Identifying Macro Causal Effects in a C-DMG over ADMGs
Authors: Simon Ferreira
Charles K. Assaad
Venue: Accepted by TMLR ()

Abstract: Causal effect identification using causal graphs is a fundamental challenge in causal inference. While extensive research has been conducted in this area, most existing methods assume the availability of fully specified directed acyclic graphs or acyclic directed mixed graphs. However, in complex domains such as medicine and epidemiology, complete causal knowledge is often unavailable, and only partial information about the system is accessible. This paper focuses on causal effect identification within partially specified causal graphs, with particular emphasis on cluster-directed mixed graphs (C-DMGs) which can represent many different acyclic directed mixed graphs (ADMGs). These graphs provide a higher-level representation of causal relationships by grouping variables into clusters, offering a more practical approach for handling complex systems. Unlike fully specified ADMGs, C-DMGs can contain cycles, which complicate their analysis and interpretation. Furthermore, their cluster-based nature introduces new challenges, as it gives rise to two distinct types of causal effects: macro causal effects and micro causal effects, each with different properties. In this work, we focus on macro causal effects, which describe the effects of entire clusters on other clusters. We establish that the do-calculus is both sound and complete for identifying these effects in C-DMGs over ADMGs when the cluster sizes are either unknown or of size greater than one.  Additionally, we provide a graphical characterization of non-identifiability for macro causal effects in these graphs.

Published in Transactions on Machine Learning Research (07/2025)
Identifying Macro Causal Effects in C-DMG over ADMGs
Simon Ferrcira
simon.ferreivt Dordoooooooooon---erssss......
Sorbonne Université, INSERM, Institut Pierre Louis d' d'Epidémiologie et de Santé Publique, F75012, Paris, France
Charles K. Assaad
charles assaad@inserm..
Sorbonne Université, INSERM, Instibut Pierre Louis d' E'Eiééémiooogie de Santé Publique, F75012, Paris, France
Reviewed on OpenReview: https: // openreview. net/ met/ forum? id= 905LEugg6R
Abstract
Causal effect identification using causal graphs a fundamental challenge causal inference. While extensive research has been conducted in this area, most existing methods assume the availability of fully specified directed acyclic graphs acyelic directed mixed graphs. However, in complex domains such medicine and epidemiology, complete causal knowledge is often unavailable, and only partial information about the system is accessible. This paper Tocuses on causal effect identification within partially specified causal graphs, with particular emphasis on cluster-directed mixed graphs (C-DMGs) which can represent many different acyclic directed mixed graphs (ADMGs). These graphs provide a higher-level representation of causal relationships by grouping variables into clusters, offering more practical approach for handling complex systems. Unlike fully specified ADMGs, C-DMGs which complicate their analysis and interpretation. Furthermore, their cluster- based nature introduces new challenges, as it gives rise to two distinct types of causal effects: macro causal elfects and micro causal effects, each with different properties. In this work, we focus on macro causal effects, which describe the effects of entire clusters on other clusters. We establish that the do-calculus is both sound and complete for identifying these effects in C-DMGs over ADMGs when the cluster sizes are either unknown or of size greater than one. Additionally, provide graphical characterization of non-identifabiilty for macro causal effects in these graphs.
contain cycles,
1 Introduction
A key challenge in causal inference for obscrvational studies, known the identification problem (Shpitser & Pearl 2008), involves determining when and how causal effect of set of variables X on another set of variables denoted
Pr(Y = do (X (x=x)) where the do).) operator represents vention can be estimated from observational data, which are typically represented as a joint probability distribution under normal, intervention-fees conditions. To address this problem, several graph-based tech- niques have been developed for causal effect identification, assuming fully specified causal structure, often represented as directed acyclic graphs (DAGs) or acyclic directed mixed graphs (ADMGs). Among these methods, the do-caleulus(Peat
external inter-
1995) stands out as a sound and complete tool for causal effect identification.
These tools were also extended to directed mixed graphs (DMGs) (Richardson 1997 Forré & Mooij 2017
2018
Forré & Mooij 2020 Bocken & Mooij. 2024)
However, constructing a fully specified causal graph is challenging because it requires knowledge of the causal relations among all pairs of observed variables. This knowledge often unavailable, particularly in complex, high-dimensional sottings, thus limiting the applicability of causal inference thcory and tools.
Therefore, there has recently been more interest in partially specified causal graphs (Maathuis & Colombo,
2013
Perkovic et. al. 2016
Perkovic 2020: Jaber et al. 2022 Wang et al. 2023 Anand et. al. 2023
Wah

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fundamental tool in do-calculus is sound and Published in Transactions on Machine Learning Research (07/2025)
An important type of partially specified graphs is the Cluster- Directed Mixed Graph that can represent many ADMGs (C-DMG over ADMGs) which provides coarser representation of causal relations through vertices that represent a cluster of variables (and can contain cycles). The partial specification of a C-DMG over ADMGs arises from the fact that each verlex represents a cluster of variables rather than single variable. Within C-DMGs, two distinet types of causal effects exist: macro causal effects, which describe the causal influence between entire clusters, and miero causal effects, which capture the causal relationships between individual variables within and across clusters. In this paper, we focus the former.
et al. 2024 Bocken & Mooij. 2024 Assaad et al. 2023 2024 Ferrcira & Assaad 2024 2025
Recent work has demonstrated that the do-caleulus can be directly applied to identify macro causal effects in certain restricted classes of C-DMG over ADMGs. Notably, these include Cluster-ADMGs, which enforce an acyclicity constraint on the cluster graph (Anand et al.) 2023: Tikka et al. 2023), and summary causal
graphs, where each cluster represents a single time series (Reiter al. 2024 Ferreira Assaad 2025). In this study, we extend this line of work by addressing macro causal effect identification in general C-DMG over ADMGs, without imposing structural restrictions. Specifically, we allow for the presence of cycles and do not impose any constraints on how clusters are formed any subset of variables can constitute cluster long as the cluster size is cither unknown or it is of size greater than 1. More specifically, our contributions are as follows:
We establish that d-separation Pearl 1988 complete in C-DMG over ADMGs.
We demonstrate that do-caleulus remains sound and complete for identifying macro causal effects in C-DMG over ADMGs.
graphical structure traditionally used to indicate the non-identifiability of causal effects does not extend to C-DMG over ADMGs. Instead, we show that the recently introduced concept of SC-hedges (Ferreira & Assaad 2025), which was originally developed for summary causal graphs (a specific subclass of C-DMG over ADMGs), is applicable general C-DMG over ADMGs.
We demonstrate that the classical concept of hedges (Shpitser & Pearl 2006
The remainder of the paper is organized as follows: In Section 2 we formally present C-DMG over AD- MGs and macro causal effects. In Section 3 we show that d-separation and the do-caleulus is sound and complete for macro causal effects in C-DMG over ADMGs and present a graphical characterization for the non-identifaabiiity of these effects. In Section E we examine the challenges that emerge when additional information about cluster sizes is available, particularly when some clusters have a size of 1. Finally in Section 5 we conclude the paper while showing its limitations. All proofs are deferred to the appendix.
2 Notation and Definitions
In this section, we introduce the key definitions and notations used throughout the paper to facilitate a clear and consistent presentation of our results. We begin by introducing the primary concept under consideration, namely the structural causal model (Pearl 2209)), also referred the probabilistic causal model (Shpitser & Pearl 2008) which is assumed to be unknown.
Definition 1 (Structural causal model (SCM) PPaal 2009 A structural causal model is tuple M (L, V, Pr()),, where
L is a set of ezogenous variables, which cannot be observed but affect the rest of the model.
V, is set of endogenous variables, which are observed and every V e V is functionally dependent some subset of Lu (VV).
F is set of functions such that for all V E V, fV is function taking as input the values of a subset of LUV(V and outputting value for V.
Pr(1) is a joint probability distribution over l.
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In this definition of SCMs, the set of functions F encapsulates the causal mechanisms governing the relation- ships among variables, while IL represents the noise, f.e., the unobserved or hidden variables that affect the observed variables, and V represents the observed variables, which are often referred to as individual variables or miero-variables or low-level variables. The uncertainty associated with these unobserved variables is char- acterized by the distribution Pr(I), which accounts for the unknown influences outside the observed system. When combined with the causal mechanisms encoded in F, this distribution gives rise to the distribution Pr(v) over the observed variables V. We also assume that each structural causal model (SCM) induces a directed acyelic graph (DAG), where every variable in V u L corresponds to vertex in the graph. In this DAG, directed edge - is drawn from one variable to another if the former serves that determines the latter. For simplicity, instead of working directly with these DAGs, we consider an alter- nalive representation known as an acyclic directed mixed graph (ADMG). In an ADMG, only the observed variables in V correspond to vertices, while hidden variables in that share common inputs are represented by bidirected edges P between the corresponding observed variables, thereby implicitly accounting for the hidden confounding. Formally, ADMGs are defined follows:
input to the function
Definition (Acyclic directed mixed graph (ADMG)). Consider an SCM M. The ADMG g =V, induced by M is a graph where:
the vertices V are the endogenous variables of the SCM; and
VX,Y V the edge X is in I Y is functionally dependent on X and the edge Y is in E if both X and Y are functionally dependent on a same ezogenous variable L e IL.
ADMGs have been extensively studied and are highly valuable in causal inference. However, in many fields such as epidemiology, constructing, analyzing, and validating an ADMG remains a 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 chal- lenging to specify the precise structure of the graph. Nevertheless, researchers can often provide partially specified version of the ADMG, which offers a more practical and compact representation of the underlying causal structure. These simplified representations, which we call Cluster-Directed Mixed Graphs (C-DMG over ADMGs), group several variables into clusters, allowing for the representation of causal relationships at a higher level of abstraction while retaining essential structural properties of the system. In a C-DMG, directed edges between clusters represent causal influences at the higher level, while bidirected edges capture hidden confounding effects that exist between clusters.
Definition 3 (Cluster directed mixed graph over ADMGs (C-DMG over ADMGs)). Let G = (V,E) E) be an ADMG induced from an SCM M. A C-DMG is a graph G (C, E*) where:
C is a partition (t.e., k disjoints sets of micro-variables) of V; and
YC2, Cr e C the edge Cx + Or (resp. Ox c.> Ox) is in Fe if and only if there exists X Cx and Cy such that X Y (resp. X +.> Y) in E.
For simplicity, we will henceforth use the terms 'C-DMG over ADMGs" and 'C-DMGs' interchangeably. The transformation from an ADMG to a C-DMG can be made explicit using the natural transformation from to C (Cx = (X1,", X, ",,,,,CCC Cr (Yi,"", Yno]]:
(I1.'" "ns," V1, Bnn) + ((21,"","==).". (11 (U..""BBr)) Bnv)).
The abstraction of C-DMG entails that, even though there is exactly one C-DMG compatible with a given ADMG, there are in general several ADMGs compatible with a given C-DMG. For example, we give in Figure D a very simple C-DMG with two of its compatible ADMGs. If all clusters in the C-DMG are of size equal to one, then the C-DMG according to our definition would be an ADMG. Notice that, C-DMG
II the literature, the bidirected edges ADMGs are generally non-dashed. However, in this work, we use dashed hidirected edges enhance visual clarity and distinction.
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Y1
X₂
X
Y2
Wi
W2
Wi
W3
W4
W3
W4
(a) ADMG
(b) ADMG
(c) C-DMG.
Figure Two ADMGs and with their compatible C-DMG. Red vertices represent the exposures of interest in and blue vertices represent the outcome of interest.
may have directed cycles and in particular two directed edges oriented in opposite directions, however, we emphasize that all cycles all C-DMGs that we consider in this paper, arise from the partial specificity. For example, Figure [ in the C-DMG, Cx - Cw and Cw - Cx form a cycle, which we often write Cx z Cw- This cycle implies that in all ADMGs compatible with the C-DMG "X; Xy Cx and Wa Wi E Ow such that X; WA and Xj- Wi.
To streamline the presentation and avoid repetitive explanations of concepts applicable to both ADMGs and C-DMGs, we will adopt the unified notation g" = (V",E") to refer to either 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* E V a graph G = refer parents graph by, Pa(V*, ,g), , its ancestors by An)) and descendants by De(V*, We consider that a vertex counts as its own descendant and as its own ancestor. In addition, in C-DMG g = (C), for every cluster Cx C, we will the notation of strongly connected components defined follows:
(whether it be an ADMG or a C-DMG), we will
Strongly Connected Component: Sco(Ox,99) An De(Cx,99)
Finally, in order to map the verlices in the C-DMG with the vertices in the ADMG, we will use the notion of corresponding cluster
Corresponding Cluster: vX e Cx. CI(X,99)=C) = Cx.
We distinguish between two types of causal effects in the context of C-DMGs, the macro causal effect AAnan
et al. 2023 Ferreira & Assaad 2025 and the miero causal effect (Assaad et al. 2024 Assaad 2025
this paper we focus on the former and we formally define below:
Definition 4 (Macro causal effect). Consider SCM on variables V and a compatáble C-DMG (C, (C,
A macro causal effect is a causal effect from set of macro-variables Cx on another set of macro-variables Cy where Cx and Cr are disjoinl. subsets of C. It. is wrillen Pr(C* Cr do tx)), where the do(.) (-) operator represents external intervention.
In the following, we will abuse the notation by writing Pr(cy do (cx)) instcad of Pr(Cy cy do (Cx = +x)) when the setting is clear.
The identification problem in causal inference aims to establish whether a causal effect of a set variables on another set of variables can be expressed exclusively in terms of observed variables and standard probabilistic notions, such as conditional probabilities. Formally, the identification problem in the context of macro causal effects and C-DMGs is defined follows:
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Definition 5 (Identifiability in C-DMGs). Let X and Y be disjoint sets of vertices in unknown ADMG g compatible with C-DMG g The causal effect of X on Y is identifiable in 9 Pr(y) | do(x)) is uniquely computable from any observational positive distribution compatible with g
Hence, there are no two ADMGs, 91 and 92 such that Pri(v) = Prz(v) where Pr1 is a positive distribution compatible with 91 and Prz a positive distribution compatible with 92 and Pri do (x)) Prz(y) do (x)).
Whenever the ADMG is known, e.e., all clusters in the C-DMG are of size equal to one, the do-calculus Peaal 1995) can be applicd to identify the causal effect of interest. Rocently, it has been shown that the do-calculus can also be applied when the ADMG is unknown but its C-DMG is known (with clusters of size greater than one) under one of the following conditions: the C-DMG is acyelic (Anand et al. 2023); or if the cluster
corresponds to a time series (Ferrcira & Assaad 2025). However, it remains unclear whether the do-calculus can be directly applied to general C-DMGs. Therefore, the primary objective of this paper is to bridge this gap by developing analogous tools tailored for general C-DMGs.
We emphasize that, throughout this paper (except in Section 4 we operate under the following assumption: Assumption 1. The identifiability problem, as defined in Definition 5 is considered under one of the following assumptions:
1. The size of the clusters is unknown, or
2. No two adjacent clusters in a cycle are both of size one.
While the first condition in the assumption may seem unrealistic- since, in practice, access to data would typically reveal cluster sizes it remains relevant in early-stage study design. A modeler may wish to assess identifiability before finalizing which specific variables will be included in each cluster. For instance, an epidemiologist planning study on the effect of salary stress may know that multiple variables may represent salary, stress, and confounders, without yet specilying their exact composition. On the other hand, if the modeler already knows the precise variables in each cluster, and thus the cluster sizes, then our results apply only when there is no cycle containing two adjacent clusters both of size one (second condition in the assumption). In Section 4 we further analyze this assumption, distinguishing the results that remain valid when it is not satisfied from those that no longer hold.
3 Identification of Macro Causal Effects in a C-DMG over ADMGs
In this section, our primary objective is to demonstrate that the do-calculus is both sound and complete for identifying macro causal effects in C-DMGs. To achieve this, we first establish in the first subscction that d-separation, fundamental tool used in do-calculus to determine conditional independencies, is also sound and complete within C-DMGs to determine "macro conditional independencies'. In the second sub- section, we present the main theoretical result of this section, providing a rigorous proof of the soundness and completeness of do-calculus for macro cansal effect identification. Following this, we introduce a graph- ical characterization of non-identifiabiiity offering insights into scenarios where causal effects cannot be determined from observational data. Finally, in the last subsection, we explore the connection between our findings and summary causal graphs, demonstrating how our results extend and apply this specifie class of graphs.
3.1 The d-separation in C-DMG over ADMGs
introduced for ADMGs. It was later extended to DMGs in Forré & Mooij (2018) and to cluster-ADMc. in Anand et al. (2023). In this subsection, we show that it is also readily extendable to C-DMGs. We start by defining blocked paths and d-scparation.
The standard definition of d-separation Pearl 1988
Definition 6 (blocked walk (Pearl 2009)). In a graph G =
(whether it be an ADMG or a C-DMG), a walk #े = (Vi',",, V is said to be blocked by a set of vertices W* g VV** if:
"In this work, we call positive distribution distribution in which Vo, Pr(+) > O, however this assumption can be loosened Hwang et. al. 2024).
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1. Vi e w Vi E EW"
2. Bilki<< such that (V + Vin) á or (Vg) + -V- S and V E <W", or 3i:1<i<n such that
f and De(V, nw =0
where + represents + or ,.,, + represents + or sn;; and +* represents any of the three arrow type +, + or 6-,, A walk which is not blocked is soid to be active.
Definition 7 (d-separation (Pearl 2009h). In a graph G (V*,E*) (whether it be on ADMG or a C-DMG), let X*, YY W be distinct subsets of V", W" is said to d-separate X and V if and only if W blocks every path from a verlex in X" to a vertez in V" It is written (x* "n"" | W")
The following theorem shows that d-separation is applicable as is to C-DMGs.
Theorem 1. (Soundness of d-separation in a C-DMG over ADMGs). Let g = (C, be a C-DMG and Cx, Cv, Cw be disjoint subsets of C. If Cx and Cy are d-separated by Cw in ge then, in any compotible ADMG G = (V, X UCEC2 C' and Ucec, are d-separated by w Ucecw C.
Theorem 1 shows that d-separation in C-DMGs guarantees finding some common macro-level d-separation in all compatible ADMGs which implies according to (Pearl 2009 Theorem 1.2.5) that it allows the detection of some conditional independencies in the underlying probability distribution directly from the C-DMG. Thus, by extending the applicability of d-separation to C-DMGs, this result allows us to infer macro-level conditional independencies even when dealing with partially specified graphs. To appreciate the above result, we give in the following an example of the application of d-scparation to C-DMGs
Example 1. Let be the true unknown ADMG and consider that its compatible C-DMG, denoted as g is one given in Pigure 2dl Using Desimition 7 we can directly deduce (Cwaaar Cx)gs. Thus according to (Pearl 20099 Theorem 1.2.5), Ow is conditionally independent of Cy given Cx in every distribution compatible with the true ADMG.
Example 2. Let G be the true unknown ADMG and consider that its compotible C-DMG, denoted as g is one given in in Figure 2d Using Desonition 7 we can diveclly deduce that (CzuaCr) Cx Ow)ge. Thus according to [Peorl, 2009 Theorem 1.2.5), Oz is conditionally independent of Cy given Cx and Ow in every distribution compatible with the true ADMG.
The following theorem shows that d-separation is also complete in C-DMGs under Assumption I
Theorem 2 (Completeness of d-separation in a C-DMG over ADMG J Let G = (C,E9) be a C-DMG, Cx, Cr, Cw be disjoint subsets of C, X = UCecx C, Y = Ucecy C and W Ucec C. Under Assumption 7 if Cx and Cy are not d-separated by Ow in Go, then there exists a compatible ADMG G = (V,E) such that X and Y are not d-separated by W.
The findings of the above theorem establish that identifying a d-separation in C-DMGs guarantees finding all common macro-level d-separation in all compatible ADMGs. This is particularly valuable in constraint- based causal discovery when the interest is to uncover the structure of the C-DMG without uncovering an ADMG. Most importantly, these findings serve critical foundation for the results presented in the next subscction.
3.2 The do-calculus in a C-DMG over ADMGs
The do-calculus initially introduced in Pearl 1995 is an important tool of causal inference that consists of three rules. It allows to express, whenever it possible, queries under interventions, í.e., those that contains do operator, as queries that can be computed from positive observational distribution, i.e., that does
The proof of this Theorem follows the same intuition the proof of Theorem 2 in Ferreira Assaad 2025 However, Ferreira & Assuad (2025) made the implicit and unnocessary assumption that the clusters were of sizes bigger Chan the number of clusters. In proof do not need this assumption and thereby our proof generalize their result t.c, clusters do not need be of size bigger than 2, nor for the completeness of d-separation in C-DMGs, nor for the completeness d-separation in summary causal graphs.
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(a)
(b)
(c)
(d)
Figure 2: C-DMGs with identifiable macro causal effects. Each pair of red and blue vertices represents the causal effect are interested in.
(a)
(b)
(c)
(d)
Figure 3: C-DMGs with not identifiable macro causal effects. Each pair of red and blue vertices represents the total effect interested in.
not contain a do (-) operator. In such cases, it said that the query containing the do is identifiable. The do-calculus was initially introduced for ADMGs so not casily extendable cyclic graphs. In this subsection, we show that it is also readily extendable to C-DMGs (containing cycles).
Firstly, define the notion of mutilated graphs PPaal 2009). Consider causal graph G* (V", E*) (whether it be an ADMG
C-DMG) and A" B" A" E V*, mutilated graph denoted by I is the graph
obtained by removing all edges coming in A and all edges coming out B*
Using the notion of mutilated graphs and d-separation we show that the do-calculus is applicable to C-DMGs. Theorem 3 (do-calculus for a C-DMG over ADMGs and macro causal effects). Let (C, be C-DMG and Cx, Cy, Cz, Cw be disjoint subsets of C. The three following rules of the do-calculus are sound.
Rule 1:Pr(cy do (c2) cx, Pr(cy/ do (c2),
if (CynaCx Czz Cw)eg if (CruaCx Cz, Ow)
Rule 2:Pr(cy do (c2) (x) (.) Pr(c,) do (cz), c2,
Rule 3:Pr(t, do (cx) do (1.) cw) Pr(c, do (c+) cw)
if (CruaCx Cz, Cw) (y
where Cx(Cw) is the set of vertices in Cx that are non-ancestors of any verter in Ow in the mutilated graph Gez
Using the soundness of the do-calculus in C-DMG (Theorem 3 easily use the rules of the do-calculus to find out that the causal effect Pr(cy/ | do (cx)) is identifialee in all C-DMGs in Figure 2 as demonstrated in the following examples.
Example 3. Both in Figure 2a and 2c one verify that (CruaOx)ge, thus Rule of the do-calculus is applicable and Pr(cy do (cx) Pr(ce cz).
Example 4. Notice that Figure 2b does not contain any cycle other than self-loops and is very similar to Figure (bb of Anand al. (2023) which corresponds to the well-known front-door criterion (Peard 2009).. Thus, using the corresponding sequence of classical rules of probability and rules of do-calculus as the one given in [Pear 2009. p.83), one obtains Pr(cy) do (cx)) Ec,, Pr(cw (x)) Ec, Pr(cy Cws GC Pr(cx))

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Example 5. Consider the C-DMG in Figure 2d containing a cycle between oe, CR. and cu and hidden confourding between Cr and ow- Let Pr(cy do Cx)) be the causal effect of interest. Using the rule of total probability we can rewrite Pr(cy| do C2) as
Pryy I do c.) (cx,
We first focus on Pr(c,) (cx. (2) aw). Notice that (Cy Cz, Cw) CW) and that (Cyl I Cx, Cw) which means by applying Rule consecutively, the expression Pr(cy do ,x), ow) can be equivalently rewritten as: Pr(cy) Cw)
0202
Now we Jocus on Pr(cw | (c), Notice that (CuLaaXx | Cr) which means by Rule 3 of the do- calculus we can completely remove do (c)) from the expression. Parthermore, we have (CuoldD2)os which means by Rule 2 we can replace do (c,) by Cz. So we can rewrite Pr(cw do (c): as Pr(cw ()).
In these three examples, the rules of the do-calculus allow macro causal effect to be expressed solely in terms of observed variables. Consequently, the macro causal effect can be estimated from data, provided that the positivity assumption holds.
In the following theorem, we show that the do-calculus is not only applicable to C-DMGs, but also is complete.
Theorem 4 (Completeness do-calculus for a C-DMG over ADMGs and macro causal effects). Under Assumption [ if one of the do-calculus rules does not apply for a given C-DMG, then there crists a compotible ADMG for which the corresponding rule does not apply-
Using the completeness of the do-calculus in C-DMGs (Theorem we can determine that the causal effect Pr(cy | do (c2)) is not identifiable in all C-DMGs depicted in Figure] and in the C-DMG in Figure by examining all possible iterations of the rules of the do-calculus. However, it is well known that exhaustively examining all possibilities for applying the rules of do-caleulus can quickly become impractical, particularly for large graphs. To address this challenge, the following subsection introduces a sub-graphical structure designed to directly determine whether it is feasible to express a macro causal effect solely in terms of observed variables using do-calculus and C-DMG.
Notice that, even though C-DMGs are defined with the inclusion of self directed edges (e.g., C C) and self dashed bidirected edges (e-g-, C C)2 we only focused on paths in this paper and thus, under Assumption 1 adding removing such edges will not influence identifiability results. However, this is not necessarily true if Assumption is not verified. In addition, as shown in Assaad et al. (2024), self directed edges and self dashed bidirected edges are very useful in the identification of micro causal effects.
3.3 Non-ldentiffabiiity: graphical characterization
In ADMGs, there exists sub-graphical structure, called a hedge (Shpitser & Pearl 2006), which is employed to graphically characterize non-identifiability as shown in (Shpitser & Pearl [0066] Theorem 4. To properly
define it for C-DMGs, it is cssential to first familiarize oncself with the two related definitions which we have adapted and provided below specifically for the context of C-DMGs:
Definition 8 (C-component, Tian & Pearl (2002)). Let G' = (V", E*)) be a graph (whether it be a C-DMG or an ADMG). subset of vertices V7 S V such that VVV", Vi e V7, 3V2, Vi E V with Visisnn n, V 6. Vi is called a C-component.
Definition 9 (C-forest, Shpitser & Pearl 2006)). ). Let g" = -V*, be a graph (whether it be a C-DMG or an ADMG) Tf G" is acyelic, G* a forest (i.e., every of ils vertices has most one child), and G is a C-component then 9 is called a C-forest. The vertices which have no children are called roots and we say C-forest is R' -rotted if it has roots R* V
Definition 10 (Hedge, Shpitser & Pearl (2006 2008)). Consider a graph g* =(V,EE" (whether it be C-DMG or an ADMG) and two disjoint sets of vertices X", W V Let F = (Vi, (F)) and IF' (Vi., Fi.) be
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(a) SC-projection of Figure 2a
(b) SC-projection of Figure 2b
(c) SC-projection of Figure 2c
(d) SC-projection of Figure 2d
(e) SC-projection of Fig- (f) SC-projection of Fig- ure
(g) SC-projection of Fig- ure
(h) SC-projection of Fig- ure 3d
3a
ure 3b
Figure 4: SC-projections of the C-DMGs in Figures and Figures 3 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.
two RR -rooted C-forests subgraphs of g such that X n V Ø, x Ø, F'sF, and R c An(Y", . Then F and IF form hedge for the pair (x*, (Y)) in g*.
A hedge turned out to be too weak to cover non-idenntifabiiity in C-DMGs. For example, the C-DMG Figure 3a contains no hedge but the macro causal effect not identifiable due the cycle between Cx and Cy. However, the SC-hedge an extension of the hedge structure introduced in Ferreira & Assaad (2025) for summary causal praphs-proves proves to be applicable to general C-DMGs. In the following, we formally define SC-hedges in the context of C-DMGs and demonstrate that this substructure serves as sound criterion for detecting non-identifiable macro causal effects.
Definition (Strongly connected projection (SC-projection). Consider C-DMG g = (C, The SC-projection He of ge is the graph that includes all verlices and edges from Ge, plus dashed bidirected edge between each pair Cx, Cr C such that See( Cz,, 9t) Sec(Cr,99) and Cx Cx.
Definition 12 (Strongly Connocted Hedge (SC-Hedge)). Consider a C-DMG g (C,E9), ; its SC-projection He and lwo disjoints sels of verlices Cx, C2 A hedge for (Cx, Cx) H an SC-hedge for (Cx, Cx) in g"
Theorem 5. Consider C-DMG g (C,E*) and two disjoints sets of vertices Cx, C2 C. Under Assump- tion if there erists an SC-hedge for (Cz, Cy) in go then Pr(c, do (cx)) is not identifiabl.
Figure illustrates all SC-projections of the C-DMGs presented in Figures 2 and Notably, the SC- projections corresponding to the C-DMGs in Figure 2 do not contain hedge, whereas all SC-projections derived from the C-DMGs in Figure 3 include a hedge. This distinction highlights the structural differences between the hedges and non-hedges and their implications for causal effect identification.
Note that Theorem states the soundness the SC-hedges or in other words guarantees the non identifi ability of the macro causal effect the presence SC-hedge. However, does not give the corresponding completeness result i.e., the absence of an SC-hedge does not imply the identifiability of the macro causal effect.

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Published in Transactions on Machine Learning Research (07 (07/202) 2025)
Cx 11
C+ 1
X1
Y1
X1
22
W2
Cw |=2
(a) C-DMG.
(b) ADMG1
(c) ADMG 2.
Figure 5: A C-DMG with the knowledge of the clusters' size and its only two compatible ADMGs that illustrates that knowing cluster sizes affects the completeness of d-separation and the do-calculus. The C- DMG in (a) contains two clusters of size and it contains an SC-Hedge. However, even in the presence of an SC-Hedge the causal effect can be identifiable since in all its compatible ADMGs (i.e., (b) and (c)) the causal effect remains identifiable by only applying Rule of the do-calculus.
4 Non-completeness of d-separation and the do-calculus when considering the size of the clusters
Up to this point in the paper, we have assumed either that the size of each cluster is unknown or that no two adjacent clusters in a cycle are both of size one Assumption D. In this section, we demonstrate that the completeness results of d-separation and the do-calculus no longer hold when cluster sizes are known and when some clusters contain only single variable (when Assumption 1 is not satisfied). In the extreme cases where Assumption I is not verified, there exists some paths in the C-DMG which do not map to any micro-path in any compatible ADMG. To illustrate this, consider the C-DAG in Figure 5a where cluster Cx has size 1, cluster Cy also has size 1, and cluster Ow has size 2. In Figure 5a the path (Cx + Cw - C+) soems to be active, however when one looks at the two compatible ADMGs Figures 5b and 5c one can see that there is actually no such path in the ADMGs. Using our SC-hedge characterization, we would conclude that the causal effect is not identifiable using the do-caleulus, as the presence of a cycle containing Cx and Oy implies the existence of dashed bidirected edge between Cxx and Cy in the SC-Projection. However, in this specific case, the only compatible ADMGs are those shown in Figures 5b and 5c where it is evident that the causal effect Pr(cy do (cx)) = Pr(s1 I do (x1)) is identifiable and equals Pr(v1 =1). The key intuition behind this discrepancy is that cycle involving X1, Y1, and W; (for é (1,2)) cannot exist, as this would violate the assumption that the true causal structure follows an ADMG, where cycles are not permitted. While Assumption [ is not required for the results concerning identifiability i.e., Theorems I and it is and
necessary for the results regarding non-identifiability hold i.e., Thcorems
5 Conclusion and discussion
In this paper, we established the soundness and completeness of d-separation and the do-calculus for respec- tively identifying macro causal effects in C-DMGs. By doing so, we bridged the gap between causal inference and many roal-world applications in epidemiology.
There are three main limitations to this work. The first limitation is that the completeness result in Theo- rem | does not take into account that there might exist different iterations of the rules of the do-calculus in different ADMGs that can give the same final identification of the causal effect. A second related limitation is Ihat we provided a graphical characterization for the non-identifiability of macro causal effects, however this characterization is not proven to be complete, even though we did not find any counter-example of its completeness. Proving it complete remains an open problem. The third limitation is that our results rely on Assumption D which is not always satisfied in practice. However, we think that we can relax it, while keeping the same results, by assuming that two adjacent nodes in cycle are both of size 1.
10

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Published in Transactions on Machine Learning Research (07 (025))
Acknowledgments
We thank Fabrice Carrat from IPLESP for his valuable insights on the importance of C-DMGs epidemi- ology. This work was supported by the CIPHOD project (ANTR22-CCP31-0222-0))
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12


Reviews:
[{"review_id": "6DQidg97Xp", "reviewer": "Reviewer_p9DU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n\n- The paper is full of illustrative examples to give intuition behind the results and discussions.\n- The authors were upfront about the limitations of their results and discussed them appropriately.\n\n# Weaknesses\n\n- Three limitations have been identified by the authors in the conclusions, and I feel that it has been discussed adequately.\n- It is not clear to me when when why Assumption 1 is necessary for the results to hold. The only time I see it referenced in the proofs is for Theorem 5. If I have missed it, please point me to the discussion. Thank you.\n- If I recall correctly, hedges were sound and complete for ADMGs. If I did not misread the paper, the authors showed that soundness of SC-hedges, i.e., \"under Assumption 1, existence of SC-hedge implies non-identifiability\". Is there a corresponding completeness result, i.e. \"under Assumption 1, non-identifiability implies existence of SC-hedge\"? If it is not possible, it should be discussed why it is not complete for C-DMGS. If I have missed it, please point me to the discussion. Thank you.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper tackles the problem of identifying causal effects when in cluster-directed mixed graphs (C-DMGs) where certain vertices in an acyclic directed mixed graphs (ADMGs) might be grouped together. The authors showed that d-separation and do-calculus remain sound and complete for identifying macro causal effects in general C-DMGs over ADMGs when the size of clusters is either unknown or when each cluster has at least two variables. Under the same assumptions, they also showed that existence of a SC-hedge (an extension of hedge introduced by prior work) implies non-identifiability in general C-DMGs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Requested changes:\n- Regarding the weakness comment about Assumption 1, can you be more explicit about it? I understand that you gave an example in Section 4 but could you add some prose to explicitly explain when and why Assumption 1 is necessary for your results to hold? Which part of your proofs explicitly relies on this assumption?\n- See the weakness comment about completeness of SC-hedge\n\nMinor typos:\n- Third bullet on Page 2: \"SC-hedge Ferreira & Assaad (2025)\" should \"SC-hedge (Ferreira & Assaad, 2025)\". Probably some a citep typo.\n- Figure 1(a) caption: Missing space between \"ADMG\" and \"1\"\n- Theorem 3, Rule 2: c^W shoudl be c_W\n- First paragraph on Page 8 starting with \"We first focus on...\": The sentences that follow have a lot of typographic errors. In your notation for conditional independence with respect to mutilated graphs, the c's in the brackets should be capitalized while the c's in the subscript should be calligraphic?", "broader_impact_concerns": "NIL", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "L35qIl4YN3", "reviewer": "Reviewer_UW5z", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths \n\n- I can see the usefulness of a partially-identified causal model (as we almost never have the complete ADMG in realistic scenarios) and so studying identification in the case of partial models, like the authors are doing, is bound to be interesting to readers. \n- Good sectioning, covering each of the contributions discussed in the intro through separate theorems in each section. \n- Multiple examples of toy graphical models provide good intuition for the results.  \n\n\n## Weaknesses \n\nThe main weakness is the format; this is an interesting theoretical work that spends a lot of time setting up definitions and stating theorems, with all proofs delegated to the appendix (similarly to the work of Anand, et al.[^1]). While the paper is appropriately sectioned, it’s possible that it would benefit from using the longer TMLR format so that there is space to talk about the proofs a bit in the main text (even if only sketches of the proofs are provided in the main text) and that appropriate time is allocated to review them in more detail. This would also make the paper seem a bit less dense. \n\n[^1]: |Anand, T.V., Ribeiro, A.H., Tian, J. and Bareinboim, E., 2023, June. Causal effect identification in cluster dags. In _Proceedings of the AAAI Conference on Artificial Intelligence_ (Vol. 37, No. 10, pp. 12172-12179).|\n\t|Vancouver||", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The work explores causal-inference identification challenges in a form of partially-defined causal graph, which the authors call “C-DMG” or “Cluster-Directed Mixed Graph”, in manner similar to the work by Anand, et al, 2023[^1] on “cluster-DAGs”. Essentially C-DMGs are a coarse version of an acyclic directed mixed graph model where each node accounts for one or more of “micro-variables”. Such models allow for causal inference to occur even when we don’t have the complete causal model. \n\n\nCluster-DAGs put specific structure to how this coarse model can look like, disallowing partitions of the micro-variables that would lead to cycles at the coarse (or cluster) level. C-DMGs do not have this restriction, allowing cycles at the cluster level, and thus require their own analysis. The authors show that standard tools of causal inference at the micro level, that is, d-separation, do-calculus, can be lifted into C-DMGs and they are sound and complete for use there. The authors also show that “hedges”, particular structures that can appear in graphical models, do not extend to C-DMGs, but a modification of them can. \n\n[^1]: |Anand, T.V., Ribeiro, A.H., Tian, J. and Bareinboim, E., 2023, June. Causal effect identification in cluster dags. In _Proceedings of the AAAI Conference on Artificial Intelligence_ (Vol. 37, No. 10, pp. 12172-12179).|\n\t|Vancouver||", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Necessary for recommendation \n\nThere’s a fair amount of typos in the text. Mentioning some of those here for the convenience of the authors. Please proofread carefully. \n\n### Typos \n\n- In Definition 8, the authors write “$\\forall V_1^*, V_n^* \\in \\mathbb{V}^*_C, \\exists V_1^*,\\ldots, V_n^*$”. Are the quantifiers correct in this? Perhaps the authors meant that “$\\exists$” should apply to the rest of the nodes apart from $V_1, V_n$?\n\n- “it **it** is also readily extendable to …”, please delete the second “it”.\n- I think the right article is “**a** hedge”, but both “a” and “an” are being used in the text.\n- Page 2, end of page: Please add “-” between the words of “micro” and variables (to make it consistent with how it is later written in the text).\n- Page 3, “is at most in two function**s** in \\mathbb{F}” (plural form for “function”)\n- “an C-DMG” –> “a C-DMG”.\n- Page 10, Conclusion, “might exist**s**” (please delete the “s” from \"exists\").\n\n\n## Not necessary for recommendation\n\n- Please consider whether a resubmission with a longer format would be appropriate. \n- Although C-DMGs are motivated as more general than cluster-DAGs when it comes to structural constraints, it would perhaps help the readers to have some example of where an acyclic Cluster-DAG is not sufficient and a more general C-DMG would be needed. This would give a bit of practical grounding. \n\nAnd some stylistic suggestions: \n\n- Page 6, top of page, “$\\exists 1<i<n$” would be more clear as “$\\exists i: 1<i<n$”.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "5gq4pGBb6h", "reviewer": "Reviewer_QfrF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "** Strengths **\n\nThis is a solid paper where I can clearly see the motivation, and the mathematical developments appear to be sound. I am familiar with causality in general but I was not familiar with this particularly line of work, but I am convinced that this paper proves natural and important extensions of famous results in identifiability to cluster graphs. Identifiability is one of the most important problems in the field of causal inference (arguably it is the domain-defining problem). Therefore, I recommend to accept this paper. \n\n** Weaknesses **\n\nIn the authors' opinion, how strong / restrictive is the assumption that each exogenous variable causally affects at most two endogenous variables? It seems quite strong to me. \n\nAlso, while this is a solid paper, the novelty is not huge, as the results appear to be straightforward extensions of well-known classical results in DAGs, especially those in (Shpitser and Pearl, 2008). However, the TMLR evaluation criteria explicitly recommends to accept papers of this kind, so I will recommend to accept.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the soundness and completeness of d-separation and do-calculus in cluster directed mixed graphs, with a focus on macro-level effects.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Minor comments**\n\nPage 3, 1st paragraph, 5th line: \"two function\" -> \"two functions\"\n\nPage 4, 1st paragraph, 2nd line: \"implies in all\" -> \"implies that in all\"\n\nPage 4, 2nd paragraph, 4th line: \"a ADMG\" -> \"an ADMG\". Also in Definition 6 and Definition 7. \n\nPage 5, Definition 5: Are $\\mathbb{X}$ and $\\mathbb{Y}$ supposed to represent clusters in $\\mathcal{G}^{\\mathbb{C}}$? I think this should be clearly stated either way. Also, what do you mean by a positive distribution? Do you mean that the densities are positive everywhere? Then you should have mentioned from the beginning that you are assuming existence of densities. \n\nPage 6, Theorem 1, 2nd line: \"separated\" -> \"d-separated\"\n\nPage 6, Example 1, 2nd line: \"in in\" -> \"in\" \n\nPage 6, Footnote 2, 2nd line: \"cluster were\" -> \"clusters were\". On the 4th line of this footnote, the first \"nor\" should be \"neither\". \n\nPage 6, Section 3.2, 2nd line: \"i.e., that contains\" -> \"i.e., those that contain\"\n\nPage 7, Theorem 3, last line: the subscript should contain a bar?\n\nPage 7 in general: \"the do-calculus\" -> \"do-calculus\"\n\nPage 8, paragraph below Example 5: Again, you never said what the positivity assumption is, nor did you give a reference for this. \n\nPage 8, section 3.3, 1st line: \"an hedge\" -> \"a hedge\". Also in Definition 10, just below Definition 10, Definition 12\n\nPage 8, Definition 10: The authors are using $\\mathbb{F}$ and $\\mathcal{F}$ interchangeably here. Please be consistent. \n\nPage 8, Definition 10: In (Shpitser and Pearl), the definition of Hedges are such that $\\mathbb{R}^*\\subset An(\\mathbb{Y}^*,\\mathcal{G}^*_{\\bar{\\mathbb{X}^*}})$, but here the bar is under the $\\mathbb{X}^*$. Is this a typo? \n\nPage 9, Definition 12 & Theorem 5, 1st line: \"an C-DMG\" -> \"a C-DMG\"", "broader_impact_concerns": "I do not think this paper needs to address broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)