG in which the corresponding o-separation does not hold-but but we also explicitly exhibited this compatible DMG being the maximal compatible DMG.
A.4 Proof of Theorem 3
Proof. Let g = E9) be a C-DMG, G a compatible DMG and Cx, Cy, Cz, Cw S C be disjoint subsets of vertices. Suppose rule of the do-calculus applies in g then Theorem T Property and Property 3 guarantees that this rule applies in g. More explicitly:
If rule 1 applies i.e., (Cy Cx Cw, do (Cz)) then using Theorem I as well as Proper- ties D and 3 one knows that (Yu W, do (2))g and thus rule 1 applies in G.
If rule applies i... (Cy Cx: Cw, do (Cz)) then using Theorem 1 as well as Properties and one knows that (Yu, X, W, do (Z))g and thus rule 2 applies in G.
If rule applies i.e., (Cylla Cw: do (Cz) ge
. then using Theorem as well as Proper- ties I and 3 one knows that (Yu, Ix W. do((( (2) )) and thus rule 3 applies in G.
Notice that because Cx and C7 are disjoint, the actions of taking the intervened graph and taking the extended graph can be done in any order without any repercussion in the o-separations of interest. In conclusion, the do-calculus using o-separation is sound in C-DMG over DMGs.
A.5 Proof of Theorem
Proof. G = (C, be C-DMG, oh be the maximal compatible DMG and Cx, Cr, Cz, Cw C be disjoint subsets of vertices. Suppose rule of the do-calculus does not applies ge, then Theorem Property T and Property 3 show that this rule does not apply in G More explicitly:
If rule does not apply ie., (Cyuj Cx Cas do (Cz))go then using Theorem 2 as well as Properties and one knows that (Y46) W. do (2) 2)) )g. and thus rule does not apply in g
If rule does not apply i.e., (Cx4/ Cx, Cw, do ( (C2)) then using Theorem 2 as well as Properties and 4 one knows that (11/bxx X, W. do (Z) )e and thus rule 2 does not apply in gr.
If rule 3 does not apply ie., (Cx Cw, do (Cz) (Cz) ges then using Theorem 2 as well as Properties 2 and one knows that (1 W, do (Z) 2)) go. and thus rule 3 does not apply in g
Notice that because Cx and C, are disjoint, considering the extended graph of the intervened graph considering the intervened graph of the extended graph does not have any repercussion in the a-connections of interest.
In conclusion, the do-calculus using o-separation is complete in C-DMG over DMGs.
14

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A.6 Proof of Theorem 5
Proof. Let G E9, be a C-DMG and take disjoint subsets Cx, Cy S C. Additionally, suppose that every cluster which is in cycle in 9 is of size at least 2. More formally, AC e C, |Sec (C,9)) 1 |C] 1, Thanks to this assumption, one can view 9 as C-DMG over ADMGs and thus use prior work(Ferreira and Assaad 2025b]. Suppose there exists SC-hedge for the pair (Cx, Cy) in g". Then, according to Theeormm Ferreira and Assaad 2025b). the effect of Cx on Cy not identifiable.
(C,
In conclusion, the SC-hedge criterion is sound in C-DMG over DMGs under the additional assumption that every cluster which is in a cycle is of size at least 2.
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
The main assumption lies in the problem setting i.e., the system we wish to study is modeled by an ioSCM. However, another additional assumption which is clearly stated is required for Theorem [] Complete proofs are available the supplementary materiel, unfortunately, due to a lack of writing space, we were not capable of adding every proof sketches in the main paper.
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Reviews:
[{"review_id": "Fn6RtR34Mv", "reviewer": "Reviewer_2YQb", "summary": "Causal identification from partially specified causal structures is a fundamental problem that is receiving increasing attention in recent years.  The paper introduces the problem of identification of macro causal effects from C-DMGs against compatible DMGs.  Below are the main takeaways from the paper:\n\n1) $\\sigma$-separation in C-DMGs implies corresponding $\\sigma$-separation in all compatible DMGs.\n\n2) Non-$\\sigma$-separation in C-DMGs implies existence of a compatible DMG where the corresponding variables are non-$\\sigma$- separated.\n\n3) The do-calculus (with $\\sigma$ separation) is sound for C-DMGs.\n\n4) Non-applicability of a do-calculus rule implies non-applicability of the same rule in some compatible DMG.\n\n5) An application of SC-Hedge for C-DMGs over DMGs, for a restricted class of graphs.", "questions": "The notion of `completeness of do-calculus' originated from the early works of Tian and Pearl, which is different from the one discussed here.  I would suggest clarifying the definition of completeness over DMGs earlier in the manuscript, possibly even in the abstract.\n\nI am trying to understand if the non-applicability of a set of do-calculus rules $\\mathcal{R}$ on a given C-DMG would imply existence of a single compatible DMG where none of the rules in $\\mathcal{R}$ are applicable?  Such a claim, if true would make Theorem 4 much stronger.\n\nBelow are some minor suggestions / typos:\n\n1. I suggest using Pa, An, De, Scc and other abbreviations in non-math mode.\n\n2. Figure 1:  Two ADMGs → Two DMGs\n\n3. Some typos in lines 160 – 164 (due to its, lead to cycle between clusters)\n\n4. Please define compatible graphs and extended graphs – the core components of the proof.\n\n5. Line 228: in in\n\n6. Line 266:  conditioned → intervened\n\n7. Please proofread Appendix.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:  The manuscript is well-written and I really enjoyed reading the paper with nice running examples.  The technical contributions appear sound and advance the state of the art in causal identification from partial causal graphs.\n\nWeaknesses:  The theorem statements are nice, my only issue is that the proofs are relatively straightforward and most of the concepts extend from previous work on ADMGs.  Proving one of the following results: do-calculus completeness for non-identifiability, or a more general graphical condition for non-identiability would have greatly improved the contributions from a technical/novelty standpoint.  The proofs in Appendix are hard to understand and they rely on two important concepts: the extended graphs and compatible graphs, whose formal definitions are missing in the manuscript.", "was_revised": true, "final_justification": "I am happy with the rebuttal and the author discussions.  With the addition of the discussed corollary, I believe the submission makes substantial contribution to causal identification from partial structures.  I would like to maintain my score.", "rebuttal": "First of all, we would like to thank for the thorough review and interesting remarks. We will take into account the typos mentioned and carefully proofread the whole paper.\n\nIndeed, for the do-calculus, our notion of completeness is different from that of Tian and Pearl as we only guarantee that if an effect is not identifiable in a C-DMG then for every identifying sequence of rules of do-calculus $\\mathcal{R}$, there exists a compatible underlying DMG in which $\\mathcal{R}$ is not applicable, or in other words there exists a rule $R_i \\in \\mathcal{R}$ and a compatible DMG $\\mathcal{G}$ for which $R_i$ is not applicable in $\\mathcal{G}$. However, it may very well be that for every individual rule $R_i$ in $\\mathcal{R}$ there exists a compatible graph in which $R_i$ is applicable. We will make sure to clarify this point in the final version.\n\nThe notions of extended graphs and intervened graphs can be found in [Forre and Mooij 2020]. However, as we use those notions in the proofs in appendix, we will properly state the definitions at the beginning of the appendix. Moreover, we will make explicit what we mean by \"compatible\" graph.\n\nThanks again for the thoughtful comments. We hope to have answered all questions and we remain available during the discussion period in case there is any additional concerns."}, {"review_id": "OpDyGSpkli", "reviewer": "Reviewer_h1gx", "summary": "Authors prove that do-calculus is sound and complete for C-DMGs over DMGs.", "questions": "- One of my major concerns is the actual instantiation of distributions for Example 4 and 5, especially given the presence of cycles in Example 5. In the introduction it is said that \"cluster graphs allow for cycles, which can arise naturally in feedback systems or time-dependent processes, complicating the analysis compared to traditional ADMGs\", but there is not explicit discussion on time-related effects, leaving the actual computation of the proposed identified effects dubious. I get the value of this contribution is to prove the correctness and completeness of the do-calculus for this class of models, but how can we use the output of the identification process in practice?", "limitations": "- Discussed in the conclusions, but I would prefer to have a dedicated section to better highlight the current limits of the proposed approach.\n- For instance, what are the actual implications of the statement: \"there might exist different sequences of rules of the do-calculus in different DMGs that can give the same final identification of the causal effect\"? Is it a limitation in the sense of \"minimality\" of sequence of rules application?", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strengths\n    - Proving the do-calculus to this class of models makes them useful to identify the so called \"macro\" causal effects.\n- Weaknesses\n    - Notation layers stratify one on top the other (e.g. E→ = E+ ∣ Vobs ∪ J, while there is an already defined V = Vobs ∪ J). I don't think the current notation choice helps the reader to follow the content of this dense paper.\n    - Some references appears to be non-peer reviewed (e.g. either arXiv or \"unpublished notes\" references).", "was_revised": true, "final_justification": "Authors addressed the issues I raised, the paper is particularly dense and it is not easy to follow due to the theoretical framework put in place, but with the additional explanations provided during the rebuttal everything is much more understandable.", "rebuttal": "First of all, we would like to thank for the thorough review and interesting remarks. We will take into account the suggested notation modifications and carefully proofread the whole paper.\n\nConcerning the question about time-related effects, the article [Ferreira and Assaad 2025] tackles similar questions in the context of time series.\nHowever, in this work we focus in settings where time is not specially of interest.\nIn this setting, cycles occur for example when there are equilibriums: take a system, in which variables are linked through feedback cycles (X->Y and X<-Y), then we wait for an equilibrium before measuring the variables.\nIn this setting, an intervention on X will affect Y and an intervention on Y will affect X, therefore, the cycle is necessary to fully express the causal relationships.\n\nRegarding the estimation process in cluster graphs: when using C-DMGs, one does not know all the causal relationships between micro variables however, one has access to all the data of micro-variables.\nTherefore, once the identification process outputs an estimand, one can estimate the causal effect exactly as one would estimate a multivariate causal effect in any classical setting described by [Pearl 2000].\n\nAs for the limitations, we are willing to discuss them more in details.\n\nLastly, the completeness of the do-calculus result has been discussed with reviewer 2YQb, we refer to this discussion for a reformulation of the explication that follows.\nThe completeness result guarantees that if a rule $R_i$ of do-calculus does not apply in a C-DMG then, there exists a compatible DMG in which $R_i$ does not apply either.\nThis result can trivially be extended to sets of rules of do-calculus: if a sequence of rules of do-calculus $\\mathcal{R}$ is not applicable in a C-DMG, then there exists a rule $R_i$ which is not applicable in this C-DMG thus there exists a compatible DMG $\\mathcal{G}$ in which $R_i$ is not applicable either and lastly the sequence of rules $\\mathcal{R}$ which contains $R_i$ is not applicable to the DMG $\\mathcal{G}$.\nHowever, one could imagine different sequences of rules that lead to the same estimand.\nFor example, suppose the corresponding $\\sigma$-separations apply, one could have $Pr(Y\\mid do(X)) = Pr(Y)$ in a DMG $\\mathcal{G}$ using rule 3 of do-calculus, and have in another DMG $\\mathcal{G}'$ $Pr(Y\\mid do(X)) = Pr(Y\\mid X)= Pr(Y)$ using rules 2 and 1 of do-calculus.\nTherefore, the completeness result of do-calculus does not guarantee that if a causal effect is not identifiable in a C-DMG, there does not exist an estimand valid in every compatible DMG.\n\nThanks again for the thoughtful comments. We hope to have answered all questions and we remain available during the discussion period in case there is any additional concerns."}, {"review_id": "vueTc3BZoE", "reviewer": "Reviewer_rE2q", "summary": "The paper proves soundness and completeness of the do calculus in a setting with macroscopic variables which are related by a possibly cyclic causal graph (resulting from coarse-graining a possibly cyclic graph with confounders into clusters of variables). The paper therefore extends known results for cluster graphs obtained from acyclic directed mixed graphs and generalizes them to cluster graphs obtained from directed mixed graphs.", "questions": "-\tPlease comment more on why generalization from C-DMG over ADMGs to C-DMG over DMGs raises serious challenges. \n\n-\t“However, Forré and Mooij [2020] showed that replacing d-separation by σ-separation in the three rules allows the do-calculus to be applicable on DMGs induced by ioSCMs.” I assume that it is crucial that the DMG is generated by an ioSCM because the semantics of interventions are not described a priori without specifying the generating model behind the DMG, right? Isn’t there just an ioSCM for the vector of cluster variables?   \n\n-\t“which may contain true cyclic causal relations”: what is a non-true cyclic causla relation? \n\n\n-\tInclomplete references: Philip Boeken and Joris M. Mooij. Dynamic structural causal models, 2024, and Patrick Forré and Joris M. Mooij. Constraint-based causal… \n\n\n-\tAbstract: “under the assumption that all clusters size are greater than 1.” Does the paper explain why this condition is necessary?", "limitations": "I like that the discussion of limitations mentions the following subtlety: “The first limitation is that the completeness result in Theorem 4 does not take into account that there might exist different sequences of rules of the do-calculus in different DMGs that can give the same final identification of the causal effect”\n\nA more serious limitation is my concern regarding relevance.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n\nThe paper contains solid math, it is carefully written. The paper seems fair with respect to citing other work, it is very transparent about its contributions. \n\n\nWeakness:\n\nThe relevance of aggregating variables into clusters of variables without dimensionality reduction is limited. Coarse-grained causal models in all applications I know work with aggregations that reduce the dimension, which raises problems that are conceptually harder than the problems solved here because interventions are no longer well-defined. I would appreciate a better motivation of clustering by convincing real-life examples. \n\n\nGiven that the results have already shown for C-DMG over ADMGs, extension to C-DMG over DMGs appears to incremental, given the concept of sigma-separation, which the paper builds upon. \n\nI know that reviewers are often too fast with judging substantial work as incremental. Sometimes because results look straightforward in hindsight only and subtleties are no longer visible once the solution is there. The authors can try to convince me that I’m falling into the same fallacy by explaining subtleties and conceptual problems they needed to solve.", "was_revised": true, "final_justification": "although one could argue the contribution to be incremental, it still is a nice and solid step forward.", "rebuttal": "First of all, we would like to thank for the thorough review and interesting remarks. We will take into account the mentioned typos and carefully proofread the whole paper.\n\nConcerning real examples of clustering without dimension reduction: this is common practice in different fields even though the soundness of this process was not guaranteed before [Anand et al 2023].\nIn particular, cluster graphs (without dimension reduction) are very useful in health and epidemiology, for example [Piccininni et al 2023] uses the front-door criterion in an acyclic cluster graph where confounders are clustered and causal relationships between the confounders are not mentioned. Other examples can be found in [Anand and Hripcsak 2025].\nMoreover, cluster graphs without dimension reduction are also used for root cause analysis in IT monitoring [Assaad et al., 2023].\n\nRegarding the difference between [Ferreira and Assaad 2025] which focuses on C-DMGs over ADMGs and our work which focuses on C-DMGs over DMGs, a first difference is the use of $\\sigma$-separation rather than d-separation as the underlying graphs might be cyclic.\nMoreover, the major difference in our opinion lies in the absence of assumption for the completeness results.\nIndeed [Ferreira and Assaad 2025] required that there is no two adjacent clusters of size 1 in a cycle to guarantee the completeness of d-separation and the completeness of do-calculus.\nIn our work, this assumption is no longer required for the completeness results, it is only necessary for the soundness of the SC-hedge criterion. Removing this assumption is the most important aspect of the paper.\n\nIndeed we assume that that the underlying DMGs are induced by ioSCMs.\nIt is also correct that a C-DMG over DMGs can be induced by a \"macro\"-ioSCM over the vector of cluster variables, just as a Cluster-DAG can be induced by a \"macro\"-SCM over the vector of cluster variables. However, the latter was not obvious until the seminal work of [Anand et al., 2023], which formally established this connection. In a similar way, we argue that the existence and formalization of a macro-level ioSCM over cluster variables was not evident prior to our work. In this respect, our contribution closely parallels that of [Anand et al., 2023], but in the context of ioSCMs.\n\nIn C-DMG over DMGs there exists 2 distinct \"types\" of cycles. We call \"true\" cycles, cycles that appear in the true underlying ioSCM, in contrast to \"false\" cycles that appear only in the C-DMG because of the clustering process.\nFor example if the true underlying DMG is $\\mathbb{V}= ( X_1,Y_1,Y_2,Z_1 ) , \\mathbb{E} = ( X_1\\rightarrow Y_1, Y_1\\rightarrow X_1, Y_1 \\rightarrow Z_1, Z_1 \\rightarrow Y_2 )$,\nand we cluster the variables according to the partition $C_\\mathbb{X} = (X_1), C_\\mathbb{Y} = (Y_1,Y_2), C_\\mathbb{Z}= (Z_1)$,\nwe get the C-DMG with edges $( C_\\mathbb{X} \\rightarrow C_\\mathbb{Y}, C_\\mathbb{Y} \\rightarrow C_\\mathbb{X}, C_\\mathbb{Y} \\rightarrow C_\\mathbb{Z}, C_\\mathbb{Z} \\rightarrow C_\\mathbb{Y})$.\nIn this C-DMG, the cycle between $C_\\mathbb{X}$ and $C_\\mathbb{Y}$ is a \"true\" cycle and the cycle between $C_\\mathbb{Y}$ and $C_\\mathbb{Z}$ is a \"false\" cycle.\nWe are willing to change the adjectives \"true\" and \"false\" if the reviewers believe it to be necessary and in any case we will clarify this in the final version.\n\nLastly regarding the necessity of the assumption for the hedge criterion, we refer to our answer to reviewer 6Zg1 where this is discussed in details.\n\nThanks again for the thoughtful comments. We hope to have answered all questions and we remain available during the discussion period in case there is any additional concerns."}, {"review_id": "ZE9or6T0jC", "reviewer": "Reviewer_6Zg1", "summary": "The paper tackles causal‐effect identification when the on a cluster-level abstraction called a C-DMG (nodes = variable clusters; edges = aggregate causal/confounding relations). Previous theory covered the acyclic case (ADMG); however, real systems often have feedback loops, motivating the cyclic DMG setting. The authors prove that, unlike ADMG setting, the do-calculus is unconditionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. They also prove graphical criteria for non-identifiability of macro causal effects established in C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs. The work builds on input-output structural causal models (ioSCMs) which generalize SCMs to allow for cycles, providing a theoretical framework for causal identification in partially specified representations of cyclic systems.", "questions": "-I think in Theorem 3 it's better to recall what is c^w.\n\n-Definition 7 is hard to follow without any explanation for each case.\n\n-Have you considered developing an algorithm for identifying macro causal effects in C-DMGs over DMGs to help practitioners apply these results?\n\nCould you elaborate on why clusters in cycles must have size greater than 1 for Theorem 5, and whether this requirement could be relaxed?", "limitations": "The authors adequately address the limitations of their work, particularly regarding the completeness of do-calculus and the soundness (but not proven completeness) of the SC-hedge criterion. They are transparent about the current boundaries of their theoretical results.\nNo guidance on how to learn C-DMGs from data, nor on the computational cost of the identification search. and no empirical validation or real-world case study.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "I found the paper’s narrative easy to follow and well organized. Although I had only passing familiarity with C-DMGs and ioSCMs, the exposition—especially the step-by-step illustrative examples—made the main ideas accessible. The identification / non-identification examples were particularly helpful. Overall, the work extends prior research in a meaningful direction and consistently provides the background needed to understand each new result. I should mention that I haven't read the proofs.\n\nWeaknesses:\n\n-The “completeness” theorem is relative: it connects C-DMGs to compatible underlying DMGs but does not establish absolute completeness for the cyclic setting, as the author acknowledged.\n\n-The paper is entirely theoretical. No algorithmic implementation or empirical demonstration is offered. Can the authors design even a small synthetic experiment to showcase how the derived formulas recover interventional quantities in practice?\n\n-The practical motivation for using C-DMGs over DMGs, rather than the acyclic C-DMGs over ADMGs studied earlier, is not fully articulated. Concrete real-world scenarios where true feedback cycles matter would strengthen the case for this extension.\n\n-I think including intuition or proof sketches would be helpful in the main body.", "was_revised": true, "final_justification": "Thank you, authors, for the rebuttal. It answered most of my concerns. However, the manuscript should be updated as promised. I increased the quality score.", "rebuttal": "First of all, we would like to thank for the thorough review and interesting remarks. We will take into account the suggested modifications and carefully proofread the whole paper.\n\nAs this is a fully theoretical paper, we are unsure as to what experiments could be relevant but we are willing to add some if the reviewer could detail what they had in mind.\nWe are considering developing a package to help practitioners apply these results (and others) but this is beyond the scope of this paper.\n\nFor real examples where feedback loops are important, we refer to the initial works on ioSCMs [Forre and Mooij 2020] where this is discusses more thoroughly.\nAn conceptual example in the context of public health is the effect of a medication on behaviors:\nIn the context of covid19, people who go party a lot are more likely to be contaminated so they will get vaccinated earlier to limit the risks, thus behaviors causes vaccinations.\nHowever, people who are vaccinated feel much safer and are more likely to go to a party, thus vaccination causes behaviors.\nTo truly understand the effect of vaccination of contamination, one has to take this kind of feedback loops into account.\n\nLastly regarding the necessity of the assumption for the hedge criterion: this assumption guarantees the existence of a compatible acyclic DMG in which one can use the classical hedge criterion.\nThis is necessary as the $\\sigma$-based do-calculus was not shown to be complete in DMGs and since the ID algorithm relies on the acyclicity assumption.\nTo remove this assumption, one would first need to find an equivalent ID algorithm and/or a hedge criterion for DMGs.\nWe will clarify this in the final version.\n\nThanks again for the thoughtful comments. We hope to have answered all questions and we remain available during the discussion period in case there is any additional concerns."}]

Author rebuttal:

Meta-review: This paper presents a theoretically grounded extension of causal identification techniques using cyclic directed mixed graphs. The manuscript is well-written and accessible, even to readers with limited familiarity with the underlying graphical frameworks, thanks to clear exposition and helpful examples. The main contribution lies in extending do-calculus to this class of models, which enables the identification of macro-level causal effects in settings that include feedback cycles.
However, the novelty and technical depth of the work are questioned. While the theorem statements are sound, the proofs are seen as relatively straightforward extensions of existing results on acyclic directed mixed graphs. The lack of formal definitions for key concepts like extended and compatible graphs in the appendix further weakens the rigor. Additionally, the paper does not offer a completeness result for non-identifiability or a broader graphical condition that could have significantly strengthened its theoretical contribution.
The notation used throughout the paper is dense and sometimes confusing, which may hinder readability. Some references are non-peer-reviewed, and the paper lacks empirical validation. Although the theoretical results are promising, the absence of algorithmic implementation or synthetic experiments makes it difficult to assess their practical utility. Including even a small-scale demonstration of how the derived formulas recover interventional quantities would enhance the paper’s impact.
The reviewers also raised the issue that the authors do not sufficiently explain why feedback cycles are essential in real-world scenarios or how their framework addresses challenges that arise in such settings. The relevance of clustering variables without dimensionality reduction is also questioned, as most coarse-grained models in practice involve dimensionality changes that complicate intervention definitions.
Despite these limitations, the paper is appreciated for its clarity, transparency, and solid mathematical foundation. It contributes meaningfully to the theoretical landscape of causal inference, but its significance would be clearer with stronger motivation, deeper theoretical results, and some empirical grounding. 
However, rebuttals and subsequent discussion that was engaing, addressed and somewhat solved many issue raised by the reviews.
Decision: Accept (poster)