Latent Variable Causal Discovery under Selection Bias
Authors: Haoyue Dai
Yiwen Qiu
Ignavier Ng
Xinshuai Dong
Peter Spirtes
Kun Zhang
Venue: ICML 2025 poster (2025)
Keywords: causal discovery
latent variables
selection bias
rank constraints
covariance matrix
Primary area: general_machine_learning->causality
TL;DR: This paper develops a graphical and algebraic tool–generalized rank constraints–that enables causal discovery among latent variables under selection bias.

Abstract: Addressing selection bias in latent variable causal discovery is important yet underexplored, largely due to a lack of suitable statistical tools: While various tools beyond basic conditional independencies have been developed to handle latent variables, none have been adapted for selection bias. We make an attempt by studying rank constraints, which, as a generalization to conditional independence constraints, exploits the ranks of covariance submatrices in linear Gaussian models. We show that although selection can significantly complicate the joint distribution, interestingly, the ranks in the biased covariance matrices still preserve meaningful information about both causal structures and selection mechanisms. We provide a graph-theoretic characterization of such rank constraints. Using this tool, we demonstrate that the one-factor model, a classical latent variable model, can be identified under selection bias. Simulations and real-world experiments confirm the effectiveness of using our rank constraints.

Latent Variable Causal Discovery under Selection Bias
Haoyue Dai Yiwen Qiu Ignavier Ng'
Xinshuai Dong'
Peter Spirtes' Kun Zhang 12
Abstract
whether involving latent variables or not, is the data be- ing randomly sampled from the underlying population. In practice, however, this is often violated due to selection bias preferential inclusion of data points based on un- known mechanisms (Heckman, 1977). Returning to the ear- lier examples, individuals with certain traits may be more willing to take a psychological survey, and methods like mail phone used to recruit respondents can systematically skew groups based on factors like economic and education level. Ignoring such bias can severely distort the inferred causal structures. Moreover, uncovering the selection mech- anisms is also crucial for understanding the data. Hence, there is pressing need for methods of latent variable causal discovery that can address selection bias.
Addressing selection bias in latent variable causal discovery is important yet underexplored, largely due to a lack of suitable statistical tools: While various tools beyond basic conditional indepen- dencies have been developed handle latent vari- ables, none have been adapted for selection bias. We make an attempt by studying rank constraints, which, as a generalization to conditional inde- pendence constraints, exploits the ranks of co- variance submatrices linear Gaussian models. We show that although selection can significantly complicate the joint distribution, interestingly, the ranks in the biased covariance matrices still pre- serve meaningful information about both causal structures and selection mechanisms, We provide a graph-theoretic characterization of such rank constraints. Using this tool, we demonstrate that the one-factor model, a classical latent variable model, can be identified under selection bias. Sim- ulations and real-world experiments confirm the effectiveness of using our rank constraints.
Despite its importance, addressing selection bias in latent variable causal discovery remains almost unexplored, the best of our knowledge. One may first recall the Fast Causal Inference (FCI) algorithm (Spirtes et al., 1999), which in- deed exploits the conditional independence (CI) constraints in data under both hidden confounding and selection bias. However, FCI typically not regarded as method of latent variable causal discovery, as focuses solely on causal relations among observed variables, with no intension or capability to identify those among latent variables. In short, though FCI can handle both hidden confounding and selection bias, and is already maximally informative under nonparametric CI constraints (Richardson & Spirtes, 2002; Zhang, 2008), is still not informative enough for latent variable causal discovery. Therefore, new statistical tools that go beyond CI constraints must be developed.
Introduction
At the core of understanding complex systems lies causal discovery, the identification of causal relations from obser- vational data (Spirtes et al., 2000; Pearl, 2009). In many real-world scenarios, the variables of interest are latent constructs that cannot be directly observed or quantized, while the observed variables serve merely as indirect measurements. For instance, in psychological political- economic surveys, measured responses serve as proxies for latent personality traits or political orientations. Recovering the causal structure among these latent variables- refeeredd to as latent variable causal discovery- is essential for un- derstanding and reasoning, yet remains a challenging task.
Many new tools beyond CI constraints have thus been devel- oped, typically by imposing additional parametric assump- tions. These include rank constraints (Sullivant et al., 2010), cquality constraints (Drton, 2018), high-order moment con- straints (Xie et al., 2020; Adams et al., 2021: Dai al., 2022: Chen et al., 2024), constraints based on matrix decomposi- tion (Anandkumar et al., 2013), copula models (Cui et al., 2018), and mixture oracles (Kivva et al., 2021). A detailed review of these tools and the latent variable causal discovery al algorithms based on them is provided in Appendix B.
Furthermore, typical assumption in causal discovery,
'Carnegic Mellon University Mohamed bin Zayed University of Artificial Intelligence.
However, all these new tools were developed for latent vari- ables solely, with none adapted selection bias. While var- ious parametric models for selection were also studied, their
Proceedings of the 4rnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Latent Variable Causal Discovery under Selection Bias
focus is either causal inference (Bareinboim & Pearl, 2012; Correa al., 2019) or bivariate orientation (Zhang al., 2016; Kaltenpoth & Vreeken, 2023). For causal discovery, the only tool currently available is still basic CI constraints.
Original rank constraints (Sullivant et al., 2010)
Latent
variables
Conditional independence constraints (Richardson & Spirtes, 2002)
handles
Selection
This creates a huge gap: while more powerful latent causal discovery methods now exist, once selection bias is intro- duced, these newer tools must be set aside, leaving us with only CI constraints-and effectively reverting back FCI.
Generalized rank constraints (this work)
bias
Figure 1. Illustration of our contribution: while CI constraints han- dles both latent variables and selection bias, original rank con- straints extends it only to latent variables. We bridge this gap by generalizing rank constraints to handle also selection bias.
Bridging this gap is the goal of our work. We aim to develop tools that go beyond CI constraints and address both latent variables and selection bias. The core challenge lies in mod- eling data under selection, which can be far more complex than handling latent variables. For instance, marginalizing linear Gaussian model over latent variables still yields Gaussian distribution with a closed-form covariance in terms of model parameters. However, under selection, even simple truncation leads to truncated Gaussian distribution, making its covariance and higher moments hard to ex express and interpret (Kan & Robotti, 2017). This situation only worsens with more complex selection mechanisms.
generalized CI constraints and enabled algorithms beyond FCI, our generalized rank constraints pave the way for algo- rithms to uncover both latent causal and selection structures.
The remainder of this paper is organized as follows. In $2, we introduce original rank constraints in settings without se- lection bias, and then provide an illustrative example show- ing how ranks can retain information under selection. In $3, we formally characterize the generalized rank constraints under selection bias, presenting a precise graphical crite- rion. We show that these ranks offer insights into both latent causal and selection structures. In $4, we apply the gener- alized rank constraints to the one-factor model, a classical latent variable model, and demonstrate its identifiability un- der selection bias. In $5, we validate the effectiveness of our method through simulations and real-world experiments, showing its ability to uncover latent causal and selection mechanisms. Finally, in $6, we discuss potential limitations.
To address this challenge, we try to avoid explicitly mod- eling the full distribution under selection and instead focus on invariant statistical patterns, much like nonparametric CI constraints with the corresponding d- d-separation graphical criterion. This leads us to rank constraints, a direct general- ization to the CI constraints, which assumes data generated by a linear Gaussian model and exploits the (low) ranks of covariance submatrices. In this case, CIs correspond to zero partial correlations, which manifest low ranks in covariance submatrices. There are also other low ranks beyond zero partial correlations. Rank constraints captures them with graphical criterion beyond d-separation, that is, I-separation. Details and examples are provided in $2.1.
Notations on matrices. For matrix M, we let Mi, be its (i. j-th entry. For two index sets A, B, we let MA,B = be the submatrix of M with rows indexed by A and columns indexed by B. For a finite set A, we denote by |A| the cardinality of A.
Notations on graphs. In a directed acyclic graph (DAG) G. for any vertices a, b, we say a is a parent of b and b is a child of a if a - is an edge in G. denoted by E pag (b) and b e chg (a); a is an ancestor of b and b is a descendant of a if a = b or there is a directed path a + ... + b in G. denoted by a E (ng and b E dag(a). These notations extend to sets: e.g., for any vertex set A, (A) = Usea ang(a).
But, do rank constraints remain informative selection- biased data, where the data no longer follows linear Gaus- sian model-or or even a linear structural equation model? Interestingly, the answer is yes. For data generated by lin- ear Gaussian model and subjected to selection, even though the biased covariance matrix becomes arbitrarily complex to express, we show that the ranks of these covariances remain well-defined and capture meaningful structural information about both causal and selection mechanisms. Illustrative examples are provided in $2.2. Specifically, we assume linear selection mechanisms, which will be formally defined later. For now, let us note that it is general, with many existing parametric selection models as special instances.
2 Motivation
In this section, we provide the background and motivation behind our approach. In $2.1, we review the basics of the original rank constraints without selection bias and illus- trate how latent variables can leave traces in the covari- ances among observed variables. Building this intuition, in $2.2, we provide an illustrative example showing how selection may also leave traces in the biased covariances.
As shown in Figure 1, the main contribution of this work is the generalized rank constraints, which, to the best of our knowledge, is the first tool beyond CI constraints handle selection, and thus enables latent variable causal discovery under selection bias. Just as the original rank constraints
2.1 Preliminaries on Original Rank Constraints
Let us provide the background to and definitions of the rank
2

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Latent Variable Causal Discovery under Selection Bias
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(b)
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Figure 2. Illustrative examples of rank constraints. White circles are observed variables. Grey squares are latent variables. Lined double circles selection response variables (i.e., only samples with specific values of them collected in data). We this color scheme throughout. (a) shows the original Tetrad structure from Spcarman (1914). The covariance terms represented by model parameters are given. Since the distribution is simply joint Gaussian, their scatterplot is omitted. (b) shows an "inverse Tetrad structure," where four originally independent variables are later truncated based on linear sum of them. An example with of = 1, λ1, A2, Aa, Aa = L, 2,,33 3, -4, and a, b 3, 10 is visualized the scatterplot, showing both selected samples ('*") and unselected samples ('x"), alongside covariance values in the selected data ('+'). The dashed box highlights one of the low-rank structures.
constraints and its corresponding I-separation graphical cri- terion, originally established in (Sullivant et al., 2010). At this stage, selection bias is yet to be introduced.
Example 1.1 (Tetrad structure in Spearman (1914)). Con- sider the graph in Figure 2a. In this graph, for any choice of model parameters, the following three low ranks hold:
We consider linear Gaussian causal model associated with a DAG G, in which random variables = Xm)
rank (E(x..x)) = 1,
follow the data generating procedure:
rank (xx,..x) =
rank(E(xx.xx)) =1,
X = = + E,
(1)
where E are exogenous noise terms that follow a jointly independent Gaussian distribution, and A is the weighted adjacency matrix that follows i.e., X; + X, is an edge in G if A O. Let & be the diagonal covariance matrix of noise terms E. The covariance matrix of Gaussian variables X, denoted by Σ, can then be written as
as be verified from the covariance terms provided in the figure. However, these low ranks do not follow from any Cls. In fact, there are no CIs among {x1,X2,44 X3.
A
Rank constraints explains where these extra low ranks come from. Similar to how Pearl & Verma (1987) used the d- separation graphical criterion to characterize CIs entailed in data, Sullivant et al. (2010) characterized these entailed low ranks using a new criterion, namely, t-separation:
Σ = (I - - A)-T.
Rank constraints investigates the ranks of submatrices in this covariance matrix Σ. It is first shown to have generalized conditional independencies as special case:
Definition 1 (t-separation; reformulated from Def. 2.7 in Sullivant et al. (2010)). Let A, B, C, D be four subsets of X that need not be disjoint. Denote by Ge the remaining graph after removing nodes C and their associated edges from 9, and similarly 9D- We then say the pair (C, D) t-separates (A, B) if ange (A\CC) n ang (B. Ø.
Proposition 1 (Conditional independencies as low ranks; Prop. 2.2 in Sullivant et al. (2010)). Let A, B, C' be disjoint subsets of X. Then the conditional independence A 1 BIC holds, if and only if the submatrix EAUG,BUC has rank |C|.
Roughly speaking, to find node sets that t-separate A from B, we are to find node sets whose removal from the graph makes A and B share no common ancestors, so that all pathways that carry shared information are blocked. Ranks in covariances reflect exactly the smallest size of such sets:
Proposition 1 is a direct reformulation of the corresponding zero partial correlations, which, since X are Gaussian, man- ifest as conditional independencies. However, there are also other low ranks beyond CI, as the following example shows.
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Latent Variable Causal Discovery under Selection Bias
Proposition 2 (Graphical criterion of rank constraints; Thm. 2.8 in Sullivant et al. (2010). In a DAG 9. for any two subsets A, B C X that need not be disjoint, the equality
to count for the smallest size of nodes needed to "choke" all these flows. Simply put, when some dependence in the data cannot be fully explained (i.e., rendered conditionally independent), rank constraints can serve to quantify the "di- mensional bottleneck" of how this dependence stems from.
rank (ank( min[|CI+|DI= D) I-separates (A, B)}
holds for generic choice of model parameters.
The question then arises: can this "dimensional bottleneck" still be reflected in ranks, when the selection mechanism is "lowdiimesioonal" We consider the following example.
Regarding the term generic, equality in Proposition 2 holds for almost all parameter choices, except for a set with Lebesgue measure O where coincidental lower ranks occur. Hereafter, when we say "assuming genericity", we exclude such coincidental cases, similar to how standard faithfulness assumes no CIs other than those entailed by d-separations.
Example 2.1 (Inverse Tetrad structure). Consider the graph shown in Figure 2b, which we call the "inverse Tetrad struc- ture". We model a simple truncation selection mechanism, where the four observed variables X1, X2, X3, X4 are orig- inally mutually independent, but then get selected by trun- cating on the value of a linear sum of them, represented by the unobserved "response" variable Y in the graph.
Let us examine this criterion in the Tetrad example:
Example 1.2 (t-separation in Tetrad structure). Continue on the graph shown in Figure 2a. We have that (Ø, {4}) or (fL).2) Ø) t-separate ({X1, X2], (X3, which explains the first rank- = 1 (= +++11 1) in Example 1.1. The other two low ranks can be explained in a similar way.
This simple truncation can introduce many complexities: as seen in the scatterplot, the remaining population no longer follows a linear Gaussian model, or even a linear structural equation model, that is, no independent exogenous noise terms can be found to generate the data. Instead, the data follows a truncated Gaussian distribution, where, unlike the closed-form polynomials in Figure 2a, the covariances and higher moments are difficult to express. Hence, we show the covariance values in a specific simulation in the scatterplot.
Rank constraints and its t-separation criterion directly gen- eralize the CI constraints and its d-separation criterion, and thus offer more structural insights into latent variables than those given by FCI. We illustrate with the Tetrad example:
Example 1.3 (Rank constraints enables latent variable iden- tification). Continue on the graph in Figure 2a. Suppose that for some reason L is latent, leaving only X2, X3, X4 observed. Using CI constraints alone, algorithms like FCI cannot distinguish this model from an alternative fully con- nected graph with the same four observed variables, as no CIs exist in both models. However, with rank constraints, this alternative is falsified, as otherwise the three low ranks cannot be satisfied. Ranks reveal that though there is conditional independence, the dependence among data is not arbitrary-ii it must stem in single-dimensional way, suggesting the presence of latent variables.
Yet, an interesting observation emerges from these values. Despite the complexities introduced by selection, the low- rank structure of covariances seems preserved. Specifically, they are the same as in the original Tetrad structure. Denote by E' the covariance matrix of the biased data, we have:
rank( rank(2((x) (xxxx, = 1,
rank (al = 1,
rank (xxx =
A
as can be verified from the values in the scatterplot (the first one is already highlighted in the dashed box).
Based on this insight, various latent variable causal discov- ery methods have been developed (reviewed in Appendix B). However, as noted, none of them address selection bias.
Example 2.1 strongly suggests that selections, just like latent variables, may leave identifiable traces about their "dimen- sional bottleneck" in the ranks of even biased data. Ques- tions naturally arise: What if the selection becomes more complex (e.g., involving randomness, unlike truncation)? What if multiple selection mechanisms act simultaneously? When low ranks occur, can we differentiate whether it is from latent variables selection bias? We formalize and aim to answer these questions in the next section.
2.2 Ranks in Intection-Biasedd Data: an Example
We now present an example to illustrate how the ranks of covariances may remain informative about the causal structure, even for the data under selection bias.
Let us first revisit the motivation behind the original rank constraints without selection bias, introduced in 12.1. The proof of the t-separation criterion consists of two key components: it first uses algebraic combinatorial tools to interpret covariance terms (which are polynomials of model parameters, as in Figure 2a) as the "flows of information" " in the graph, and then uses the max-flow-min flow-min-cu principle
3 Generalized Rank Constraints
In this section, we formalize the generalized rank constraints for handling selection bias. In $3.1, we define the linear se- lection mechanism, a model both general and necessary for rank constraints to work. In $3.2, we provide the graphical
4

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Latent Variable Causal Discovery under Selection Bias
criterion for ranks in the biased covariances. Finally, in $3.3, we discuss the identifiability of distinguishing between la- tent variables and selection bias using rank constraints.
Having defined the linear selection mechanism and demon- strated its connection existing models, we now turn the graphical criterion of rank constraints under selection.
3.1 Linear Selection Mechanisms
3.2 Graphical Criterion of Ranks under Selection
In this part, we introduce the linear selection mechanism. Similar to how original rank constraints generalize nonpara- metric CI constraints within linear Gaussian models, in this paper, we focus on specific class of selections called linear selection mechanisms, defined below.
We are now ready to provide the graphical characterization of the covariance ranks in selection-biased data.
Since selection can also be viewed as causal process in the data generating procedure, we incorporate it into the causal graph. This leads to the definition of the selection- augmented graph, following Bareinboim & Pearl (2012):
Definition 2 (Linear selection mechanism). For a set of vari- ables X, a linear selection mechanism is described by con- figuration S, which consists of tuples {(V,, 3a, c, (1.) f--1- Each tuple specifies single selection condition, where:
-aggetttnn- graph). Consider a DAG G with nodes X that represents the original data generating process for X, and linear selection configuration 8 {(V, Bi, e,, Ni) with k single selection mechanisms. The selection- -atrtiion-agmeeneedd graph is new DAG, denoted g(5), obtained by augmenting 9 with the following:
Definition
V S X is the subset of variables from X directly in- volved in the i-th selection;
is a vector of nonzero linear coefficients that specifies how variables in V2 contribute to the selection; e; is an independent noise term that models selection randomness. It may follow arbitrary distribution, in-
3. E #00
TRIV
Additional selection response nodes Y = (Yi) and Additional edges (X, + Y2 Vi = 1, ... ,.....,,,,, X, E ViJ.
cluding non-Gaussian, or be degenerate to a constant; V& & RR is the set of admissible values, a proper subset of R. which may consist of a single value, multiple values, an interval, or a union of intervals, etc.
The post-selection data X can then be viewed as generated from g(s), with each response Y, restricted within its ad- missible values V. When there is no selection, i.e., S = Ø. the g(Ø) reduces to the original g.
For each single selection, we call Y; = V; + c as its response variable. Finally, sample of X is included in the selected data if and only if Y E V1 for all i = I,...,k. k.
Note that, unlike conventional notation, use the letter "Y" instead of "S" to denote selection variables. This is because, conventionally, selection refers to conditioning on a single value (often Boolean in nonparametric settings) of a variable, whereas here we allow Y, to take multiple values.
The linear selection mechanism is versatile, allowing mul- tiple selections to act simultaneously, as commonly seen in real world like multi-criteria admissions. Each single selection is also flexible, with various existing parametric models for selection bias fitting as specific instances:
With the celetion-auugmenee graph, we now provide the graphical criterion for generalized rank constraints:
Theorem 1 (Graphical criterion for generalized rank con- straints). Let G be a DAG and X the variables generated from G with a linear Gaussian model specified in Equa- tion (1). Suppose X then undergoes linear selections spec- ified by S = {(V,, Bi, es, Vi){ with k single selections. Let Σ(S) be the population covariance matrix of X after selection. For any two subsets A, B C X which need not be disjoint, assuming genericity, we have:
Example 3 (Existing parametric models fit in the linear selection mechanism). Let us first explain what is modeled by a single linear selection. For a sample X, the probability for it to be selected in the i-th selection is:
P(Y E 31 1x) [-- Pe, (u) du.
(3)
where } - 3,, V= = fu e u VTV, E V:). Then,
rank(E/A.)) min[IC] + + C.D.XUY,
with different choices of E; and V., many existing common selection models can fit as instances, including:
(C, D) t-separates (AUY,BUY) in g( kk
6 = and )/ = (a, b) reduce hard truncation model, as illustrated in Figure 2b;
where Y denotes the additional selection response variables introduced to the selection- augmeetted graph g(s).
LoogistccO,, 1) and ) = (a, oo) reduce to a logistic
selection model (Dubin & Rivers, 1989);
The proof of Theorem 1 is provided in Appendix A. Note that when there is no selection (i.c., S = Ø and k = 0), the theorem reduces to Proposition 2. That is, our graphical criterion generalizes the original rank constraints to accom- modate selection bias, hence the name "generalized rank
E ~ (0, 1) and }/ = 0o) reduce to a probit selection model (Heckman, 1977);
e (0, 1) and Vd {a} reduce to stabilizing selection model (Lande & Aroold, 1983).
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Latent Variable Causal Discovery under Selection Bias
Table The spider example and its variants with selection bias. In the selection-augmentie graphs, cach node represents group of non non-adjacee adjacent variables (e.g., C)2, with the lowercase letter indicating its cardinality (e.g.. c |C|). Edges between nodes represent fully connected directed edges from every variable in one group to every variable in another. We assume a, b >> 1 > c, The table presents the rank values (upper rows) and corresponding I-separation sets (lower rows) among observed variables in A, B, derived from Theorem 1. Here, A1, Ag denote large enough disjoint subsets of A, and B1, Bg similarly. For brevity, unions like A₁ U B1 are written as A By. For example, in the second graph, the rank of the covariance matrix between A and B is 2c + d, with (CD, CD) t-separating (AD,BD)
R
R
A
B
A
B
Graphs and covariance submatrices
C
A
B
A
B
l+c
1+c
[+c
l+c (LRC, LC)
A2
(Ø, LC)
(Ø, LCD)
(LR, LC)
r (0, (C)
r
r
r+c
B1, B2
(CD)
(IR,, RC)
(LRC,
2c
+ d (CD,CD)
2c
A,B
(C,C)
(RC, LC)
(RC,
1rr+e e RL
tt (LC,
1+r
Itr+c (LRC,
A₁ A,Bi,A22
(LR, LR)
constraints". This bridges the gap beyond basic CI con- straints to handle selection bias, as illustrated in Figure 1.
3.3 Identifiability between Latent Variables and Selection Bias: Examples
Let us examine this criterion in the inverse Tetrad example:
Now, we discuss the possibility of distinguishing between
Example 2.2 (t-separation inverse Tetrad structure). Let us revisit the model in Example 2.1. The corre- sponding dilectio--augmenter graph shown in Fig- ure 2b. In it, we have that ((Y), (Y}) t-separates ([X1, X2, Y), (X,, XJ)Y)). YJ), rank- = 1 (= 1 11+1 1) in Example 2.1. The other two low ranks can be explained in a similar way.
latent variables and selection bias, using rank constraints from data that may or may not be selected.
For two Intection-augmentee DAGs over the same ob- served variables X, but different and possibly empty la- tent variables and selection response variables, say they are "rank equivalent" if their covariance submatrices for any A, B C X have identical ranks. This parallels the concept of "CI equivalence", where, consider for exam- ple two measured variables, whether they are confounded (X1 < + X2). directly causal related (X, ~ X2), or selected (X; + + + X2) indistinguishable by CI, as all three have Xi y X2. We then answer the following question: for every graph with selection, is there always an alternative graph-free of selection but may involving latent variables-that is rank equivalent to it, and vice versa?
which explains the first
Generalized rank constraints reveal that when dependence in the data cannot be fully conditioned out, the "dimen- sional bottleneck" of how this dependence stems from-be it latent variables or selection bias-can leave traces in ranks of covariance matrix of even biased data. This provides powerful graphical tool for identifying causal structures that involve both latent variables and selection bias.
Before applying this tool for latent variable causal discov- ery, a question arises. Recall the Tetrad and inverse Tetrad structures in Examples 1.1 and 2. Though one has latent variables and the other has selection bias, their rank con- straints among four observed variables are identical. That is, while ranks reveal single-dimensional bottleneck in the observed dependence, they cannot distinguish whether this arises from latent variables or selection bias. Then, does ev- ery selection structure have an alternative structure- -free of selection but may involving latent variables that is equiva- lent in ranks? We explore this in the next part.
Interestingly, the answer is no, contrary to the intuition suggested by Examples 1.1 and 2.1. We first note that latent variables and selection bias can sometimes be distinguished using only CI constraints, as shown below.
Example 4 (CI constraints distinguish between latent and selection variables). Consider the following two graphs.
X
L
X2
X
X
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Latent Variable Causal Discovery under Selection Bias
In the first one, with X1 L X4, X2 L X3, X1 L. X2|X3, and X2 1 X4, it can be concluded that latent variables must exist between X3 and X4. In the second one, with the only two CIs among observed variables being X1 TL Xa X2, X3 and X2 1 X X4. it follows that selection bias involving all four variables must be present.
The data are further subject possible selection S (,, to latent variables, i.c., U V3 C L.
Using the graphical criterion in Theorem 1, we show that CI among I,, even unobserved and biased, can still be recovered from the rank constraints of the observed data:
A
Proposition 3 (Ranks recover CIs among L under selection bias). For any disjoint subsets A, B.C c L, A L B C holds in the selection- biased data, if and only if the rank
These examples are originally introduced as illustrations for the FCI algorithm (Zhang, 2008). The derivation uses v-structure orientation rules, omitted here for brevity.
of the population covariance mutrix between XA U (0) and X3 U UXX is |Cl. Here, x(I and x!) are disjoint partitions of Xe such that |X
We then note that beyond CI. rank constraints can sometimes also distinguish between latent variables and selection bias, even in CI equivalent graphs, as shown below.
N CC|.
Example 5 (Rank constraints to distinguish between la- tent and selection variables). Consider the graphs in Ta- ble 1. The first column shows the original "spider" structure in (Sullivant et al., 2010), and the other three are its variants with selection bias, as detailed in the table caption. In these graphs, only and B are observed, with no CIs among them. However, low ranks exist and differ across graphs.
Proposition 3 is a direct consequence from the graphical criterion, showing that the d-separations among I. persist as low-rank structures in X, even under selection bias. By re- covering these CIs, constraint-based algorithms such FCI can then be applied to L as if L were directly observed. The next section presents experiments that leverage this insight.
5 Experiments and Results
In the first column, A and B share latent variables C. yet the rank between them is 2|C| instead of |C), property unique to the original spider structure (up to indeterminacies inside groups). In contrast, in the second and third columns, while other ranks remain unchanged, either the rank increases between A, B or decreases between A B1, B2, which can not be achieved by any graph without selection. A
Having introduced the generalized rank constraints and pro- posed method for one-factor model, we conduct experiments synthetic and real-world data, showing that our method effectively recovers the causal structure under selection.'
5.1 Experiments on Synthetic Data
We conduct empirical studies on synthetic data to evaluate our method against existing ones. Specifically, we simulate linear SEMs under the truncated probit model. We first gen- erate a random Erdös- Rényi graph (Erdös & Rényi, 1959) among the n E {5, 10, 15, 20) latent variables, with an av- crage degree of 2. Each latent variable has either 2 or 3 observed variables as children. The linear coefficients of the edges are sampled uniformly at random from 1-2, 0.55 [0.5, 2]. For latent variables, we simulate n/5 selection variables using linear selection mechanisms with error terms, where each selection variable has an average of [0.3n] par- ents chosen from the latent variables. We then retain only the samples where these selection variables fall within the 40th to 60th percentile of their values. Here, we consider both Gaussian and exponential distributions for the error terms.
Beyond these examples, we have to note that the complete- ness in distinguishing latent variables from selection bias requires characterizing the rank equivalence class, anal- ogous to maximal ancestral graphs (MAGs) for CI con- straints (Richardson & Spirtes, 2002). This remains an open challenge and is beyond the scope of this paper.
Now, having introduced generalized rank constraints and its graphical criterion, let us apply it to latent causal discovery with selection bias in specific model class.
4 One-factor Model under Selection Bias
In this section, we illustrate how the generalized rank con- straints helps identify latent causal structure under selection bias. For case study, we consider the one-factor model.
Since the goal is validate Proposition 3, we evaluate the partial ancestral graph (PAG) (Spirtes et al., 2000) among the latent variables estimated by FCI. Thus, we assume ac- cess to the oracle clustering in the one-factor todel-that is, knowing which observed variables correspond to the measurements of which latent variables. We compare our method against FCI (Spirtes et al., 2000), PC (Spirtes & Gly- mour, 1991), and BOSS (Andrews al., 2023). We include
The one-factor model from Silva al. (2003) captures cases where latent variables are indirectly measured, such as questionnaires. Since selection bias is also common in such data (e.g., personal traits influencing survey participation), we extend the model to incorporate it, as defined below.
Definition& 4 (One-factor model with selection bias). Let G be a DAG generating latent variables = (L)...., Lm). )
Each latent variable L; has measurements X, (i.e., chil- dren that has no other parents than I.)) with [X|] 2 2.
An implementation of our method is available at https: // iigithub.ccm com Maaaaaaaaaaaaaaaaaaaaaaa Latent- selection.
7

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Latent Variable Causal Discovery under Selection Bias
Indt
Corrgt
Sucuity
Miguttan
Comprio
Migrako
Prelic
(a) Output PAG in Canada
(b) Output PAG in China
(c) Output PAG in Germany
Figure 3. Estimated PAGs on the World Value Survey dataset by country. Arrows (-+ indicate ancestral causal relations, double tails (-) and double heads (++) suggest selection bias and latent confounding, respectively, and open circles (o) denote uncertainty orientation.
100
ity² (BIG5) dataset. Both datasets are in questionnaire for- mat, and thus, can be effectively captured by the one-factor model. As for (I), WVS is the largest non-commereial academic social survey program, organized in waves and conducted every five years. The survey provides time- series data spanning 39 years (1981-2020), includes over 600 indicators (questions), and covers 120 countries, en- suring a global scope. WVS is devoted to the scientifie and academic study of social, political, and cultural val- ues of people in the world, nevertheless, the relations be- tween these values remain under-researched. Here, we fo- cus on 259 indicators of 13 core variables of interest. The core variables include Social Norms, Happiness, Security Values, Ethical Values and so on. For suceinctness, single keyword employed for each variable represented in the graph. Data are collected across different nations, with varying sizes (500 4000 samples). Detailed information for the nation-specific data and the preprocessing scheme are provided in Ap- pendix C. As for (2), the five personality dimensions are Openness, Conscient iiientioousess.. Extraversion, Agreeablenesee and Neuroticiss im (O-C-E-A-N), are measured with their own 10 indicators, with total of 50 questions and approximately 20000 samples. This dataset has been closely examined by Dong et al. (2023), however, the possibility of selection bias has not been considered.
#Difference Edgenaris
Ours
80
BOSS
60
FCI
40-
20
10
15
20
#Lateen Variables
Figure 4. Results with Gaussian error terms. The y-axis shows the total number of differing edge marks between the ground-truth PAG and the estimated PAG (lower is better).
BOSS in the comparison as it has been shown to outper- form classical score-based methods such as GES (Chicker- ing, 2002). Since these three methods learn structures over observed variables rather than directly modeling the rela- tionships among latent variables, we randomly select one observed child (i.e., measurement) as the representative for each latent variable and use these methods to infer the struc- ture among these chosen representatives. For PC and BOSS, we convert their output to a PAG. To evaluate the estimated structure among the latent variables, we report the differ- ences edge marks for the estimated PAG and the true one.
The results for Gaussian error terms are presented in Fig- ure 4, while those for exponential error terms are provided in Figures 5 and 7 in Appendix C. Notably, our method performs the best across all settings, with the performance gap widening as the number of variables increases. This demonstrates the effectiveness of our approach in recover- ing the latent PAG under selection bias and further validates Proposition 3. All experiments are from random runs with 2 CPUs and 16 GB of memory. Each run with number of latent variables of 5, 10, or 15 requires less than one second, and with 20 latent variables requires less than five minutes.
Results. Figure 3 presents the results corresponding to the estimated PAG over three countries in different continents: Canada, China, and Germany. Note that some variables are missing in certain countries due to the low response rate. Finding (A): We observe a similar (potential) selec- tion pattern across three countries. In particular, the tail at node Social Trust is indicative of the node's potential role as an ancestor of selection. The interpretation is that people's propensity to complete a questionnaire is based on their degree of trust towards others. Finding (B): Different countries also exhibit nation-specific potential selection pat- terns. As shown in Figure 3b, we can observe a potential selection on Percepti
5.2 Experiments on Real-World Data
Datasets. In this part, we present experimental results on real-world datasets. We examine the (1) World Value Survey² (WVS) dataset and the (2) Big Five Personal-
of Science. This potential selection can be interpreted as follows: a positive perception
httpps:/ /ww. www.wwwwwwllrrr.oo ldva luessurvey.crg wwSDocumentationnVy jsp
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Latent Variable Causal Discovery under Selection Bias
of scientific studies is associated with higher probabil- ity of participation in this social science study. Moreover, in Figure 3c, we observe that selection bias involving five variables is present. Finding (C): For the BIGS, a selection based on Agreeabl eness is present, where a hypothesis could be made that higher levels of agreeableness may fa- cilitate higher levels of responsiveness. We leave the result for BIG5 Appendix C. These findings not only serve as intuitive validation of our main theorems and method, but also help to recover the whole ground-truth underlying data structure by recognizing the potential selection.
shrink trees. In Thirty-seventh Conference Neural Information Processing Systems, 2023.
Bareinboim, and Pearl, J. Controlling selection bias in causal inference. In Artificial Intelligence and Statistics, pp. 100-1000000000000000000000000001 100. PMLR, 2012.
Bareinboim, E. and Pearl, J. Causal inference and the data- fusion problem. Proceedings of the National Academy of Sciences, 113(27):7345-77
2016.
Bareinboim, E. and Tian, Recovering causal effects from selection bias. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015.
6 Conclusion and Limitations
In this work, we develop the generalized rank constraints, providing tool to identify latent causal structure under se- lection bias. A limitation is that complete rank equivalence characterization has yet to be developed.
Bareinboim, E., Tian, J., and Pearl, J. Recovering from selection bias in causal and statistical inference. In Prob- abilistic and causal inference: The works of Judea Pearl, pp. 433-450, 2014.
Acknowledgment
Bareinboim, E., Tian, J., and Pearl, J. Recovering from selection bias in causal and statistical inference. In Prob- abilistic and causal inference: The works of Judea Pearl, pp. 433-450, 2022.
We would like acknowledge the support from NSFAward No. 2229881, AI Institute for Societal Decision Making (AI-SDM),tt) National Institutes of Health (NIH) under Contract R0IHL159805, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program. IN ac- knowledges the support of the Natural Sciences and Engi- neering Research Council of Canada (NSERC) Postgraduate Scholarships Doctoral program.
Bhattacharya, R. and Nabi, R. On testability of the front- door model via verma constraints. Uncertainty in Artificial Intelligence, pp. 202 212. PMLR, 2022.
Chen, W., Huang, Z., Cai, R., Hao, Z., and Zhang, K. Identi- fication of causal structure with latent variables based on higher order cumulants. In Proceedings of the AAAI Con- ference on Artificial Intelligence, volume 38, pp. 20353- 20361, 2024.
Impact Statement
This paper presents work whose goal is to advance the field of causal discovery. Our method has implications for ana- lyzing e.g., personality traits, political attitudes, and cultural differences in social science related datasets. While our work enhances such tasks, responsible application is crucial to avoid misinterpretation on selections in the results.
Chickering, D. M. Optimal structure identification with greedy search. Journal of Machine Learning Research, 3 (Nov):507-554, 2002.
Choi, M. Tan, V. Y., Anandkumar, A., and Willsky, A. S. Learning latent tree graphical models. The Journal of Machine Learning Research, 12:1771-1812 2011.
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Latent Variable Causal Discovery under Selection Bias
A Proofs of Main Results
Theorem (Graphical criterion for generalized rank constraints). Let & be a DAG and the variables generated from with a linear Gaussian model as specified in Equation (1). Suppose X then undergoes linear selections specified by S {(V,, Bi, e, )=)=== with k single selections. Let ( be the population covariance matrix of X after selection. For any two subsets A, B C X which need not be disjoint, assuming genericity, we have:
rank(2
A,B min[[C]+ [P]: C,D XUY,
(C, D) I-separates (AUY,BUY) in g(
where Y denotes the additional selection response variables introduced to the selection-augmente graph G(S)
Proof of Theorem I. To prove this graphical criterion, show equality between ranks in selected and unselected covariances:
(
rank (5(S) rank (Tu,(B)A |
where E denotes the original global joint covariance matrix of (X,Y). before the selection happens.
The proof consists of the following main steps. First, we establish the rank structure when each selection variable Y; is Gaussian and is conditioned on a single fixed value (i.e., each 3/ is a singleton subset of R). Then, we extend the result to the case where the selection noise terms & are still Gaussian but 3 may contain multiple values. Finally, we generalize to the case where the selection noise terms E; are necessarily Gaussian.
Step 1: Rank structure under pointwise Gaussian selection. Suppose & is Gaussian and each V = {yy a singleton. Let Σ denote the joint covariance matrix of (X,Y) Since (X, Y) is jointly Gaussian, the conditional covariance of X Y = y at all different values of y can be expressed as a fixed term:
Cov(X Y y) = Ex.x - Ex,y Ex,
Now show how the ranks of submatrices the conditional covariance matrix Cov(X Y== the original global covariance matrix E. For any subsets A, B c X, consider the original covariance submatrix:
= y) correspond those in
F F
G
EAUY.BUY
H1
where === EA,,, G= EAY. F= Σy,B, H = Eyy. We then have
rank (
E
G H
rank (66
-
E G
T
F
H
([
E GH-IF 0
rank
F H
rank (E GF + [Y)
where the first step is due to multiplying an invertible unit upper triangular matrix, and the second step is because that H invertible, and thus upper rows (containing at columns) must be linearly independent with lower rows.
In it, E -H- F is AA SyyEEy' Σy.B. which is exactly the covariance xx()) A, in the selection-biased data. This completes the proof of this part, showing that ( A,B) rank(E.uur [V].
Note that in this part of pointwise selection, the above rank equality actually holds for all joint Gaussian X, Y random variables, not necessarily requiring that Y are outcome- mateome-dependent selection response variables-Y can also be X's causes or ancestors. However, for the following multi-valued selection sets, we will explicitly require that Y are X's effects.
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Latent Variable Causal Discovery under Selection Bias
Step 2: Extension to multi-valued Gaussian selection sets.
Now consider the case where each is Gaussian and is
restricted measurable subset IR, and let y V1 Vk- The conditional covariance of X Y y is given by the law of total covariance:
Co E|Cov(X e (E M E)).
We analyze each term:
Due to Gaussianity, Y = =yis Gaussian with constant covariance under different y values, and thus we have:
E[Cov(X Y) Y Ex.x -
The conditional mean is linear:
B[X px + P(Y - -uy),
where p= Ex,r
Thus,
Cov(E|X Y Y e Y
Putting this together:
Cov(X Y E)) Σx,x - Pεr, PT + Cov(( Y YEy)PT.
Let A, B C X. Then the corresponding submatrix can be decomposed to two terms:
(()) A, MA.B + KA.B,
where let PA and PB be the submatrices of P indexed by rows of A and B, we have:
MA.B CA.B PA KA. = PA Y E y)P.
Note that MA.B is the same matrix appearing in the pointwise selection case. So we are now show that adding the correction term KA,B does not alter the rank of MA.B.
Write Y = 3 + e, where ,3 E x is the loading matrix of the selection response variables. We then have that in P. the term Σx,y equals TT. Thus, the columns of P [RI x lie the column space Σx.x, i.e., images Tm(K A. C Tm (EA,) Im(EA,x). Since also both MA.B and KA,B lie within the row space of EAuY,BUY, we have:
rank(E A,B) rank(Ma,n KA,B) tank(MM,,n rank((AAY,,LU)> -Y]-
That is, the correction term KA,n, although nonzero, cannot increase the rank. Further, since each V is proper subset of R. i.e. not all values are admissible, Nyy and Cov(Y Y E y) will not cancel each other under generic assumption, and thus the equality holds, which completes the proof for this part, showing rank( A.B = rank(EAuv,BuY)) - |Y].
Extension to non-Gaussian selection noise. Our goal is to prove that rank (Cov(X,,X) XB Y E ))) = rank (E i holds, even when for each Y, = AT X + G, 6 may be non-Gaussian, as long as X themselves are still joint Gaussian. We first note that the law of total covariance still holds:
Step 3:
Cov(X Y )) E[Cov(X Y) e y) Cov YY|Y e)),
where the since the linear structure still holds in the global generating process, we still have:
Σx,y T
E[X YJ = px + P(Y - -uy), where P Σx,y Exy
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Latent Variable Causal Discovery under Selection Bias
The only difference is that, now conditioning on each singleton y value, the covariance term Cov y) is not constant anymore. However, since X themselves are still Gaussian, conditioning X = y is equivalent to restrict the noise terms e y- Bx, where the right hand side is Gaussian. Therefore, with X independent to e, this imposes affine constraints on X, which are still linear in X. Since the rank reductions are exactly due to those affine constraints, we have that still
rank(() B Y) MY Y e 31)
|Y]-
Then, using the similar argument as in Step 2, we have the final rank equality, concluding the proof.
Note that when X is non-Gaussian, even when is Gaussian, the above properties do not hold, because in that case, X| = y is no longer affine transformation of a Gaussian, and Cov(X y) may reflect other nonlinear constraints and have higher rank than expected.
B Related Work
In this section, we provide a comprehensive review of the relevant literature, focusing on two key areas: latent variable causal discovery and causal discovery causal inference under selection bias.
Statistical tools for latent variable causal discovery As discussed in $1, standard conditional independence (CI)-based approaches to causal discovery fail to provide sufficient information for recovering latent structures. To address this limitation, a range of alternative statistical tools have been developed, typically by introducing additional parametric or structural constraints. These include rank constraints (Sullivant et al., 2010), which generalize the Tetrad representation theorem from Spirtes et al. (2000) and provide algebraic conditions on covariance matrices; equality constraints derived from Gaussian structural equation models that even has rank constraints as a subclass (Drton, 2018); and high-order moment constraints (Xie et al., 2020; Adams al., 2021; Robeva & Seby, 2021; Dai al., 2022; 2024b; Chen et al., 2024), which exploit non- Non-Gaussianit for identifiability. Additionally, matrix decomposition methods (Anandkumar et al., 2013), copula-based constraints (Cui et al., 2018), and mixture oracles (Kivva et al., 2021) were also developed.
Constraints in linear Gaussian structural equation model Among the various statistical tools, rank constraints and their associated graphical criteria are particularly well-known. Recent work has extended these constraints to nested rank constraints, which characterize additional algebraic polynomial varieties in covariance matrices beyond zero-determinant conditions (Drton et al., 2020). These extensions relate e.g., the Verma constraint, originally formulated in the nonpara- metric setting (Pearl & Verma, 1995) and later explored with graphical criteria (Evans & Didelez, 2015; Richardson et al., 2023; Bhattacharya & Nabi, 2022). In the context of conditional distributions, the most closely related work is Silva & Shimizu (2017), which primarily aims to validate instrumental variables, rather than establishing more general graphical criterion for causal structure recovery in the linear non-Gaussian setting.
Latent variable causal discovery methods Building on these statistical tools, a variety of latent variable causal discovery algorithms have been proposed. Many of these methods fall within the constraint- based framework, leveraging CI tests and algebraic constraints to infer causal relations. Notable examples include approaches based on rank or tetrad constraints (Silva et al., 2003; 2006; Silva & Scheines, 2004; Choi et al., 2011 1; Kummerfeld & Ramsey, 2016; Huang et al., 2022: Dong al., 2023; 2024). While the majority of these methods fall within the constraint- based paradigm, recent efforts have attempted to formalize score-based methods for latent causal discovery (Jabbari et al., 2017; Ng et al., 2024).
Causal modeling with selection bias Classical approaches to deal with selection bias primarily fall into two categories: (1) nonparametric methods that leverage selection mechanisms to correct for selection effects, and (2) parametric approaches that introduce explicit selection models into causal inference frameworks.
For the purpose of causal discovery, following the foundational work of the FCI algorithm, nonparametric methods have been developed to identify causal relations using conditional independence constraints (Hernán et al., 2004; Tillman & Spirtes, 201 1; Evans & Didelez, 2015; Versteeg et al., 2022). There are recent works that deal with selection bias in interventional studies (Dai et al., 2025) and in sequential data (Zheng et al., 2024; Qiu et al., 2024). Several parametric approaches have been developed for bivariate causal orientation (Zhang et al., 2016; Kaltenpoth & Vreeken, 2023).
For the purpose of causal inference (bias adjustment), the nonparametric perspective builds on the graphical representation of selection mechanisms, as introduced in the selection diagram framework (Bareinboim et al., 2014; Bareinboim & Pearl,
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Latent Variable Causal Discovery under Selection Bias
2012: Bareinboim & Tian, 2015; Bareinboim & Pearl, 2016; Bareinboim et al., 2022), which characterizes conditions under which causal effects remain identifiable despite selection bias. Subsequent works extended this approach by developing testable implications of selection mechanisms (Correa et al., 2019) and providing adjustment criteria. From a parametric perspective, selection bias has been studied extensively in economics (Heckman, 1977; 1990; Robins et al., 2000).
Structure learning and inference results can follow different methodological 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 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 et al., 2024a).
C
Supplementary Experimental Details and Results
C.1 Additional Result on Synthetic Data
The difference of Edgemarks with non-Ganee Gaussian error terms is shown in Figure. 5. We also report the structural Hamming distance (SHD) of the skeleton to capture the method's ability to recover the structure up to the Markov equivalence classes (represented as PAGs). The result for Gaussian and Exponential error terms are shown in Figure. 7.
#Difference of
100
75
50
25
5
10
15
20
#lattent Variables
Figure 5. Difference of Edgemarks with exponential error terms.
C.2 Additional Result on Real-world Data
To ensure diverse and representative evaluation, Australia and India selected as part of our experiment encompass all continents and include more developing countries. The results, illustrated in Figure 8, indicate a potential selection bias related to social norms, which aligns with the findings from Germany (Figure 3c). This suggests that the survey design or respondent characteristics may systematically favor certain perspectives on social norms. Additionally, the results reveal a potential selection bias concerning happiness, as shown in Figure 9. This bias can be interpreted as an overrepresentation of individuals who are happier and boatt mentally and physically-within the survey sample.
C.3 World Value Survey Dataset Details
The main research instrument of the World Value Survey project is a representative comparative social survey which is conducted globally every 5 years. Extensive geographical and thematic scope, free availability of survey data and project findings for broad public turned the WVS into one of the most authoritative and widely-used crossnnational surveys the social sciences. At the moment, WVS is the largest non-commercial cross-national empirical time-series investigation of human beliefs and values ever executed. In addition, WVS is the only academic study which covers the whole scope of global variations, from very poor to very rich societies in all world's 's main cultural zones. The WVS has over the years demonstrated that people's 's beliefs play key role in economic development, the emergence and flourishing of democratic institutions, the rise of gender equality, and the extent to which societies have effective government. The full list of 13 variables that we focus on in this study, with their corresponding indicators, are as follows:
Social Values, Norms, Stereotypes (Q1-Q45)
Happiness and Wellbeing (Q46-Q56)
15

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Latent Variable Causal Discovery under Selection Bias
40
Ours
PC
30
BOSS
SHD
FCI
20
0-
5
10
15
20
#Latent Variables
Figure 6. SHD results with Gaussian error terms.
Ours
40
PC
30
BOSS
FCI
20
10
#Latent Variables
Figure 7. SHD results with Exponential error terms.
Social Capital, Trust and Organizational Membership (Q57-Q105)
Economic Values (Q106-Q111)
Perceptions of Corruption (Q112-Q120)
Perceptions of Migration (Q121-Q130)
Perceptions of Security (Q131-Q151)
Index of Postmaterialism (Q152-Q157)
Perceptions about Science and Technology (Q158-Q163)
Religious Values (Q164-Q175)
Ethical Values (Q176-Q198)
Political Interest and Political Participation (Q199-Q234)
Political Culture and Political Regimes (Q235-Q259)
We also present below some examples of the raw questions in the section of Happiness and Wellbeing:
Q46: Feeling of happiness: Taking all things together, would you say you are?
Q47: State of health (subjective): All in all, how would you describe your state of health these days?
Q48: How much freedom of choice and control: Please use seale where means "none at all" and 10 means "a great deal" to indicate how much freedom of choice and control you feel you have over the way your life turns out.
Q49: Satisfaction with your life: All things considered, how satisfied are you with your life as a whole these days?
16

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Latent Variable Causal Discovery under Selection Bias
SocialNomms
Eiltics
Happiness
Science
Security
Politics
Culture
Mignation
Religion
Figure 8. Output PAG in India.
SocialNorms
Happiness
Ethics
Corruption
Migration
Culture
Security
Politics
Trust
Figure 9. Output PAG in Australia.
Agrecableness
Neuroticism
Extraversion
Conseientiousness
Openness
Figure 10. Output PAG for BIG5,
17

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Latent Variable Causal Discovery under Selection Bias
Q50: Satisfaction with financial situation of household: How satisfied are you with the financial situation of your household? If "I" means completely dissatisfied and '10' means completely satisfied, where would you place your satisfaction?
Q51: Frequency youlfamily (last 12 months): Gone without enough food to eat.
Q52: Frequency you/family (last 12 months): Felt unsafe from crime in your own home.
Q53: Frequency youffamity (last 12 months): Gone without needed medicine or treatment that you needed.
Q54: Frequency you/family (last 12 months): Gone without a cash income.
Q53: Frequency you/family (last 12 months): Gone without a safe shelter over your head.
Q56: Standard of living comparing with your parents: Comparing your standard of living with your parents' standard of living when they were about your age, would you say that you are better off. worse off. or about the same?
C.4 Prte-peepg
In the WVS dataset, the data preprocessing consists of two main components: sample choice and measurement variable choice. For sample choice, we remove those with data missingness in entries (e.g., some questions left unanswered or "preferred not to answer"). For measurement variable choice, though the questions are already categorized into different aspects, to ensure that the rank method works, we find subsets from each latent variable's corresponding measurements that share rank-1 I structure against others, i.e., they indeed appear as masurements to same latent variable.
18


Reviews:
[{"review_id": "ZilPg8UdbO", "reviewer": "Reviewer_nj2K", "summary": "This paper extends the rank constraint and t-separation criteria in latent variable causal discovery to scenarios involving selection bias. The authors demonstrate that the rank constraint retains its informativeness even under selection bias, despite the potential invalidation of the linear Gaussian assumption. Additionally, they generalize the t-separation criteria to the augmented selection graph, enhancing its applicability. Leveraging these developed tools, the authors investigate the problem of causal discovery in a one-factor model under conditions of selection bias.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer's constructive comments and helpful feedback. Please see below for our response.\n\n---\n**(Q1)** The reviewer asks about intuitions for the rank equivalence class.\n\n**A:** Thank you for this constructive question. We fully agree that the following three problems are fundamental:\n\n1. A graphical criterion to decide rank equivalence,  \n2. A formal object to describe the rank equivalence class,  \n3. An algorithm to enumerate all members of the class.\n\nHowever, as the reviewer noted, **all three remain open and challenging.** To illustrate the hardness, we may consider e.g., the spider graph and the (2,6)-factor analysis graph, and see how two seemingly so different graphs can still, counterintuitively, be rank equivalent [[SSK10]](https://tinyurl.com/5z8x7td5).\n\nIndeed, despite almost four decades of work on rank-based methods since [[G+86]](https://tinyurl.com/2s48ba2v), no general characterization of rank equivalence has yet been developed, even in settings without selection bias. This is why these methods, including ours, typically rely on structural assumptions (e.g., one-factor models), rather than arbitrary graphs.\n\nThat said, we see two potential directions:\n\n1. **Algebraic direction**: Rank constraints can be seen as encoding flow capacities between node sets. Reformulated, the problem becomes: _Given unknown graph structure, but known flow (e.g., min-cut) values between certain node groups, what can we recover?_ Algebraic geometry may offer tools to approach this.\n\n2. **Algorithmic direction**: Even without a full characterization, one might still build a sound but incomplete algorithm first, by relaxing existing methods gradually. This also mirrors the trajectory of CI-based methods: while FCI was first introduced in 1993 [[SGS93]](https://tinyurl.com/yyzy32f3), the direct graphical criterion for equivalence came in 2002 [[RP02]](https://tinyurl.com/3afjpm98), the full class characterization in 2008 [[Zha08]](https://tinyurl.com/2s47c6yc), and enumeration algorithms are still under study [[WDZ24]](https://tinyurl.com/yeys53vb).\n\nFinally, even with a full characterization, distinguishing latent variables from selection bias–as in our Section 3.3–may still require **algorithmic reasoning over sets of rank constraints rather than graphical patterns.** This mirrors the CI setting–such as in our Example 4, the presence of latent variables or selection is implied by FCI's output, not by any obvious graphical patterns.\n\nWe thank the reviewer again for highlighting this important and exciting problem.\n\n---\n**(Q2)** The reviewer wonders whether the rank-constraint theory can be extended beyond the linear Gaussian setting.\n\n**A:** This is a very important question. We address it from the following three folds:\n\n1. **Theoretical scope**: Our results indeed rely on properties of the linear Gaussian model. Specifically, Theorem 1 relies on the closure of Gaussians under point conditioning, so that linear structures in covariances are preserved under selection. This does not hold in general non-Gaussian or nonlinear settings.\n\n2. **Empirical robustness**: Nonetheless, simulations on both linear non-Gaussian and nonlinear data show that our method remains empirically effective (consistently the best among competitors). Due to space limits, please kindly refer to our response to Reviewer nKfC for results.\n\n3. **Broader outlook**: Finally, we note that this work serves as a (first) proof of concept showing that ranks can remain informative under selection bias. Just as original linear-Gaussian rank results were extended to, e.g., nonlinear, cyclic [[Spi13]](https://tinyurl.com/25y2wx4s), and discrete models [[Gu25]](https://tinyurl.com/n6jm5jry), we believe the insights here may generalize as well, though the tools may differ.\n\n---\n**(Q3)** The reviewer wonders whether the personality trait data satisfies the linear Gaussian model.\n\n**A:** Thank you for this insightful question. **Strictly speaking, no**–they are discrete questionnaire responses. However, some literature interprets such ordinal variables as discretized versions of Gaussians, in which case covariance ranks can still be informative [[LN08]](https://tinyurl.com/2423ea4d).\n\nMore broadly, **model testing in our setting is difficult:** Even if the data were first generated by a linear Gaussian model, selection can then induce strong non-Gaussianity and nonlinearity. Hence, even with Gaussian tests like Mardia's, we cannot easily tell whether a non-Gaussianity arises from model violation or simply selection distortions.\n\nHowever, from another view, note that our method detects many low-rank constraints in the Big Five dataset. Since major model violations would typically destroy such low ranks in the covariances, these low ranks, together with the meaningful causal structure recovered from them, offer a **partial empirical validation for model adequacy.**\n\n---\nOnce again, we thank the reviewer for the insightful feedback.", "claims_and_evidence": "Yes. The two primary theoretical contributions of this paper are Theorem 1 and Proposition 4. Specifically, Theorem 1 extends the t-separation criteria introduced by Sullivant et al. (2010) to scenarios involving selection bias, thereby broadening its applicability. Proposition 4 builds on this extension to address the causal discovery problem in a one-factor model under selection bias. Both results are supported by rigorous, well-constructed proofs and are complemented by intuitive illustrations that enhance their clarity and accessibility.", "methods_and_evaluation_criteria": "Yes. The major proposal of this paper is that the rank constraint remains valid for causal models under selection bias. The authors have successfully demonstrated their points using intuitive examples and rigorious theorems. It is convincing that the proposed method can work under latent causal discovery with selection bias.", "theoretical_claims": "Yes, I have checked the correctness of the proofs.", "experimental_designs_or_analyses": "Yes, the experiment design and analyses are sound.", "relation_to_broader_scientific_literature": "This paper extends the rank constrain and t-separation criteria introduced by Sullivant et al. 2010 to cases with selection bias.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "A weakness of this paper is the lack of a comprehensive characterization of rank equivalence graph, as also acknowledged by the authors. This can affect the application of their theory to real world causal discovery problems.", "other_comments_or_suggestions": "No", "questions_for_authors": "1. I am particularly interested in Sec. 3.3. While a detailed characterization of the rank equivalence graph is not yet available, can the authors provide some intuition or potential ideas to this problem? Further, is it possible to delevep an algorithm like the FCI to recover the rank equivalent graph? How is this rank equivalent graph related to the causal graph over latent variables (which is our target)? \n\n2. Can the rank-constraint theory be extended beyond linear Gaussian data? (I presume not) How this method work empirically when the linear Gaussian assumption does not hold, e.g., with linear nonGaussian data or nonlinear data?\n\n3. Does the personality trait example in the introduction section satisfies the linear Gaussian model?", "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": "AwxNKTpm00", "reviewer": "Reviewer_6VTP", "summary": "In this paper, the authors propose the use of rank constraints to infer causal structure of the latent variables underlying a set of measurements, when these latent variables are themselves subject to selection bias (e.g. more conscientious individuals are more likely to fill out a full Big 5 questionnaire).\n\n### update after rebuttal ###\n\nI thank the authors for their answers. My assessment remains positive.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's constructive comments. Please see below for our response.\n\n---\n**(Q1)** The reviewer wonders how to identify latent structure and selection bias when the structural assumption of one-factor models are violated–such as when the number of latent variables is unknown, or when selection acts on observed variables as $L \\to X \\to Y$.\n\n**A:** Thank you for this insightful question. Moving beyond structural assumptions has long been a goal–but also a challenge–for rank-based methods. Below, we address this from two complementary perspectives: why the problem is inherently hard with rank constraints alone, and how additional information beyond ranks may help.\n\n1. **Using only second-order information, this is a hard problem:**\n   - Distinguishing selection from confounding via ranks in arbitrary graphs requires characterizing the **rank equivalence class**–the set of all graphs (with possibly different latent and selection variables) that entail the same rank constraints.\n   - However, this characterization **remains open and challenging, even without selection.** Due to page limit, please kindly see our response to Reviewer nj2K's Q1 for details.\n   - In short, despite four decades of work on rank-based methods, no such characterization has been developed, which is why current methods, including ours, typically rely on structural assumptions (e.g., one-factor models).\n   - Even with such a characterization, distinguishing selection and confounding may still **require algorithmic reasoning** over constraint sets rather than direct graphical patterns (see Example 4).\n\n2. **With higher-order information and additional parametric assumptions, certain distinctions may become feasible:**\n   - Consider two measured variables. Whether they are confounded ($X_1 \\leftarrow L \\to X_2$), directly causal ($X_1 \\to X_2$), or selected ($X_1 \\to Y \\leftarrow X_2$) is indistinguishable by rank or CI constraints, since all imply $X_1 \\not \\perp X_2$.\n   - But they may be distinguishable under additional assumptions. For instance, in linear non-Gaussian models, direct causation can be identified via residual independence.\n   - For selection bias, suppose it acts through truncation, scatterplots may show clear cut-offs (see Fig. 2). Reproducing such patterns under the other scenarios would require **more complex**–and arguably less plausible–functional forms. With a **preference for simplicity**, such patterns can guide model selection.\n\nThough (2) lies beyond this work's scope, we appreciate the reviewer’s question, which points to a valuable direction: merging the two lines to better identify selection and confounding in practice.\n\n---\n**(Q2)** The reviewer asks for clarification on several experimental details:\n\n- **(Q2.1)** Why in Fig. 3, the graph for China contains more nodes than for Canada and Germany?\n\n  **A:** Thank you for your careful reading. As noted in Sec. 5.2, \"some variables are missing in certain countries due to the low response rate.\"\n  \n- **(Q2.2)** In Big 5, to what extent do results depend on the \"objective\" existence of the measured traits?  \n\n  **A:** Although traits are labeled for the personality they are designed for (e.g., \"Openness\"), **we do not use such side information.** Instead, we cluster traits using rank constraints: under one-factor models, two observed traits share a latent parent iff the rank between them and all others is one. These inferred clusters indeed align well with the labels.\n \n- **(Q2.3)** How do we know our results on Big 5 are not due to preprocessing (e.g., varimax rotation)?  \n  **A:** Thank you for raising this. While preprocessing like varimax rotation is common in factor analysis, here we directly use the raw, discrete data from [link](https://tinyurl.com/4brackf6)–**no preprocessing is applied.**\n\n  That said, directly verifying model assumptions is still difficult due to selection-induced distortions. However, from another view, the many low-rank patterns–which would typically be destroyed under model violations–are detected, and the recovered graph appears reasonable. This offers a **partial empirical validation** for model adequacy. See Reviewer nj2K’s Q3 for more.\n\n- **(Q2.4)** The comparisons are mainly to basic methods.\n\n  **A:** We do have included BOSS [[A+23]](https://tinyurl.com/3saupn8x), a recent SOTA outperforming classical constraint and score-based methods. No other latent-variable methods are used, because they require different structural assumptions and the outputs are incomparable.\n\n- **(Q2.5)** It would be interesting to see the kinds of errors made by the algorithm.\n\n  **A:** Thank you–this is a valuable point. In the 20-node setting of Fig. 4, our method greatly improves selection-edge ('--') recovery (F1: 0.12 to 0.77). Though '--' edges are few, their correct detection aids propagation, improving direct-edge ('->') recovery as well (F1: 0.52 to 0.80).\n\n---\nWe want to thank the reviewer again for all the valuable feedback.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I checked the proofs and they seem correct.", "experimental_designs_or_analyses": "The setup of the empirical evaluation is sound, including experiments on synthetic and real-world data. As the authors primarily compare to basic rather than state-of-the-art methods, there is a risk for an illusion of progress.", "relation_to_broader_scientific_literature": "The paper related its key contributions well with regard to the literature.", "essential_references_not_discussed": "The most relevant papers are discussed.", "other_strengths_and_weaknesses": "- Strengths\nThe approach is novel and interesting.\nFew methods can effectively deal with both latent variables and selection at the same time.\nThe paper is clear and straightforward.\n\n- Weaknesses\nIt's unclear how the results here can be used to deal with cases where we don't know how many latent variables there are\nIt's not quite clear to me how Theorem 1 can be used when we don't know what the selection variables are", "other_comments_or_suggestions": "None.", "questions_for_authors": "- Can you explain why in Fig. 3, the graph for China contains more nodes than for Canada and Germany?\n- In the experiment of the Big 5, can you explain to what extent the results depend on the \"objective\" existence of the measured traits? Big 5 often involve various preprocessing steps (such as varimax rotation), how do we know our results are not due to these preprocessing steps?\n- In Definition 4 we assume that the parents of each $Y_i$ are among the latent variables $L$. Does this matter, when the paths from $L \\to X \\to Y$ would also all be linear?\n- In Fig. 4 (or in a followup in the Appendix), it would be interesting to see the kinds of errors made by the algorithm. Does it mostly miss selection, confounding, or something else?", "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": "iqWpeG5bPH", "reviewer": "Reviewer_nKfC", "summary": "The authors address the problem of causal discovery with latent confounders when the data has a selection bias. Here, the authors address this via generalized rank constraints that extend beyond conditional independencies. The ranks of covariance submatrices in biased data can reveal information about the latent causal structure and the selection mechanism. The proposed method has been evaluated in artificial and real-world data.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the insightful comments. Below, we provide detailed response.\n\n---\n**(Q1)** The reviewer suggests several additional experiments.\n\n**A:**  Thank you for the constructive suggestions. In light of them, we conducted new experiments to evaluate our method under violations of **Gaussianity** and **linearity**, under **small sample sizes**, and regarding **time complexity**. Results are summarized below and detailed in [this anonymous link](https://tinyurl.com/ydmznrxm).\n\n1. **Non-Gaussianity**: We simulate linear SEMs with noise types: Gaussian, Exponential, Laplace, Uniform, and Gumbel. These choices test the sensitivity to skewness, tail behavior, and asymmetry. The numbers of edgemark differences on 10 and 20 nodes and 5 random seeds (same below) are reported. Despite the violations, our method consistently outperforms others:\n   |#Nodes|10||||20||||\n|-|-|-|-|-|-|-|-|-|\n|**Noise**|Ours|PC|BOSS|FCI|Ours|PC|BOSS|FCI|\n|Gaussian|**19.0±8.3**|28.2±9.1|27.6±12.6|35.6±12.2|**39.4±6.4**|76.0±7.2|65.4±4.3|88.8±6.9|\n|Exponential|**19.8±7.8**|27.4±8.1|28.4±10.4|35.0±11.8|**40.6±5.8**|69.2±6.2|60.0±9.6|80.2±11.4|\n|Laplace|**18.0±7.9**|28.4±7.9|22.2±6.4|33.2±11.3|**39.6±3.0**|70.0±11.2|62.8±11.0|84.8±8.5|\n|Uniform|**17.0±8.6**|28.8±9.6|30.6±11.3|34.6±11.9|**35.4±6.9**|64.0±9.8|72.6±12.7|77.4±10.7|\n|Gumbel|**18.4±8.1**|25.8±6.2|24.6±7.4|33.8±10.5|**38.8±6.0**|69.6±10.3|64.0±10.8|80.4±11.7|\n    \n2. **Nonlinearity**: We simulate additive noise SEMs with functions: Linear, Leaky ReLU, Tanh, Cubic, Quadratic, and Since. All but the last two are monotonic. We do see a performance drop on the last two, while comparing to other methods, ours remains the best:\n    |#Nodes|10||||20||||\n|-|-|-|-|-|-|-|-|-|\n|**Function**|Ours|PC|BOSS|FCI|Ours|PC|BOSS|FCI|\n|Linear|**19.0±8.3**|28.2±9.1|27.6±12.6|35.6±12.2|**39.4±6.4**|76.0±7.2|65.4±4.3|87.6±4.2|\n|Leaky ReLU|**20.0±7.7**|28.4±8.6|27.6±12.5|32.6±11.5|**42.4±6.1**|69.6±7.2|71.6±10.0|87.0±7.1|\n|Tanh|**20.8±6.8**|28.8±9.0|24.0±8.8|30.0±13.3|**38.2±6.4**|66.6±9.1|50.6±9.8|77.8±4.6|\n|Cubic|**25.0±4.3**|28.0±5.0|27.8±4.7|34.0±3.6|**54.0±5.7**|75.8±12.4|64.0±8.8|81.5±13.4|\n|Quadratic|**25.8±3.5**|26.2±3.7|28.6±4.8|27.8±5.0|**55.2±3.6**|61.4±8.8|59.6±10.4|70.6±5.7|\n|Sine|27.4±2.7|28.2±4.4|**25.8±6.4**|30.8±6.2|**47.6±6.5**|60.0±9.0|48.8±6.6|66.0±9.6|\n\n3. **Small sample sizes**: Using linear Gaussian SEMs, we vary sample size. With fewer samples, our method continues to detect low-rank patterns reliably and outperforms others:\n   |#Nodes|10||||20||||\n |-|-|-|-|-|-|-|-|-|\n |**#Samples**|Ours|PC|BOSS|FCI|Ours|PC|BOSS|FCI|\n |100|**24.6±4.5**|24.8±7.7|25.0±5.7|26.0±5.4|**53.8±4.2**|54.8±5.6|55.6±6.4|56.4±3.7|\n |500|24.0±6.9|**23.8±6.5**|27.4±8.3|29.6±7.5|**46.2±11.7**|55.4±10.1|48.8±10.3|60.2±11.0|\n |1000|**22.0±6.2**|26.8±8.1|22.4±7.5|27.4±8.2|**53.0±5.9**|62.2±7.0|61.4±6.2|74.2±10.1|\n    \n4. **Time complexity**: Our algorithm follows the same structure as FCI. Hence, like FCI, it has a worst-case complexity exponential in the number of variables. However, if the graph is sparse–a common and reasonable assumption–the runtime becomes polynomial. We report running time in ms below:\n   |#Nodes|10||||20||||\n |-|-|-|-|-|-|-|-|-|\n |**#Samples**|Ours|PC|BOSS|FCI|Ours|PC|BOSS|FCI|\n |100|**12±1**|479±15|453±19|447±15|**43±4**|488±29|462±12|493±41|\n |1000|**28±8**|493±19|475±35|500±30|**110±31**|542±29|503±13|606±53|\n |10000|**110±48**|629±31|604±42|665±54|**434±113**|976±106|776±32|4364±3554|\n\n---\n**(Q2)** The reviewer wonders how one might distinguish between selection bias and latent confounding in practice.\n\n**A:** Thank you for this insightful question. Below, we try to briefly address it from two complementary views. Due to space limit, for a more detailed discussion, please kindly refer to our response to Reviewer KB6X's Q1.\n\n1. **Using only second-order information (rank in covariances), this is a hard problem**:\n   - Ideally, **one would need to characterize the rank equivalence class**, which is an open and challenging problem (see, e.g., our response to Reviewer nj2K's Q1).\n   - Even with a full characterization, distinguishing the two may still **require algorithmic reasoning** over constraint sets, rather than any obvious graphical patterns.\n\n2. **With higher-order information and parametric assumptions, some distinctions become feasible**:\n   - Even for two rank-equivalent graphs, data generated from one graph with a natural SEM may require a much more complex, unnatural SEM to reproduce under the other graph.\n   - Such distinctions with a **preference for simplicity can guide model selection.**\n\nThough (2) is beyond this work's scope, we sincerely appreciate the reviewer's question that highlights a valuable direction: seeing how the two lines can be merged to better identify selection and confounding in practice.\n\n---\nWe want to thank the reviewer again for all the valuable feedback, and we hope the reviewer's questions are properly addressed.", "claims_and_evidence": "Under the given assumptions (linear and Gaussian), Theorem 1 supports the claim that ranks preserve structural information under selection bias. However, distinguishing between latent confounding and selection bias is only discussed through examples.", "methods_and_evaluation_criteria": "The run experiments make sense, although real-world validations rely heavily on qualitative assessments.", "theoretical_claims": "Theorem 1 appears to be sound and Proposition 3 directly follows from applying the theorem to one-factor models.", "experimental_designs_or_analyses": "The experiments are fair, but use a simplistic selection mechanism and the real-world data lacks quantitative validation.", "relation_to_broader_scientific_literature": "Fair discussion of how related literature is lacking in the area of latent confounders under selection bias. While some discovery methods that support latent variables are discussed, it is only a small (but sufficient) selection.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "- Good motivation\n- Fair discussion of related work and pointing out the novel aspect\n- Clear notation and figures\n\n- Limited to linear Gaussian models and linear selection mechanism\n- Distinguishing between latent confounding and selection bias is unclear", "other_comments_or_suggestions": "See questions.", "questions_for_authors": "The paper addresses an interesting problem and is novel in this regard. While the theoretical contribution is great (even though it requires some limiting assumptions), the experiments could be more insightful. Some questions:\n\n- It is unclear how robust your approach is against a violation of linearity and the Gaussianity assumption. Experiments with data that explicitly violates your main assumptions would be insightful.\n- In practice, how could one determine if selection bias is present in a dataset versus just latent confounding?\n- The complexity of the proposed method is unclear. It generally appears to be fast (especially compared to PC/FCI etc.). Can the authors briefly comment on this?\n- How robust is the rank determination under small sample sizes, do you have some insights on this?", "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": "GBa0pLitzk", "reviewer": "Reviewer_KB6X", "summary": "This paper investigates the problem of Latent Variable Causal Discovery in the presence of Selection Bias and proposes a new statistical tool, Generalized Rank Constraints, to simultaneously handle latent variables and selection bias within linear Gaussian models.\n\n## update after rebuttal\nI have decided to raise my score from 3 to 4. The authors have provided thorough and thoughtful responses to my concerns. For Q1, they addressed the challenge of distinguishing selection bias from latent confounding by discussing both the limitations of second-order information and the potential of higher-order information with additional assumptions. For Q2, they clarified the computational complexity of the generalized rank constraints compared to conditional independence tests, highlighting the efficiency of their method. For Q3, they acknowledged the omission of the RFCI reference and included it in the revised manuscript. These detailed and constructive responses demonstrate the authors' commitment to addressing feedback and improving the quality of their work, leading me to adjust my score upward.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the reviewer's insightful comments and suggestions. Please see below for our response.\n\n---\n\n\n**(Q1)** The reviewer wonders how one might distinguish between selection bias and latent confounding in practice.\n\n**A:** Thank you for this insightful question. We try to address it from two complementary perspectives:\n\n1. **Using only second-order information (covariances), this is a hard problem:**\n   - Distinguishing selection bias from latent confounding based **solely on rank constraints is, in general, a hard problem.** Ideally, one would need to characterize the rank equivalence class–the set of all graphs (with potentially different latent and selection variables) that entail the same rank constraints. Then, the presence of selection or confounding could only be confirmed when all rank-equivalent graphs contain them.\n   - However, a general characterization of rank equivalence **remains open and challenging,** even in the setting without selection bias. Due to space limit, please kindly refer to our response to Reviewer nj2K's Q1 for detailed discussion.\n   - Moreover, even with a full characterization of rank equivalence, distinguishing between selection and confounding–as in our Section 3.3–may still **require algorithmic reasoning** over sets of rank constraints rather than straightforward graphical patterns. This is similar to the setting with only CI constraints. For instance, in our Example 4, the presences of latent variables or selection are implied by FCI's output, not by any obvious graphical patterns.\n\n\n2. **With higher-order information and additional parametric assumptions, certain distinctions may become feasible:**\n   - Let us consider the example of two measured variables. Whether they are confounded ($X_1 \\leftarrow L \\rightarrow X_2$), directly causal related ($X_1 \\rightarrow X_2$), or selected ($X_1 \\rightarrow Y \\leftarrow X_2$) are indistinguishable by rank or CI constraints, as all three have $X_1 \\not \\perp X_2$.\n   - However, they may become distinguishable with more parametric assumptions. For the direct causal effect, for instance, in a linear non-Gaussian setting, it can be distinguished from the other two cases via independence of regression residuals.\n   - For the case of selection bias, suppose the selection acts through truncation, and the scatterplot of observed variables may exhibit a clear truncation pattern (e.g., as in Figure 2 in our draft). Reproducing this pattern under the other two cases **would require more complex**–and arguably less plausible–functional forms, often by reverse-engineering rather than natural functions.\n   - Hence, with a proper **definition of simplicity and the preference for it**, the model can be further identified.\n\nNote that observations in (2) fall outside the scope of this work, which serves mainly as a (first) proof of concept to show that rank constraints alone can remain informative under selection bias.\n\nHowever, we sincerely appreciate the reviewer's question, which points to an important future direction: exploring which additional assumptions are needed, and **how the above two lines can be combined** to enable a more practical identification of selection bias and latent confounding in real-world scenarios.\n\n---\n**(Q2)** The reviewer asks for the time complexity comparison between the generalized rank constraints and conditional independence tests.\n\n**A:**  Thank you for the question. Please let us note that the generalized rank constraints do not change the computational complexity of the statistical tests themselves–they simply **provide a characterization** of the covariance terms for recovering the graph.\n\nSpecifically, testing rank constraints (e.g., via canonical correlation analysis) and testing for conditional independence (e.g., via Fisher's Z-test) both involve computations on the covariance matrix. The underlying operations–matrix inversion, or eigenvalue decomposition–are **of comparable complexity.**\n\nAs for the algorithmic procedure used to recover the graphs, ours follows the same structure as FCI. Hence, like FCI, our algorithm has a worst-case complexity exponential in the number of observed variables. However, if the underlying graph is sparse–a common and reasonable assumption–**the runtime becomes polynomial.**\n\nWe also report our empirical running times. Though with a same complexity, our method is **consistently the fastest** among competitors, possibly thanks to some implementation speedup. Due to space limit, please kindly refer to our response to Reviewer nKfC's Q1 for the results.\n\n---\n\n**(Q3)** The reviewer notes that a relevant work (RFCI) was not discussed.\n\n**A:** We thank the reviewer for pointing out this relevant reference [[C+12]](https://tinyurl.com/4mw7h5ku). We have now included it in the updated manuscript and revised our discussion accordingly.\n\n\n---\n\nOnce again, we thank the reviewer for the insightful feedback, and hope the questions have been properly addressed.", "claims_and_evidence": "The claims submitted are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed method and/or evaluation criteria (e.g., benchmark datasets) are meaningful for the problem or application at hand.", "theoretical_claims": "No", "experimental_designs_or_analyses": "I believe the experimental design or analysis is reasonable and valid.", "relation_to_broader_scientific_literature": "This solves the problem of eliminating the influence of selection bias to discover hidden variables and restore causal relationships when hidden variables and selection bias exist at the same time.", "essential_references_not_discussed": "There are some latent variable causal discovery algorithms not discussed and compared in this paper, for example:\n[1] Colombo D, Maathuis M H, Kalisch M, et al. Learning high-dimensional directed acyclic graphs with latent and selection variables[J]. The Annals of Statistics, 2012: 294-321.", "other_strengths_and_weaknesses": "Strength:\nThis solves the problem of eliminating the influence of selection bias to discover hidden variables and restore causal relationships when hidden variables and selection bias exist at the same time.\nWeakness:\n1. The paper mentions that certain latent variable structures and selection bias structures may be equivalent under rank constraints. How can they be distinguished?\n2. There is a lack of discussion on the time complexity comparison between the generalized rank constraints method and conditional independence tests.", "other_comments_or_suggestions": "No", "questions_for_authors": "1. The paper mentions that certain latent variable structures and selection bias structures may be equivalent under rank constraints. How can they be distinguished?\n2. There is a lack of discussion on the time complexity comparison between the generalized rank constraints method and conditional independence tests.", "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:

Meta-review: All reviewers agree that this is a solid nethodological contribution to the field of causal discovery; in the presence of latent variables.
Decision: Accept (poster)