Characterization and Learning of Causal Graphs with Latent Confounders and Post-treatment Selection from Interventional Data
Authors: Gongxu Luo
Loka Li
Guangyi Chen
Haoyue Dai
Kun Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Post-treatment selection
Selection bias
interventional causal discovery
Primary area: causal reasoning

Abstract: Interventional causal discovery seeks to identify causal relations by leveraging distributional changes introduced by interventions, even in the presence of latent confounders. Beyond the spurious dependencies induced by latent confounders, we highlight a common yet often overlooked challenge in the problem due to post-treatment selection, in which samples are selectively included in datasets after interventions. This fundamental challenge widely exists in biological studies; for example, in gene expression analysis, both observational and interventional samples are retained only if they meet quality control criteria (e.g., highly active cells). Neglecting post-treatment selection may introduce spurious dependencies and distributional changes under interventions, which can mimic causal responses, thereby distorting causal discovery results and challenging existing causal formulations. To address this, we introduce a novel causal formulation that explicitly models post-treatment selection and reveals how its differential reactions to interventions can distinguish causal relations from selection patterns, allowing us to go beyond traditional equivalence classes toward the underlying true causal structure. We then characterize its Markov properties and propose a $\mathcal{F}$ine-grained $\mathcal{I}$nterventional equivalence class, named $\mathcal{FI}$-Markov equivalence, represented by a new graphical diagram, $\mathcal{F}$-PAG. Finally, we develop a provably sound and complete algorithm, $\mathcal{F}$-FCI, to identify causal relations, latent confounders, and post-treatment selection up to $\mathcal{FI}$-Markov equivalence, using both observational and interventional data. Experimental results on synthetic and real-world datasets demonstrate that our method recovers causal relations despite the presence of both selection and latent confounders.

Published as a conference paper at ICLR 2026
CHARACTERIZATION AND LEARNING OF CAUSAL GRAPHS WITH LATENT CONFOUNDERS AND POST- TREATMENT SELECTION FROM INTERVENTIONAL DATA
Gongxu Luo', Loka Li', Guangyi Chent-2, Haoyue Dai', Kun Zhang 1 Mohamed bin Zayed University of Artificial Intelligence, Carnegie Mellon University (gongxu.luo, kun.zhan kun.zhang) @ @mbzuai.ac.ac
ABSTRACT
Interventional causal discovery seeks to identify causal relations by leveraging distributional changes introduced by interventions, even in the presence of latent confounders. Beyond the spurious dependencies induced by latent confounders, we highlight common yet often overlooked challenge in the problem due to post- treatment selection, in which samples are selectively included in datasets after inter- ventions. This fundamental challenge widely exists in biological studies; for exam- ple, in gene expression analysis, both observational and interventional samples are retained only if they meet quality control criteria (e.g.. highly active cells). Neglect- ing post-trees tetttreatment selection may introduce spurious dependencies and distributional changes under interventions, which can mimic causal responses, thereby distorting causal discovery results and challenging existing causal formulations. To address this, we introduce a novel causal formulation that explicitly models post- testttreatment distinguish selection and reveals how its differential reactions to interventions causal relations from selection patterns, allowing us to go beyond traditional equiv- alence classes toward the underlying true causal structure. We then characterize its Markov properties and propose a Fine-grained Znterventional equivalence class, named FZ-Markov equivalence, represented by new graphical diagram, J-PAG. Finally, we develop a provably sound and complete algorithm, F-FCI, to identify causal relations, latent confounders, and post- treattrettmnt selection up to FI-Markov equivalence, using both observational and interventional data. Experimental results on synthetic and real-world datasets demonstrate that our method recovers causal relations despite the presence of both selection and latent confounders.
INTRODUCTION
Causal discovery from interventional (and observational) data, often referred to interventional causal discovery, aims to identify causal relations by exploiting distributional changes induced by interventions (Spirtes al. 2000 Pearl 2000). Despite progress in interventional causal discovery in handling latent confounders, pre-treatment selection (Dai et al. 2025). and biological constraints (Luo al. 2025), we highlight a common yet often overlooked problem, post-treamentt selection, which refers to the selective inclusion of samples after interventions (Heckman) 1978). For example, in gene perturbation studies, only perturbed cells (intervention) that pass the quality control (selection) are profiled (Norman et al. 2019). In Clinical Trial Per- Protocol Analysis, only participants completing over 80% of scheduled visits (e.g.. up to week 12) are included in the final analysis (Detry & Lewis 2014). Failure to account for post-treatment selection introduces spurious dependencies and intervention-drive distributional changes that mimic causal responses, thereby leading to incorrect statistical inference and challenging existing interventional causal formulations.
Specifically, existing interventional formulations neither distinguish causal relations from post- treatment selection nor detect where the selection is present. Mainstream frameworks identify causal relations and characterize interventional Markov equivalence classes (ECs) on DAGs by exploiting irossiintervnntion pattern: after interventions the cause, marginal distribution p(effect) changes, and conditional distribution p(effect)cand cause) remains (Tian & Pearl 2001: Hauser & Bühlmann, 2012) 2015). When latent confounders present, variations in pl p(fffect[cause) are further utilized to

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L
X
X1
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X1
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(b)
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Figure 1: Motivation examples. (a) & (b) exhibit same dependence with tails from X1 and arrowheads
into X2, regardless of direct causation; (c) & (d) exhibit same dependence with tails on both X and X2, regardless of direct selection. Existing methods cannot distinguish these cases, whereas ours can.
characterize the interventional ECs involving latent confounders (Faton & Murphy 2007; Ghassami al. 2017; Kocaoglu et al., 2019 Zhou al. 2025). However, post-treatment selection is non- identifiable within these frameworks because it yields the same pattern, variant p(effect) and invariant p(effect cause) before and after the intervention, as causal relations. For example, under post- treatment selection, Figure 1 a exhibits the same pattern (variant p( p(X2) and invariant p(X2 X1) after intervening on X1) as (b). Subsequently, current frameworks place (a) and (b) in the same EC (same representation), regardless of whether a direct causal link exists between X and X2, thereby failing to identify causal relations from post- post-treatment selection. An analogous non-identifiability arises for direct selection, as illustrated in Figure cc) and (d). This representational gap challenges existing frameworks and motivates a new formulation that explicitly models post-treatmen selection.
In this paper, we examine the causal structure among intervened variables in the general setting involving latent confounders and selection bias, explicitly handling post-treatment selection without imposing graphical or parametric assumptions. First, we demonstrate how causal relations, latent confounders, and post-treatment selection differ in structural symmetries (e.g.. selection structure with both tails on endpoints, while causation is not) and distributions (variability and invariance) after intervention, which are characterized by conditional independence (CI) patterns. Second, building on these observations, we propose Fine-grained Znterventional equivalence class (c.g.. distinguishing Figure a from (b), and (c) from (d)), named FII-Mrrovv equivalence, and provide detailed characterizations. In graphical representation, partial ancestral graph (PAG) edges encode ECs with a broad range of possible structures and thus prevent the unique graphical representation. To obtain a more concise and expressive graphical representation for FZ-Markov equivalence, we introduce F-PAG, extension of the PAG diagram that incorporates novel edge types. Third, we present a sound and complete algorithm F-FCI for recovering the FI-Markov equivalence class.
Contributions. In this paper, focus fundamental yet largely overlooked problem, the post- treatment selection that lies beyond the scope of existing interventional causal discovery frameworks. First, we introduce causal formulation that models post-treatment selection in the presence of latent confounders, and we define the novel FI-M equivalence and F-PAG accordingly. Second, build- ing on this formulation, patterns with tailored orientation rules. Theoretically, we prove its soundness and completeness. Third, we validate our approach on both synthetic and real-world datasets, demonstrating its effectiveness. Collectively, these contributions provide a principled framework for distinguishing post-treatmen selection from true causal relations. thereby broadening the scope of interventional causal discovery.
develop new algorithm F-FCI, which integrates intervention-based CI
PRELIMINARIES AND MOTIVATION
In this section, we first introduce the graphical causal model that involves both latent confounders and selection bias ($ 2.1 We then review the standard paradigm for interventional causal discovery (details in Appendix and demonstrate why it fails to handle post-treatment selection 2.2).
2.1 GRAPHICAL CAUSAL MODEL. WITH LATENT CONFOUNDERS AND SELECTION BIAS
We begin with the general problem setup: DAG with latent confounders and selection bias. Let the DAG G on vertices with index [N] = {1.... ,N encode the structure of the underlying causal model where vertices correspond to observed random variables X (X,) For any subset A C [N]. let XA := (Xt)seA and by convention Xe = O. Apart from the observed ones, L = ( accounts for the confounders that affect X but remain unobserved (latent confounders), and the exogenous selection variable S { generally represent both pre-treatment selection (preferential inclusion of samples before intervention) and post-treatment selection (arising after intervention) (Heckman, 1978 Elwert & Winshipl 2014). In this paper, specialize in
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Figure 2: Examples of graphical representations. (a) Augmented DAG with explicit intervention indicators (16). (b) Extension of the augmented DAG to include latent confounders.
(c) Modeling
post-treatment selection using the augmented DAG, with toy examples of selection on observational data (d) and selection after intervention (c), where the positively invariant pX) is marked in red.
post-treatmen selection and assume selection works on at least two observed variables. Throughout, analyses are conducted conditional on (i.e., within the selected sample).
To represent the general graph with latent confounders and selection bias, the ancestral graph is defined by mixed graph without direct and indirect cycles (detailed in Definition [8) To investigate limits of the CI test on observational data, the Markov properties of ancestral graphs are examined. Analogous to the d-separation (Defini- tion 7 criterion used for DAGs, the m-separation (Definition (9) blocks the paths in ancestral graphs. Under the pairwise Markov property, in which the absence of edges reflects conditional independence, the ancestral graph that satisfies this property, a.k.a. maximality assumption (Definition 10), is the Maximal Ancestral Graph (MAG). Given that the global, local, and pairwise Markov properties enable the recovery of causal structures via CI tests, under the representation of the MAG diagram, different graphs that entail the same m-separation form the Markov equivalence class (Definition
the learnability of graphical models and the information-theorre
2.2 LIMITATIONS OF EXISTING INTERVENTIONAL CAUSAL DISCOVERY PARADIGM UNDER SELECTION
Interventional causal discovery aims to learn the structure of G from data collected under multiple (hard and soft) intervention settings, each with an intervention larget I C [N]. meaning variables X₁ are intervened on. Iet (7(0), T(1) ..... {(K) denote the collection of intervention targets across K interventions, and (p (1)) ..... .....p(KK) p(K}} indicate the corresponding interventional distributions over X. We assume throughout f(0) = Ø. i.e., the pure observational data is available.
Hard interventions remove all incoming edges to the vertices in the intervention target ( in g while all other edges remain. Soft interventions do not break any arrows incident the intervention target; instead, they only change the conditional distribution (Eberhardt & Scheines) 2007). Rather than analyzing each interventional setting separately, more effective approach is to exploit changes and invariances across settings: intervening on cause alters the marginal distribution p(effect), while the conditional p(effect]caus remains invariant. Conversely, intervening on an effect leaves p(cause) unchanged, but p(cause/effectt changes (Hoover 1990, TTian & Pearl (000). ). Such invariance is
exploited the invariance causal inference framework (Meinshausen et al.] [016]] Ghassami et al.
2017) and has been extended to settings with latent confounders (Jaber al. 2020
To formally exploit such invariance analysis and model "the action of changing targets", Newey & Powell (2003); Korb et al. (2004) introduce the augmented DAG, denoted by aug(G, 2), which, as shown in Figure 2(a), a), extends the original G by adding exogenous binary vertices 1h = {x) }K lk=1
as intervention indicators, each pointing to its target 7(,). Whether the k-th intervention alters a marginal density, i.e., (XA) * p (kk) (XA) or equivalently p(XA A 1/r00 = 0) * p(XA
2 1). is then nonparametrically represented by the CI relation p0 V6 XA. and graphically represented by the d-separation 20 La XA in II. where r7t denotes d-separation and A. d-connection. Moreover, latent confounders can also be incorporated into augmented DAGs, as shown in F The invariance in marginal distributions (p( (X2) = (X)) represented by 1i IL X2) and variability in conditional distributions (x2) + (X2 represented by 1/1 4 X2) after intervention help distinguish latent confounders from causal relations. Causal discovery algorithms like PC (Spirtes et al. 2000) and FCI (Spirtes al. 2000 Zhang 2008b) have been applied in this context (Zhang 2008a] Huang et al. [2000] Maghacane et al.) 2016;; Kocaoglu
et al. 2019), and augmented MAG have been developed the corresponding graphical representation.
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Building the established framework of interventional causal discovery, when selection appears after intervention, the post-treatment selection induces changes in the marginal distribution p(effect) while keeping invariant in the conditional distribution p(effect|cause) as illustrated in Figure 2 (X2 5 1) + (X2 S 1) and p pl) (X2 S 1) pl (X2 X 1) with examples (d) and (e). Although post-treatment selection can be represented within the augmented framework, its invariant and variant characteristics are indistinguishable from those of causal relations, rendering it non-identifiable as discussed in Figure |
This motivates a new formulation that models post-treatment selection and identifies true causal relations.
NEW FORMULATION: INTERVENTION MEETS POST-TREATMENT SELECTION
Based on the exploration of the interventional causal discovery paradigm, in this section, we extend the paradigm to design new formulation for post-treatment selection in the presence of latent con- founders ($ (.1.1,, characterize the Markov properties (S (..).). and provide the graphical criteria for de- termining whether two augmented DAGs are Markov equivalent given the same interventions ($ 3.3)
3.1 MODELING POST-TREATMENT SELECTION
The first step is model the post-treatment selection explicitly. Since the variant and invariant characteristics are consistent with the Markov assumption, post-treatment selection can be naturally modeled within the augmented DAG (see $ (22)) by adding a selection variable S. Accordingly, we adopt the augmented DAG to coherently unify observational and interventional data by introducing an intervention indicator vb. Under this model, the joint distribution over the observed variables X in the k-th intervention, conditioning on post-treatment selection denoted by ps (X), factorizes as
Ps (k) (X)
II 610))10))
(X, pao (1), S.SS,.SS 1)
II (5H5))Z
p (Xyl&pao S =1,
(I)
where X (fg (4) C X U T. indicates the parents of X,, and S = indicates the presence of
post-treatment selection. To represent the details of the structural causal model, the graph involving S
and L is represented by the augmented DAG, which is redefined as follows (details in Definition 12
Definition 1 (Augmented DAG). For DAG & over U S and intervention target I C
the augmented graph Augr(g) is a DAG with vertices t U X U T. U UUSUU Uc, where: 1o = {w2200) } k=
K
set of exogenous binary indicators for the representation of marginal changes between two environments (observation-intee interventionnnnnonn or sneeventiioniiterventtoo
pointing to the corresponding
intervened variable X x is exogenous noise for variables, whether observed or hidden.
An illustration of the augmented DAG is shown in Figure 3 depicting the data gen- eration process. In Augz(9), only X and & are measurable, forming the basis of the representation of different environments, such as observational data p( v/ = 0,5 = 1) and interventional data pf X 1). is conditioned on, meaning all individuals, whether observed intervened, are selected at the outset. Moreover, following the diagram of MAG for marginalized representation Zhhng 2008b), each Augmented DAG can also be formally represented by the corresponding Augmented MAG.
E
E
E3
Figure 3: Illustration of structural causal model (SCM) represented by an augmented DAG.
3.2 CHARACTERIZING MARKOV PROPERTIES
Our ultimate goal is to learn the causal structure from both observational and interventional data. On the rationale modeling post-treatment selection and marginal changes between observational and interventional data using the Augmented DAG Augr(G) in $3.1 the standard Markov properties, ie., the global Markov property (formulated via d-separation) and the local Markov property (each node is conditionally independent of its non-descendants given its parents), hold exactly they do in conventional DAGs. These properties provide the theoretical foundation for using CI tests in structure learning and offer an information upper bound for the CI implementations.
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CI conditions
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Figure 4: Characterizing Markov properties with CI patterns (i) of augmented DAGs (a)-(h). Red dashed lines indicate that CI patterns persist regardless of whether X1 - (b) or SSe X2 (f), whereas 4/ J X2 provides evidence of the presence of direct causal link selection.
By leveraging the Markov properties, CI tests can recover causal structure from data. particular, three classes of statistical signals are informative: 1) Interventional distribution changes: Variability in marginal observational and interventional distributions manifests as conditional dependencies between the intervention indicator v and affected variables X. For example, in Figure 4 a), 1i J X2
indicates that perturbing X1 propagates a distributional change to X₂. 2) Invariant relations: Equality of conditional distributions across observational and interventional data signals invariance. Specifically, 1/1 II. X2 X1 indicates the invariance of (X) X2)
=p''
X1). 7 (1)-
3) Structural symmetries: Certain structures exhibit characteristic symmetry in their CI patterns. For instance, symmetric selection shown in Figure 4[) yields d4 IL X2 X1,, 12 Il X1 X2. 1/1 J. X2, the J X1. Below, we formally define these relations implied by the model.
Theorem 1 (CI and invariance implementation). For positive interventional distributions ( (X) and observational distribution p!)) (X) generated from the DAG G in the presence of latent confounders L and selection S with intervention targets (I(k) Tek [O]u[K), let {Augras (G) (M]u[K) be the corresponding augmented DAGs. For any disjoint A, B, C E (N], the following statement holds:
For any & E {0} U [K], if X lll XBl(Xc, S} holds in Aug (9), then XA XBl(Xc, in p(k)
For any l E [K Thy Ha XA holds in Augros (9), then p (X
X B) = (Xx For any & E [ if dha Ad XA Ø holds in Augnun then p (XA) * p (XA).
Remark 1. ya) generally marginalizes changes between different environments. The difference between the two interventions on the same T(a) also follows the statements in Theorem where the hard-hard intervention changes the causal diagram, providing additional information that is only used to identify the structures of unblocked paths.
Theorem shows that invariance and variability in marginal and conditional distributions are implied by graphical conditions, namely d-separation among vp U X|S = 1 in augmented DAGs. Previous studies leveraged this statistical information distinguish causal effects from associations induced by latent confounders. However, it is known that selection bias also introduces spurious dependencies:
Lemma (Additional dependencies induced by selections). For any DAG on X UUUS S, targets E [N], and disjoint A, C E [N], XA 14, X & S holds in the augmented DAG Augr(g), then XA la X holds in the original DAG. The reverse is not necessarily true.
With the characterization of global and local Markov properties ready, differences in graphical properties, such as asymmetry, captured in Cl patterns as shown in Figure help distinguish different structures. Specifically, Figure 4i) shows that (a) and (c) exhibit different CI patterns, with (e) being symmetric. We further observe that although (a) and (b) share the same CI patterns between X, and X2 regardless of whether a direct causal link exists in (b), they differ in their underlying causal structures. This is due to the Y-structure at X₂ forming an unblocked path, which exhibits the same characteristics as causation. Beyond focusing only on cause and effect targets, hard interventions on X3 open the path by blocking the selection effect on the latent confounder L. The variation in two different hard interventions on X3 can be modeled by thy for representation. Then, d/s 1 X₂ allows us distinguish case (b) and assess whether there direct causal link between X1 and X2. Similarly, direct selection can be identified in the same way for (c) and (f). This, in turn, goes beyond traditional ECs and can identify concrete causal structures at the DAG level.
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3.3 FI-MARKOV EQUIVALENCE
In $ 3.2, we characterized the Markov properties implied by the true model of the data. Now, to identify the true model from data, in this section, we shall understand to what extent the true model is identifiale,, as different models may share identical CI implications, namely, being Markov equivalent. To characterize the equivalence class under the general setting with MAG representation. the Markov equivalence with corresponding m-separation on observational data is discussed. However, with the help of interventional data, we propose novel Fine-grained Znterventional Markov Equivalence, named FI-Markov equivalence (Definition 2), and characterize the EC under the augmented DAG framework based on the Markov properties. Different from the graphical rep- resentation of the structural causal model, the representation of learned ECs is only over observations X. We subsequently extend the Partial Ancestral Graph (PAG) framework (Definition 4 with novel edges for the representation of FI-Markov ECs, which are more informative compared with PAG.
Because the Markov property allows us to distinguish structures that existing formulations cannot, for instance, whether direct causal link exists in Figure 4b.. we define
FI-Markov equivalence under the augmented DAG framework for a more precise structural representation. Two different aug- mented DAGs with the same intervention targets entail the same CI patterns in the data. Formally,
Definition 2 (FI-Markov equivalence). Two Augmented DAGs, Augr(G1) and Augz Augr(92), are FI- Markov equivalent with the same intervention targets I, if and only if they have the same d-separation (the same skeleton and v-structure in the description of the corresponding MAG representation) among X [iJjz,, and the same CI patterns between yp and any intervened variable X; E XUz.
3.3.1 GRAPHICAL CRITERIA FOR FZ-MARKOV EQUIVALENCE
When learning the EC based Markov properties, causal discovery methods only recover whether the unblocked paths have tail or an arrowhead at each endpoint over [ not the full structure with and S (Kocaoglu al. 2017). These unblocked paths are named inducing paths, defined follows: Definition 3 (Inducing path). In augmented DAGs, X,, X; are any two vertices, and L, S are disjoint sets of vertices not containing X,. A path p between X₁, X, is called an inducing path relative to S) if every non-endpoint vertex on p is either in L or collider, and every collider on p is an ancestor of either X;, Xj. or a member of S.
Example. In Figure 4 (b), the path between X, and X2 is the inducing path with unblocked X3-
To characterize the ECs via graphical features over X X[N]- we still need to borrow the MAG representa- tion for marginalization. Following the procedure of MAG construction introduced in (Zhang (008b),
every augmented DAG corresponds to an augmented MAG M ((Aurr( (9) in graphical representation. Then, we show the rules to construct M(Augr(G)) by presenting the following lemmas.
Lemma 2 (When are two variables dependent in observational data?). For any i, E [N], X, and X; are adjacent in M ((Augr(9 (G)). if and only if X; and X, have at least one inducing puth in Augr(g).
The adjacencies in Lemma 2 capture all dependencies induced by indueing paths, forming the foundation for constructing the skeleton. Then, interventional data help further identify the structures:
Lemma 3 (When does intervention always alter marginal distribution?). For any i.j E [N], i I, X, and X, are adjacent in M(Aug (G)) with a tail X,, if and only if every inducing path between X; and X, begins with tail in Augz(G), i..,, X, is the ancestor of X or is ancestrally selected.
With Lemma 3 the variant marginal distribution characterizes the ECs of inducing paths starting with tail. The variability in the conditional distribution is utilized to characterize the ECs follows:
Lemma (When does intervention always alter conditional distribution?). For any i, E [N], i E I, X; and X, are adjacent in M(Augi (9)) with an arrowhead at X,. if and only if every inducing path between X; and X, begins with an arrowhead in Augr (G), i.e. X₃ is descendant of X or L.
With the foundational graphical criteria that are consistent with the MAG construction ready, the graphical criteria for the FI-Marrkvv equivalence learned from data are as follows:
Theorem 2 (Graphical criteria for FI-Markov equivalence). Two augmented DAGs Augz (91) = X U S. F. and Augz(G2 (1/ U UUXUUUUS US" E²) are FI-Markov equivalent for a of

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X
MAG:
PAG:
X
X
JF-PAG:
(8)
(b)
(d)
(c)
Figure 5: Illustrations of F-PAG graphical representation for the FI-Markov equivalence class.
targets if and only if the corresponding MAGs Mr -M(Aur (Gi) and M₂ = M(Augs (92) have the same skeleton and v-structure, and have the same marks and edges among intervened nodes.
3.3.2 F-PAG: GRAPHICAL REPRESENTATION FOR FI-MARKOV EQUIVALENCE
Based on the Markov properties of the augmented DAG in the presence of latent confounders and post-treatment selection, the CI patterns precisely characterize when two such distributions Markov equivalent in Section 3.3.1 This alignment with our learning objective allows us to recover the EC directly from data. For the graphical representation of the ECs, we follow the conventional approach of using the Partial Ancestral Graph (PAG), defined as follows:
Definition 4 (Partial ancestral graph). Let [M] be the Markov equivalence class of MAG M. A partial ancestral graph (PAG) for [M] is graph P' with possibly six kinds of edges; -, - +> :0 o, 0-t, such that (1) P has the same adjacencies
M does; (2) every non-cirele mark in P is an invariant mark in [M]. If it is furthermore true that (3) every circle in P corresponds to variant (indeterminate) mark in [M], is called the maximally informative PAG.
Although PAG is general framework for the graphical representation of DAGs under selection bias and latent confounders, it designed for ECs, which maximally informative in CI relations from observational data (y-structure induced independence). This limitation usually results in dependencies induced by broad inducing paths represented by properties in Section 3.2 PAG is too broad to represent FZ-Markoy equivalence. For example, in Figure 5)b) and (c), the presence absence of a causal link between X, and X₂ results in the same PAG. However, Figure (ic) can be distinguished with interventional data as discussed in Figure (b). To describe the reduced FI-Markov equivalence, we proposed the J-PAG defined as follows:
With the discussed characterizations of Markov
Definition 5 (F-Partial Ancestral Graph). A F-Partial Ancestral Graph, denoted Gp. is graph- ical representation derived from DAG with latent and selection variables. captures conditional independence relationships and consists of four types of marks (tail arrowhead square a, and circle o), and eight types of edges
47, D-;
O 0-0, o-)).
The mark n denotes a node with at least one tail and at least one arrowhead, + (Figure ()) and (Figure (()) represent inducing paths that have the same CI patterns with -, but without direct causal link and selection separately in between. These two types of inducing paths can be identified only through the graphical conditions involving Type inducing nodes defined as follows:
Definition (Inducing Node). In an F-PAG, the nodes are referred to as inducing nodes if and only the non-endpoint nodes on the inducing path are characterized either by an incoming arrowhead into a square (++ Type I) or by adjacent two squares (co Type II).
For example, in Figure 5 b)),, non-endpoint node X3 is Type I inducing node. With the advanced graphical representation in place, FI-Markov equivalence can be expressed more clearly, allowing us distinguish whether the observed dependence arises from genuine causal relations (a) and (b), direct selection (d), or from equivalent inducing paths (c) and (e) as shown in Figure
ALGORITHM: F-FCI
In this section, we propose a novel Algorithm I named Fine-graind FCI (F-FCI). Using Markov properties of the augmented DAG in Section F-FCI learns causal relations, latent confounders,
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Algorithm 1: F-FCII Algorithm for learning F-PAG
Input: Observational and interventional data (p( RR-O over X[N] with interventional targets I. Output: A fine-grained partially ancestral graph (F-PAG 9p) over vertices X.
Step 1: Get skeleton from pure observational data. g <- FCT (pl
Step 2: Get F-PAG orientation over Xuz from interventional data. for siijjs K do Step 2.1: Capture CI patterns between & and Xuz.
foreach condition set C S {X,t n X E AllaathsSy X20,, X26))
do Cls CI (1re, X (o) C) CI (Mra; Xso) X,0, C)2 CI (1y6n X (C),
CI (176n, X66) X ro), }) If no more paths can be blocked then break;
Step 2.2: Orient the skeleton between X and X KG.
if Cly == (JC, 11, 1 then Orient X + X
if Cly [1L 1 +
then Orient Xy() ++ X
if CIs = [LL, K,, 11,
then Orient X x X(()
if Cls (LK, H 11)
then Orient X10) Xy60
if Cls - -J,,,, 11,11,,,,, (1) then Orient X X
if Cls =- KK,
then Orient X
Step 2.3: Refine the orientation. Identify causal relations in between for X X X
foreach inducing path between the node pairs with interventional data (X 10), X1Gs) marked with
from Step 2.2 do
Detect if the path has non-endpoints vertex and Type induuing nodes. If Type I inducing node X,, with X + 3x5-X,9 then CI (tin- X If th X then update X -X10) to X ((0) -Xys)-
If Type inducing node X, with X -n,, ++ X X16) then CI (Wh X
If th (6)) then update X X160- - X to Xie 4 X15).
Step 2.4: Update the F-PAG. gf t ((k-
Step 3: Get F-PAG orientation over XIN//UZ- Apply the orientation rules of FCI among
[[N//Uz and apply the invariance rules in Theorem (the see-see, do-see, and do-do rules
Kocaoglu al. 2019h) between Xuz and XIM/UU in g.Kk) Update Op gk
return 4p
and post-treatment selection up to the FI-Markov equivalence class, from both observational and interventional data. We assume faithfulness, i.e., no Cls beyond those implied by the graph.
The first step is to recover the undirected skeleton from observational data y(") since it yields the spars- est graph encoding all inducing paths. We then adopt the standard constraint-based skeleton discovery procedure (e.g., as in FCI) under our general graphical assumptions to obtain this skeleton. Step 2 con- sists of four sub-steps. Step 2.1 captures CI patterns reflecting marginalized changes between observa- ) In Step 2.2, we then leverage the captured CI patterns to orient edges incident to the intervened variables Xuz, using the orientation rules summarized Fig- u In particular, the rule o-> uses instead of a because the existence of an inducing path beginning with tail in between is uncertain when based solely on the marginalized changes between observation and intervention. For example, the structure t1 + X1 + L + X2. When X1 is under selection, one observes v/n V. X₂ regardless of whether X, conditioned on. Then, this structure cannot be distinguished from the structure of the latent with causal relation in Figure d), represented by o->.
tional p(o) and interventional data {p(*)) 1==1-
To go beyond CI patterns and identify real structure, in Step 2.3, F-FCI firstly addresses the uncer- tainty of o-+ by blocking selection on latent confounders via marginalized changes from two hard interventions. Then, Type inducing nodes along inducing paths between intervened variables are detected to disambiguate cases that CI patterns of endpoints alone cannot distinguish. This procedure is valid for inducing paths containing more than one non- non-endpoint vertex and including at least one Type I inducing node. For example, graph (b) in Figure 4 share the identical CI patterns regardless of whether a direct or indirect causal link exists between X, and X2. By utilizing changes of hard intervention the Type I inducing node Xs. we can test for the II. X₂ S to determine whether true causal relation exists. Likewise, direct selection (Figure becomes identifiable under the same rationale. Specialized edge marks A and are established represent the inducing paths in Figure
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(OURS)
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Figure 6: Comparison results in identifying causal relations under DAG Precision and DAG SHD metrics. All values are averaged over 10 graphs. Error bars represent the 95% confidence interval.
By explicitly orienting edges between intervened node pairs, the core contribution of our algorithm, we subsequently apply the standard FCI orientation rules and rules of invariance to all remaining edges: those between intervened and unintervened nodes, well as those among unintervened nodes in Step 3. The extra orientations recovered from interventional data furnish richer structural information than v-structures identified from only observational data.
! with oracle Cl tests on multi-distributtinn data (pi*)) kzzo given by (G, 9p is consistent with augmented DAG Augz(9) (G) in arrowhead, tails, square, and structures of paths 4, among intervened variables.
Theorem 3 (Soundness of F-FCI), Let Gp be the oulput J-PAG of Algorithm
Theorem (Completeness of F-FCI). Let 9p be the output of Algorithum with oracle CI tests on milti-distribution data (p( KKK. given by (I,I). Each type of substructures represented by tail, arrowhead, square, and between pair of intervened nodes in the corresponding augmented DAG of 9p can be identified by different types of CI patterns.
EXPERIMENTS
In this section, we present empirical studies on simulations and real- world data to demonstrate that F-FCI identifies causal relations in the presence of latent confounders and post-treatment selection.
5.1 SIMULATIONS
We conduct simulations to validate the effectiveness of our proposed F-FCI. We compare F-FCI against strong baselines in interventional causal discovery, including GIES (Hauser & Bühlmann 2012), IGSP (Wang et al. 2017), UT-IGSP (Squires et al. 2020). JCI-GSP (Mooij et al. 2020). FCI with interventional data (FCI-interven) (Kocaoglu al.
2019). and CDIS (Dai et al. 2025).
Following the data-generating procedure in Definition I we begin by randomly sampling Erdös-Rényi graphs with an average degree of 2 as the ground truth DAG for [X,) We then randomly generate 2 or 3 selection variables, each with two randomly chosen parents from {X,) and or latent confounders, each with two randomly chosen children. Finally, we simulate general SEMs X; = f(X ((xpag(t) + fg. is sampled from Unif((o f[0, 2) U [2, 4)), and select samples with Ee((X, that fall (ag))) within a predefined interval, where f and fa are randomly drawn from linear, square, sin and tanh.
To evaluate the effectiveness of F-FCI identifying causal relations despite the presence of latent confounders and post-treatment selection, we report the main Precision and Structural Hamming Distance (SHD) metrics compared with baseline methods in Figure (the F1-score and recall are given Figure i0 Appendix(D) DD. The experimental results demonstrate that F-FCI outperforms baselines with an average precision of over 5% in most configurations and lower SHD. These observations validate the effectiveness F-FCI in identifying true causal relations, whereas baselines may infer spurious ones induced by latent confounders and post-treatment selection. Moreover, the robustness of F-FCI under different noise levels is evaluated Figure [2] the scalability is evaluated in Figure II and its ability to distinguish post-treatment selection assessed in Table
'A Python implementation of F-FCT is available at httttsst/////t thub. whw..ccom//ongguuuooo////

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5.2 REAL-WORLD APPLICATIONS
We evaluate the gene regulatory networks (GRNs) of genes using large-scale single-cell gene per- turbation data collected from Human Lung Epithelial Cells (HLEC), i.e., Norman datasets (Norman) et al. 2019). We report both the regulatory (causal) links and the spurious dependencies induced by post-treatment selection, as identified by F-FCI in Figure[[3] and detailed analysis can be found in Ap- pendix D33 Experimental results are evaluated using prior knowledge provided by Enrichr, tool that compiles extensive libraries from enrichment experiments (Kuleshov et al. 2016 Xie al. 2021).
6 CONCLUSION AND LIMITATIONS
We introduce a fundamental yet underexplored challenge for causal discovery: post-treatment selection, particularly the often-overlooked quality control constraint that shapes dependencies. We show why existing models fail to handle such bias, propose a new formulation to model post-treatment selection, establish criteria for novel fine-grained interventional Markov equivalence, and define a corresponding graphical representation. Building on this formulation, we develop a sound and complete algorithm, named F--FII that uncovers causal relations and post-treatment selection. Empirical analyses on synthetic and large-scale real-world datasets demonstrate the effectiveness of F-FCI in accurate and robust causal discovery.
The identification of direct causal links and selection structures depends critically on the presence of Type I inducing nodes. One future direction is how to identify the causal structure along inducing paths composed solely of Type II inducing nodes. In addition, as discussed in (Luo et al., 2025), biological constraints filter out cells and introduce extra dependencies; another extension can be how to distinguish biological constraints from post-treatment selection.
ACKNOWLEDGEMENTS
We would also like acknowledge the support from NSF Award No. 2229881, AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Quris AI, Florin Court Capital, MBZUAI -IIUUAI-IISAA. Joint Program, and the Al Deira Causal Education project.
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A DEFINITIONS
Over the past decades, comprehensive causal discovery frameworks have been established. We now introduce the basic concepts and fundamental definitions for the general setting involving latent confounders and selection bias. To represent the general graph with latent factors, selection bias, and loops, the mixed graph is first designed with direct ->, undirected -, and bidirected t+ edges for all structural representations. Given a mixed graph GM and two adjacent vertices X,, X, E X. X; is parent of X, and X, is child of X;, if X; + Xy. X; is called a spouse of X, if X; A direct path from X1 to X, is sequence of vertices where each S i < n - 1, X; is the parent of X,+1. X, is called an ancestor of X, and X, is the descendant of X; if X; = X, or there is directed path from X; to Xj. Let Ansgar (X;) denote the set of ancestors of X; in GM- A directed cycle occurs in Ias when X; X, is in GM and X, Ansg,r (X;). An almost directed cycle occurs when X; + X, is in GM and X3 E Ansg,, (X,). The ancestral graph is defined in Definition without considering cycles.
X, in Gas-
Definition 7 (d-separation). If every path from node in X to a node in Y is d-separated by Z, then X and Y are always conditionally independent given Z.
Definition 8 (Ancestral graph ZZhnng 2008b)). A mixed graph is ancestral if and only if the following conditions hold: (1) there is no directed cycle; (2) there is no almost directed cycle; (3) for any undirected edge X,-X,, X, and X, have no parents or spouses.
Xj,
The other detailed description of the definitions that characterize the graphical causal model is introduced as follows:
Definition 9 (m-separationn) In mixed graph, a path p between disjoint subsets of vertices X, and X3 is active (m-connecting) relative to a set of vertices Xz Xx & X2) if and only if (1) every non-collider on p is not in Nz: (2) every collider on p has descendant in Xz. X A and X n said to be m-separated by Xz if and only there is no active path between them relative to Xz.
Definition 10 (Maximality). An ancestral graph is said to be maximal if any two non-adjacent vertices, there is a set of vertices that m-separates them.
Definition 11 (Markov equivalence). Two MAGs Mi, M2 are Markov Equivalent if for any three disjoint sets of vertices Xa. XB, Xc. XA and X3 are m-separated by X in M2 if and only if they are m-separated by X in M2 as well.
Definition 12 (Augmented DAG). For a DAG 9 over X U LULUS S.S.Sa and intervention target C [N] the augmented graph (ugr(G) is DAG with vertices &p UXULUSU U.C. where:
1/ fo 1 is a set of exogenous binary indicators for the representation of marginal changes between two environments (observation obbervattioniineerventio or intervention-intrveniii two hard interventions are a special case), pointing to the corresponding intervened variable X ;
X = (X)) are the observed variables, pure observational or interventional;
L LL,,L,, LR) and {S1, S2,.... .... Sr represent the sets of latent confounders and post-treatment selection respectively in the hidden world that influence the observations;
e denotes the exogenous noise term for variables, whether observed or hidden.
With the defined variables, Augr(9) consists of the following edges:
Direct causal effect edges: for each X; X, E g with i [N], add X; + X, E GAug for each X, E G with i [N] is directly influenced by L,, r [R]. add L, + X;;
Selection edge: for each X, with E [N] is directly selected by St, 1 E (T), add X; - St:
Edges representing common exogenous influences: {Er X.J.EIM)
Edges representing mechanism changes due to the intervention: (wi Xi)ier.
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L
L
X
X
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X
X
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(d)
Figure 7: Illustrative examples of inducing paths with X1 is an ancestor of X₂ in (ubfigures(a)&((c) or is ancestrally selected, shown in subfigures (b)&
B PROOFS OF MAIN RESULTS
Since the technical development of this paper is fundamentally built upon the characterization of the augmented DAG, we focus on proving only those results that do not directly follow from the basic properties of augmented DAGs. Specifically, the results stated in Theorem I Lemma [ and
Lemma are immediate results of the definition of augmented DAGs (Spirtes et al. 2000 Korb et al.
2004
Zhang 2008b Kocaoglu al. 2019). Therefore, their proofs omitted.
Lemma 3 (When does intervention always alter marginal distribution"). For any E [N] i I, X; and X, are adjacent in ((Aug (9)) with a tail at X,, if and only if every inducing path between X; and X begins with tail in Augz(g), i.e., X; is the ancestor of X is ancestrally selected.
Proof of Lemma 3 Before proceeding, we note a special case where the marginal distribution changes: when E with v + X₁- In this case, is straightforward that (X,) + ((k) (X.). 11h) = {i}, as X; is intervened upon. We omit the proof as it is self-evident.
('=' direction) For any i.j E [N]. the paths between X; and X, are inducing paths that start with tails directed outward from X, toward X, in the augmented DAG Augz(G). Because the inducing path starts with a tail and its definition that every collider the ancestor of either X;, Xj. or S, this implies that X; is either an ancestor of X, or directly selected, as illustrated in Figure Otherwise, the path will be blocked, conflicting with the definition of the inducing path. Furthermore, only the ancestral relationship and selection propagate distribution changes. Therefore, when X; is intervened upon, distribution changes propagate along the inducing paths via the outgoing tails from X,, thereby altering the marginal distribution of Xj- Whin in the augmented DAG (g) th + Xi, as all inducing paths orient outward from X; with tails,
the 14. X,. Then, p (X;) * 2. (*=' direction) By contradiction, suppose X; (Ans (o) (X₃) U Ans (ugz(g)) (S)), the paths between X; and X, are inducing path, compromising by inducing nodes, that can not be blocked, which requires both collider and tails occurs.
(X,), T(R) = {i).
When the arrowhead emerges outward from X; -, the formation of colliders requires another arrowhead at the same intermediate node from e.. Then, according to the definition of the inducing path, whether the tail of the middle node from causation or selection, it violates the condition X, & [AAnAAugz (X,) U Ans (g)z
When the collider formation from X; ++ and 4- X,, the tail between the intermediate node and X, only origin from selection. When the collider formation is from X1 < and +> X,, the edge with tails at the collider toward X, can arise from both causation and selection. Then, the inducing paths starting from X; with the one inducing node are represented by X, ++ a X, and X; ++ o-oX, separately. However, in both structures, ()) (X,) = (()) (X;), 7(k) = (i). which contradicts with the changes in marginal distribution.
Lemma (When does intervention always alter conditional distribution?). For any i, j E [N], X, and X, are adjacent in M(Augi (9)) with arrowhead at X,, if and only if every inducing path between X; and Xj begins with an arrowhead in Augr(g), i.e. X; is descendant of X; or L.
Proof of Lemma 4
1. (+=' direction) For any i.j E [N], E I, the paths between X; and X, are inducing paths in the augmented DAG A9g If X; is the descendant of X, I., there must be an
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L
UP
X
X
X
X2
X
X
(b)
(c)
Figure 8: Cases of when selection introduce Algorithm 1 (a) is not distorted by Y structure V/ + X; +- Xj. X; - S. due the causation between X1 and X2. However, a direct causation can not be identified (b) and (c).
arrowhead pointing toward X,, Then, with inducing paths containing one middle node as examples, the path may take the form X; <o <ox or X; p-o Xj When X, is intervened upon, the changes in p(X,) cannot propagate along the inducing path as the arrowhead pointing to X, blocks the propagation, resulting in the changes of the conditional distribution
p(X, X,). Accordingly, in the augmented DAG Augr (9), we have th J X, X,, and thus (0) (X; X,) * (X, X,). ((k) = {i}.
('+' direction) By contradiction, suppose X; & (ugg() (X,) Des Augz (G) (L)). To satisfy the requirements for inducing paths between X; and Xj. every non-endpoint node must be collider with a tail oriented outward to X, or S. Consequently, the arrowhead only can arise from X; -, yielding an inducing path of the form X; D Xy. In this case, however, pX)) X;) = p)() X.), (()) = {i], which conflicts with the requirement that conditional distribution changes.
Theorem 2 (Graphical criteria for FI-Markov equivalence). Two augmented DAGs (G1) (vb U ULUS US E) and Augr(92) ( targets I if and only if the corresponding MAGs M1 = M(Agi (A)))(91)) and M2 = M(Agg (9z)) have the same skeleton and v-structure, and have the same marks and edges among intervened nodes.
E²) are FI-Markov equivalent for a set of
Proof of Theorem 2 If two augmented DAGs Augı (91) and Augı (92) are FI-Markov equivalent, they have the same CI patterns for the paths between any pair of nodes X; and Xj. where i.j I. In total, there are three types of node marks: tail, arrowhead, and both; these can be uniquely learned from data based the Lemmas 2 to 4
Arrowhead. If the arrowhead points into X₁, no matter from causation latent confounders, this implies that X; cannot be the ancestor of X, or the ancestor of S on the inducing paths. Therefore, following the construction rules of MAG (Zhang) 2008b), must be oriented as X; 4 in MAG representation.
Tail. If tails are oriented outward from X,, as all the paths between X; and X, must be the inducing path, when the tail arises from X; -,, the formation of an inducing node like Xg in Figure 7 d) needs least one tail and one arrowhead. When the tail originates from selection, then X; E Ans (ugz(G) (S), resulting in a tail pointing out X; in MAG representation. When the tail comes from causation, then X; E Ans Angz(C) (X,). In either case, the orientation is consistent with MAG construction rules for tails.
Both arrowhead and tail. When both an arrowhead points to X; and tail points outward from X, only two types of combinations are possible: selection with arrowhead, or causation with latent confounders. In both cases, the MAG representation assigns tail at X,. This is because, according to the formation of the inducing node, causation and latent confounders provide only arrowhead. Regardless of thether the tail from causation selection, X; e {Ans (G) (X,) U Ans (o)) Then, both are presented by tail in MAG representation.
The consistency in the MAG construction rules of augmented DAGs ensures that Mj and M2 have the same skeleton and y-structure in their MAG presentations, as well as the same marks for intervened nodes.
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(b)
(c)
(d)
Figure 9: Inducing path between X1 and X2 with Type inducing node (a,c) or not (b,d).
Theorem 3 (Soundness of F-FCI), Let 9p be the output F-PAG of Algorithm ! with oracle CI tests on multi ditti-ddsttibuttin data {p(k) AK given by (G.I). 9, is consistent with augmented DAG among intervened variables.
Ag) in arrowhead, lails, square, and structures of paths
The soundness of Algorithm indicates that the output arrowhead, tail, square, and 4, are consistent with the corresponding structures in the augmented DAG.
Proof of Theorem
Tail. The F-PAG is learned according to the CI results that align with d-separation in the augmented DAG Augr( (G) as analyzed in Figure 4 under the faithfulness assumption. With the theoretical guarantee proved in Lemma 3 and Lemma 4 changes in marginal distribution
p(o) (X;) 5) + p(k) (X,), (()] {i} and invariance in conditional distribution p!"((x, X:) {i} appear, if and only if the all inducing path between X, and X, start with tail point outward from X,, resulting in e JL X, and d/ 1L Xj X, The CIs can uniquely identify the tail; others will exhibit different Cls.
[(s, (X,
Arrowhead. Similarly, the unique invariant marginal distribution p) (X,) (x)) 106
{i} and variant conditional distribution p()) (X, Xx)) # (X,, X:), 1(k) = {i} only from inducing paths start with the arrowhead in the augmented DAG.
Square. When both tails and arrowheads appear, the node is marked with a square, indicating variation in both marginal distribution p (X,) * (*)) (X,), l(k) = {i} and conditional distri- bution (0)) X;) * (**) (X, X2). ((k) {i). This allows the square mark to be uniquely identified. However, there is special case that needs further discussion. In the augmented DAG Augr (9) with 1 + X1, if X1 - and 1 +., they form the Y-structure at X₁ such as Fig- ure 8 a). Here, as the existence dependence induced by X2 - X1, the extra dependence induced by selection in the Y-structure does not distort the final results, so the resulting edge remains Siiillrly,, the extra dependence induced by Y-structure does not affect the identifiability of c-o as well, as the existence the inducing paths starting from both tail and arrowhead in between. However, when the arrowhead into X1 <- arises from latent confounders shown in F the causal relation becomes onniddntifiable regardless of whether X; + X₂ holds. In this case, selection contributes tail and makes the path unblocked; the d-connection between L and gù; results in spurious causation, but no inducing path begins with tail between X1 and X2. This because of the effect of selection on L. Fortunately, we found that hard intervention can block the selection on latent confounders. The marginalized changes from different hard interventions help us identify if there exis the causal link between. Therefore, the soundness is up to the square of c- and D-O.
3 The identification of + needs every inducing path have at least one Type I inducing node with corresponding interventional data available, such as Xa of Figure Bla. If all the nodes are Type II inducing nodes as shown in (b), although interventional data available, both the marginal distribution p(X,) and p(X₂) change no matter X1 + X2 or not, contradicting the identification 4.
As the identification rule condition of is the same node like X3 of Figure cc. the existence of direct selection can be identified by intervening on Xg. If there is only a Type II inducing node like X3 in (d), the change of p(X₂ do(Xs)) can be due to the selection between X₂ and X3, and cannot identify the existence of direct selection of X, S X2.
If there is Type inducing
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Theorem 4 (Completeness of F-FCI), Let Sp be the output of Algorithm with oracle CI tests on multi-distributionn data (p(8) ) given by (G,I). Each type of substructures represented by tail, arrowhead, square, 4, and between a pair of intervened nodes in the corresponding augmented DAG of 9p can be identified by different types of CI patterns.
To establish the completeness, we need to show that the F-PAG returned by F-FCI is maximally informative when interventional data is available. Based on the proof of Lemmas 2 to the identifiability of learning the fundamental node marks, i.c., arrowhead, tail, and square, from data is guaranteed. Therefore, among the intervened variables, all the node marks can be identified. Moreover, with the identification guarantee of node markers, the identification of 4 and - can be guaranteed as well, this is because the condition of Type I is detectable based on the identification of node marks. For the node marks without interventional data, the completeness was established with arrowheads in (Ali et al. (012)) and tails in (Zhang 2008b). For the node marks for node pairs where one of them is intervened, a.k.a. do-see rules, the completeness was established in (Kocaoglu et al. 2019).
Proof of Theorem
C RELATED WORK
In this section, we provide a comprehensive review of the relevant literature.
When only pure observational data is available. There constraint-based causal discovery algorithms (Spirtes et al. 2001 Luo et al. 2025 Li et al. 2024b), score-based algorithms (Chickering 2002, Li al. 2024a| Ziu al. ). and methods that utilize properties of functional forms in the
underlying causal process (Shimizu et al. 2006
Hoyer et al. 2008 Zhang & Hyvarinen
2012). The
corresponding Markov equivalence characterization can referred to (Verma Pearl 1991 Meek
1995:
Andersson et al. 1997. Robins et al. 2000; Friedman et al. 2000 Brown et al.
2005).
Two kinds of interventions. When interventional data is available, two types of interventions have been proposed to model data generation: hard (or perfect) interventions and soft (or imperfect) interventions, also referred to as mechanism changes. Hard interventions disrupt the dependence between target variable and its direct causes, either by deterministically fixing its value or by stochastically assigning values drawn from an independent distribution (Pearl 2009 Korb et al.. 2004). In contrast, soft interventions preserve the dependence but alter the functional form governing the target variable's ' causal mechanism (Tian & Pearl 2001 Eberhardt & Scheines 2007
Early Attempts in Interventional Causal Discovery. The earliest Bayesian approaches (Cooper & Yoo, 1999. Eaton & Murphy] 2007) estimated the posterior distribution of DAGs using both observational and interventional data. However, these methods did not address key challenges such as identifiability or equivalence class characterization. Tian & Pearl (2001) was the first to explore identifiability and Markov equivalence in interventional causal discovery. They focused on single-variable interventions with mechanism changes (soft interventions) and provided a graphical criterion for determining when two DAGs are indistinguishable, though no formal representation of the resulting equivalence class was introduced.
Advancements in Hard Interventions. Hauser & Bühlmann (2012) first characterized Markov equivalence classes (MECs) under hard stochastic, multiple-variable interventions. Their graphical criterion, based on mutilated DAGs (as introduced in Section 2), provides an equivalence class representation using Z-essential graphs. This criterion aligns with TTan & Pearl (2001). though the latter focuses solely on single-variable interventions. The proposed GIES algorithm (Hauser & Bühlmann 2015) integrates conditional independence (CI) relations from different experimental settings. Following this paradigm, related methods have been developed (Tillman Spirtes) 2011 Claassen & Heskes 2010). However, Wang et al. (2017) identified consistency issues in GIES when faithfulness assumptions are violated and introduced permutation- based algorithms as an alternative solution.
When latent confounders are involved. In the pure observational data and nonparametric causal discovery setting, the frameworks of MAG and FCI have been well established (Richardson &
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FCHNTERVEN
(OURS)
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Figure 10: Comparison results in identifying causal relations under DAG FI score and DAG Recall metrics. All values are averaged over 10 graphs. Error bars represent the 95% confidence interval.
Spirtes 2002; Zhang 2008b). For interventional causal discovery, various methods have been proposed to address latent variables based on measuring overlapping variables across different interventions (Hyttinen et al. 2013 Triantafilluu & ssmaardinoss 2015 Cao et al. 2024) and invariance (Kocaoglu et al., 2017 Eaton & Murphy, (007;; Magliacane et al., 2016). They are either lying under the umbrellas of FCI and the augmented DAG frameworks or using parametric
assumptions.
When selection is involved In purely observational and nonparametric causal discovery, selection is typically constrained by structural limitations (Zhang et al.) 2016). However, in the context of interventional causal díscovery, various methods have been developed to explicitly address selection bias. These methods leverage interventional data to disentangle the effects of selection mechanisms from genuine causal relations Li et al. (2023); Mooij et al. (2020). Similarly, these methods are still limited to equivalence classes under the umbrella of FCI. Moreover, Dai et al. (2025) discussed how selection interacts with intervention, and built twin interventional graph to model the selection that happens before intervention.
When both latent confounders and selection are involved. The presence of latent confounders and selection bias introduces spurious dependencies among observed variables, undermining key assumptions in causal discovery and leading loss of identifiability. In the pure observational data, the Fast Causal Inference (FCI) algorithm (Spirtes et al., 2001] Zhang 2008b) was an early attempt infer ancestral relationships, but it is constrained identifying an equivalence class limited by the structural information of v-structures. Moreover, ambiguities remain in handling selection effects. Other approaches have similarly characterized ancestral equivalence classes based on graphical properties, but without fully addressing these challenges Rohekar et al. (022I.
D SUPPLEMENTARY EXPERIMENTAL DETAILS AND RESULTS
D.1 SIMULATION
We use the implementation IGSP, UT-IGSP, and JCI-GSP from the causaldag package (Chan- aler Squires 2018), and the implementation of CDIS from ht /ttp::///gthhs com/ MarkDana/
CDIS We use the Kernel-basee CI test (Zhang et al. 2012 to examine nonlinear conditional relations for both baselines and F-FCI. The significance level is set to 0.05.
To comprehensively evaluate the effectiveness of F-FCI in identifying causal relations compared with baselines, besides the experimental result on simulation shown in Figure experimental results on FI-score and recall are shown in Figure T0, All experimental results show that F-FCI can identify the spurious dependence induced by post-treatment selection by inducing paths illustrated in the motivation examples Figure I while baselines cannot.
The simulation results on graphs with 10-25 nodes in the non-parametric setting evaluate the scalability of F-FCI to moderately large graphs. We further conduct experiments with 50 variables, of which 30 have interventional data, while 4-7 selection variables and latent confounders are randomly generated. The resulting F-PAG is shown in Figure IT In addition, experiments on over 5000 genes further demonstrate the scalability of F-FCI to high-dimensional biological data.
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B
Figure 11: The output J-PAG generated by F-FCI on 50 variables.
Theoretically, the complexity of constraint-based causal discovery methods, including F-FCI, scales with the graph size. However, because the CI tests are independent, the computational burden can be substantially reduced through parallelization. In particular, Fast FCI (Ramsey al. 2017) can handle millions of variables, and we leverage this implementation for large-scale gene expression data. Based the previous analysis, these results support the scalability of F-FCI and its practicality for real-world applications.
Table 1: Accuracy % of F-FCI in identifying the mean and variance values of accuracy across 10 independent graphs for each configuration.
selection on synthetic data. We report
X
10
15
20
25
10
15
20
25
Hard intervention
Soft intervention
500
15.7
64.8
75.9+ 10.5
11.7++5.6
60.4+
$7.3+17.3
66.74 15.0
$7.2+ 17.0
1,000
86.5
72.5+9.11
73.1+2.9
10.2+10.1
63.0+22.7
56.7+17.3
74.7+
$2.1+ 17.5
1,500
93.8
80.9
6..0
70.649..
70.0+2100
60.8-16.
75.049.6
63.9
2,000
91.6
85.0.11.8
7..61
71.3
80.01 16.0
63.9-13.1
76.31
7.9
Moreover, to verify the effectiveness of F-FCI in handling the post-treatment selection, we report Table[[ihe accuracy identifying post-treatment selection on simulated data across different configurations. Experimental results demonstrate that under each setting with 2 or 3 selection variables, the accuracy of F-FCI increases with sample size. most configurations with more than 1000 samples, the accuracy exceeds 70%.
The performance of constraint- based causal discovery methods relies on the accuracy of the condi- tional independence test. To verify the robustness of F-FCI, conduct experiments across different noise levels of X, where the parameters of the uniform distribution are randomly sampled from the following settings, i.e., X ~ ( 1] U [1, 2),, X U( 2] U 4)), and X U 3] U [3, 6)). Experimental results on DAG Precision, DAG F1 score, DAG Recall, and DAG SHD are reported in Figure 12 Experimental results show that F-FCI is stable in precision across different noise levels. The low noise level (green line) results in relatively higher recall. This is because when the noise level is low, in X U U(0, 1] U [1, 2)), the variance smaller than in other settings. Consequently, dependence patterns are more easily detected, leading to the recovery of more edges and thereby yielding higher recall.
D.2 EXPANDED EVALUATION AND ROBUSTNESS ANALYSES OF F-FCI
To further assess the ability of F-FCI to handle post-treatment selection, we compared F-FCI with FCI-INTERVEN that can only handle latent confounders but not post-treatment selection while increasing the number of selection variables. Experimental results Table 2 show that across both hard- and soft-intervention regimes, F-FCI achieved precision gains exceeding 15%, and this advantage grew as the extent of post-treatment selection increased, confirming that explicitly modeling selection yields tangible benefits for structure recovery.
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O00 (1.2
-0-00.0. 2).
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DAG
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Figure 12: Robustness analysis of F-FCI on X with different noise levels under four metrics: DAG Precision, DAG Fi. DAG Recall, and DAG SHD (Structural Hamming Distance). All values are averaged over 10 runs with different random seeds. Error bars represent the 95% confidence interval.
Table 2: Comparison of F-FCI and FCI-INTERVEN on graphs with 20 variables, 2-3 latent con- founders, and 5-6 randomly chosen selection variables. All values are averaged over 10 runs.
F-FCI
ECI-INTEVVE
Precision
SHD
FI
Recall
Precision
SHD
F1
Recall
Hard intervention
500
60.1 + 0.6
12.6 1.8
56.7 = 0.2
55.7 + 1.0
46.7 0.1
3.2
54.8 0.1
69.0 2.1
1500
62.1 0.6
12.4 4.2
58.2 0.7
57.1 1 2.1
46.8 0.1
1.66
56.8
1 0.2
74.5 1 2.2
2000
64.2 1 0.4
10.9
64.7 0.5
66.4 1.3
43.7 0.1
6.7
55.2 1 0.1
77.8
Soft intervention
500
67.6 0.6
9.8 3.9
63.0 1 0.8
61.6 1.0
49.0
14.2
54.1.0.3
62.2
1.3
1500
68.7 1.710.4 0.4
4.713.2
69.1
- 0.4
70.2 0.4
49.8
14.1=5.49
58.3
72.2
1.1
2000
7 1.5
8.6 7.2
68.8 + 0.9
68.9 0.7
48.4
5.0
57.1
72.1 0.6
To further assess the robustness of F-FCI in more challenging and realistic settings, we conducted two stress tests: (i) richer nonlinear mechanisms and (ii) alternative noise families. For the nonlinearity test, in each run the structural function f on every edge was sampled uniformly from
F { (in((T)) + 0.2 sin x², tanh(x). x, De xe=====> // log + +e c22=1) + 0.052²)
Experimental results in Table show modest decline in precision as functional complexity increases, while overall performance remains competitive. For the noise test, we replaced Uniform noise with Laplace and Gumbel noises: for each variable in X, the location and scale were drawn independently as μ ~ (0, 3) and ~ (22 4) (Laplace: s = b; Gumbel: s = 3). Table the experimental results of F-FCI with Laplace noise; Table 5 reports the results with Gumbel noise. Across these noise families, F-FCI's performance remains stable and in some settings slightly improves, indicating robustness to heavy-tailed and asymmetric error distributions.
D.3 REAL-WORLD
The Norman dataset comprises gene expression profiles of 91,205 lung epithelial cells measured across 5,045 genes. Among these, 7,353 samples correspond to purely observational data (control group), while the remaining samples are derived from gene perturbation experiments. In total, 105
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Table 3: Experimental results of F-FCI under more complex nonlinearities on the graph with 15 variables. All values are averaged over 10 runs.
F-FCI
F-FCI (complex non-linearities)
7
Precision
SHD
F1
Recall
Precision
SHD
F1
Recall
Hard intervention
500
61.7
12.5 5.9
57.9 0.7
56.3 =
57.9 = 2.2
4.6
5.4
55.5
0.3
1500
62.6 +
12.1
57.4 0.7
53.8 - 0.9
56.3 = 0.9
4.4
58,2
60.5 1.0
2000
60.9 +
12.5 6.6
57.8 0.8
56.8 1.0
57.6 1.1
8.3
67.6 6..
Soft intervention
500
72.7
8.3
65.7 = 0.6
60.5 = 0.3
64.0 = 2.6
8.2
63.7 L.
1500
75.6 +.7
4.
70.3 + 0.8
66.4 +
66.8 - 2.6
6.1
1.2 L6
2000
1.
6.8
72.2 = 0.6
70.9 0.8
71.4 1.2
5.0
75.9 1.
Table 4: Comparison of F-FCI under Uniform noise vs. Laplace noise graph with 15 variables.All values are averaged over 10 runs.
Uniform
Laplace
n
Precision
SHD
F1
Recall
DAG Precision
DAG SHD
DAG F1
DAG Recall
Hard intervention
(15 variables)
500
01.71.0
12,5
57.9 0.7
56.3 53.8 0.9
59.9 1.5
8.6 5.6
58.6 68.6
58.7 1.6
1500
62.6 = 1.1
57.4
64.1 2.4
7.7 = 6.6
62.3 = 1.5
63.0 = 2.2
2000
60.9 = 1.6
12.5 6.6
57.8 + 0.8
56.8 + 1.0
61.7 1.9
7.7 + 7.6
60.6 + 1.7
60.4 + 2.2
Soft intervention (15 variables)
500
72.7 + 1.6
9.5 8.3
65.7 0.6
60.5 + 0.3
69.8 + 2.2
9.1 4.1
59.2 1.2
57.3 + 1.6
1500
75.6 = 0.7
7.9 4.3
70.3 - 0.8
66.4 1.2
72.7 + 1.6
7.4 4.8
65.3 0.6
63.7 0.7
2000
75.7 =
8.0 6.8
72.2 0.6
70.9 = 0.8
73.4 1.8
6.7 4.0
67.5
0.9
63.5 - 0.9
Table 5: Comparison of F-FCI under Uniform noise vs. Gumbel noise on graph with 15 variables.All values are averaged over 10 runs.
Uniform
Gumbel
Precision
SHD
Recall
DAG Precision
DAG SHD
DAGFI
DAG Recall
Hard intervention (15 variables)
500
61.7 = 1.0
12.5 5.9
57.9 0.7
56.3
60.9 0.8
9.6 4.0
55.5 + 0.9
52.6
1500
62.6
- 1.1
12.1
57.4 0.7
53.8 1 0.9
57.6 0.8
9.5 5.3
57.0
57.3
2000
60.9 = 1.6
6.6
57.8 0.8
56.8
61.9 1.8
9.1
62.7
66.5 2.2
Soft intervention (15 variables)
500
72.7
8.3
60.5 -
69.4 3.1
7.4344
58.6 1.2
52.6
1500
75.6 0.7
7.9 4.3
70.3 0.8
66.4
73.8 1.7
6.7 4.2
62.8
6.06
2000
75.7 + 1.8
6.8
72.2 + 0.6
70.9 + 0.8
76.9 2.6
6.0 4.0
66.4 1.6
62.0 +.10
genes were perturbed individually. Following the procedure of Algorithm T parallel algorithm Fast-FCI (Ramsey et al., (017)) with BIC score first utilized to get the skeleton over 5045 genes, due the large number of genes. Then, the rules of CI patterns are utilized to orient the skeleton among perturbed genes.
To evaluate the effectiveness of F-FCI in identifying causal relations, latent confounders, and post- treatment selection, we conducted experiments on gene perturbation (interventional) data, with the results summarized in Figure 13 Several identified causal links are consistent with established biological evidence, including IGDCC3 - FEV is aligned with biological evidence provided by ARCHS4 (Lachmann et al.) 2018) and Enrichr (Chen et al. 2013). CEBPA -> FEV is supported by Enrichr, SET + FOX L22 and SET - KMTT2 are supported by GEO (Clough & Barrett 2016), SET + PRDMI 1 is supported by ChEA2022 (Keenan et al. 2019) and ENCODE-TF-CHIP seq (Consortium et al., 2012). HOX C13 + SET is supported by ChEA2022, 2C3HAV1 - TSC22D1 is supported by Enrichr, LY - KLF1 is supported by ChEA2022 and ARCHS4, CELF2 FOX L2 is supported by ARCHS4, GEO, and Enrichr. Moreover, F-FCI flags several genes, such as ZNF318, CDKNIC, CDKNIA, and RREBI, as being affected by post-treatment selection. Here, post-treatment selection refers the scRNA-seq quality-control (QC) filtering step
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that retains only transcriptionally active, viable cells, typically those with high nFeature and high nounnt together with a low fraction of mitochondrial transcripts (percent.mt). To contextualize this selection effect, prior work links CDKNIA (p21) and CDKNIC (p57) DNA-damage responses and cell-cycle inhibition. CDKNIA is a canonical DNA-damage-responsivv CDK inhibitor that enforces cell-cycle arrest and is implicated in growth suppression, transcriptional reprogramming. and apoptosis under genotoxic stress (El-Deiry et al., 1993 Harper et al. 1993). CDKNIC is another CDK inhibitor (acting primarily at GI) that negatively regulates proliferation and helps maintain quiescent/non-preee http:///oooooooooorrtiiieo proliferative states (Matsuoka et al. 1995 Creff & Besson, 2020). Cells exhibiting elevated CDKNI
activity are therefore plausibly less transcriptionally active, consistent with lower nFeature/nCount and a higher likelihood of being excluded by QC. In addition, RREB 1 and ZNF318 have been associated with mitochondrial-relatee programs. RREB1 is Ras-responsive transcription factor that has been implicated in the regulation of nuclear-encoded mitochondrial respiratory-chain components (Han et al. 2024), while ZNF318 (also reported as ZFP318 18 in some contexts) is linked to transcription/splae
regulation and has been connected to mitochondrial eene-expression programs in specific immune states (e.g., memory B-cell-related programs) (Wang et al. 2024). Together, these associations consistent with QC-based post-treatment selection that is sensitive to mitochondrial transcript burden and overall transcriptional activity.
E DISCUSSION
Pre-treatment selection vs. biological constraints vs. post-treatment selection.
CDIS Dai et al. (2025) addresses pre-treatment selection, where samples filtered before inter- ventions (e.g., screening drug-trial participants prior to treatment assignment). GISI. Luo et al. (2025) identifies biological constraints, which happen before intervention and continuously filter out non-viable cells. However, F-FCI targets post-treatment selection, where samples are retained after treatmenn/meessrrmmentt (e.g., gene expression analysis, cells that pass quality control in both observational and interventional groups are included in the dataset for analysis). This difference in when intervention interacts with selection yields opposite retss-setting awariaat/variabiliig pat- terns for variable pairs under selection: under pre-treatment selection, murginal distributions are invariant, while conditional distributions change; under biological constraints, both marginal and conditional distributions change; under cost-treatment selection, marginal distributions change (via collider/selection effects) while the conditional distribution remains invariant. Methodologically, to model the different interaction patterns between intervention and selection, CDIS and GISL model interventions via an interventional twin graph, whereas F-FCI uses an augmented DAG with an explicit selection node S and intervention indicators to leverage these invariance/variiabiiity patterns. Theoretically, F-FCI is sound and complete, and characterizes a finer-grained Markov equivalence class.
F THE USE OF LARGE LANGUAGE MODELS
An LLM used refine writing for clarity and readability but did not contribute to the research design, experiment, or analysis. All intellectual work was independently conducted by the authors, and any suggestions from the LLM were critically evaluated before use. The authors bear full responsibility for the research, and the LLM is not listed as a contributor or author.
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Figure 13: The causal structure among genes with interventional data identified by F-FCI.
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Reviews:
[{"review_id": "vQmJQLz0Gb", "reviewer": "Reviewer_uvY9", "summary": "The paper addresses learning a Markov Equivalence class amongst causal graphs what admit latent variables in the Semi-Markovian model, interventional distributions based on soft mechanism changes and unknown post treatment selection variables. It is assumed that the identity of the interventions are known. \n\nPaper makes a case that equivalence classes under interventions and under selection for semi markovian models has been characterized. But the Markov equivalence classes become very different when both are present - interventions and selection variables on post interventional distributions. Authors define a generalized notion of Markov equivalence based on marginal invariance tests, conditional invariance tests and also conditional independence under selection. \n\nAuthors characterize the Markov Equivalence class under this new setting, providing also a sound and complete constraint based discovery algorithm.", "questions": "1) Can authors explain how one would extend to the setting where invariances across two interventional sets can be incorporated ?\n\n2)  Why is the performance curves in the synthetic data very close to FCI-Interven (FIgure 6) ? Is it that latent variable presence dominates the selection variables effect ? Is it possible to introduce more selection variables than latents to see experimentally if the separation is bigger ?\n\n3) Is there a typo in Line 357 where two squares case needs to be denoted $\\\\square - \\\\square$ with an edge in between ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1) Paper tackles a novel problem in equivalence class characterization when interventions and selection variables are involved. Authors demonstrate how post interventional selection could give different invariance signatures of p(effect), p(cause) and p (effect|cause) across interventions with and without selection.\n\n2) The characterization and sound and complete algorithms are very novel and elegant contributions as well.\n\n3) Reasonable experimental demonstration that this works both with synthetic data and real world datasets.", "weaknesses": "I have only one principal concern - it is with respect to the definition of the Markov equivalence. It is defined only with respect to marginal invariance, conditional invariance between observational distribution and interventional distributions.\n\n However, it is possible to incorporate invariances between two interventional distributions (not involving the observational one) as in Kocaoglu et.al. 2019 by including augmented variables, one for each pair of interventional target sets. This is possible when a soft intervention on a node could makes the same causal mechanism change irrespective of the interventional collection it is in. For some reason this has not been incorporated in the current set up. This means even with respect to invariance tests, there is a bit more leeway in terms of constraining the equivalence class.  \nrct64e3", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "eMDchZ9irw", "reviewer": "Reviewer_eoQR", "summary": "The authors present a new formulation for interventional causal discovery which allows for explicit modeling of so-called post-treatment selection and for considering the extent to which differential responses to interventions may distinguish causal relationships from selection patterns. They propose a new definition of Markov equivalence, called FI-Markov equivalence, to characterize equivalence classes for the new setting, and provide a new graphical representation for an FI-Markov equivalent class, called F-PAG. They then modify the canonical FCI learning algorithm to develop the F-FCI algorithm for learning F-PAGs and prove its correctness and completeness.", "questions": "L. 106: Conditioning on S = 1 encodes selection for both pre-treatment selection \nand post-treatment selection. Q: Why don't you differentiate between these types of selection? Later, e.g., L.187 you write: S = 1 indicates the presence of post-treatment selection.\n\nEq.(1): what does the subscript s in p_s^{(k)} mean? Moreover, in the first product the probability p^{(i)} does not make sense. \n\nL. 134: Do you make any assumptions about the collection of intervention targets, other than that I^{(0)} is empty?\n\nDef. 1 is not clear: explain I, {\\mathcal I}, I^{(k)}.\n\nL.213: improve the punctation.\n\nL. 234: what do you mean by Aug_{I_K} ? Should be Aug_{I^{k}}?\n\nL. 235: the corresponding augmented DAG --> you mean DAGs?\n\nDefinition 2 is not clear: what does X_i ∈ X_{\\cup I} mean?\n\nL. 332:   R-PAG --> F-PAG", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "“Post-treatment selection bias” is a well-known phenomenon in empirical research and is an important issue in causal inference, with implications for research in medicine, the social sciences, and other fields. To my knowledge, this work presents the first systematic study of the interventional causal discovery, taking into account post-treatment selection. The introduced new concepts of Markov equivalence (FI-Markov equivalence) and graphical models for equivalence classes (F-PAGs) constitute an interesting contribution to the field. An important achievement is the development of a provably sound and complete algorithm, F-FCI, that enables the identification of cause-effect relationships, latent confounders, and post-treatment selection.", "weaknesses": "The relationship of this article to the paper introducing pre-treatemt selection framework presented at the ICLR by Dai et al. (2025) is not clear and should be further discussed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "muEv2Lz66I", "reviewer": "Reviewer_1GEE", "summary": "The authors investigate the issue of post-treatment selection in the context of causal discovery. They develop a thorough framework for characterizing post-treatment selection and introduce a new class of observationally equivalent causal structures in this context.  Furthermore, a new algorithm based on their framework is discovered and  investigated theoretically (soundness and completeness) and experimentally on synthetic and real-world data.", "questions": "Please feel free to reply to the weaknesses above.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "The problem that the paper addresses is well motivated and I agree that it is an overlooked yet critical issue. All claims are well supported by evidence. \n\nI am also not aware that this problem has been addressed (in this way) before. \n\nThe conceptual and theoretical analysis of post-treatment selection is rigorous and extensive. The paper is clearly written, well motivated and supported by effective figures. \n\nThe proposed algorithm is reasonable and the provided experimental results seem very sound and support the contribution by the paper.", "weaknesses": "The experiments are somewhat limited: \n(a) the synthetic data is rather simple, it would be interesting to see how the method performs with different non-linearities, forms of noise and larger graphs; including and ablation over those factors. It would be especially interesting to see how well the proposed method can perform on realistic simulators introduced by Robertson et al. (https://arxiv.org/abs/2506.06039). \n\n\nThe discussion of limitations and future work for the method is fairly short and could benefit from being extended. \n\n\nNIT: There is an extra space at the end of line 166", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "uF51NEJOBz", "reviewer": "Reviewer_SifF", "summary": "This paper addresses an important problem: interventional causal discovery under post-treatment selection. The authors systematically analyze how post-treatment selection affects causal identifiability and summarize invariant and variant distributional patterns that arise in such settings. Building upon existing interventional frameworks, they propose a novel causal formulation that explicitly models post-treatment selection, i.e., FI-Markov, F-PAG and  a new algorithm F-FCI. Experiments on both synthetic and real-world datasets demonstrate the effectiveness and robustness of the proposed approach.", "questions": "1.\tHow does F-FCI handle Type II inducing nodes?\n2.\tWhy does GIES outperform F-FCI under hard interventions in terms of F1 and recall?\n3.\tShould “R-FCI” in Figures 10 and 11 be corrected to “F-FCI”?\n4.\tCan the authors provide a case study showing spurious edges caused by post-treatment selection that existing algorithms falsely discover, and that F-FCI correctly identifies?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1.\tThe characterization of invariant/variant rules under post-treatment selection and the introduction of the causal formulation are novel. \n2.\tThe paper provides a solid theoretical foundation for F-FCI.", "weaknesses": "1.\tI did not observe any discussion in the paper regarding how F-FCI handles Type II inducing nodes.\n2.\tFigure 10 shows that GIES outperforms F-FCI under hard interventions in terms of F1 and recall, but the paper does not analyze why this occurs. A brief discussion would help clarify the underlying cause of this performance gap.\n3.\tThere is a typo in Figures 10 and 11, R-FCI should be corrected to F-FCI.\n4.\tIt would strengthen the paper if the authors could include a case study showing which edges existing causal discovery algorithms falsely infer due to post-treatment effects, and how F-FCI correctly identifies and removes these spurious edges.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)