When Selection Meets Intervention: Additional Complexities in Causal Discovery
Authors: Haoyue Dai
Ignavier Ng
Jianle Sun
Zeyu Tang
Gongxu Luo
Xinshuai Dong
Peter Spirtes
Kun Zhang
Venue: ICLR 2025 Oral (2025)
Keywords: causal discovery
selection bias
experiments
interventions
Primary area: causal reasoning

Abstract: We address the common yet often-overlooked selection bias in interventional studies, where subjects are selectively enrolled into experiments. For instance, participants in a drug trial are usually patients of the relevant disease; A/B tests on mobile applications target existing users only, and gene perturbation studies typically focus on specific cell types, such as cancer cells. Ignoring this bias leads to incorrect causal discovery results. Even when recognized, the existing paradigm for interventional causal discovery still fails to address it. This is because subtle differences in _when_ and _where_ interventions happen can lead to significantly different statistical patterns. We capture this dynamic by introducing a graphical model that explicitly accounts for both the observed world (where interventions are applied) and the counterfactual world (where selection occurs while interventions have not been applied). We characterize the Markov property of the model, and propose a provably sound algorithm to identify causal relations as well as selection mechanisms up to the equivalence class, from data with soft interventions and unknown targets. Through synthetic and real-world experiments, we demonstrate that our algorithm effectively identifies true causal relations despite the presence of selection bias.

Published as a conference paper at ICLR 2025
WHEN SELECTION MEETS INTERVENTION: ADDI- TIONAL COMPLEXITIES IN CAUSAL DISCOVERY
Haoyue Dai
Ignavier Ng
Jianle Sun!
Zeyu Tang'
Gongxu Luo"
Xinshuai Dong'
Peter Spirtes
Kun Zhang 1:2
'Carnegie Mellon University
2 Mohamed bin Zayed University of Artificial Intelligence
ABSTRACT
We address the common yet often-overlooked selection bias in interventional stud- ies, where subjects are selectively enrolled into experiments. For instance, partici- pants a drug trial are usually patients of the relevant disease; A/B tests on mobile applications target existing users only, and gene perturbation studies typically focus on specific cell types, such as cancer cells. Ignoring this bias leads to incorrect causal discovery results. Even when recognized, the existing paradigm for interven- tional causal discovery still fails to address This is because subtle differences in when and where interventions happen lead to significantly different statistical patterns. We capture this dynamic by introducing graphical model that explicitly accounts for both the observed world (where interventions are applied) and the counterfactual world (where selection occurs while interventions have not been applied). We characterize the Markov property of the model, and propose a prov- ably sound algorithm to identify causal relations as well as selection mechanisms up to the equivalence class, from data with soft interventions and unknown targets. Through synthetic and real-world experiments, we demonstrate that our algorithm effectively identifies true causal relations despite the presence of selection bias.
INTRODUCTION
Experimentation is often seen as the gold standard for discovering causal relations, but due to its cost, alternative methods have been developed to infer causality from pure observational data (Spirtes et al.) 2000, Pearl 2009). Many real-worle scenarios fall between these two extremes, involving passive ob- servations collected from interventions. Interventional causal discovery addresses this, with methods for hard interventions (Cooper & Yoo| 1999; Hauser & Bühlmann] 2015), soft interventions (Tian & Pearl 2001 Eberhardt & Scheines) 2007). and those with unknown targets (Eaton Murphy 2007 Squires et al. (0200). A detailed review provided in 22. with further related works in Appendix D
While significant progress has been made in interventional causal discovery, existing methods over- look critical issue: selection bias (Heckman) 1977, 1990, Winship & Mare 1992). Though ideally, experiments should be randomly assigned in the general population, practice, subjects are usually pre-selected. For example, drug trial participants are typically patients with the relevant disease; A/B tests target only existing users, and gene perturbation studies often focus on specific cell types like cancer cells. Ignoring this bias leads to incorrect statistical inferences. While various methods address
selection bias for causal inference (Didelez et al., 2010 Bareinboim & Pearl 2012] Bareinboim
al. 2014) and for observational causal discovery (Spirtes et al.) [995] Borboudakis & samardinos
2015
Zhang et al. 2016), no existing work tackles selection bias in interventional causal discovery.
Then, one may naturally wonder whether existing well-established paradigms from both interventional causal discovery and observational causal discovery with selection bias can solve the problem. However, will illustrate in Examples I and 2, these methods still fail to characterize data given by intervention under selection, and will thus lead to false causal discovery results. This is because subtle differences in when and where interventions happen
lead to significantly different statistical patterns, demanding a new problem setup and model. This is exactly what we address in this paper.
Contributions: We introduce new problem setup for interventional causal discovery with selection bias. We show that existing graphical representation paradigms fail to model data under selection, since the when and where of interventions have be explicitly considered ($2). To this end, propose a new graphical model that captures the dynamics of intervention and selection, characterize its Markov properties, and provide graphical criterion for Markov equivalence ($3). We develop

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mut(g, Ø): + 2
3
G
L
mut(g, (2))
2
3
mut(G, (8)) 2 3 1
2
3
(a)
(b)
(c)
(d)
Figure I: Examples of the existing graph representations. (a) and (b) show mutilated DAGs (Hauser &
Bühlmann 2012) and the augmented DAG (Yang et al. 2018) for G - + 2 - 3 with targets Z
10, [2) {3}}. Solid nodes represent the intervention indicators. (c) is the augmented DAG for G = 1 + L +2 2 where L, in square, is latent, and I {Ø, {1}} MMgliacane al. 2016). (d) shows a seemingly natural representation for selection bias, 9 = 1 - S f 2 where S, in double circles, is selected, and I {05, (1))- But does (d) truly capture the underlying process? See Example|] |
sound algorithm to identify causal relations and selection mechanisms up to the equivalence class, from data with soft interventions and unknown targets ($ We demonstrate the effectiveness of our algorithm using synthetic and real-world datasets on biology and education ($5).
2
MOTIVATION
In this section, first revisit the established paradigm graph representations for interventional causal discovery ($2.1). then use illustrative examples to demonstrate how directly extending this paradigm fails to address the selection bias issue (.2.)) followed by an analysis of why this occurs (2.))
2.1 REVISITING THE ESTABLISHED PARADIGM OF INTERVENTIONAL CAUSAL DISCOVERY
We start with the problem setup. Let the DAG g on vertices [ = (1..... D represent a causal model where vertices correspond to random variables X = (X,) For any subset A [D], let XA = (Xi) and by convention Xg = O. Interventional causal discovery aims to learn the structure of from data collected under multiple intervention settings, each with an intervention target I [D], meaning variables Xy are intervened on. Let I = {1 1()) ,..,, [(K) denote the collection of intervention targets, and (p(), (00,p(2) p22) ,..,, ................................ .......p(k)) {(K) the corresponding interventional distributions over X. We assume throughout 7(0) Ø0 i.e., the pure observational data is available.
For hard interventions, Hauser & Bühlmann 2012) consider each (*) as factoring to mutilated DAG over [D] denoted by mut (()), where edges incoming to target (k) in & are removed and other edges remain, as in Figure Ta They show that two DAGs are Markov equivalent under I if and only if vk 0, 0.................... K, their corresponding mutilated DAGS have the same skeleton and v-structures.
For soft interventions, however, mutilated DAG representation fails, as interventions may not remove edges, and all settings may factor to a same G. Instead of checking each setting individually, better approach is then to compare changes and invariances across settings: intervening on a cause changes the marginal p(effect), but the conditional p(effect) remains invariant. Conversely, intervening on an effect leaves p( causs) unchanged, while p( ccause/ changes (Hoover) [990 Tian & Pearl 2001). Such invariance is exploited in the invariance causal inference framework (Rothenhäusler et al. 2015
Meinshausen et al. 2016 Ghassami et al.] 2017). typically as parametric regression analysis.
Invariance also be understood nonparametrically, To model "the action of changing targets", Newey & Powell (0003); Korb et al. (2004) introduce the augmented DAG, denoted by aug I). which, as shown in Figure Tb extends the original g by adding exogenous vertices Ç [GA k= as
intervention indicators, each pointing to its target / (k), Whether the k-th intervention alters con- ditional density p( A ) is then nonparametrically represented by the conditional independence (CI) relation ÇK L X X in the pooled data of p' and p**), and graphically by the d-separation Sk LAACC (1 in aug(g.I) I). Yang al. (2018) show that two DAGs are Markov equivalent un-
der soft interventions I if and only if their augmented DAGs have the same skeleton and v-structures.
Such invariance analysis, together with the augmented DAG representation, offers unified way to understand interventional data or, more generally, data from multiple domains with changing mechanisms. For example, unknown target is no longer a challenge; I (i.e., where changes occur) can be learned by discovering the adjacencies between ( and X (Zhang et al.) 2015: Huang al. 2020 Mooij et al., 22200] Squires et al. 2020). Hidden confounders can also be incorporated by introducing latent variables into augmented DAGs, shown in Figure Ic| Algorithms like FCI (Spirtes et al. 2000
Zhang 2008) are then used to for discovery (Magliacane et al.) [0166] Kocaoglu et al. 2019). Despite
different specifics, these problems share same core concept under the augmented graph paradigm.
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Now finally, let us consider the issue of selection bias. Although, to our knowledge, no prior work has addressed it, a seemingly natural solution is to follow the paradigm and introduce selection variables into augmented DAGs, shown in Figure ld This seems intuitive, especially given Figure Ic] as the FCI algorithm can indeed handle both hidden confounders and selection bias. However, does this augmented graph truly capture the underlying dynamics? The answer is no or at least, it is not straightforward. Let us examine the following Examples |
and 2 that illustrate these complexities.
2.2 How THE AUGMENTED DAG PARADIGM FAILS IN THE PRESENCE OF SELECTION
We start with an example of clinical study where only patients with a specific disease are involved.
Example 1. Consider clinical study fo- cusing on two variables: X, (Blood Glu- cose Levels) and X2 (Hearing Ability). Unknown to the doctor, these variables are independent with no causal relations or confounders. This independence is shown in the scatterplot (both 'x' and 'e') in [f]oo of Figure (hereafter omitted), where X, and X2 are independently drawn from U11, 2]. However, the study somehow only includes Alzheimer's Disease (AD) patients. Since both variables contribute to AD, only individuals with a high combined value of the two (say, X1 + X2 2 3) were included in the study, represented by the 'e's in (a)
XG
Figure 2: (a) Scatterplot of X1; X2 in general population (both 'x' and 'o'), with only 'e' individuals involved into study pl)), (b) and (c) show p(!) after two distinct but both effective interventions on X1, applied to from (a).
In the trial, patients are randomly assigned to either a placebo Glucose Levels, i.e., I = {Ø0 {1)}. The control group y(u) follows the 'e's distribution in (a) To model the intervention, we can use a hard stochastic one, randomly assigning X1 a lower va lue, shifting each 'o' X2) in (a) i.e., individuals before their treatments. to (X) in (b) with X ~ |00.5, 1.5] and importantly, X2 remains unchanged, as indeed X1 does not influence X2. Alternatively, 'soft' intervention can be modeled, reducing X1 relative to its current value, c.g.. to (X1 0.5, X2), resulting in scatterplots (c) The interventional distribution p!1) follows (b) or
treatment controlling Blood
Then, can we directly apply the augmented graph Figure Id since exactly independent X1 and X2 selected to the trial, and X1 is intervened on? According to it, two statements should hold:
The marginal p( X2) should change by the intervention, since Ç1 X,2|5:
2. The conditional p(X² (x1) should remain invariant, since Çi 1 1, S;
However, our observations contradict both. Comparing (a) with (b) or (c) p(X2) remains unchanged, as seen in the marginal density plots, while p(X2) Xi) changes. For example, at Xi 1.25. X2 in (a) follows Uf1.75.2 2], in (b) is non-uniform, more concentrated at higher values, and in follows U1.55 2). This discrepancy suggests that the graph in Figure Id does not fit the data here. One aug- mented graph however indeed fits is (i - <- 2, meaning that if directly applying existing standard algorithms, the doctor would falsely conclude that Hearing Ability causes Blood Glucose Levels. A
We show that simply augmenting the DAG with selection variables fails to model interventional data under selection. In Example there is at least another (incorrect) augmented DAG that fits the data, In contrast, in Example 2 no augmented DAG fits the data, suggesting a failure of this paradigm. Example 2. Let X,, X2, and X3 denote the number of bird-nesting shrubs, predatory birds, and pests, respectively. An ecologist is studying pest issues in several fields across the state. After collecting data, denoted (9)), the ecologist finds that X, and X3 are conditionally independent given X2. This aligns with the true causal relations + 2 - 3, where shrubs attract birds, birds reduce pests, but shrubs do not directly affect pests. To control pests, the ecologist plants more shrubs (intervening on Xi) in these fields and, after some time collect data, denoted p()), As expected, there is increase in X2 and decrease in Xy. However, further analysis reveals a surprise: conditioning on X2, X, and X3 now appear dependent in p p1)), with more shrubs associated with less pests. This confuses the ecologist- find a new type of shrubs that directly reduce pests, i.e., added a direct edge 1 - 3?"
"Selection may be at play!", suggested student, noting the study focused only on fields with pest issues, i.e., high X3 values. The initial (0) follows DAG 1 2 + - S where d-separation 1 1 3/2, S indeed holds, consistent with the observed CI in (0), But after intervening on X,, shouldn't
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p()) still be Markovian to this DAG? Why did the CI disappear? No augmented graph can explain this anomaly, as it suggests that each p(k), conditioned root variables Ç. should still be Markovian to the original DAG. This puzzle is unsolved until explained in Example 3 there is no specialty with
the shrubs. A simulation this example, similar to Figure aabove, is provided in Appendix C.1 A
2.3 WHY THE AUGMENTED DAG PARADIGM FAILS WHEN SELECTION PRESENTS
The core reason why augmented DAG paradigm fails, illustrated in 2 lies in the timing and context when and where interventions are applied. In real-world scenarios, interventions are usually applied after selection, as experiments are typically designed for specific scopes of study. When selection is interpreted as survival, this means an individual must first survive itself, before undergoing any observations or interventions (e.g.. 'e's in Figure 2 We consider this setting in our work.
Simply extending the augmented graph with selection variables (as in Figure Id however, models a different scenario: one where interventions are applied from scratch. There, individuals are not selected when they receive interventions (and non-interventions) and then undergo the same selection mechanisms afterwards, until observed. This scenario is much rarer, with examples like social pro- grams applied to newborns and observed later in life, or medical trials newly generated stem cells.
This fundamental distinction in when and where interventions occur is often overlooked. This is because when selection bias is absent, and even with latent variables, these two forms produce the same interventional data at the distribution level. However, now with selection it is different; the selective inclusion of individuals and their pre-intervention world must be carefully modeled.
CAUSAL MODEL INVOLVING SELECTION AND INTERVENTION
In 2 we demonstrated that the when and where of interventions matter. Building this motivation, in this section, we define the causal graph on how interventions are applied under selection ($3.T). characterize the Markov properties ($..). and provide the criteria for determining whether two DAGs, possibly under selection, are Markov equivalent given possibly different interventions ($3.)
We follow the notation in 22. with the key difference being that the DAG G is now over vertices [D]US, where S = (S: = values. W.l.o.g. let each S, be binary, has no children, and has parents only from [D]. Let 1 be the vector of all 1s. A sample is observed if and only if it satisfies all selection criteria, denoted by S = 1.
represents unobserved selection variables conditioned upon their specific
In the DAG G, for any vertices j E [D] S. is a parent of j and j is child of i if + E G. denoted by E pāg(j) and j chg (i); is an ancestor of j and is a descendant of if i = j or there is directed path -.... -.. in g, denoted by i (jn and j E dcg(1). These notations extend to sets: e.g., for any vertex set C c [D] U S, PC,(C) = Usec pag(1). For any vertex i e [D]. we say i is directly selected if E pag (S), and ancestrally selected if E ang(S).
3.1 CAUSAL GRAPHICAL MODEL FOR INTERVENTIONS UNDER SELECTION
Building on the principle that enrolled individuals are already selected before interventions, we introduce the following graphical model, with examples and explanations given afterwards.
Definition 1 (Interventional twin graph). For DAG 9 over [D] U S and intervention target
I C [D], the interventional twin graph (()) is DAG with vertices {c)UXUX,m UEn wherd
( is an exogenous binary indicator for whether a sample is intervened (C = 1) or not (C = 0); X {,,,,, are variables in the observed reality world, pure observational interventional; X (X: i E ()) are variables in the unobserved counterfactual basal world" representing the variable values before the intervention. As indicated in its name, only variables affected by the intervention, i.c., those in deg are split into these additional vertices; unaffected variables retain identical values in both worlds and can be represented solely by X;
East fc í ( are common exogenous noise terms shared by the two worlds; S { 1== represent the selection status before the intervention in the counterfactual world.
g()) consists of the following four types of direct edges:
Direct causal effect edges in both worlds: for each + E 9 with i,j E [D), add X; - Xy to g()), Additionally, if deg add X; > Xj: otherwise, if E (T)() add X; - X3:
'The word "twin" is to echo the swin network from (Balke & Pearl 1994). J. See discussions in Appendix D
'The word "basal" is borrowed from biology, referring natural state of cells prior to any perturbations.
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(a) (b) g(tL)
(c) H
(d) H((L))
(c) H(13))
Figure 3: Examples interventional twin graphs (Definition I (a) and (c) are two DAGs for clinical and pest control Examples| [ [and respectively; (b) and (d), (c) are their corresponding interventional twin graphs under different targets. The white X nodes and solid Ç node are observed, forming the reality world (enclosed by solid frames), where observations or interventions are conducted. The grey nodes are unobserved, which squares (Xar and Eatt) are latent variables and double circles (5*) are selection variables. The counterfactual basal world is enclosed by dashed frames.
Selection edges in the counterfactual basal world: for each -> Sj E 9 with i (D), j [T]:
i E dIş add X; + Sj g(1; otherwise, add X; + Sj:
Edges representing common exogenous influences: {e; - e;, + X; he(Dnddeuu
Edges representing mechanism changes due to the intervention: {C Xihee-
Illustrative examples for interventional twin graphs (Definition D are shown in Figure In what follows, we explain several key structural and statistical insights of this causal graphical model.
What is modeled by the interventional twin graph? In g(),, only X and Ç are observed, repre- senting pure observational data, p(Xlç S" 1), and interventional data, p(X|p 1, 1). Crucially, S* = 1 is conditioned on, meaning all individuals, observed or intervened, were selected at the outset. The key difference from the augmented graph (Figure Id) is that, here g()) explicitly mod- els each individual's unobserved pre-intervention values: non-affected variables retain their value: X, while affected variables, whose values change, are modeled by extra vertices X X The pre- and post-intervention worlds share ε, common external influences like individual-specifie traits. Specif- ically, selection is applied in the pre-intervention world, while observation is made post-intervention
According to the graph, what is changed by the intervention? Individuals in pure observational and interventional data are not matched, which reflects common interventional studies (e.g.. RNA sequencing destructive, preventing measurements to a same cell both before and after gene knockout). Instead, the two datasets are related at the distribution level: directed edges from Ç to X indicate changes in generating mechanisms for targeted variables. For affected but not directly targeted variables i E deg T, as suggested by the graph, their generating functions x(Xx]Xpag.he Gi) remain invariant for = 0, 1. However, unlike in augmented graph paradigm, this invariance no longer extends observed conditional distributions (Xpgg For instance, Figure 3b intervening on X1 alters not only p(X)) but also p( (1) and p( (x((X2 (see Example 3
3.2 THE MARKOV PROPERTIES
Our ultimate goal is to discover true causal relations and selection mechanisms from data. To this end, we must understand the CI implications in the data. Hence, what follows, we define the Markov properties, showing how the interventional twin graph model serves identify these CI implications.
To start, let us revisit Example (clinical study) with the defined interventional twin graph (())), as shown in Figure 3bb becomes clear that there Ç 1 S and X,, consistent with the invariant p( (Xx) 2)) and the changed p(X2|X) resolving the earlier discrepancy given by Figure ld
For discovery from data, two types of statistical information can help: 1) conditional (in)dependencies among variables within each interventional distribution, and 2) the (in)variances of conditional distri- butions across different interventions. Below, we formally define these relations implied by the model.
Theorem 1 (CI and invariance implications). For interventional distributions (x) (X) (aju[k]
generated from DAG G with targets {1(K) tike tek [o]u[k], let (g(t()) Tke (o)u[]] be the corresponding interventional twin graphs. For any disjoint A, B,C C [D), the following two statements hold:
For any le E (0) U [K], if XA Le St. S* holds in girlo), then XA 1 X XB|Xe X in (())
2. For any k E [K]. if L.XAlXo,S S" holds in g(rtt), then (XAlXc) (X
'For reasons why not to also split unaffected variables across both worlds, see Appendix C.2
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Theorem I shows that both types of statistical information are implied by the graphical conditions, namely d d-separations among X U {SH in interventional twin graphs. The two conditions charac- terize the CIs in each intervention, and the conditional invariances across interventions, respectively.
We now show how selection and intervention specifically lead to the first type of information, CIs in each distribution. First, selection is known to introduce spurious dependencies in pure observations:
Lemma (Additional dependencies induced by selections). For any DAG G on [D]US and disjoint A, B, C C [D) if A LBB|C S holds, then A L,B[C C also holds. The reverse is not necessarily true.
Further, interventions can introduce even more dependencies, making interventional distributions no longer Markovian the original DAG & (recall the post-pest-control distribution in Example 2
Lemma (Even more dependencies induced by interventions). For any DAG 9 on [D] US target C [D), and disjoint A, B,C C [D), if X LaXB Xc,S", , hoold in the twin graph g(n, then A C, holds in the original DAG G. The reverse is not necessarily true, except when 1 ø.
Lemma 2 offers a counterintuitive insight that contrasts with cases without selection or where selection is not applied before interventions, as those modeled by the augmented graph paradigm. In those cases, interventions add no dependencies and may only add independencies (e.g., via hard interventions). In our setting, however, an intervention may indeed add more dependencies. Now, let us examine Theorem and Lemma 2 ecologist's concern, and summarize the motivations behind the Markov properties in this subsection.
in the context of Example 2 (pest control), address the
Example 3. Continuing from Example 2 the original DAG H. and the interventional twin graph H(( are shown in Figures 3e and 3d respectively. To explain the X1 y Xg. X2 observed after in- tervening on shrubs, graphically we indeed see the d-connection X, XaXa/ X2, S*.C in H((1))), with an open path X1 61 Xi XX X3 E3 + X3 where the collider X3 has its descendant ST conditioned on. Statistically, X3 is combination of exogenous noises 61, <2, 63, and selection on X3 renders E1, E2, E3 not independent anymore. Such dependent noises also leave trace in conditional invariances: though only X₁ is targeted, not only p(X,) but also (,) and ((Xa(X2) are altered, as graphically implied by the Ç K LaX2 X1, S and C X St, respectively.
If, however, the ecologist somehow directly targets X3 (pests), the corresponding H((33) (see Fig- ure Beb shows that X1 1 X2 holds in the interventional distribution this time, and the marginals of X1, X2 remain unaltered. More details on this pest-control example are in Appendix C.1 A
3.3 MARKOV EQUIVALENCE RELATIONS
In $3.2 we characterized the CI relations implied by the true model in the data. Now, to identify the true model from data, in this section, we must understand to what extent the true model is identifiable, as different models may share identical CI implications, namely, being Markov equivalens (Definition (). To establish graphical criteria for this equivalence, graph (MAG) framework. In I we introduce MAG basics. In $3.3.2 we characterize the MAGs of interventional twin graphs, and accordingly, present the graphical criteria for Markov equivalence.
leverage the maximal ancestral
We first define the Markov equivalence. Two different DAGS with different intervention targets (since allow unknown targets) can entail the same CI and invariance relations in the data. Formally,
Definition 2 (Markov equivalence). Let G and H be two DAGS over vertices [D] U S and [D] U respectively, i.e., defined over the same variables [D] but possibly with different selections. Let I and J be two collections of intervention targets with a same size 1 1+ K. The pairs (G, and (H, J) are said to be Markov equivalent, denoted by (G, T) ~ (H, ). if they imply the same set of Cls in each distribution and the same conditional invariances across distributions, as given by Theorem D
3.3.1 MAG BASICS: REPRESENTING Cls WITH LATENT AND SELECTION VARIABLES
We now introduce the MAG framework to simplify the graphical representation for the CI implications. Only the essentials are covered here; for more details, see (Richardson & Spirtes 2002 Zhang 2008).
MAG is a mixed graph with three kinds of edges: directed (-)) bi-directed (++), and undirected (-). Given any DAG G over vertices partitioned O (observed), L (latent), and S (selected), correspond- ing MAG over O, denoted by Mg, can be constructed. Before presenting the construction rules, let us recap the motivation: capture the CI relations among O marginalized over I. and conditioned on S.

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"
c
(a) MoMga Mggan
(b) M Mggtt
(c) M
(d) Mager
(c) M
Figure 4: Examples of MAGs of interventional twin graphs. Readers may reconstruct these MAGs from DAGs in Figure using either general rules (Definitions 3 4 or interventional twin graph- specific rules (Lemmas 4 O and verify if they match. Readers may also verify if the CI implications (Theorem match between d-separations on DAGs and m-separations (Definition 5 on MAGs.
First, for two observed variables i.j E O, when are they always d-connected given C' S for any subset of observed variables C' C O'i,, The answer is given by the adjacencies in the MAG:
Definition 3 (MAG construction step I: adjacencies; |Richardson & Spirtes (2002)). For each pair of observed variables i E Q, i and j are adjacent in Mg if and only if 2 are adjacent in G. or there exists a path p between and j in G where every non-endpoint observed or selected vertex on p is a collider on and every collider on p is an ancestor of or j or member of S.
Next, we orient these adjacencies fully capture the d-separation relations implied by 9 on O|S: Definition 4 (MAG construction step 2: orientations; Richardson & Spirtes (2022).
). For each two adjacent vertices i defined by Definition 3 orient the edge in MM as i + if
} and j & ang [i) US:): i +- j if E ang ( US) and i & ang U S); ++ j if & ang (()
ang {{1) US)
and & ang (i) U S): and - j otherwise, i.e., if E (nn US) and j E (nn
The two steps above give general MAG construction rules. In a MAG, a vertex j is a descendant of i i or i - ... - j. An edge between i, j is into j it i + j or i ++ j. A vertex is collider on a path if both edges incident to on are into i. A path p as (i, j) ) is a v-structure if i and j are not adjacent, and le is collider on p. Using these notations, the MAG's CI implications are as follows:
Definition 5 (m-separation). In the MAG Mg. a path between vertices and j is open relative p is not in C, and every collider
to vertex set C (i,j & C), if every non-collider descendant in C. and j are m-separated by C, denoted by i Il mm, if there is no open path between j relative to C. i LC holds in MAG Mg, if and only if Laj|C U S holds in DAG g.
p has
3.3.2 GRAPHICAL CRITERIA FOR MARKOV EQUIVALENCE
The MAG framework is useful, as it represents Cls in distributions under latent variables and selection, and determines Markov equivalence between such distributions, precisely aligning with our goal. Ac- cordingly, we construct MAGs of interventional twin graphs, with vertices partitioned into observed (XU(C)). latent (Xaa UE and selected (S*) ones. Examples of such MAGs are shown in Figure While general MAG construction rules are outlined in $3.3.1 to better understand the CI implications graphically, we present the following construction rules specific interventional twin graph.
Given a DAG G over [D] USUSS S and a target I C [D], denote by Mgins the MAG constructed on X O [c} from the interventional twin graph g()) We show the specific rules to construct Mgin, by presenting the following lemmas and the questions they aim to answer.
Lemma 3 (When are two variables always dependent in pure observational data?). For any i, e (D), X; and X, are adjacent in Mguan, if and only if i and j are adjacent in G, or chg n chę(j) n (S) * Ø, ie., they involve in same selection, or have common child that is ancestrally selected.
The adjacencies in Lemma 3 reflect additional dependencies due to selection bias (Lemma [ Furthermore, interventions can introduce even more dependencies, as shown in Lemma -aagain, recall the pest control example. We now generalize Lemma to characterize all these dependencies.
Lemma 4 (When are two variables always dependent under an intervention?). For any j E [D], X, and X3 are adjacent in Mgin, if and only if X; and X, are adjacent in the observational MAG Mgizy, or there exists a path between X; and X, in Mginj, where all non-endpoint vertices are affected (i.e., in dog(T) and they, together with i, are all ancestrally selected (i.e., in aSg(S)).
Further generalizing the adjacencies in Lemma 4 to the indicator variable Ç. we have: Lemma 5 (When does an intervention alters all of variable's conditional distributions?). For any j E [D), Ç and X, are adjacent in Maan, if and only if is directly targeted (i.e. j E 1), or j is both indirectly affected and ancestrally selected (i.e., j E deg (T) (S)(
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iimpliss implies the unidentifiability of direct intervention targets when they are unknown. This is different from the case without selection, where "unknown target is no longer a challenge" ($2.1)
Lemma
After constructing the adjacencies of MMgi) in Lemmas too 5 we conclude with edge orientations:
Lemma (MAG of interventional twin graphs). The MAG Mgan consists of the following edges:
"Pseudo" " direct intervention target edges (by Lemma 5): {s + Xi)iir (n)))
Edges among X. Let Mg be the MAG of G over [D]. For each adjacent X,. X5 (by Lemma 4:
- if + j is the original MAG Mg. then orient X; + X, in interventional MAG Mgas;
OOtherwise (i.e., i-j -M ori,j are not adjacent in Mg):
i & deg and j & deg(T); orient X; Xxx
If j E dIg and (i & deg(I) or ang(j)): orient X; + Xy:
* if E deg and (j & deg or E ang(t)): orient X; x -x;
Otherwise, iee., both i.j E (T)( but neither is another's ancestor: orient X₃ <> X--
Finally, we present the graphical criteria for Markov equivalence, using MAG construction rules defined above. By enforcing same CIs in each setting and same invariances across settings, we have: Theorem 2 (Graphical criteria for Markov equivalence), For two DAGs G and H with two col- lections of targets I = {1(9), I0),.. ] and J = ((1)) alence (G,I)= ~ (H, J) holds, if and only if for each k = 1, K, the corresponding MAGs of interventional twin graphs, Marts) and M Masth have the same adjacencies and v-structures"
J(K),, the Markov equiv-
Below let us examine such Markov equivalence and its graphical criteria on Examples L and
Example 4. In the clinical example, let 9 be the DAG in Figure 3a and G' the DAG 2 + 1 without selection. For I {2, { the equivalence (G.I) ~ (g' I) holds, i.c., intervening on one variable cannot identify if the two variables are causally related or just correlated by selection. However, adding an intervention
(g, [) of (I', I), for I = (0, [1) {2)]. In the pest control example, let H. be the DAG in Figure 3c and H' the DAG 1 2 + S1 3. For 1, 2. (H, (0, {i})) 7 (H', {ø, {i))), i..., interventions on upstream vari- ables can distinguish the two, but not on downstream Xg. With unknown targets, however, upstream interventions may still leave the two indistinguishable, e.g., (H. {0, (1}3) ~ (H', f2, (1,3)1). A
the other variable can identify this distinction:
4 ALGORITHM: INTERVENTIONAL CAUSAL DISCOVERY UNDER SELECTION
In this section, we develop Algorithm [ named Causal Discovery from Interventional data under potential Selection bias (CDIS). Using the twin graph framework and Markov properties from $ this algorithm learns causal relations and selection structures up to the equivalence class, from interventional data with soft interventions, unknown targets, and potential selection bias. We assume causal sufficiency and faithfulness, i.e., no Cls beyond those implied by the graph (Theorem I
A first thought might be to obtain adjacencies from observational data (0), as it provides the sparsest skeleton (Lemmas D and (2. Then, one could form v-structures involving intervention indicators by checking conditional (in)variances, and use these v-structures for further orientation on the sparsest skeleton. However, as we will show, this seemingly intuitive approach can lead to false discoveries:
+ + Si + 3 with targets I = (0, [1]}. CIs in p()) first yield skeleton 1 - 2 - 3, and the target is identified as - (1,2). Since p(Xs) does not change (C JL Xa), the v-structure Ç + 2 4 3 is formed. Now, we directly apply orientation rules (Meek) 1995 on the skeleton, the non-collider + 2 + 3 will be oriented, leading to a false edge 1 + 2. A
Example 5. Consider the DAG G
Example highlights the pitfalls of directly applying orientations on adjacencies obatined from pure observational data, even if they are the sparsest and closest to truth. Instead, orientations must be applied on denser adjacencies from interventional data to prevent false propagation, and then used to refine edges in the sparsest skeleton. Our CDIS algorithm is built on this principle.
The pseudocode for CDIS is detailed in Algorithm I in Appendix B due to page limit. Below we provide high-level summary. CDIS consists of the following three steps:
Step 1. Maximal orientation from pure observational data. Run FCI (Zhang 2008 (0), we
obtain the maximal information possible from p()) only, represented by a PAG M(0)
4
"Another condition for general MAG equivalence is not needed here, due to twin graphs' specific structures.
Partial ancestral graph (PAG) is graph to represent class of MAGs. See definitions in Appendix B
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Step 2. Maximal orientation from interventional data. For each hs = 1,... ,K, orientations are derived from pooled data (p("), (()), represented by a PAG M( over [D] u (C). Significant pruning is applied since conditional dependencies from y(") must hold in p(k). Step 3. Refinement using interventional twin graph-specific criteria. Guided by the specific con-
struction rules Lemmas 4 to information from M( are used to orient uncertain edges in _M i... to narrow down the equivalence class about the true model. Updated M(() then guides further refinement on MM((), iterating until no new orientations are possible.
We show that CDIS correctly identifies and distinguishes causal relations and selection mechanisms: Theorem 3 (Soundness of CDIS), Let (9) be the output PAG of Algorithm ! with oracle CI tests on interventional data (p!k) Jk=n given by (I, Then, M is consistent with MAG Mg. Specifically,
1. For any j (D), there is also + j G, and is not ancestrally selected in G.
2. For any i- E (0) both i and j are ancestrally selected in the true DAG g.
The soundness of CDIS follows from the soundness of FCI rules, but its completeness- invariant edges in the equivalence class are identified-remains uncertain to us. Through brutal search on ~ 50, 000 (G.I) pairs with up to 15 vertices, we have not found any incomplete example, and thus we conjecture its completeness. However, proving this is challenging, since the refining step relies on our graphical criteria (Lemma (6) as background knowledge, for which no complete rules
all
exist yet (Andrews al. 2020 Wang al. 2024 Venkateswaran & Perković 2024). This technical difficulty mirrors also to FCI itself: though FCI has been widely adopted since Spirtes et al. (1999),
its completeness result (without background knowledge) was not established until Ali et al. (2005)
(partially) and Zhang (208) (fully). Notably, in our scenario, FCI is guaranteed complete on pure
observational data, while CDIS provides more information than that.
5 EXPERIMENTS AND RESULTS
In this section, we present empirical studies on simulations and real-world data to demonstrate that our algorithm effectively identifies true causal relations despite the presence of selection bias.
5.1 SIMULATIONS
We conduct simulations to validate the soundness of our proposed method, CDIS, and explore the validity of its conjectured completeness. We compare CDIS against existing methods such as GIES (Hauser & Bühlmann, 2012), IGSP (Wang et al.] 2017), UT-IGSP (Squires et al.] [2020).
CD-NOD (Huang et al.] (020). and the JCI-GSP used in Squires et al. (2020) as an extension of JCI (Mooij et al. 2020) with GSP (Solus et al., 2021). Further details are described in Appendix D
We follow the data generating procedure outlined in Definition [ Specifically, we begin by randomly sampling Erdös-Rén Rénnii (Erdös Rényi (959) graphs with an average degree of 2 as the ground truth
1D DAG for {X; 1==1- Next, to represent how multiple selection mechanisms operate simultancously,
we introduce [D/5] selection variables Si...., .,. S l///a where each S is randomly assigned one or
two parent variables from {X;) Note that we do not allow the last [D/2] variables in the causal ordering to be involved in selection, because doing so could result in the corresponding PAGs having an excessive number of "a-o" edges, making them overly uninformative.
We then simulate linear SEMs for HX with exogenous noise terms [ Sample selection performed based on each ST. as linear sum of its parents in X and an independent noise, falling within a predefined interval and ensuring the desired sample size (5, 000 samples after selection). Here, the linear edge coefficients are sampled from Unif -0.5] U [0.5. 2)). and the variances of exogenous noise terms are sampled from Unif[1, 4]. Finally, using the exogenous noise terms {c of the selected samples, we simulate interventional data over { under different targets. We simulate total of [D/2] interventions, each with one variable being intervened on.
In order to show how CDIS reliably identifies correct causal relations, we report the precision, recall, and Fl score of the PAG "-," edges estimated on the simulated data, as compared to the true '->' edges that can be theoretically discovered, indicated by Theorem B Furthermore, we report the accuracy of the edgemark of the estimated PAGs, well the structural Hamming distance (SHD) of its corresponding skeleton, as compared to the true PAGs. For the CPDAGs estimated by the baselines, we treat their directed and undirected edges as "->" and "o-o" edges, respectively.

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The three main metrics are given in Figure 5 while the remaining ones available in Figure T2 in Appendix E The experimental results demonstrate that CDIS outperforms the baselines across most settings and metrics, especially for precision. Notably, the average precision of our algorithm exceeds 0.7 across most configurations. In contrast, the baselines generally achieve precision below 0.65 (with the exception of UT-IGSP in some instances). These observations validate the soundness of CDIS in identifying true causal relations, while suggesting that other methods may falsely infer spurious causal relations, possibly due to selection bias, illustrated in Examples and 2
Fl score of --" edles
Acc. of odgeeaarkk
0.8
0.8
0.8-
0.6
0.6
Precisions
0.4
0.6
0.2
10
15
20
10
15
20
10
15
20
Number of variables
Number of variables
Number of variables
-0- CDIS
GIES
IGSP
UT-GGSP
JCI-GSP
CD-NOD
Figure 5: Empirical results across different numbers of variables. Here, the first figure shows the accuracy of the edgemark of the estimated PAGs. The rest two figures show the precision and Fl score of the '2' edges in PAGs. The error bars illustrate the standard errors from 20 random simulations.
5.2 REAL-WORLD APPLICATIONS
We evaluate the gene regulation networks (GRNs) of 24 previous reported essential regulatory genes encoding different transcription factors (TFs) ( (Yang et al. (018)) using a single-cell perturbation data, ie.. sciPlex2 (Peidli et al., 2024). where the A549 cells. human lung adenocarcinoma cell line, are either exposed to one of the four different transcription modulators, including dexamethasone, nutlin-3a, B.MS-345541, and suberoylanilide hydroxamic acid (SAHA), or simply treated with dimethylsulfoxide vehicle control (Srivatsan et al. 2020). As shown in Figure 9 some validated regulatory relationships like RELA + RUNXI and JUNB - MAFF, since RELA has been implicated as RUNX3 transcription regulator (Zhou al. 2015) and Maf family was upregulated by JunB (Koizumi et al,, 2018), despite the extensive co-regulation between these two genes (Kataoka et al. 1994). A more detailed analysis of these results is provided in Appendix [E]
we discovered
We also apply CDIS to an educational dataset (Table [ from a random controlled trial evaluating the effects of incentives and services college freshmen's 's academic achievements (Angrist et al. 2009). First-year undergraduates are randomly assigned to either the control group or one of the three treatment arms: service strategy (Student Support Program, SSP) with both peer-ade advising service and supplemental instruction service in facilitated study groups: an incentive strategy (the Student Fellowship Program, SFP) with the opportunity to win merit scholarships based on academic achievements; and an intervention offering both named SFSP. As depicted in Figure TO, subgroup analysis stratified by genders indicate that SSP only improves the women's 's performance while SFP shows effects only on men (see Figure IT). The results indicates that interventions exhibit genuine heterogeneous treatment effects on college students' academic performance, with the moderating effect of gender rather than being attributed to selection bias based on gender.
6 CONCLUSION AND LIMITATIONS
We introduce a new problem setup for interventional causal discovery with selection bias. We show how and why existing models fail to represent the data, propose a new model to capture intervention and selection dynamics, characterize Markov properties and criterion for equivalence, and develop a sound algorithm called CDIS for identifying causal relations and selection mechanisms.
One could naturally extend the graph by introducing S also the reality world to model round of post-intervention selection, e.g., lost to follow-up (Akl et al. 2012). Another extension model. While provide the graphical criteria to
be the causal inference results based determining equivalence, graphical representation for the equivalence class is to be developed. Also, the completeness guarantee of the CDIS algorithm, though hypothesized, yet be proven.
'A Python implementation of CDIS is available httpt::t////g thub. com /Ma wwwaaaakaaaaaaaaaaaaaaaa
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ACKNOWLEDGMENTS
We would like to acknowledge the support from NSF Award No. 2229881, AT Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract RO1HL159805, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program. ZT is supported by the National Institute of Justice (NIJ) Graduate Research Fellowship, Award No. 15PNIJ-24-GG
REFERENCES
Raj Agrawal, Chandler Squires, Karren Yang, Karthikeyan Shanmugam, and Caroline Uhler. ABCD- Strategy: Budgeted experimental design for targeted causal structure discovery. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 3400-3409. PMLR, 2019.
Elic A Akl, Matthias Briel, John J You, Xin Sun, Bradley C Johnston, Jason W Busse, Sohail Mulla, Francois Lamontagne, Dirk Bassler, Claudio Vera, et al. Potential impact on estimated treatment effects of information lost to follow-up in randomised controlled trials (lost-it): systematic review.
Bmj, 344, 2012.
Ayesha R Ali. Thomas S Richardson, Peter L Spirtes, and Jiji Zhang. Towards characterizing markov equivalence classes for directed acyelic graphs with latent variables. arXiv preprint arXiv: 1365, 2005.
Steen A Andersson, David Madigan, and Michael D Perlman. A characterization of markov equiva- lence classes for acyclic digraphs. The Annals of Statistics, 25(2):505-544 541, 1997.
Bryan Andrews, Peter Spirtes, and Gregory F Cooper. On the completeness of causal discovery in the presence of latent confounding with tiered background knowledge. International Conference on Artificial Intelligence and Statistics, pp. 4002-4011. 1. PMLR. 2020.
Joshua Angrist, Daniel Lang, and Philip Oreopoulos. Incentives and services for college achievement: Evidence from randomized trial. American Economic Journal: Applied Economics, 1(1):136-163, 2009.
Alexander Balke and Judea Pearl. Probabilistic evaluation of counterfactual queries. In Proceedings of the Twelfth AAAI National Conference on Artificial Intelligence, AAAI'94, pp. 230-237. AAAI Press, 1994.
Elias Bareinboim and Judea Pearl. Controlling selection bias in causal inference. In Artificial Intelligence and Statistics, pp. 100-108. PMLR, 2012.
Elias Bareinboim, Jin Tian, and Judea Pearl. Recovering from selection bias in causal and statistical inference. In Probabilistic and causal inference: The works of Judea Pearl, pp. 433-450, 2014.
Giorgos Borboudakis and loannis Tsamardinos. Bayesian network learning with discrete case-control data. In UAI, pp. 150- 2015.
Laura Brown, Ioannis Tsamardinos, and Constantin F Aliferis. comparison of novel and state-of- the-art polynomial bayesian network learning algorithms. In AAAI, volume 2005, pp. 739-745, 2005.
Chandler Squires. wwwsaldaaaaao... dag: creation, manipulation, and learning of causal models, 2018. URL httts: wtt.://c./whhhll.eeee causaldagi
David Maxwell Chickering, Optimal structure identification with greedy search. Journal of Machine Learning Research, 3(Nov):507-554 2002.
Tom Claassen and Tom Heskes. Causal discovery in multiple models from different experiments. Advances in Neural Information Processing Systems, 23, 2010.
Gregory F Cooper and Changwon Yoo. Causal discovery from a mixture of experimental and observational data. arXiv preprint arXiv:1301.o0 361.6666.66666 1999.

---

Published as a conference paper at ICLR 2025
Juan D Correa, Jin Tian, and Elias Bareinboim. Identification of causal effects in the presence of selection bias. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 2744-2751, 2019.
Haoyue Dai, Ignavier Ng. Gongxu Luo, Peter Spirtes, Petar Stojanov, and Kun Zhang. Gene regulatory network inference in the presence of dropouts: causal view. ICLR, 2024.
Vanessa Didelez, Svend Kreiner, and Niels Keiding. Graphical models for inference under outcome- dependent sampling. 2010.
Daniel Eaton and Kevin Murphy, Exact bayesian structure learning from uncertain interventions. In Artificial intelligence and statistics, pp. 107-114, PMLR, 2007.
Frederick Eberhardt. Almost optimal intervention sets for causal discovery. In Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence, pp. 161-168, 2008.
Frederick Eberhardt and Richard Scheines. Interventions and causal inference. Philosophy of science, 74(5):981-995, 2007.
Frederick Eberhardt, Clark Glymour, and Richard Scheines. On the number of experiments sufficient and in the worst case necessary to identify all causal relations among n variables. In Proceedings of the Twenty-Fins First Conference on Uncertainty in Artificial Intelligence, pp. 178-184, 2005.
Frederick Eberhardt, Patrik Hoyer, and Richard Scheines. Combining experiments discover linear cyclic models with latent variables. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statisties, pp. 185-192 192, 2010.
Erdös and A Rényi. On random graphs I. Publicationes Mathematicae, 6:290-297, 1959.
Robin J Evans and Vanessa Didelez. Recovering from selection bias using marginal structure in discrete models. In ACI@ UAI, pp. 46-55, 2015.
Nir Friedman, Michal Linial, Iftach Nachman, and Dana Pe'er er. Using Bayesian networks to analyze expression data. Proceedings of the fourth Annual International Conference on Computational Molecular Biology, pp. 127-135, 2000.
AmirEmad Ghassami, Saber Salchkaleybar, Negar Kiyavash, and Kun Zhang. Learning causal structures using regression invariance. Advances in Neural Information Processing Systems, 30, 2017.
AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash, and Elias Bareinboim. Budgeted experiment design for causal structure learning. In International Conference on Machine Learning. pp. 1724-1733. PMLR, 2018.
Alain Hauser and Peter Bühlmann. Characterization and greedy learning of interventional markov equivalence classes directed acyclic graphs. The Journal of Machine Learning Research, 13(1): 2409-2464, 2012.
Alain Hauser and Peter Bühlmann. Jointly interventional and observational data: estimation of interventional markov equivalence classes of directed acyclic graphs. Journal of the Royal Statistical Society Series B: Statistical Methodology, 77(1):2913318, 2015.
Yang-Bo Bo He and Zhi Geng. Active learning of causal networks with intervention experiments and optimal designs. Journal of Machine Learning Research, 9NNv):2523-25447, 2008
James Heckman. Varieties of selection bias. The American Economic Review, 80(2):313-318, 1990.
James Heckman. Sample selection bias specification error (with an application to the estimation of labor supply functions), volume 172. National Bureau of Economic Research Cambridge, MA, 1977.
Miguel A Hernán, Sonia Hernández-Diau and James M Robins. A structural approach to selection bias. Epidemiology, 15(5):615-6255 2004.
12

---

Published as a conference paper at ICLR 2025
Kevin D Hoover. The logic of causal inference: Econometries and the conditional analysis of causation. Economics & Philosophy, 6(2):207- -244,, 1990.
Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. In Advances in Neural Information Processing Systems, volume 21, 2008.
Biwei Huang, Kun Zhang, Jiji Zhang, Joseph Ramsey, Ruben Sanchez-Romero, Clark Glymour, and Bernhard Schölkopf. Causal discovery from heterogeneousshoostttionnyy data. Journal of Machine Learning Research, 2198)):1-555 2020.
Antti Hyttinen, Frederick Eberhardt, and Patrik O Hoyer. Experiment selection for causal discovery. The Journal of Machine Learning Research, 14(1):00113071, 2013a.
Antti Hyttinen, Patrik O Hoyer, Frederick Eberhardt, and Matti Jarvisalo. Discovering cyclic causal models with latent variables: A general sat-based procedure. arXiv preprint arXiv: (et:::309666 2013b.
Amin Jaber, Murat Kocaoglu, Karthikeyan Shanmugam, and Elias Bareinboim. Causal discovery from soft interventions with unknown targets: Characterization and learning. Advances Neural Information Processing Systems, 33:9551 1-6611,, 2020.
David Kaltenpoth and Jilles Vreeken. Identifying selection bias from observational data. In Proceed- ings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 8177-8185, 2023.
Kohsuke Kataoka, Makoto Noda, and Makoto Nishizawa. Maf nuclear oncoprotein recognizes sequences related to ap-1 I site and forms heterodimers with both fos and jun. Molecular and cellular biology, 14(1):700-712, 1994.
Murat Kocaoglu, Alex Dimakis, and Sriram Vishwanath. Cost-optimal learning of causal graphs. In International Conference on Machine Learning, pp. 1875-1884. PMLR, 2017.
Murat Kocaoglu, Amin Jaber, Karthikeyan Shanmugam, and Elias Bareinboim. Characterization and learning of causal graphs with latent variables from soft interventions. Advances in Neural Information Processing Systems, 32, 2019.
Shin-ichi Koizumi, Daiki Sasaki, Tsung-Har Han Hsich, Naoyuki Taira, Nana Arakaki, Shinichi Yamasaki, Ke Wang, Shukla Sarkar, Hiroki Shirahata, Mio Miyagi, et al. Junb regulates homeostasis and suppressive functions of effector regulatory t cells. Nature communications, 911):334,, 2018.
Kevin B Korb, Lucas R Hope, Ann E Nicholson, and Karl Axnick. Varieties of causal intervention. In PRICAI 2004: Trends in Artificial Intelligence: 8th Pacific Rim International Conference on Artificial Intelligence, Auckland, New Zealand, August 9-13, 2004. Proceedings 8, pp. 322-331. Springer, 2004.
Sara Magliacane, Tom Claassen, and Joris M Mooij. Ancestral causal inference. Advances Neural Information Processing Systems, 29, 2016
Christopher Meek. Causal inference and causal explanation with background knowledge. In Proceedings of the Eleventh conference on Uncertainty artificial intelligence, pp. 403-410, 1995.
Nicolai Meinshausen, Alain Hauser, Joris M Mooij, Jonas Peters, Philip Versteeg, and Peter Bühlmann. Methods for causal inference from gene perturbation experiments and validation. Proceedings of the National Academy of Sciences, 113(27):7361-73 2016.
Karthika Mohan. On handling self-masking and other hard missing data problems. AAAI Symposium, 2018
Karthika Mohan, Judea Pearl, and Jin Tian. Graphical models for inference with missing data. Advances in Neural Information Processing Systems, 26, 2013.
Joris M Mooij. Sara Magliacane, and Tom Claassen. Joint causal inference from multiple contexts. Journal of Machine Learning Research, 1-099) 2020.
13

---

Published as a conference paper at ICLR 2025
Kevin Murphy. Active learning of causal bayes net structure. 06 2001.
Whitney Newey and James Powell. Instrumental variable estimation of nonparametric models. Econometrica, 7165):1565-15777 2003.
Judea Pearl. Causality: models, reasoning and inference. Cambridge University Press, 2009.
Stefan Peidli, Tessa D Green, Ciyue Shen, Torsten Gross, Joseph Min, Samuele Garda, Bo Yuan, Linus J Schumacher, Jake P Taylor-Kiine King. Debora Marks, et al. scperturb: harmonized single-cell perturbation data. Nature Methods, 21(3):531-544 2024.
Jonas Peters, Peter Bühlmann, and Nicolai Meinshausen. Causal inference by using invariant prediction: Identification and confidence intervals. Journal of the Royal Statistical Society Series B: Statistical Methodology, 78(5):94-11012, 2016.
Álvaro Ribot, Chandler Squires, and Caroline Uhler. Causal imputation for counterfactual sems: Bridging graphs and latent factor models. In Causal Learning and Reasoning, pp. 1141-1175.
PMLR, 2024.
Thomas Richardson and Peter Spirtes. Ancestral graph markov models. The Annals of Statistics, 30 (4):962-1030, 2002.
Thomas S Richardson and James M Robins. Single world intervention graphs (swigs): A unification of the counterfactual and graphical approaches to causality. Center for the Statistics and the Social Sciences, University of Washington Series. Working Paper, 128(30):2013, 2013.
James M Robins, Miguel Angel Hernan, and Babette Brumback. Marginal structural models and causal inference in epidemiology, 2000.
Dominik Rothenhäusler, Christina Heinze, Jonas Peters, and Nicolai Meinshausen. Backshift: Learning causal eyclic graphs from unknown shift interventions. Advances in Neural Information Processing Systems, 28, 2015.
Safiye Sarper, Toshihiro Inubushi, Hiroshi Kurosaka, Hitomi Ono Minagi. Koh-ichi Kuremoto, Takayoshi Sakai, Ichiro Taniuchi, and Takashi Yamashiro. Istat- signaling regulates the epithelial stem cells in continuously growing incisors. Scientific Reports, 8(1):10906, 2018.
Comelia Johanna Franziska Scheitz, Tae Scung Lee, David James McDermitt, and Tudorita Tumbar. Defining a tissue stem cell-driven runx signalling axis in epithelial cancer. The EMBO journal, 31(21):4124 4139, 2012.
Karthikeyan Shanmugam, Murat Kocaoglu, Alexandros G Dimakis, and Sriram Vishwanath. Learning causal graphs with small interventions. In Advances in Neural Information Processing Systems, volume 28, 2015.
Shohei Shimizu, Patrik O Hoyer, Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non-gaussian acyclic model causal discovery. Journal of Machine Learning Research, 7(10), 2006.
Anil K Singh, Mahamudul Haque, Bhanupriya Madarampalli, Yuanyuan Shi, Benjamin Wildman, Abdul Basit, Sadik A Khuder, Bhagwat Prasad, Quamarul Hassan, Madhu M Ouseph, et al. Ets-2 propagates il-6 trans-signaling mediated osteoclast- like changes in human rheumatoid arthritis synovial fibroblast. Frontiers in Immunology, 12:746503, 2021.
L. Solus, Y. Wang, and C. Uhler. Consistency guarantees for greedy permutation- based causal inference algorithms. Biometrika, 108(4):795-814, 2021.
Spirtes, C Meek, and T Richardson. An algorithm for causal inference in the presence of latent variables and selection bias (vol. 1), 1999.
Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search. MIT press, 2000.
14

---

Published as a conference paper at ICLR 2025
Peter L Spirtes, Christopher Meek, and Thomas S Richardson. Causal inference in the presence of latent variables and selection bias. arXiv preprint arXiv:1302.4983 1995.
Chandler Squires, Yuhao Wang, and Caroline Uhler. Permutation-based causal structure learning with unknown intervention targets. In Conference on Uncertainty in Artificial Intelligence, pp. 1039-1048. PMLR, 2020.
Sanjay R Srivatsan, José L. McFaline-Figuerra, Vijay Ramani, Lauren Saunders, Junyue Cao, Jonathan Packer, Hannah Pliner, Dana L Jackson, Riza M Daza, Lena Christiansen, et al. Massively multiplex chemical transcriptomics at single-cell resolution. Science, 367(6473):45-511 2020.
Scott Sussex, Caroline Uhler, and Andreas Krause. Near-optimal design for causal structure learning. In Advances in Neural Information Processing Systems, volume 34, pp. 777-788, 2021.
experimental
Jin Tian and Judea Pearl. Causal discovery from changes. arXiv preprint arXiv: 2001.
Panagiotis Tigas, Yashas Annadani, Andrew Jesson, Bernhard Schölkopf, Yarin Gal, and Stefan Bauer. Interventions, where and how? experimental design for causal models at scale. In Advances in Neural Information Processing Systems, volume 35, pp. 14130-24144 2022.
Robert Tillman and Peter Spirtes. Learning equivalence classes of acyclic models with latent and selection variables from multiple datasets with overlapping variables. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 3-15. JMLR Workshop and Conference Proceedings, 2011.
Simon Tong and Daphne Koller. Active learning for structure bayesian networks. International Joint Conference Artificial Intelligence, volume 17, pp. 863-869, 2001.
Sofia Triantafillou and Ioannis Tsamardinos. Constraint-based causal discovery from multiple interventions over overlapping variable sets. The Journal of Machine Learning Research, 16(1): 2147-2205, 2015.
Aparajithan Venkateswaran and Emilija Perković. Towards complete causal explanation with expert knowledge. arXiv preprint arXiv:2407.07336 2024.
Thomas S Verma and Judea Pearl. Equivalence and synthesis of causal models. In Uncertainty in Artificial Intelligence, 1991.
Philip Versteeg, Joris Mooij, and Cheng Zhang. Local constraint- causal discovery under selection bias. In Conference on Causal Learning and Reasoning, pp. 840-860. Pmlr, 2022.
Tian-Zuo Wang, Wen-Bo Du, and Zhi-Hua Zhou. An efficient maximal ancestral graph listing algorithm. In Forty-fistt International Conference Machine Learning, 2024.
Yuhao Wang, Liam Solus, Karren Yang, and Caroline Uhler. Permutation-base causal inference algorithms with interventions. Advances in Neural Information Processing Systems, 30, 2017.
Christopher Winship and Robert D Mare. Models for sample selection bias. Annual review of sociology, 18(1):327-350, 1992.
Karren Yang, Abigail Katcoff, and Caroline Uhler. Characterizing and learning equivalence classes of causal dags under interventions. In International Conference on Machine Learning, pp. 5541-5550. PMLR, 2018.
Jiaqi Zhang, Louis Cammarata, Chandler Squires, Themistoklis P Sapsis, and Caroline Uhler. Active learning for optimal intervention design in causal models. Nature Machine Intelligence, 5(10): 1066-1075, 2023.
Jiji Zhang. On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias. Artificial Intelligence, 172(16-17):183-1188 2008.
Kun Zhang and Aapo Hyvärinen. On the identifiability of the post-nonlinear causal model. In Uncertainty in Artificial Intelligence, 2009.
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Kun Zhang. Biwei Huang, Jiji Zhang, Bernhard Schölkopf, and Clark Glymour. Discovery and visualization of nonstationary causal models. arXiv preprint arXiv:1509.000 080560 2015.
Kun Zhang, Jiji Zhang, Biwei Huang, Bernhard Schölkopf, and Clark Glymour. On the identifiability and estimation of functional causal models in the presence of outcome-dependent selection. In UAI, 2016.
Kun Zhang, Biwei Huang, Jiji Zhang, Clark Glymour, and Bernhard Schölkopf. Causal discovery from nonstationaryhheerogeneee data: Skeleton estimation and orientation determination. In IJCAI: Proceedings of the Conference, volume 2017, pp. 1347. NIH Public Access, 2017.
Yujia Zheng, Biwei Huang, Wei Chen, Joseph Ramsey, Mingming Gong, Ruichu Cai, Shohei Shimizu, Peter Spirtes, and Kun Zhang. Causal-leam: Causal discovery in python. arXiv preprint arXiv:2307.16405. 2023.
Yujia Zheng, Zeyu Tang, Yiwen Qiu, Bernhard Schölkopf, and Kun Zhang. Detecting and identifying selection structure in sequential data. arXiv preprint arXiv:2407.00529, 2024.
Hufeng Zhou, Stefanie CS Schmidt, Sizun Jiang, Bradford Willox, Katharina Bernhardt, Jun Liang, Eric C Johannsen, Peter Kharchenko, Benjamin E Gewurz, Elliot Kieff, et Epstein-barr virus oncoprotein super-enhancers control b cell growth. Cell host & microbe, 77(2):0552166 2015.
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A PROOFS OF MAIN RESULTS
Since the technical development of this puper is fundamentally built upon the characterization of maximal ancestral graphs (MAGs), focus proving only those results that do not directly follow from the basic properties of MAGss Specifically, the results stated in Theo- rem T Lemma Lemma Lemma and Lemma immediate consequences of the definition of MAGs(Richardson & Spirtes 2002 Zhang 2008 Therefore, their proof' are omitted.
Lemma 4 (When are two variables always dependent under an intervention?). For any i.j E [D], X; and X3 are adjacent in Mggn, if and only if X; and X, are adjacent in the observational MAG Mg or there exists a path between X; and X5 in Mggn, where all non-endpoint vertices are affected (i.e., in deg(1) (T)), and they, together with i, are all ancestrally selected (i.e., in an)(S)).
Proof of Lemma For any i,j E [D]. X, and Xj are adjacent in the MAG Mg if and only if there is an inducing path between X, and X, in the interventional twin DAG g(), We now consider this DAG g(), We consider by cases where j are affected by the intervention or not.
1. Consider cases where both and j are not affected, i.e., X; and X, are not splited in the DAG g(n) When X, and Xj are adjacent in the observational MAG Mggas, according to Lemma 3 either and j are adjacent in G or they have a common child that is in (Sg. Therefore, they still have this inducing path in between in the DAG g()), When the two are not adjacent themselves, but there is a path Xi,, Xlm. X,) in Mgias where ".......,mm....
C (S) and (1,.... .mm C deg( then in the DAG G(T) all of (4..... .m. are splited into (' X J. For each two adjacent vertices this path, know that are either adjacent or have a common child that is an ancestor of S g()), We can then consider this path connecting them on g(¹) (note that we can safely assume the common children, if any, are different to each other, as if, for example, 1 12 and 122 la have a same common child to render them adjacent, then we can always build the path (71, l3 ... ) to bypass (2, where 4,, 43 must also be adjacent in Mg(e:). On this path, except for the two end X; and Xj. all the vertices are not observed. All the colliders must be ancestor of S Therefore, this is an inducing path between X, and X, in g())
In above, we have proved the "=" direction. For the " direction, consider the inducing path between X; and Xj. When there are more than two nontrivial vertices on it, they must be all in X*, i.e., all the intermediate vertices are affected and splited. Otherwise, suppose there an observed X. on the path. For it to be a valid inducing path, X. has to be collider, X,... + X. + ... Xj. If this path consists of only X; + X t X,. X. must be ancestrally selected, leading to the case of adjacent X,, X, in Mg(aa. Otherwise, when there are more than X, on the path, the intermediate vertex next to X. must also not be affected (and is thus observed), as it is parent of X.. However, this observed variable cannot be a collider, making this not an inducing path anymore. Similarly, we can show all the intermediate vertices and X,, X, must be ancestrally selected.
2. Consider cases where both and j are affected, i.e., X; and X, are both splited in the DAG g(). Consider the path (X; < 6, > X; X X X; + Ej + X,) in the DAG g(), all the colliders must be in the '*' part, then similar to 1., this also be an inducing path.
For the "=' direction, since the right reality world is not ancestor of the S* variables, the inducing path, if any, must route back to the '*' world (otherwise, there are observed but non-collider vertices). The rest is similar to that of [1.].
For cases where one of and j is affected and the other is not, similar to 1, and 2., the inducing path can also be built. Also, the "=" direction goes in a same way.
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Lemma 5 (When does an intervention alters all of variable's conditional distributions?). For any j E [D], ( and Xj are adjacent in Magn, if and only if j is directly targeted (i.e. j E I), or j is both indirectly affected and ancestrally selected (i.e.. j E deg (I) n aSg(
Proof of Lemma 5
('=² direction)
If j E I, there is direct edge < + Xj in g(),, and so this directed edge is also in Mgun. Hence, (cannot be m-separated from Xj given any Xc.
Othhewiiee, j E (!g n (S). Consider the directed path from Ç to X, on the DAG g(). For any Xc C {(xx),
* If X contains no vertices on this path, then this path remains open, and thus Ç and Xj are not d-separated by Xc and S
* Otherwise, let X₁ be the first vertex on this path that is in Xc. Consider the path < X₁ f- él + Xi + X; t ej + Xx. The collider X, is conditioned on, and the collider X; has its descendant in S conditioned on. All the noncolliders from ( to X₁ are not conditioned on. All the noncolliders from X; to X; are either not conditioned on (since they are changed by intervention). Therefore, this path remains open, and thus Ç and X, are not d-separated by Xe and S 2. ('+' direction) By contradiction, suppose j & I U (deg(I) n ang(S)). j & deg (f), i.e., intervention ( doesn't 't change Xj, then X; and Ç are d-separated by empty set (X, is not even splited). Contradiction.
Otherwise, j E (Tg but j is not an ancestor of any S in G. Let C' be pag(i). We now show that this Xc m-separates ( from X,. Consider any path p between Ç and X₁ in g(1), If p contains no vertices in the counterfactual world, then by Markov condition (X, is d d-separated from its non-descent non-descendant by its parents), is blocked by Xc.
Otherwise, let X2 be the last vertex on p that comes out from the counterfactual world, i.e., p is in the form of - ................................ ....; Xi f él least the part between X, and X; should be open. So, similarly by the Markov condition, X₁ must be Xy 's descendant. Now obviously on p between (- and + e there must be a collider. Let A" be first collider counting from El. i.c., ... - A <.... + Xi + et- We use A because it must be in the counterfactual world, and can be either among X* or S A can be X7 itself. To remain p open, A" has to be an ancestor of S So X7 is ancestor of S* So X3 is ancestor of S contradicted to our presumption that j aSg( So such an open p does not exist.
Xj.. j. For this path to be open, at
Theorem 2 (Graphical criteria for Markov equivalence). For two DAGs G and H with two col- lections of targets I = (1) 11)).. {I( and J = (1)) JKK),, the Markov equiv- alence (I ~ (H, J) holds, if and only if for each = 1, ,,, the corresponding MAGs of interventional twin graphs, M gists and M Mygis, have the same adjacencies and v-structures
Proof of Theorem 2 The Markov equivalence (G.I) (H, holds if and only if the MAGs on X U {{} of each interventional twin graphs g1 and Hi(h) encode the same m-separations. According to the MAG characterization, this means that they have the same adjacencies, the same v-structures, and if path p is a discriminating path for a vertex V in both graphs, then V is collider on the path in one graph if and only if is a collider on the path in the other. Our condition covers the first two cases, and now we show that we do not need to consider the third case.
We show that any discriminating path (X, W1, W... V. Y) in a MAG of a interventional twin graph (for detailed definition about the discriminating path, please refer to Definition 7 in (Zhang 2008). Here we abuse the notation X to denote one vertex, to be consistent with the definitions there), V must not be collider, i.c., it must be V + Y. Suppose that V is a collider, it must be > Wm + V <+ Y, as otherwise there will be an almost directed cycle W + Y + V ++ W.. Then, according to Lemma 6 with those "+>" edges, all of W.. W W. V, Y are ancestrally selected, and are influenced by the intervention. Consider two cases:
Another condition for general MAG equivalence is needed here, due twin graphs' specific structures.
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1 If X is ancestrally selected, according to Lemma the path (X, W1.
Wm. V.Y) will render X and Y induced adjacent, violating the definition for the discriminating path.
2 Otherwise (X is not ancestrally selected), it must be true directed edge X + Y in the original 9. as the condition for (nontrivial) induced adjacency is violated. Then, since W1 is already ancestrally selected, it is impossible for X to be not selected.
Then, we have shown that all V nodes on discriminating paths must be non-colliders. Therefore, the equivalence of MAGs is given by the first two conditions: same adjacencies and v-structures.
Theorem 3 (Soundness of CDIS). Let (0) be the output PAG of Algorithm ! with oracle CI tests on interventional data {p(*) Ik=0
Ik=0 given by (g. T). Then, M consistent with MAG Mg. Specifically, 1. For any i j M there is also + j G, and is not ancestrally selected in G. 2. For any i j E (M), both i and j are ancestrally selected in the true DAG g.
Proof of Theorem 3 The orientation rules of Algorithm T directly follow from the characterization of MAGs in Lemma 6 The soundness of the algorithm follows from the soundness of FCT's orientation rules. The only aspect requiring further elaboration is the use of "FCI+ where all o-t edges are further oriented as + edges. This step relies on the fact that the PAGs contains no + edges.
For the purely observational PAG, this follows straightforwardy from the fact that in the MAG MG there are no bidirected edges, as our framework assumes no latent variables beyond those indexed by [D]. However, the case of PAGs derived from the MAGs Mgen needs additional explanation.
According to Lemma 6 the corresponding MAGs Mgnr may indeed contain ++ edges. However, we show that such edges cannot be identified in the PAG.
Suppose there exists an edge X; ++ X; in the MAG Mgn,, according to Lemma characterization on bidirected edges, there must be X, < + Xy. For the arrowheads to be identifiable in the PAG, another v-structure must be present.
Assume, for instance, that there is also an edge X ++ XXy Then, by Lemma 4 the existence of an inducing path through X, implies that X, must also be adjacent to X₃ in the MAG. This results in a triangle X,, Xx) whose orientation remains unidentifiable.
Consider the other scenario where X₁ + Xy. In this case, X₂ is also reachable from X; and subsequently from X; to X,. This inducing path still renders X₁ and X, adjacent, to form triangle where definitive determination of orientation becomes impossible.
B PSEUDOCODE FOR THE CDIS ALGORITHM
Before detailing CDIS. introduce key notations. Like CPDAG for DAGs, ancestral graph (PAG (Spirtes et al., ) to represent a class of MAGs with same adjacencies. A PAG is a graph with six kinds of edges marks shared by all MAGs in the class. FCI Zhang 2008) is an algorithm that identifies the true PAG from data. In the pseudocode, FAS (fast adjacency search) denotes a function that takes data and returns PAG. Adjacencies are determined by CIs, with orientations as ia-kt-oj for v-structures and io oj otherwise") FCI" denotes a function that takes in PAG, recursively applies the ten FCI rules and an extra rule to orient all a-t -, and returns the PAG when no further orientations can be made.
partial
o-, o-o, o->) where non-circle marks indicate
The pseudocode for the CDIS algorithm is provided below:
SPAS can be implemented by the PC algorithm followed by additional step of skeleton refinement, For further details, see (Spirtes 2000), p. 144.
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Algorithm 1: Causal Discovery from Interventional data under potential Selection bias (CDIS)
Input: Observational and interventional data {p(k) k==0 over [jd] with unknown targets.
Output: A partially ancestral graph (PAG) over vertices [D].
Step 1: Get maximal orientation from pure observational data. M <- FCI (FAS(p
Step 2: Get MAG adjacencies from interventional data. for k: +- 1,... do
M(K) + FAS(p(o) p(()) running PC on pooled data of p! and p(k) over {[]U[D), with Cls in forms of Theorem Pruning can be made: any adjacency in (1) must appear in (
Step 3: Refine M using graphical criteria on MAGs of twin graphs. repeat
Step 3.1: Orient ( based on current knowledge. for k + 1, K do foreach i j E (0) do Orient i j in (M)), suggested by Lemma 6
foreach adjacent Ç, do Orient ( in M((k, as intervention indicator exogenous; ((R) FCII (FM(((M)))) further orientation with above background knowledge edges;
Step 3.2: Update (7 using information from (M) foreach adjacent i,j (0) do
3k such that - j E ((() then Orient i -j in Mi
3k such that + E (R) and io-j E (0) then Orient - j in M
3k such that + E (h) and (0) (X,) * p(a) (X.) then Orient i + j in M
if 3k o K2 such that + j E ((k) and + j E (itss) then Orient i - j in M
Step 3.3: Further orient M based on current knowledge. M + FCI+(MM))))))
until no further orientations are found for (0) in Step 3.2; return M0))
C MORE ELABORATIONS ON EXAMPLES AND MOTIVATIONS
C.I SIMULATION FOR THE PEST CONTROL EXAMPLE
0.66
0.25
$7
0.50
2 03
2 0.20
0.45
0.2
0.40
X1
Xi
X1
X:
(a) p (X,,
(b) (X,, X2 m2)
(c) (X, Xa)
(d) (X, XslX, 22)
Figure 6: (a) shows the scatterplot of X,; X3 in general population (both 'x" and 'o'), with only individuals involved into study (0)) (b) shows the scatterplot of X1, X3. in ())) with X₂ conditioned on its mean value 22, illustrating the conditional independence X, IL Xx/X, in (()) Applying an intervention to X, on the 'e' individuals (a), we get the interventional distribution (11), (c) shows the scatterplot of X1; X3 in (1)), (d) shows the scatterplot of X X3 in (0)) with X₂ conditioned on its mean value 22, illustrating that the intervention destroys the original condition independence, i.e., X1 J X1 X2 holds in p!
Following Example
let the DAG 9 be + 2 + 3 > St. Data is simulated as follows:
E1, E,, Eg U -1,11 independently;
X1 = En;
X2 = sigmoi xxx + E2);
X; = sigmoid( -222 + Ex);
S 1(x; 0.4).
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Only individuals with S1 = 1 values are involved into the study and get observed, corresponding to the 'o' markers in p()) in Figure 6a For these individuals, an intervention to lift X1 is made:
X1 x E', with E' (4.
That is, individuals are expected to gets a lift of 4 in its X1 values, while a variance of 1 is also given model the randommess in applying real interventions. X2, X3 are then generated using the same generating functions and their same E2, E3 values above. The scatterplots of the resulting
interventional distribution p()) are shown in Figure 6c
To illustrate the condition independence relation X1 T X3/X2, we show in Figure 6b and Figure 6d the scatterplots of X1 X3 with X2 conditioned on their specific mean values in p!o) and p I respectively. Clearly, X1 Il. X3l X2 holds in p!0) (though to be rigorous, the scatterplot on a single conditioned 72 value is not enough), while this condition independence no longer holds in the interventional data p(!).
C.2 WHY NOT SPLIT EVERY VARIABLE INTO COUNTERFACTUAL AND REALITY VERTICES?
In our definition of the interventional twin graph (Definition T). the graph is defined for each single intervention with target 7, instead of on a collections of targets I. This is different from the usual augmented graph setting where only one augmented graph is defined, with multiple exogenous intervention indicators (1,-.., .'.. The specific reason why we do not choose to use only one combined graph is that, our definition of the graph depends on each specific L, namely, only variables that are affected from an intervention target are split. Questions then naturally arise: can we split all variables into the two worlds, as in typical twin graph models (Balke & Pearl [994)) In this way, can we formulate single graph that represents for the whole I, which seems simpler?
In what follows, we show that a single graph with all vertices split is doable. However, in contrast to our expectation, it introduces more unnecessary complexities.
First we define this alternative model. Let 1 and 0 be vectors of all Is and all Os, respectively, and 1 be the vector with 1 at its k-th entry and Os elsewhere (by convention, 1, 0).
Definition 6 (Alternative one-for-all interventional twin graph). For a DAG G over [D] U S and a collection of targets I = (7(K)) K k= the alternative one-for-all interventional twin graph g is a DAG with vertices X S U UUUU U UUG where:
<= [=(Sk are intervention indicators: Ç= 1 denotes a sample from the k-th interventional distribution (k)), Specifically, < <= 1o 0 denotes for the pure observational samples from (()),
X [ are variables in the observed reality world, pure observational or interventional;
X = [X;] and 5* = (S } are variables in the unobserved counterfactual basal world, representing the corresponding variable values and selection status before the interventions;
= (c are exogenous noise variables capturing common external influences on X and X*.
GD consists of the following four types of direct edges:
Direct causal effect edges in both worlds: {X; + X;, + Xjjr-se9, weii:
Selection edges in the counterfactual basal world: [X; + hj,- je(t)
Exogenous influence edges: {er + X;, + X; heepp:
Edges representing mechanism changes due to interventions: (GA- + Xi))iemn,Ke((H ME(K).
Using structural equation model (SEM) notation, denote by f the generating function that maps (X* (1) <4) to X; and ( = the functions (if exists) that maps (X (t) 6:) to X, in corre- sponding pure observational or interventional distributions (where ç= 1x). We assume the counter- factual basal world operates the same as the pure observational world. For each intervention targeted
at (k), we also assume non-targeted variables' generating functions are invariant to this intervention:
Vk E [K], Vi (*))
f/(²) exists and f(()) fr
(C.1)
The graphical structure g with the invariance constraint Equation C.1 completes our definition.
An illustrative example of the alternative one-for-all interventional twin graph is shown in Figure Readers may compare it with the ones shown in Figure 3
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D
X
Counterfactual Basal World
CK
Reality World
Figure 7: The alternative one-for-all interventional twin graph (Definition 6). and solid ( nodes are observed, forming the reality world, where observations interventions conducted. The grey nodes are unobserved, of which squares are latent variables and double circles are selection variables. The unobserved X and S* variables form the counterfactual basal world where interventions have not been applied. The two worlds are linked by E, latent exogenous noise variables that commonly influence X and X*
The white nodes
X
X
X
(d)
CK
Figure 8: Examples showing how d-separations in one-for-all twin graph can lose information. (a)
DAG g. (b) G for arbitrary I. (c) G constructed from (b) in Lemma 7 where all d-separations among X Ç remain the same. (d) Another DAG H. whose H is also (c).
As we have shown in Lemma referring to the original DAG G is not Markovian for interventional distributions. Then, does the one-for-all model G fully capture the interventional distributions? The answer is still no: G may be unfaithful, i.c., there CIs not implied by d-separations in G A trivial example is that, one could construct the G for the pest control example where G = - 2 + - Si and I {Ø, {1}}- 9 encodes d-connection X1 Xx S,S", which is indeed the case for X1 IL X3 X2 in ()), However, if we let I {2, {3)), the d-connection Xı Xa/ S* still holds, while this time, in p()) it is actually X, 1L X3|X2. To explain this, one may notice that intervention on X3 does not change the values of X₂ for individuals, and therefore this case, when conditioning on X2, the counterfactual value X3 is automatically conditioned on, blocking the open path.
We have shown a major issue of the one-for-all twin graph, or, the issue of splitting every vertices into worlds: the d-separations over the graph do fully capture the conditional independencies implied. Then, one may wonder, what is exactly the d-separations in this one-for-al twin graph? We show the following lemma characterize these d-separations, and also to illustrate why the onc-for-all twin graph misses key distributional information:
Lemma 7. For each one-for-all twin graph G², construct another DAG G by removing the X* nodes and associated edges, and adding edges {c; + S Hienne(SS) <E(D), je(m) iee. selection now applies ancestrally on exogenous noises. Then, G and G entail the same set of d-separations over X X
Examples illustrating Lemma 7 are presented in Figure For each one-for-all graph GT. the counterfactual values X can be further marginalized, allowing the construction of a new graph, GT. This new graph excludes X and includes edges from & ancestrally pointing to S*. Importantly, the d-separation patterns remain unchanged. Subfigures (c) and (d) show this construction. Then, what is the consequence of this equivalence? Consider the following example:
Let DAGs and 7. shown in (a) and (b). They share the same d-separations among X U U in their respective GI and HI for arbitrary I, as both G and H are equivalent to the GT shown in (d). Then, does this imply that the G and H are indistinguishable under any I? The answer is no. Actually, they can already distinguished in (0), where 1,3/2, 4, S1 holds in & but not in H.
From above, have shown that the d-separations alone of the one-for-all interventional twin graph can fail to characterize the distributions. There are two specific losses: one is determinism, i.c., when some unaffected variable is conditioned on, its counterfactual basal variable will also be conditioned
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on; the other is the loss of sparsity of selection mechanisms, i.e., it falsely represents selection as being applied ancestrally on exogenous noise terms instead of on X variables. While we can solve these issues by defining Markov properties in technically heavier way, choose to interventional twin graphs as defined in Definition where the d-separations are exactly all the conditional independence implications.
D RELATED WORK
In this section we give a more comprehensive review of literature.
When only pure observational data is available. There are constraint-based causal discovery algorithms (Spirtes et al., 2000), score- based algorithms (Chickering 2002), and methods that utilize
properties of functional forms in the underlying causal process (Shimizu al. [2066] [Hoye et al.)
2008 Zhang & Hyvärinen 2009). The corresponding Markov equivalence characterization can be
referred (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 experimental data is available, previous literature has considered two types of interventions to model how experimental data is generated: hard (or perfect) interventions and soft (or imperfect) interventions, also known as mechunism change. Hard interventions destroy the dependence between targeted variables and their direct causes, either by deterministically fixing the target variables to specific values, or by stochastically setting them to values drawn from independent random variables (Pearl 2009 Korb al. 2004). In contrast, soft interventions do not destroy the aforementioned dependence, and they modify the functional form that characterizes the causal generating mechanism of targeted variables (Tian & Pearl 2001 Eberhardt Scheines) 2007).
The earliest attempts on interventional causal discovery. The earliest Bayesian methods are introduced by (Cooper & Yoo| T999. Eaton & Murphy] 2007). compute the posterior distribution of DAGs using both observational and interventional data. These methods however did not address critical challenges like identifiability or equivalence class characterization. (Tian & Pearl 2001) is the first to consider identifiability and the Markov equivalence for interventional causal discovery. They single-variable consider the single- variable interventions with mechanisms change (soft interventions). graphical criterion for two DAGs being indistinguishable is given, but no graphical representation for such equivalence class characterized.
Treatments on hard interventions. (Hauser & Bühlmann 2012) first considers the characterization
of MEC with hard stochastic, multiple-variabl interventions. There graphical criterion is based on the mutilated DAGs as introduced in $2] and the graphical representation for equivalence class is given by Z-essential graphs. Their graph criterion actually is consistent with (Tian & Pearl "s though the latter focuses on single vvrrible interventions only. The provided algorithm GIES Hauser & Bühlmann (2015) is basically utilizing the CI relations in each experimental setting and integrate results together. Under such paradigm, methods such as (Tillman & Spirtes) 2011] 1. Claassen & Heskes 2010) are also developed. (Wang et al.) 2017) show the consistency issue of GIES when certain Taithfulness assumptions are violated, and propose the new permutation-basse algorithms (Wang
ct al. 2017).
For soft interventions, exploiting invariance in mechanisms. The basic idea of using the (in)variance of causal generating mechanisms and the asymmetries among them to identify causal relations can root back to (Hoover 1990) with its application economics. The invariant causal inference framework (Meinshausen et al. 2016 Rothenhäusler et al.) 2015 GGassami et al. 2017 Peters et 2016) is developed, though they typically require parametric assumptions such as linear models and the problem is transformed to regression analysis. Such invariances is later seen as conditional independencies between an augmented exogenous domain index variable and the other variables, which further can be related to the d-separation conditions graphs. The interventional causal discovery can then be unified with pure observational causal discovery, by viewing domain indices causal variables. Such representation has been discussed in e.g., (Korb et al. [2004] (Nweyy & Powelll 2003). In the causal discovery field, it is proposed by (Zhang et al. 20155) and got for- Mooij al. 2020), providing a unified way of
malized in (Zhang al. 2017 Huang al. 2020
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seeing data from multiple domains with mechanism change. The graphical criterion for Markov equivalence under soft interventions is given in (Yang et al. 2018). As with the earlier consistency between (Hauser & Bühlmann 2012) and (Tian & Pearl 2001). it is shown that long the pure observational data is available, the equivalence condition for hard interventions and soft interventions the same. The issue of unknown intervention targets, also known "fat hand" issue, is also directly solvable from the augmented graph (Squires et al. 2020 Jaber et al. 2020).
When latent variables are involved. In the pure observational data and nonparametric causal discovery setting, the frameworks of MAG and FCI have been well established (Richardson & Spirtes 2002] Zhang 2008). For interventional causal discovery, various methods have been proposed to address latent variables (Hyttinen et al. 2013b Triantafillou & I'samardinos] 2015 Kocaoglu et al.
cstabisshedRRichardson
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.
Another parallel line of study: active experimental design. Active experimental design is a closely related area of research, but with distinct focus. In the interventional causal discovery setting, the interventional data are passively observed. In active experimental design however, we have control over experiments. The aim is then select specifie targets for intervention in sequence of steps to efficiently uncover the final DAG. It can be roughly drawn into two lines. The first line
is graph based methods, such (He & Geng 2008 Eberhardt 20088 Eberhardt et al. 2005 2010 Hyttinen et al. 2013a 3a Shanmugam et al.) 2015 Kocaoglu et al. 2017 Ghassami et al. 2018 which characterize the equivalence at each step, and considers counting the DAGS in each equivalence class so as to find the next step interventions that can possibly maximally reduce the DAG search space.
The second line Bayesian based methods, such as (Tong Koller [00T] Murphy [00T] Agrawal
et al.) 2019 Sussex et al. 2021 Tigas et al.) 2022 /Zhan et 2023). which treat the problem as an optimization problem.
Modelling the interaction between reality and counterfactual world. At first glance, our model seems similar to the twin network defined by (Balke & Pearl 1994), as both link reality and counterfactual worlds through exogenous noise. This is where name "twin' is echoing. However, key differences exist. Twin networks, or single world intervention graphs (SWIGs (Richardson & Robins 2013)), are used as diagrams to guide counterfactual queries when the structure is known, while we discover structure from data. Their CI queries are in forms of X X3 Xc,, so as to find Cls to calculate counterfactual quantities from the reality quantities, while we check the (in)variances of pC))) Also, they only model hard interventions while we handle soft ones. The most relevant model we find is from (Ribot et al. 2024) with a different focus on imputation. They assume linearity and have non-fixed S while we explore Markov properties nonparametrically.
Causal discovery in the presence of selection bias. For causal discovery from selection biased data, one of the earliest approaches is the Fast Causal Inference (FCI) algorithm (Spirtes et al.) 1995). which leverages conditional independence relations to infer causal structures while accounting for both latent confounders and selection bias. Following this foundational work, most subsequent methods have also relied on conditional independence constraints to recover causal relationships (Hernán el al., 2004: Tillman& & Spirtes [011 Evans & Didelez 2015 Versteeg et al.) 2022 Zheng 2024). Several parametric approaches have been developed for bivariate causal orientation (Zhang et al.. 2016, Kaltenpohh & Vreeken 2023). Other research efforts with selection bias have focused on causal inference (Bareinboim et al., 2014 Correa et al., 2019). Structure learning and inference results can follow different methodologica directions depending on the problem setting. A particularly relevant area is the study of data missingness, which shares similarities with selection bias. For instance, in cases involving se self-masking missingness, the true data distribution and model parameters may be unidentifiable, making causal inference infeasible (Mohan et al. 2013 Mohan 2018). However, the causal structure may still be recoverable (Dai al. 2024).
E SUPPLEMENTARY EXPERIMENTAL DETAILS AND RESULTS
We the implementation of IGSP, UT-IGSP, and JCI-GSP from the causaldag package (Chan dler Squires 2018), and the implementation of CD-NOD from the causal- package (Zheng al. 2023). We use the Fisher 7. test to examine conditional relations. The significance level is set
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Published as a conference paper at ICLR 2025
to 0.05 for 5 and 10 variables, and 0.01 for 15 and 20 variables.
In what follows, demonstrate the resulting graphs real datasets. In these graphs. the red square nodes and their outgoing edges represent the intervention targets across different interventions. Among observed variables, X; - X, indicates causal edge from X; to Xj and X, is not ancestrally selected; X; -x indicates that both X; and Xj are ancestrally selected; X,o-XX indicates that each endpoint may vary in the equivalence class. For sim- plicity in showing the DAG and the estimated intervention targets at the same time, we put multiple intervention indicators, observed variables, and the selection variables, if any, all into one graph. This is merely for presentation ease; according to Example [ and Lemma 7 such graphs do not characterize the CI relations in data. The edge interpretation applies to all figures below.
RNNI
Figure 9: Causal structure estimated by our CDIS method on single-cell perturbation data, i.e., sciPlex2 (Peidli et al. 2024). We discovered some validated regulatory relationships like RELA - RUNXI and JUNB MAFF, since RELA has been implicated
RUNX3 transcription regulator (Zhou et al. 2015) and Maf family was upregulated by JunB (Koizumi et al.) 2018). despite the extensive co-regulation between these two genes (Kataoka et al. 1994). Besides, the link between ETS2 or RUNXI and STAT3 may encounter the risk of spurious correlation induced by selection (i.e., conditioning on particular cell line). It is supported by previous studies since it reported Runx1 and Stat3 synergistically driving stem cell development in epithelial tissues trough Runx I/Stat3 signalling network (Scheitz et al. 2012 Sarper et al., 2018). while TF Ets2 together with P-STAT3 activation induce cathepsins K and B expression in human rheumatoid arthritis synovial fibroblasts (RASFs) (Singh et al. 202T).
Category
Variable
Meaning
GPA yearl
1st year GPA
goodstandi~~1
Good standing in year
Main outcome
prob_yearl
On probution in year I
poddit__cav))
Credits carned in year
mathsci
Number of math and science credits attempted
female
Sex (Female dummy)
Personal backgrounds
age
Age
english
Whether Mother tongue is English
gpa0
High school GPA
heom
Whether Lives at Home
chouseUTM
Whether first choice schoul
work1
Whether Plans work while in school
dad_cet edn
Father education
mom edn
Mother education
Other covariates
Im rarely
Whether Rarely puts off studying for tests
Im never
Whether Never puts off studying for
lastmin
how often do you leave studying until the last minute for and exams
graddeg
Whether Wants more than BA
finish4
Whether Intends to finish in years
Table 1: Description of variables in the educational dataset
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Published as a conference paper at ICLR 2025
chouseUTM
12
mam
II
B3
mitriuec
graddeg
Jnrrrrely
jevvrr
yearl
workl
finiish4
lastmin
gpan
goodstanding yearl
have
fémale
credit camed!
prob
Figure 10: Causal structure estimated by our CDIS method on an educational dataset over all students. The interventional indicators represents SSP, represents SFP, and 13 represents SFSP.
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Published as a conference paper at ICLR 2025
AgP
chooseTMM
momm edn
gracdlgg
im never
gport
moongue
mathsci
goodstanding_oor year)
hoom
prob
credioo_camnel
(a) Results male students.
lcoo
lastmin
I2
gaodearding_gen
gradieg
claoseLTTM
finishé
GPA yearl
mathsci
ago
moze edn
Ino. rarely
creitts camedl
upob
prob yearl
(b) Results on female students.
Figure 11: Causal structure estimated by our CDIS method educational dataset conducted with subgroup analysis stratified by gender.
Rocall of edges
SHDD of skeletton
30
0.75
20
0.50
10
0.25
0
10
15
20
5
10
15
20
Number of variables
Number of variables
CDIS
IGSP
JCI-GSP
GIES
UT-IGSP
CD-NOD
Figure 12: Empirical results across different numbers of variables. The two figures show the SHD of the PAG skeletons, and the recall of the "-2" edges in PAGs. The error bars illustrate the standard errors from random simulations.
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Reviews:
[{"review_id": "HQXAnIR0si", "reviewer": "Reviewer_VRoQ", "summary": "The work shows that current theory for dealing with interventional causal discovery is insufficient under selection bias, as there are cases where the selection mechanism takes place before the intervention. The authors then propose a twin graph framework where the selection happens before the intervention and then define a Markov property on this twin graph that implies certain independences in distributions when conditioned on the selection mechanism. The authors then propose a method that constructs the graph based on the Markov property up to an equivalence class.", "questions": "- In example 2: This will be much more convincing if there is a numerical example that actually shows this (like example 1). Otherwise, it's not obvious to me why selection might imply that $X_1, X_3$ are dependent when conditioning on $X_2$.\n- In section 3.2: It is claimed that interventional distributions are not Markovian due to the original dag due to $X_1$ not being independent of $X_3$ when $X_2$ is conditioned on. The open path here goes through $X_2^*$. However, in example 4 it is claimed that conditioning on $X_2$ automatically conditional on $X_2^*$, This seems contradictory to the previous statement, what am I missing here?\n- Thm 1: If this is contrasted directly with the regular Markov property in DAGs, it will help clear up what the extra condition here is. Reading this section, the extra condition has not been motivated with respect to the examples shown. For example, how does this theorem differentiate DAGs $\\mathcal{G}$ and $\\mathcal{H}$.\n- I don't really understand how Lemma 1 \"misses key distributional information\", could this be made more explicit?\n- L291: Why does the twin graph have fewer degrees of freedom with the invariance constraint? And why does it lead to CIs not implied by d-separations? The claim in L293 is also not clear at all to me.\n- Section 3.3: Intuitions behind the lemmas and why they are needed would greatly help here. Right now they are hardly motivated and tough to understand given the dense notation.\n- Algorithm 1 step 2: how is this done? What method is used to test if the conditional distributions are the same or not?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- The authors have found an interesting flaw in previous methods in the presence of selection bias\n- The paper is well motivated with clear examples. Some examples can be made clearer (see below)", "weaknesses": "Some bits of the exposition are unclear (see below).", "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": "Lbih37V0tX", "reviewer": "Reviewer_riv8", "summary": "This paper addresses the issue of selection bias in interventional causal discovery, highlighting how existing methods fall short when subjects are selectively enrolled in experiments. To address this, the authors introduce a new graphical representation called \"interventional twin graph\" that explicitly represents both the observed world (where interventions are applied) and the counterfactual world (where selection occurs). They characterize the Markov properties of the proposed graphical model and develop an algorithm built upon FCI based on the proposed graphical model and Markov properties, named CDIS (Causal Discovery from Interventional data under potential Selection bias). The authors prove the soundness of CDIS. Experiments conducted under selection bias conditions demonstrate that CDIS outperforms baselines on synthetic datasets and uncovers novel causal relationships in real-world applications.", "questions": "1. Please refer to the questions mentioned in Weaknesses.\n\n2. I think theoretically the proposed framework (the proposed graphical model + algorithm) should be able to handle interventions without selection bias, but how does the proposed algorithm perform empirically compared to existing methods in scenarios without selection bias?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The selection bias in interventional experiments is an under-explored but crucial issue in causal discovery, and the authors use two clear examples to illustrate why this problem matters and why the existing methods and simply augmenting DAG fail.\n\n2. The authors provide a solid theoretical foundation in this paper by (1) rigorously defining the interventional twin graph and characterizing its Markov properties and (2) proving the soundness of the proposed algorithm.\n\n3. Synthetic experiments show that the proposed method outperforms baselines in handling selection bias and remains robust as the number of variables increases. It also uncovers novel causal relationships in real-world applications.", "weaknesses": "1. While the introduction of the interventional twin graph and its Markov properties is rigorous, it may be challenging for readers to grasp at first glance due to its complexity. Providing a high-level explanation to offer an intuitive understanding would greatly benefit readers.\n\n2. The interventional twin graph is more complex than a simple DAG and involves additional nodes. It would be helpful if the authors discussed the computational cost of the proposed model compared to the simpler DAG, including an analysis of the algorithm's computational complexity under the new graphical model.\n\n3. The authors did not address the identifiability guarantees of the proposed method. It would be useful to know if the method can reliably identify the selection variables and under what conditions the true interventional twin graph can be identified.\n\n4. Minor typos:\n    * Line 48: \"We show that existing existing graphical representation paradigms\" --> \"We show that existing graphical representation paradigms\"\n    * At the end of line 169: \"models a completely different scenario\" needs to be revised", "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": "jz41Nr15z8", "reviewer": "Reviewer_5bYr", "summary": "This paper addresses the limitations of current graphical models and causal discovery methods in handling pre-intervention selection bias in interventional causal discovery. To overcome these limitations, the authors propose a novel twin graph model that effectively captures both the observed world and the counterfactual world where selection occurs. The paper establishes the Markov properties of this new model and introduces a provably sound algorithm for identifying causal relationships. The effectiveness of this approach is demonstrated through synthetic and real-world experiments.", "questions": "1. What are the main challenges that affect the precision and completeness of the algorithm? How sensitive is it to the unmeasured confounders?\n2. If selection bias is detected through diagnostics, how can this information be leveraged to help causal discovery?\n3. Any data points on the computational cost of the algorithm?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The problem of selection bias is important yet often overlooked in existing interventional causal discovery. The setting is general.\n2. This paper is technically sound, with clear formulation of the causal DAG and Markov properties. \n3. The illustrative examples enhance the paper's clarity, helping readers better understand the concepts.", "weaknesses": "More comprehensive results and explanations of the empirical studies would be beneficial to support the effectiveness of the proposed algorithm. For example:\n1. For the simulation, can you report the proportion of true causal edges estimated as directed edges by the algorithm? Additionally, a comparison of the output graphs with the ground truth would illustrate how the new algorithm perform differently from other methods under selection bias. \n2. For the gene application, can you provide more comprehensive analysis of the result? \n3. For the education dataset, can you explain why the pre-intervention selection bias is a potential issue? Highlighting and interpreting key information of the resulting graphs would be helpful, as the current graph and variable names are difficult to follow.\n\nMinor comments about clarity:\n- The notation in this setting is dense and improvements of readability can be helpful for readers less familiar with the area. For example, \"CI\" in line 93 and different types of arrows in Example 7 can be clarified before their first appearance.\n- Typos: line 48 \"existing\", line 295 \"false\".", "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": "zjH4Ii6fII", "reviewer": "Reviewer_kjX2", "summary": "This paper considers the problem of interventional causal discovery in the context of selection bias. The paper first explains why simply augmenting the causal DAGs with a selection variable is insufficient. The paper then goes on to introduce the concept of a twin graph, in which every node is replicated in a counterfactual/\"basal\" world, and there is a defined set of rules by which the relationships of the basal world are constructed. Based on this construction, the paper goes on to define the set of d-separations that are implied by this model and establish the graphical criteria for Markov equivalence. The paper provides an algorithm to recover the Markov equivalence class and evaluates it on both simulated and real-world data.", "questions": "Can you think of any ways in which you could evaluate the completeness of your proposed algorithm in the simulations? In a causal discovery paper, we should be concerned with making true discoveries, not just avoiding false discoveries.\n\nDoes the paper's approach generalize to arbitrary latent variables?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper clearly lays out the problem of selection bias in causal discovery and why certain natural approaches to the problem are not sufficient. The paper also puts forward a very general solution to the problem and considers its consequences. Overall the paper is well-written, despite being notation-heavy.", "weaknesses": "One thing that was unclear to me is how complete is the paper's characterization of Markov equivalence classes in the given model. I would think that the Markov equivalence class would encode all DAGs with the same conditional independence structure (i.e. the right-hand side of the implications in Theorem 1). However, the equivalence structure is defined with respect to the left-hand side of the implications in Theorem 1. This would seem to imply that the equivalency classes are not as fine-grained as they potentially could be.\n\nThe algorithm provided also suffers from this issue, in that the authors point out that it may not be complete. It is not clear to me how useful it is to have a causal discovery algorithm that is sound but not complete. The trivial algorithm that says there are no causal relationships is sound but not useful.\n\nThe simulation study not reporting any information on completeness is disheartening. While I understand that the paper does not contain any guarantees on completeness, in the simulations there is access to the ground truth. So it is hard to see how there is no way to evaluate the ability to discover some fraction of those relationships.\n\nAt a higher level, I'm not sure how much of the framing of the paper is specific to the selection problem. It seems like the approach of the paper is tackling the more general problem of causal discovery with unobserved latent variables. If that is not the case, then the paper should explain how their methods do not generalize to the latent variable setting.", "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": "olunTGthvI", "reviewer": "Reviewer_9cf1", "summary": "This paper addresses the important but overlooked problem of selection bias in interventional causal discovery, where subjects are selectively enrolled in experiments (e.g., drug trials focusing on patients with specific conditions). The authors show that existing methods for interventional causal discovery fail to properly handle selection bias, as subtle differences in when and where interventions occur can lead to inconsistent conditional indepenence relations. To address this, they introduce a novel graphical model called the \"interventional twin graph\" that explicitly accounts for both the observed world (where interventions are applied) and the counterfactual world (where selection occurs before interventions), along with characterizing its Markov properties and equivalence classes. They develop a provably sound algorithm called CDIS (Causal Discovery from Interventional data under potential Selection bias) that can identify causal relations and selection mechanisms from data with soft interventions and unknown targets. Through experiments on both synthetic data and real-world applications in biology and education, they demonstrate that their method achieves higher precision in identifying true causal relations compared to existing approaches when selection bias is present.", "questions": "-", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- This paper studies a relevant and interesting problem - both mathematically and philosophically. It considers the question: \"What does selection bias actually mean\" and proposes a sound answer and the necessary mathematical framework to deal with such situations.\n- Based on the framework to treat selection bias, a sound and complete causal discovery algorithm is proposed.\n- The method is evaluated not only on synthetic data, but on real-world examples as well. This provides some confidence that it may be useful in practical applications.", "weaknesses": "The biggest weakness I see is the presentation of the paper. The first two sections are dense, but give a good introduction and motivation to the problem, based on good illustrations in Examples 1 and 2.\n\nHowever, Section 3 is the most painful piece of text I have read in a while. It relies mostly on mathematical notation to bring across the main points and lacks the contextualization in prose. I appreciate that examples are given in Section 3, but even those are a bit cryptic and fail to provide an accessible intuitive understanding. I suppose there are three main reasons for this: (1) Writing about complex SCMs is inherently difficult and a certain level of formalism is necessary - not much you can do here. (2) The amount of content in the main paper, given the page limit might a bit too much. Some of the more technical parts could be relegated to the appendix and exchanged for more contextualization. (3) The text could consider the reader's state of mind more. Some examples:\n\n- L211f: introducing the functions $f^*$ uses the mathematical symbols for the corresponding variables in the counterfactual basal world to introduce them, but does not use the word \"counterfactual\". That means as a reader, I either have it in working memory, or I have to go back to the definition and jump back again to the sentence to parse it.\n- As far as I can tell, abbreviations like \"CI\" and \"MAG\" aren't defined, or used before they are defined, e.g. \"PAG\".\n\nSuch presentation choices add unnecessary mental effort for understanding, and I would think twice if I'd go back to this paper and build on it for future work - not because it's wrong, but because of the mental effort to access the information.", "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 (Oral)