Causal Effect Identification in a Sub-Population with Latent Variables
Authors: Amir Mohammad Abouei
Ehsan Mokhtarian
Negar Kiyavash
Matthias Grossglauser
Venue: NeurIPS 2024 poster (2024)
Keywords: Causal Effect Identification
Selection Bias
Latent Variables
Primary area: causal_inference
TL;DR: We consider the s-ID problem in the presence of latent variables.

Abstract: The s-ID problem seeks to compute a causal effect in a specific sub-population from the observational data pertaining to the same sub population (Abouei et al., 2023). This problem has been addressed when all the variables in the system are observable. In this paper, we consider an extension of the s-ID problem that allows for the presence of latent variables. To tackle the challenges induced by the presence of latent variables in a sub-population, we first extend the classical relevant graphical definitions, such as c-components and Hedges, initially defined for the so-called ID problem (Pearl, 1995; Tian & Pearl, 2002), to their new counterparts. Subsequently, we propose a sound algorithm for the s-ID problem with latent variables.

Causal Effect Identification in a Sub-Population with Latent Variables
Amir Mohammad Abouei
Ehsan Mokhtarian Negar Kiyavash 2, Matthias Grossglauser
School of Computer and Communication Sciences, EPFL
Cooleeee of Management of Technology, EPFL
(amirabouei,, ehsan. mokhtarian, negar.kiyavash, matthias suathias.goossgausserr @epfl.ch
Abstract
The S-ID problem seeks compute causal effect in specific sub-population from the observational data pertaining to the same sub-population [AMK24]. This problem has been addressed when all the variables in the system observable. In this paper, we consider an extension of the S-ID problem that allows for the presence of latent variables. To tackle the challenges induced by the presence of latent variables in a sub-population, we first extend the classical relevant graphical definitions, such C-components and Hedges, initially defined for the so-called ID problem [Pea95] [PO2), to their new counterparts. Subsequently, we propose a sound algorithm for the s-ID problem with latent variables.
Introduction
Causal inference, i.e., understanding the effect an intervention in stochastic system, is a key focus of research in statistics and machine learning [Rub74] Pea00 [Pea09] [SGSH00). Scientists, policymakers, business leaders, and healthcare professionals must understand causal relationships to move beyond correlations and make informed, wridence-based decisions. To perform causal inference tasks, it is crucial differentiate between two types of data: observational and interventional [PM88
Obtaree
unitccod Sompler
P,00
Samples from popuiasion
Samplis from the populition
Blndd swcoler
Sumpler
Suer
PyOns 11)
Populition
Samples from sub-poputation
Samplic from posulation
(a) Observational data.
(b) Interventional data.
Figure I: A population consists of sample space for the study of the causal effect of intervention. While the unbiased sampler draws samples uniformly at random, the biased sampler selects samples based on certain criteria, forming a sub-population.
Observational Data. Figure illustrates a population that pertains to the entire sample space for study of the causal effect of some intervention. A sampler draws samples from the population. The sampler is unbiased it draws samples at random such that each individual in the population has
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Table 1: Various causal effect identification problems. X is the set of intervened variables, Y the set of outcome variables, and S = corresponds to sub-population. ID, c-ID, and S S-Recoverability have been addressed in the presence of latent variables. S-ID problem has only been studied in causally sufficient cases where all variables are observed.
Problem ID c-ID
Given distribution
Target distribution
Presence of latent variables
P(V)
Px(Y) Px(Y
P(V)
S- S-Recoverability S-ID
P(VI==1) P(VIS=1)
Px
Px
an equal chance of being selected. As a result, the obtained sample is representative of the entire population. In contrast, a biased sampler selects samples based on certain criteria forming a sub- population. For each extracted sample, we collect data from set of observed features denoted by V. As depicted in Figure lal when the sampler is unbiased, the observational data comes from the joint distribution P(V). For a biased sampler, the observations can be modeled as drawn from a conditional distribution P(VIS- = 1). where S indicates that the sample belongs to a sub-population.
Interventional Data. An intervention on a subset X C V assigns specific values to the variables in the subset. If performing an intervention results in changes in other variables of interest, it suggests a causal relationship. apart from mere correlation. Interventions are often represented with the do() operator, highlighting the deliberate change of a variable [Pea00] Pea09]. For the sake of simplicity in notation, we use (-) to denote the distribution of the variables after an intervention on X. Figure 1b dejectcsss the population after an intervention on subset X S V. where we seek to understand how changes in X would affect set of outcome variables Y C V \ X. To analyze this causal effect across the entire population, we must compute the distribution Px (Y). On the other hand, if we are merely interested in the results of the intervention on a specific nub-population, it suffices to compute the conditional distribution Px = 1) pertaining to the sub-population.
Causal Effect Identification. Performing interventions in populations can be challenging due to high costs, ethical concerns, or sheer impracticability. Instead, researchers often use observational methods, leveraging the environment's 's causal graph, graphical representation that depicts the causal relationships between variables [Pea09] [SGSHOO), and observational data to estimate interventional distributions of interest. Various causal effect identification problems in the causal inference literature are concerned with this issue.
Related Work. Table I lists four causal effect identification problems. The most renowned among them is the ID problem, introduced by (Pea95), which seeks to determine causal effect for the entire population using the observational distribution pertaining to the entire population. Specifically, it aims to compute Px Px(Y) from P(V). The e-ID problem, introduced by [SPO6a]. extends the ID problem to handle conditional causal effects, i.c., compute the conditional causal effect Pxx (Z) from the observational distribution P(V) pertaining to the entire population, [BTP14] introduced the s-Recoverability problem that focuses on inferring the causal effect of X on Y for the entire population using data drawn solely from a specific sub-population. (AMK24) introduced s-ID, which asks whether a causal effect in a sub- aub-ooulation such as PY = 1) can be uniquely computed from the observational distribution pertaining to that sub- aub-population, i.e., P(VIS 1). Another direction of research considers learning a causal effect from multiple datasets [LCB99 KMEK22) CLB21 KEK23 LGS244
all aforementioned causal inference problems, the causal graph is assumed to be known. Some recent work relax this assumption [JZB19] [RZB22) or introduce additional conditions on the causal graph with the goal of identifying a broader range of causal effects (THK19 MJEK22 JAK23]. Settings where data samples dependent introduce new challenges to
causal inference, which have been explored in another line of research [SS18 BMS20 ZMP23].
S-ID Is Not ID. It is worth emphasizing that the s-ID problem is not special case of ID problem, where the population restricted to the target uub-population. The presence of selection bias S introduces additional dependencies among variables, and ignoring S in the graph invalidates the application of the rules of do-calculus [Pea00] (which are the main tools used to tackle the ID problem) on input distribution, i.e., P(VIS = 1). Consequently, there are many instances where causal effect is identifiable in the ID setting but not identifiable in the s-ID setting. In particular, when all the variables in causal system are observable, all causal effeett are identifiable in the setting of

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the ID problem [Pea0O]. This is not the case in the S-ID setting, and some causal effects become non-identifiable, as noted by [AMK24]. Moreover, even when causal effeet is identifiable both the ID and S-ID) settings, using the expression from the ID algorithm can lead to erroneous inference. Example T illustrates this case.
Consider an example pertaining to study of the effect of a cholesterol-lowering medication on cardiovascular disease. Figure ddepicts the causal graph of this example, where X is the medication choice that directly affects Y. cardiovascular disease. Variable Z represents the diet and exercise routine of person. In this scenario, X and are confounded by the person's socioeconomic status (e.g., income). It can be shown that the causal effect of X on Y (in the entire population, for instance, the people around the globe) is identifiable (ID) from P(X, Y, Z) and can be computed as Px (Y) = P))
Example 1.
However, we might instead be interested in study that focuses on the people of a specific region. In this case, the target sub-population would correspond to individuals who are biased toward particular diet and exercise rou- tines and possibly have higher genetic predisposition for heart disease. Let S be an indicator node for this sub- population, has a directed edge toward S, and S and Y are confounded by the latent genetic predisposition of the people of this group. We will show in Section | that the causal effect in this sub- sub-population, PY 1),
X
Y
S
Figure ADMG G in Example
s-ID and equals Ez P(Y|X. Z. = P) =
this example, the presence of S introduces a spurious correlation between X' and Y through the path involving Z and Therefore, if we were to ignore the presence of and apply the ID algorithm to the input P(X, Y, Z|S = 1), it would result in incorrect inference: P(Y
1) as opposed to the correct value (Y JS = 1). In Appendix D we empirically compare the differences between ID and s-ID settings. We present another example in Appendix A where a causal effect is ID but not S-ID.
In this paper, we consider the s-ID problem in the presence of latent variables. Our main contributions are as follows.
We extend the classical relevant graphical definitions, such as c-components and Hedges, initially defined for the ID problem so that they inherit the key properties of their predecessors but are applicable the s-ID setting in the presence of latent variables (Section 4).
We present a sufficient graphical condition to determine whether causal effect is s-ID (Theorem 5.1 Accordingly, we propose a sound algorithm for the S-ID problem (Algorithm D
We show a reduction from the s-Recoverability problem to the s-ID problem (Theorem 6.1 indicating that solving S-ID can also solve the s-Recoverability problem.
Organization. In Sections 2 and for the ID problem. In Section we formally define the s-ID problem in the presence of latent variables and present the proper modifications of the classical graphical notions of interest for the s-ID problem. We present our main results in Section 5 In Section 6) we introduce a reduction from s-Recoverability s-ID. The appendix includes proofs of our results, as well as a numerical experiment.
cover the preliminaries and review key definitions and results
2 Preliminaries
Throughout the paper, we use capital letters to represent random variables and bold letters to represent sets of variables. Furthermore, to facilitate case of reading, we have summarized the key notations in
Table 2
Acyelic directed mixed graphs (ADMGs) consist of a mix of directed and bidirected edges and have no directed cycles. Let g = (V, E,, Ea) an ADMG, where is a set of variables, Ej is a set of directed edges (-),, and Eg is a set of bidirected edges (++). The set of parents of variable X E V, denoted by (X), consists of the variables with directed edge to X. Similarly, the set of ancestors of X E V. denoted by Ancg(X), includes all variables
Graph Definitions.

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Table 2:
Table of notations.
Notation
Description
V. U
Sets observed and unobserved variables
Auxiliary vertex (variable) used to model sub-population
g
Augmented ADMG over V U [S} Parents of vertex X in graph ç
(X)
Aneg (X), Annca
Ancestors of vertex X (including X); the union of ancestors for all X X
V VAS, NS
V ' AnAng (S) and its complement, V (S) Subgraph of ç induced by the vertices in X
9xz
Subgraph of & after removing incoming edges to X and outgoing edges from Z col-pooulation distribution. ie.. P(V =) Causal effect of X on Y, ie.. post-interventional distribution
(V(V
Px
P (Y)
Causal effect of X on Y in the sub- cobb-opulation, i.c., Px Pv (H)
QH
Q' H'H
P H (H Anogs (S) (S).8 1), 1),
directed path to X, including X itself. For a set X C V, we define Anco (X) = (Ncg In an ADMG over V, a subset S V is called ancestral if Anog (X) X.
path is called bidirected it only consists of bidirected edges. A non-endpoint vertex X, on path (X1, X2,..., . Xx) called collider if one of the following situations arises:
X - X; + X1+1, X,-1 + X, + X X + X; + X X + X, + X
Let X, Y, W be three disjoint subsets of variables in ADMG G. A path P = (X, 2I.... Zu, Y) between E X and Y Y in 9 is called blocked by W there exists s k such that Z, is
collider on P and Z; & Ancy (W), Z₁ is not a collider P and Z₁ E W.
Denoted by (X IL g) we say W m-separates X and Y for any X X and E Y. W blocks all the paths in & between X and Y. Conversely, (X J. Y/W), if there exists at least one path between a variable in X and a variable Y that is not blocked by W.
For X, S. V, Ixz denotes the edge subgraph of G obtained by removing the edges with an arrowhead to a variable in (including bidirected edges) and outgoing edges of Z (excluding bidirected edges). Moreover, G[X] denotes the vertex subgraph of G consisting of X and bidirected and directed edges between them.
SCM. A structural causal model (SCM) is tuple (V, U. F, P(U)), where V is set of endogenous variables, U a set of exogenous variables independent from each other with the joint probability distribution P(U), and F {fx)xev is set of deterministic functions such that for each X E V.
X (PaX U*),
where Pax C V [X} and U S U. This SCM induces causal graph over V such that (X) x and there is bidirected edge between two distinct variables X, Y e V when U nu # Ø. Henceforth, we assume the underlying SCM induces a causal graph that ADMG, ie., it contains directed cycles.
An SCM M = (V, U, P(U)) with causal graph & induces unique joint distribution over the variables V that can be factorized as
Pm (V)
ΣI XEV
II
PM P (X)) II PM(().
UEU
This property is known as the Markov factorization [Pea09] Note that Σx denotes marginalization, ie.. summation (or integration for continuous variables) over all the realizations of the variables in set X. We often drop the M in p.M(.) when it is clear from the context.
We model a sub-populatiin using auxiliary variable S and a biased sampler from a causal environment akin to [BP12 AMK24). Suppose A the underlying SCM of an environment with the set of observed variables V. In this causal environment, an unbiased sampler produces samples drawn from P(V). When the sampler biased, draws samples from the conditional distribution (V(V) = P(VIS S 1), where S is auxiliary variable defined as
Modeling a Sub-Population

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= fs ) where fs is a binary function, Pa S V, and U is the set of exogenous variables corresponding S. Note that U can intersect with U, but the variables in UUU are assumed be independent. In this model, S = indicates that the sample is drawn from the target sub-suuut aub--opulttion. Furthermore, we define the augmented SCM M (VU (S), U UUUS,
FU P(U U ) obtained by adding S to the underlying SCM M. We denote by go, the causal graph of M which is an augmented ADMG over V [ Note that in g variable S does not have any children, but can have several parents and bidirected edges.
Intervention. An intervention on a set X S converts M to a new SCM where the equations of the variables in X are replaced by some constants. We denote by QIV X = Px(V \ X) the corresponding post-interventional distribution. i.e., the joint distribution of the variables in the new SCM. The causal effect of X on Y refers to the post-interventional distribution PY(Y), where X and Y are disjoint subsets of V. Accordingly, the causal effect of X on Y in a sub-population denoted by Px P 1
Problem Setup. Let (V, U, F, P(U)) be an SCM with ADMG G representing its causal graph. Additionally, let S be an auxiliary variable representing specific sub-population. In this paper, given the augmented graph G and two arbitrary, disjoint subsets X and Y. we address the following question: Can the causal effect P (Y) ) be uniquely identified from the observational distribution Ps (V)? Please refer to Definition 4.1 for the formal definition of the s-ID problem.
3 ID, C-component, and Hedge
Our proposed approach to address the S-ID problem extends certain definitions and properties from the classic ID problem [Pea95]. For the sake of completeness and pedagogical reasons, in this section, we review some definitions and the main results in the ID problem [T022 HV06 SPO6b]
Definition 3.1 (ID). Suppose G is ADMG over V and let X and Y be disjoint subsets of V Causal effect Px (Y) is said to be identifiable (or ID) for short) in G if for any two SCMs M1 and M2 with causal graph G for which p.M (V) wwd (V) > O, then Px (Y) P (Y).
Px
Next, we review C-components, a fundamental concept address the ID problem.
Definition 3.2 (C-component). Suppose is an ADMG over V. The C-components of G are the connected components in the graph obtained by considering only the bidirected edges of g. Furthermore, G is called single C-component if it contains only one C-component.
There exist a few different definitions for Hedge, another central notion in the ID literature. Here, we provide a somewhat simplified definition that not only suffices to present the main result of the ID problem but also allows to extend it in the next section to the s-ID setting.
Definition 3.3 (Hedge). Suppose G is an ADMG over V, and let Y C V such that Y is single C-component. A subset H C V is called a Hedge for Y in G, Y & H. 9 is a single C-component, and H = Ancg.H] (Y).
Example 2. Consider the ADMG & over V = (X, Y2) depicted in Figure 3a In this case, G. G[X X2), and G[),,Y2] are single C-components. The c-components of G X2, Yil are X2) and {Yi). Furthermore, H (X1, Y2 is Hedge for Y = {Y, Y23 since H[H and G are single C-components and Ance (Y) = H. Similarly, V is a Hedge for Y, but there exists no Hedge for either (Yi) or (Y2).
The following theorem, restating the results in [SP06b] and [VV96), outlines a necessary and sufficient condition to determine the identifiabiliyy of a causal effect in an ADMG.
Theorem 3.4 (ID). Let G be an ADMG over V, and X and Y be two disjoint subsets V. Causal effect Px(Y) ID in G if and only if [[]] is ID in G, where = Ancoiv (YY). Furthermore, let {D.)+= be the C-components of G DD., then Q[D] is ID in G if and only if there are no Hedge in G for any of the C-components {D
Example 3. Following Example Theorem 3.4 implies that Px, (Yi,, Px,(Y)), Pix, (Yi), and P(x,,x,)XXx (Y2) are ID since no Hedge for either (Yr} or (Y2} exists. However, Px, (Y),))) is not
NNoe that in this notation, the order of operations is intervention first, then conditioning.
5

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X
Z1
Xi
X2
X
22
Y1
Y2
Y2
S
(a) ADMG 9 in Examples 2-3.
(b) Augmented ADMG G in Examples 4-9.
Figure 3: ADMGs in Examples of Sections
and 5
ID because D = (Yi, 2,, X2) and the c-components of Q Q are Dj (Y,, Y2) and D2 = (X2). and {X1, Y1, Y2 (or Vs Hedge for Dj. Similarly, can show that P P(x..x)) (Y1, Y2) is not ID.
S-ID, S-component, and s-Hedge
We begin by providing formal definition of the S-ID problem in the presence of latent variables, i.e., when the causal graph is an ADMG. Then, we present modifications of the graphical notions from the previous section so that they inherit the key properties of their predecessors and can be applied to the S-ID setting.
To avoid repetition, henceforth, we denote by V the set of observed variables and by G an augmented ADMG over V (S}. Furthermore, we denote by V As and ns the ancestors and non-ancestors of S in V, i.e.,
Vs V n Ancgs (S), Ns V Ancgs (S).
Definition 4.1 (s-ID), Let X and Y be disjoint subsets of V. Conditional causal effect Px (Y = 1) (or PX(Y) 1 is s-ID in g if for any two augmented SCMs M and M with causal graph g for which pMi (V S 1) = p.M (VIS = 1) > 0, then P.M (Y = 1) = MM (YIS 1).
Next definition extends Q[-] and introduces Q*1-). Definition 4.2 (+)..)) For C VNs. we define ( == PV H AHcgn (S) (S), S = 1).
The next definition extends C-components (Definition 3.2 and introduces S-components.
Definition 4.3 (S-component). For subset H C V NS. let Ci,..., C2 denote the C-components of U Ancgs (S)). We define the s-components H G as the subsets H, C, n H which are non-empty. Furthermore, H is called single s-component in G if it contains only S-component.
Note that [*[+] and S-components are only defined for the subsets of VNs- Figure 4a visualizes the structure of s-components of a subset H Vxs- In this figure, each blue subset (e.g., M,) represents a c-component, which means all the nodes within them are connected via bidirected edges. Therefore, according to Definition 4.3 all nodes inside S S-components (c.g.. H2) of H are connected via bidirected edges in Go H U VAs Figure 4b shows the structure of a single s-component, where all the nodes of H connected via bidirected edges in Go VAs).
Example 4. Consider the ADMG go in Figure 3b over V U {S), where V (X,, X2, Y1, Y2, Z1, Z2). Since (nc V = X2, Y,, )2). In this case, the s-components of Vas are X,} and {Y, Y2). Moreover, the s-components of (X1, Y1, Y2) are (X1} and (Y,, Y2).
(Z,, S}. we have VAS {Z,, Zz} and
We now provide two crucial properties for @*1-..
Lemma 4.4. Let W, W be two subsets of V Ns such that W C C. W' is an ancestral set in G2 () then
= Σ @'W..
(1)
W

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And(S)
H
And(S)
Anc(S)
H
H
(a) s-components H (b) is single are H H2. S-component.
(c) H is an s-Hedge for
(d) Hedge
Figure 4: Visualization of the graph structures defined in Sections and
Lemma 4.5. Suppose H C
Vss and let H H2 denote the S-components of H in g Then,
[H] decomposes
Q' H = Q'HHIQQ'HH Q..].
(2)
Let m be the number of variables in H, and consider topological ordering of the variables in
graph g [H), denoted as Vh, <...< Vh.. Let H 0 and for each S S m.
H denote
the set of variables in H ordered before Vn, (including Vn.). For every 1 S S k, (*[ can be
computed from Q3 H|] by
@HO
Q QHH11-
II (IV,,HH,)
(3)
Q9H60111
where Q$ [s can be computed by
Q [H
Σ Q'H
(4)
HAHO
The affreementioeed lemmas are extensions similar lemmas for Q[- TP03 to Q+[-).
Example 5. Following Example since {Y1} is ancestral in [ Y2), Lemma 4.4 implies that
('[i Ex @' Y2). Furthermore, since the s-components of VNs are (X1, X2) and (Yi, Y2}.
@'Iv Lemma 4.5 implies that [V Y2] My,y, V.. Thus, [V( can be computed from Q$ [s]]
Finally, we define S-Hedges, which extends Definition 3.3 for Hedges.
Definition 4.6 (s-Hedge). Suppose Y C Vss is single s-component in go. A subset H C V is called an s-Hedge for in Y Ç. H, H is a single s-component Go, and AACcg|HI (Y).
When H C Vxs is single c-component, it is also a single s-component. Therefore, if H is Hedge for Y. will also be S-Hedge for Y. Thus, Hedges can be seen as special cases of s-Hedges when H C V Figure 4d shows the structure of H, which is single c-component and forms Hedge for Y. Moreover, Figure presents the structure of an s-hedge H for Y. Note that s-hedges are more complex graph structures compared to Hedges. This complexity is required for to be able to determine whether a causal effect S-ID.
Example 6. Following (xamplessandd) 5 (X1, X2) is an s-Hedge for (X2), because both {X,, X2) and {X2} are single s-components and (X1. = (x.,, Similarly, (Y1, Y2) is an s-Hedge for (Y2).
Main Results
In this section, we provide sufficient graphical condition for causal effect to be S-ID in an ADMG. This extends the condition presented [AMK24), which assumes that the causal graph is DAG. Accordingly, propose sound algorithm for the s-ID problem in the presence of latent variables.
7

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Recall that 9 is an augmented ADMG over the set observed variables V and auxiliary variable S, and we defined As V Ancg: (S) and V us =V \ (Ancg (S).
Theorem 5.1. For disjoint subsets X and Y of V, let XAS = XOVA VAS. XNS == X n V and Ys== YNNNs-- Ns-
1. Conditional causal effect Px (Y) is S-ID in G if and only if
(Xas Y| S) S)
(5)
and Px.. (Y, XAS) is S-ID in ga.
Suppose D = A X Y us) and let (D;) denote the s-components of D in g³. Conditional causal effect Px (Y, Xss) is S-ID in G if there are no S-Hedge in G for any
(D
Remark 5.2. If either Xss or ys an empty set, then Px (Y) is S-ID in G if and only if Equation (5) holds.
In the absence of latent variables, i.e., when G is directed acyclic graph (DAG), there are no s s-Hedge in G since all the edges are directed. Therefore, Theorem 5.1 states that Px (Y, XAs) is always s-ID, and Px( (Y) is s-ID in g if and only if Equation (5) holds. We note that this is consistent with the condition presented in [AMK24] Theorem 2] for the s-ID problem in the absence of latent variables.
Example 7. Consider again ADMG G in Figure 3b where we want determine whether Pf [X,,,, 2, (Y1, Y2) is S-ID in G.. In this case, XAS = Z1]. Xxs = X2), Yss = (Y1, X2).
and (Z) 1L (Yi, (2), X2) X2, 5) X1.2 This shows that Equation 5 holds. Hence, we need to determine whether x, (Y, Y2, Zi) is S-ID in go. In this case, D Ancgi(y, (Y), Y2) (Yr,Y2) which is a single s-component. Since there exists no s-Hedge for (Yi, Y2,, Theorem 5. implies that P (x.,x2.7.) (Y,, Y2) is s-ID g
Algorithm for S-ID. So far, we have presented a graphical condition to determine whether a causal effect s-ID. In this section, we propose recursive algorithm that returns an expression for P (Y) in terms of ps (V) when the condition of Theorem 5. holds, and otherwise, returns FAIL.
In the proof of Theorem 5.1 presented in the appendix, show that when Equation
5) holds, then
( = Σ (Y AS, W(X (Y VAs),
(6)
where W VAS \ (XAs U Y A.) Thus, it suffices to find expression for P (Y Ns VAs) in terms
of Ps (V). Let Ancgs(v xxx) (Y NY))) and (Dike be the s-components of D in g5. From Lemmas 4. and 4.5 we have
P PY AAS = IIOD.
DIY
Therefore, it suffices to find an expression for each D in terms of is (V). Note that D is a single s-component in gS. We can now propose Algorithm | for computing Ps (Y) from ps (V)..
Function sID takes disjoint subsets X and Y of V along with an augmented ADMG g and conditional distribution Ps(V) input. After defining the required notations in lines 3-5, it checks
Equation (5) in line 6. If this condition is met, it defines D and its s-components (D.) )= in GS. For each S 7 k, it finds the corresponding s-component T, of VAs in GZ that contains D.. Note that T; is well-defined as D; cannot partially intersect with the s-components of V Ns in G.. Next, the algorithm seeks to compute Q* ['[ from Q* [T.] J by calling Function sID-Single. If Function sID-Singt succeeds in returning expression for each then the algorithm uses Equations (6) and to return an expression for P (Y) in terms of ps (Y). Otherwise, the algorithm returns FAIL.
Function s sID-Single takes two single S-components C and T in go such that C C T and aims to drive an expression for Q* [C] in terms of ().. The procedure is recursive and uses Lemmas 4.4 and 4.5 In each recursion, the algorithm reduces T to smaller subset T such that T' is still single le S-component G and C C T' (lines 7-1)). Eventually, the function either returns FAIL or an expression for Q* [C].
8

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Algorithm 1 Computing P (Y) from (V()
1: Function SID(X, Y, g (S(V))
1: Function sID-Single(C, T, Q'
2: Output: Expression for Px (Y) in terms of
2: Input: Two single S-components C and T in g
ps or FAIL
such that T
3: VAs f VnnAnngs (S). V 8s V Ancgs
Output: Expression for [*[C] in terms (T)
4: XAs & XAVAS ASS
XNs f Xn V NS
or FAIL
4: A < AnegirTj(C) (C)
5: YAS & Yn YnVVAS YNS NS Yn NS
5: if A C: Return Ere (T(T)
6: if (XAS H.. Y|XNs, S)
then
Xax
6: if A = T: Return FAIL
7: Return FAIL
7: ifC A Ç T, then
8: D AV xxu] Ns)
T
The s-component of A in G* that contains
9: (D,,..., D& S-components of in Gs
10: for in [1: k] do
9: Compute Q'[ from Q using Lemma 4.5
10: Return sID-Single(C. T,, (T)))
T, The s-component of Vss that con-
tains D,
Compute @ T. using Lemma 4.5 Q*[D.] & SID-Single(D; T;, (T):))
13:
Q = FAIL then
15:
Return FAIL
16:
W Vss (Xas Y
17:
Return) > ps (YAs, wiXAs) E IQD"
W
DIY
Example 8. Consider again ADMG G depicted in Figure 3b where we want to apply Algorithm
I for causal effect PX, (Y2). Herein, Xxs {X2). Yss (Y2) and X,S AS 2. Function
S-ID passes the condition in line 6 and defines D = {X1, Y1, Y2). leading to D {X1} and
D2 = (Y,, 12 It then defines T, 's in line 12 as T, = (Xi,X2/ and T2 (Y1, Y2). In line 13,
uses Lemma 4.5 compute [T[ En.y [*[V and [T Yy V (See Example
[V
5 It then calls Function sID-Single, which returns Q*[D,] Ex, Q* T and Q' [T)).
Finally, in line 20, the function returns
Ps.
Σ Pd P',, Z1.22
Σ Q'(x110'M X1.Y
where
Z2) and Q* Y2] = (Y,, Y, X1 X2, Zz, 2z)
Example 9. Following the previous example, suppose we want to apply Algorithm for computing causal effect PX, (Y,,Y2). In this case, the algorithm needs to compute [* ,, Yg] and [x[]2]-
However, when the algorithm calls sID-Single (X2, (X1, X2), Q (2)).. Function sID-Single returns FAIL. Accordingly, Function sID returns FAIL for PX (2.
Remark 5.3. Algorithm I is sound for the s-ID problem in the presence of latent variables. We con- jecture that this algorithm is also complete, meaning that whenever returns FAIL, the corresponding causal effect is not S-ID.
6 Reduction from s-Recoverability to S-ID
Recall that the objective in S- s-Recoverability is to compute PY from P*(V) [TP14], while s-ID aims to compute (Y) from (V).. [TT5 proposed RC, sound algorithm for the Recoverability problem. Subsequently, [CBB19 19 proved that RC is complete. In this section, we
present reduction from the s- RReooverabbllity problem to the s-ID problem. This indicates that solving s-ID can solve the s- R-Recoverability problem (but not the other way around).
Theorem 6.1. For disjoint subsets X and Y of V, Px can be uniquely computed (V) in the augmented ADMG Go and only
(Y 11 5
(8)
and PY( S-DD in g

---

and Algorithm 2
X1
Z1
Reduction from s-Recoverability to s-ID
Input: X, Y, G,, ps
2: Output: Expression for Px(Y) in terms
of P or FAIL
3: if (Y μ. S|X)az then
X2
Z2
4:
Return FAIL
Figure The augmented ADMG G in Example 10
5: else
6: Return sID(X, Y, 7,9
Remark 6.2. Equation 8 is very restrictive condition, and when it holds, Rule of do-calculus implies that
Theorem 6.1 implies that when Equation does hold, then PY( not S-Recoverable. However, this causal effect might be identifiable in the target sub-population, i.., PX (Y) is S-ID.
As consequence of Theorem 6.1 algorithm takes as input two disjoint subsets X and Y of V along with an augmented ADMG over VU and the conditional distribution ps (V). It first checks Equation (8) in line 3, and then calls Algorithm| as a subroutine to compute P (Y) from Ps(V) when is s-ID in go.
propose Algorithm 2 for computing Px PY from (V).. The
Example 10. Consider the augmented ADMG G in Figure In this graph, (Y ./. S|X1)s thus,
Theorem 6.1 implies that Px, (Y) cannot be uniquely computed from ps (V).. On the other hand, Px, (Y) can be identified from P*(V) since (Y LL S( and due to Theorem 5.1 PX, (Y) is S-ID in g In this case, Algorithm returns the following expression for Px, (Y) in terms of ps
(* X2 Z2)
PY) = Σ (' M2)
Σ
X1, X,, 71,72)
21.22
X1
7 Conclusion
The S-ID problem, introduced by [AMK24], asks whether, given the causal graph, causal effect in a sub-population can be identified from observational distribution pertaining to the same sub-population. [MM24]] addressed this problem when all the variables in the causal graph are observable. In this paper, we studied the s-ID problem the presence of latent variables and provided sufficient graphical condition determine whether causal effect is S-ID. Consequently, proposed sound algorithm for S-ID. While this paper proves the soundness of our proposed method, also conjecture that our approach is not only sound but also complete. Finally, by presenting appropriate reduction, we showed that solving s-ID can solve the S S-Recoverability problem.
Acknowledgments
thank anonymous reviewers for their feedback. This research was part supported by Swiss National Science Foundation under NCCR Automation, grant agreement Swiss SNF project 200021 20000022200355//1111
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[KMEK22] Yaroslav Kivva, Ehsan Mokhtarian, Jalal Etesami, and Negar Kiyavash. Revisiting the general identifiability problem. In Uncertainty in Artificial Intelligence, pages 1022-1030. PMLR, 2022.
[LCB19] Sanghack Lee, Juan David Correa, and Elias Bareinboim. General identifiability with arbitrary surrogate experiments. In Conference on Uncertainty in Artificial Intelligence, 2019.
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Appendix
The structure of the appendix as follows. Appendix A includes an additional example of S-ID problem in the presence of latent variables. In Appendix B we provide some preliminary lemmas used throughout our proofs. The proofs for the main results, namely, Lemmas (.4 4.3] and Theorems 6.1 6.1 are presented in Appendix C In Appendix [ will conduct experiment to compare the outputs of s-ID algorithm and the classic ID algorithm.
A Additional Example
X
U1
S
Figure 6: ADMG G of the example of Appendix A
In this section, provide example where ignoring S results in an identifiable effect, whereas the target causal effect is, fact, not s-ID.
Consider the following two SCMs. SCM Mı:
U1 ~ Bern(0.5)
U2 Bern(0.5)
Ey ~ (ern(0.3)
X U1
Y X e U2 5
S U1 U
where @ denotes the XOR operator, and Bern(p) denotes Bernoulli random variable with parameter
SCM M2:
U1 ~ (ern(0.5)
U2 Bern(0.5)
Ey ~ Bern (0.3)
X U1
Y = Sy
S = 1
According to the above equations, we have
PMi (x- = x, Y = y/. = 1) = P(U = (r)) = y) = 0.5 x Ple, =1) > 0.
Similarly for M2 have
MMa (X x, = y/S = 1) = pMa x, Y = y)
>MMa x) pMa = y)
= P(U, = x) Ple, = = 0.5 x Pley = > 0.
Thus, P.M. (X, X 1) PM (X, Y/S = 1) > 0. Furthermore, we have
P.M 1|$ 1) ( + Ey 1|s 1) p. (U2 Ey = = 0.5.
13

---

Similarly, for SCM M2:
p = 1|$ = 1) PM = 1|5 = Pley = =] =
This shows that Px (Y) is not s-ID in this causal graph, pMi (x, (XS - p Y|S 1) > 0, but (Y = 1 # PY 1 1).
Note that ignoring the sub-population, the causal effect (Y is clearly identifiable from P(X, Y), but as we showed above, the causal effect of X on Y is not identifiable in the sub-population from the observational data of that sub-population.
B Technical Preliminaries
Pearl's do-calculus rules [Pea00]: Let Y, Z, W be four disjoint subsets of V. The following three rules, commonly referred to as Pearl's do-calculus rules [Pea00], provide tool for calculating interventional distributions using the causal graph.
Rule 1: If (Y X W)gx then
P W) P)(YW).
Rule 2: If (Y I1 Z W)
Xx then
Px (Y, W). XZ where Z(W) = Z Anncg (W), then
Rule 3: If (Y IL ZX, W)
Px,z(Y(W) PY(Y)
Lemma B.1 (PP3). For two sets W c W. if W' is an ancestral set in G[W), then
Qw1 = Σ Q[W].
(9)
W.w
Lemma B.2 (PO3). Let C S V, and assume that C is partitioned into C-components C1, in the subgraph G[C).
C.n
Then we have
Q[C] decomposes
Q[C] II Q
(10)
Let denote the number of variables in H. and let us assume topological order of variables in C as Ve, Vo, <... < Ve, in Gc. Let C² be the set of variables in C ordered before Ve, (including Ve,), for i = 1,2, ... k, where C is an empty set. Then each Q Cj},
j 1,2,..., m, is computable from Q[C] and is given by
QIC()
Q[C,I= = II
(11)
Q[C
(V EC,}
where each Q[C*) is given by
Q[C(0]=
Σ Q[C).
(12)
C(C
Corollary B.3. Ler H, L. HH, L.... Hm be a partition of set C, where for each C, there exists 1 1 j S such that C; C Hy. Then, we have
QIC] = (HH
Lemma B.4. Let H be a subset of ks, we have the following equation
QHH Q U Anc(S))
(13)
Q[Anc(S))
14

---

Proof. According to definition of Q², we have
Q' [H PV. (H)nA(((())))(((() = PV.AH (H,
Q Anc (H
Pv H (Anc(S))
Note that (S) is an ancestral set in 9; Hence, for any H, we have
Pv ((Ann(()) P(Anc(S) - Q[Anc(S)).
This implies Equation
13
Lemma B.S. Let X and Y be disjoint subsets of Vss V V.AAn Annco (S). We have
Pxu (Y Ns Anc(())
Σ [Di]...Q'[DAA,
DYY
where D AncgiV X.x) Ns), and Dj's are S-components of D.
Proof. Using marginalization over Vus \ (Xxs U Yss). we have
Px (Y (ns(Anc(S))
E
Pxa (Vis (xs(Anc(S))
E
@'IVxs (Ns].
Val((xuUYn))
Since D is ancestral set in G[VNs Xxs], according to Lemma 4.4
we have
Px Anc
E
Xxs] = Σ Σ
Q'(Vis)XNsh E QDD)
(XAS(X))
DIYsa VaS (X,,U
D
Therefore, the first property of Lemma 4.5 implies that
Px.. Ns Anc(S))
Σ
Q'DiI.... QD[....
DAYM
Lemma B.6. If Function sID-Single returns FAIL the inputs and T, then T is an S-Hedge for C.
Proof. When the algorithm returns Fail,
1. C and T are both single s-components since the inputs of the algorithm have to be S- components.
2. If A Ancgir (C), then T A.
According to the definition of s-Hedge 4.6 T is an s-Hedge for
C Proofs of Main Results
Lemma 4.4 Let W, W be two subsets of V us such that W C. W. If W' is ancestral set in G[W], then we have
Q'(w-= Σ Q!W].
(14)
W!W"
Proof. According Lemma B.4 we have the following equations
Q*[w] Q[W Anc
Q[Anc(S))
W'] Q(W' Anc(
Q[Anc(S))
15

---

Therefore, by replacing the above equations in Equation ① it is sufficient to show that
Q' UAc((S]] Q[Anc(S))
Q[w U ASn
Q[Anc(S)]
w.w
Since P(Anc(S)) > 0, this is equivalent to
Q ( Anc(SS]
Σ W
Q[W Anc((S)
Note that W Anc(SS in an ancestral set in g[W U Anc( and W W (W (WU Uncc(S))
(W' An)( Hence, Lemma B.I concludes the proof.
Suppose
C Ns and let HI,..., H2 denote the s-components of H in Go. Then,
Q" H decomposes as
Q H QH ... H'H)))
Let m be the number of variables H. and consider a topological ordering of the variables
in graph G H), denoted Vn, <... < Vhn.. Let H and for each S S m, ( denote the set of variables in H ordered before Vs, (including Vn,). For every S j S k. Q* []]]] can computed from Q* [H] by
Q' H
II
(15)
(H(
dd eH,}
where Q* (*(((()) can be computed by
Q @ Σ H'H).
(16)
H
Proof. First part: according to the definition of 111,1,, we have
PV H (H, Anc()) Q U Anc(S)] Q (S))
@ H Pv.. (VAAH(((AnC(S))
Pv (Anc((()
Note the last equality holds because AS) is an ancestral set in G. Now, we have Similarly, for each i E [1: : k). we have
QS)] = ( = P(Anc(S) = Pv (AAnC(A
H U Anc((S)
Q H.
Q (S)]
Therefore, the above equations imply that
Q Anc Q[Anc(S)]
Q Anc(S))
QH U An)(S))
(' = Q² H ... [..)
[S)]
Q[Anc (S)
(17)
Let C,'s be the corresponding C-component of H, (i.e., H, C, and C, is a C-component of H, Unn((S)). According the definition of C, and Corollary B.3 we have
QH; U An) = Q[C,]Q[Anc((S C.].
Moreover, since C, is a C-component of Anc(S) U H. then C, \ H, should be union of some
C-components of Anc(S) (S). Therefore, for each Corollary B.3 implies
Q[Anc(S)) Q(C, H.(((Anc(S) (C, H.) Q(C, H.]Q[Anc(S) C.]
16

---

Putting the above equations together, we have
QHH Q Anc Q[Anc(S)]
Q(ccs) | C2]
Q
(18)
QIC, H Anc (S) C.
Q(C, H.
Hence,
QH, Un) h II
C
Q[Anc(S)]
Q H.]
Consequently, have
Q' [* (HQ Q" Hb) 4 Q[H = Q[Anc(5)
R
Q[..
II. QIC, H. ==1
Note that C; H. disjoint subsets of Anc(S), where each of them is the union of some components of Anc(S) If Ck+1 = Anc(S) \ Ur C₁ then Corollary B.3 implies the following.
Q(Anc(S) Q[C++1]
[[Anc(S)] = II Q H.AHHH/Q+H
I Q H.]
Finally, applying Corollary B.3 for H Anc(S), we have
Q (HUUAc((S)] Q[C II [[C.].
This concludes the first part.
Second part: the proof is similar to proof of Lemma B.2 Define C H (UAnc(S). Let C1. C... be C-components C (w.lo.g. assume that H; S C; for [1 k)). Consider a topological order of Vh, Vha <... Vn,, for H. and Vs, <... < Vs,, for Anc(S). Since any Vs, is not ancestor of Anc (S), Vs, <... Va,, < Vhy < Vna < ... Vn,, is topological order for H U Anc(S).
According to Lemma B.2
have
Q
Q[C,] =
II
Q[Cki-1j
EC,)
where (c1, C2,..., Cn+n') = (s Sn' h,, For each E [1 n). H'
(Vi,, Vhai..., Vr.3 and H = Ø. For each i >> 0, we have
Q Q(((+)))
Q U Anc(S))
LAee (8)
Q[Ane (S)
H
Q[C(++n'-D]
Q H 1-11) U Anc(
Q 1) Wn
Q H
[SAn
where the last equality holds according Lemma B.4 It suffices show that
Q Q'((()
HHH
II
Q [i(
(V EH,
Equation 18
implies that
C[C,]
@HHH,
C H
Hence, according to this equation and Lemma B.2 have
Q
QIC, H, = Q[C,]= =
II
QIC--D
(I,,,
Q
H
II
QC--1)
Q (00
(1V, (dVV,,,,,AAe((S)
(I EH,)
QC
Q"
= II
II
(19)
Q[CU-1
Q [H
UIV, (H,
(i EH,)
17

---

Based on the definition C and H, C \ H2 is the union of some C-components of (n). Denote the e-components of C H by Ap....y A.. Applying Lemma B.2 to C Anc(S) with topologiaal order Va, < ... < V s we have
Q|C0o]
Q[Ai] II
Q[C(i-1]
(ilV,, EA1)
Q|C01
Q[A[] II
Q[C(i-11
(A,)
By multiplying all Q[A.], we have
Q
QIC, Hj
U2 U. = II
Q[A,]
II
Q[C-11] ")
d=1
EC, (H
By substituting this Equation
99
we obtain
QHH Q [(H
Q' H =
II
(i EH,)
To complete the proof, it suffices to show that Q H [ are computable from [ Since we have used the topological order, H' is an ancestral set in g H.. Therefore, Lemma 4.4 implies that
Q'11-=
Σ @"HH.
H.H
This shows that (H.] is uniquely computable from Q"
Theorem 5.1 For disjoint subsets X and of V, let
XAS == xx= ASs X = Xn VNs, and Yss == Y n V
1. Conditional causal effect P (Y) s-ID g if and only if
(X 11 Y|Xss, S) xnxs'"
(20)
and Px (Y, XAs) is S-ID in gs.
2. Suppose D = Ancgs X ( (ss) and let (Dr)ez denote the S-components of D in G³. Conditional causal effect Px XA) is S-ID in g if there are no s-Hedge in G for any of (D.
a necessary condition. Suppose (X.s + 1 Y Ns,, S) X.. Denote by G' the equivalent DAG of go, obtained by adding the unobserved variables. Theorem 2 in [AMK24) shows that there are two SEMs M1 and M2 compatible with g' such that
Proof. We first show that (XAs IL Y X S) 5)
MMi (V, USS = MM (V, US = 1), and
Px ( S 1) P (Y 11)
We use these SEMs to construct SCMs M₁ and Mr compatible with go, in which we have
p.i (V. U|s = = p.M (V, U|S 1) = pMi = (V 1).
Note that all causal effects in both models are the same for g5 and G. Hence when (X,s / holds, then PX (Y) is not s-ID. It shows that this condition is necessary
1 Y Xss, S) XxXx
condition.
18

---

Now suppose that (Xss 1 X|Nss S) holds. According to Rule of do-calculus, we have
Px Px.. As,
The above equation shows that Px 1) s-ID in 9 if and only if Px.. (Y|X S 1) is s-ID in g Moreover,
XY =) Pr X A
Since XNs n V As = 2, acccording to Rule 3 of do-calculus, we have Px (Xis S = 1) = P(XAs|5= 1). Hence,
Px.. X 5) Px (Y, x
(21)
P( -= =1
This shows that s-Identifiability of Px.. (Y|X,s, S = 1) and Px.. (Y, (Y. = 1) are equivalent
(note that P P(Ass 1) > 0 due the positivity assumption Definition 4.1)
Let W = Vs (Xss As)) then
(Y S = =) Σ Px. (Y, S
W
Px,. YASs, W/X S (Yss| YAS, W,S= Σ
W
Σ
Px. (YAS, W/X == 1 S
The above equalities have been obtained by marginalizaaton over W, chain rule, and replacing the definition of W, respectively. According to Rule 3 of do-calculus, since Xws n VAs Ø, we have
Px (YAs, W XAs
AS, S = 1) = P(YAs. W
Moreover, according to Lemma
B.5 we have
Pxss Y Ns V AS S = =) Σ
D
Combining the aforementioned equations with Equation
21
we get
Pxu (Y,
Σ P(A.s. YSS. WIS =) Σ DIDi)...Q''PA. W
(22)
DiYM.
Now, note that according to Lemma B.6 if there not any s-Hedge D,s, then Function sID-Single will compute Q3 [D[D. Therefore, we can compute Px.s (Y, VAsAs which concludes the proof. As a result, Equation (5) holds, we have
using the above equation,
(Y) = Px, (Y Σ P(YAs, Wss 5 Σ @'D..)..Q'(.....
(23)
D\Y
Theorem 6.1
For disjoint subsets X and Y of V, Px(Y) can be uniquely computed from ps (VVV in the augmented ADMG go if and only if
(Y S
and (Y) is S-I in g..
Proof. If (Y IL S|X)sx then [BT15] Theorem 2] implies that PY is not S-recoverable. If (Y I. S|X) then according to Rule of do-calculus, we have
Px PY(). Therefore, the identifiability of Px(Y) equivalent to Px 1), which concludes the proof.
(24)
D
19

---

X
Z
S
Figure 7: A DAG, where U1 and U2 are unobservable, and represents the auxiliary variable for modeling col-population.
D Numerical Experiment
We conduct numerical experiment to demonstrate the significance of the s-ID problem and assess the output of Algorithm|]] [
Note that the experiment is simple and can be run on a system with any level of computational power.
Consider the following structural causal model (SCM) with the causal graph depicted in Figure
U1 (er)(
U2 ~ Bern(0.5)
X U1 En Ex Bern(0.2)
Y = X e U2
U1 Ez, E, ~ Bern(0.2)
Herein, @ denotes the XOR operation, all the variables (U1, U2, E2, Esi Eay, are indepen- dent, and Bern(p) denotes a Bernoulli random variable with parameter p. Now, consider the sub-pon cobbpopulation with the following mechanism:
$ = (Z x Es, @ (U2 X Es,) ① Ezgs (Es, Eag Exg) ~ (Bern(0.6), Bern(0.9), (ern(0.1))
We now consider the problem of estimating the causal effect of X in this sub-population. Particularly, our goal is to calculate Px=o(Y 1|5 = =).
D.I Theoretical Analysis
To analyze and compare the s-ID and ID algorithms, first determine the exact values of Px=o(Y 11s = 1) and Px= = 1). According to the equation of Y in the SCM, we have
Px= (Y y) Px= U2) P(U2 = y + x).
Since U2 is Bernoulli random variable with parameter 0.5, the above probability always equals 0.5. For Px (Y|S SS = 1), we have
P S|S 1) PY @ U2|S P(U2 + y/ = =))
Therefore, for X = 0 and Y = 1, we need to compute P(U₂ = 1 = 1). By using the equation of S in the model,
= 1,5 1, 1
P(U, = 1 =1
0, 0,
= 1|U2 1)
1 = 1) P(U, = P) = 1 0) + P(U, = 1)
Note that P(U, = 0) = P(U, 1) = 0.5, hence, we have
P 1 1) P(U 1|5 P = 0) + = 1 =)
Since Esj and E.g and Z are independent variables, let W be Z Eeg Egg then, have
20

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W (0r1 (1 -11-0.6 0.6 0.6 0.5) + 0.9 0.5 0.6) (0.3()
Note that W and U2 Eag independent, and 5 W + +U2 Exyi thus.
P(S 1112 - 0) P(W -) 0.34
P(S = 1|U2 = =1)= =PW = =1) + P(W =1) 1)
=0.34 0.1 0.66 0.9 0.628
Hence,
0.628
Px=o(Y 1/5 1) 0.628 0.34
0.648
(25)
D.2 Empirical Analysis
The ID algorithm returns the following simple expression for Px Px(Y)
Px(V) = P(M
(26)
On the other hand, the s-ID algorithm returns
Px 1) -
(27)
Z
Next, generated 3000 samples from the population. We then computed S for each generated sample and collected the samples where S 1, resulting in 1469 available samples from our target nub-population. Recall that our goal was to estimate Px= 1|5 1). Consider the following two approaches.
If consider the s-ID algorithm and the existence of the selection bias S, applying the simple plug-in estimator to the formula in (27) results in
Px=o(Y 1|5 1) PP 1 = 0, 2 = 0) P(Y 1| = 0, =
0.641
(28)
Suppose we ignore the presence of sub-population and apply the II) algorithm. In this scenario, we have estimate Px (Y) using the empirical distribution of variables, i.e.,
P(V). If we use the ID algorithm, we need to estimate the quantity mentioned in equation The empirical estimate for this case is
26
Px(Y) R P(Y[X) 0.725.
(29)
A comparison between the estimation results of our proposed method in Equation 28 and the classical ID problem in Equation (29) with the true underlying value Equation (25) shows that our approach accurately computes the target causal effect. On the other hand, ignoring the subtleties related to sub- populattiiion can lead to erroneous estimation.
21

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28


Reviews:
[{"review_id": "5y6dUiSqKz", "reviewer": "Reviewer_rYoX", "summary": "The paper presents a sound algorithm for checking the s-identifiability of causal effects under sub-populations. This work complements earlier work on s-ID by generalizing the causal graphs to allow hidden confounders (no causal sufficiency). Specifically, the paper introduces the notions of s-components and s-Hedge, in parallel to the classical notions of c-components and c-hedge, and derives theoretical results based on those. The main theorem (Theorem 5.1) summarizes the condition under which a causal effect is s-ID, and a detailed algorithm is also proposed for deriving an identifying formula. Moreover, a reduction from s-recoverability to s-ID is mentioned which provides yet another approach to solve the s-recoverability problem.", "questions": "- Are there any insights on the difference between having $P^s(V)$ vs. having $P(V)$ for identifiabilty? For example, would more variables become dependent so they now belong to the same s-components when collecting data under $P^s(V)$?\n- Is there any evidence (counterexample) proving that Algorithm 1 is not complete?\n- Regarding the reduction from s-recoverability to s-ID, I'm wondering if there is any impact of this reduction besides theoretical interests. For example, will the reduction-based approach for s-recoverability be more computationally efficient?", "limitations": "OK.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- I think the problem is quite meaningful since selection bias can be common in the data collection process.\n- It is great that the paper not only introduces novel notions such as s-components and s-hedges but also thoroughly reviews the classical notions of c-components and c-hedges, so we can make a comparison. The lemmas and theorems are also in parallel (but different ) to the previous ones for classical identification in [Tian, Pearl], which makes these profound concepts easier to penetrate.\n- I found the examples helpful, especially Examples 4 and 6, in aiding my understanding of definitions. \n- In general, a hard work that contains valuable theoretical contributions.", "weaknesses": "- It seems that the s-ID method is sound but not complete, but I guess the paper already contains enough contributions and the completeness part can always be the future work.\n- More intuitions can be provided on the difference between s-ID and ID at the end of page 2 - it would be helpful to provide a more intuitive explanation for Example 1 (in addition to an explanation based on identifying formulas) for readers to see the importance of the problem.\n- The definition of $Q[]$ seems to be ambiguous. In Section 2 last subsection, $Q[X]$ is defined as the interventional distribution $Q[X] := P_{x} (V \\setminus X)$, but in Theorem 3.4 $Q[D]$ seems to mean $Pr_x(D)$. It may be helpful to clarify the formal definition of $Q[D]$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed feedback and appreciate the positive comments on the contributions and presentation of the paper.\n\n-----\n\n> Are there any insights on the difference between having $P^s(V)$ vs. having $P(V)$ for identifiabilty? For example, would more variables become dependent so they now belong to the same s-components when collecting data under $P^s(V)$?\n\nWe confirm your intuition. The presence of selection bias ($S$) injects additional dependencies among variables. Consequently, the rules of do-calculus cannot be directly applied to $P^s$ because of the influence of $S$. As a result, the ID algorithm is no longer applicable to the s-ID setting.\nTo tackle this challenge, we defined s-components over the non-ancestors of $S$ to capture these additional dependencies among c-components.\nWe further defined s-Hedge based on the s-component and derived our identification results based on that.\n\n-----\n\n> Is there any evidence (counterexample) proving that Algorithm 1 is not complete?\n\nWe have not found any examples where the algorithm produces a false negative. In fact, we have evidence to support that the s-ID algorithm always returns \"Fail\" when the target causal effect is not s-ID. This suggests that there are no false negatives and that the algorithm is complete. We have tried to prove this conjecture, and we have succeeded in proving it for some specific cases. However, proving it in the general case has evaded us so far due to the complexity of s-component and s-Hedge structures and the positivity constraint. This is definitely a future work direction for us, but also for the community at large.\n\n----\n\n> Regarding the reduction from s-recoverability to s-ID, I'm wondering if there is any impact of this reduction besides theoretical interests. For example, will the reduction-based approach for s-recoverability be more computationally efficient?\n\n\nThe goal of our reduction was to demonstrate that the s-ID algorithm can also address the s-Recoverability problem, thus, algorithm 2 might not be the optimal choice for the s-recoverability problem. We also note that computational complexity is not a limitation for either approach. However, in cases where the causal effect is identifiable, our method could provide a different expression. Having different expressions for the target causal effect could potentially be beneficial for developing an estimator that numerically computes the causal effects using a finite set of samples. This can be further studied in future work.\n\n----\n> More intuitions can be provided on the difference between s-ID and ID at the end of page 2 - it would be helpful to provide a more intuitive explanation for Example 1 (in addition to an explanation based on identifying formulas) for readers to see the importance of the problem.\n\nWe thank the reviewer for the suggestion. We will include a more detailed discussion in the revised version, where we have an additional page.\n\n----\n\n### Regarding the definition of $Q[]$:\n\nThank you for pointing out this typo - we will fix it.\nIn graph with observable variables $\\mathbf{V}$, for each $\\mathbf{D} \\subseteq \\mathbf{V}$ we have $Q[\\mathbf{D}] = P_{\\mathbf{V} \\setminus \\mathbf{D}}(\\mathbf{D})$. $Q[\\mathbf{D}]$ follows this definition in Theorem 3.4.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9NJ7xhSVOZ", "reviewer": "Reviewer_APnQ", "summary": "This paper extends the S-ID problem, which asks if a causal effect within a specific sub-population can be identified using only observational data from that group. The authors consider the scenarios where some variables are latent. They provide a  sufficient graphical condition to determine whether a causal effect is S-ID and propose an algorithm based on this criterion. While the paper proves the algorithm's soundness, it suggests it might also be complete. Finally, they show that solving S-ID can solve a related problem called S-Recoverability.", "questions": "- Although the work provides rigorous theoretical contributions, it would have been nice to evaluate how it would also work empirically, especially in a close-to-real-world scenario.\n- There are many variables and notations used throughout the paper. A table summarizing these notations and their definitions could improve readability.\n- How often do real-world problems satisfy the restrictive condition in Equation (8)​​?", "limitations": "The limitations of the proposed approach have not been discussed.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper addresses an important problem in causal inference.\n- The paper is very well-written and covers the prerequisites very well.", "weaknesses": "See the Questions section below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer and appreciate the positive feedback on the importance of the problem and the clarity of the presentation.\nBelow we answer the questions.\n\n---\n\n> Although the work provides rigorous theoretical contributions, it would have been nice to evaluate how it would also work empirically, especially in a close-to-real-world scenario.\n\nComputing a causal effect numerically typically involves several steps:\n\n- **Causal Discovery (a.k.a., Causal Identification)**: Learning the causal graph from the available data.\n\n- **Causal Effect Identification**: This step focuses on determining whether the causal effect is identifiable and, if so, deriving a formula for it.\n\n- **Numerical Estimation**: Given a finite set of samples, compute the expression derived in the identification phase numerically.\n\n- **Evaluation**: This step measures and evaluates the quality of estimators proposed in the estimation step, which often includes sensitivity analysis.\n\n\nSimilar to [1-5], this paper focuses on addressing the identification part. Still, we provided an example with real-world variables in Example 1 and also carried out a simple numerical experiment in Appendix C using a finite set of samples. However, designing an end-to-end pipeline is beyond the scope of this paper and is an interesting future research direction. This would involve collecting a real-world dataset, conducting rigorous causal discovery, designing a proper estimator based on the variable distribution, and applying various evaluations.\n\n\n[1] Tian, Jin, and Judea Pearl. \"A general identification condition for causal effects.\" In Proceedings of the Eighteenth National Conference on Artificial Intelligence (2002)\n\n[2] Shpitser, Ilya, and Judea Pearl. \"Identification of conditional interventional distributions.\" Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, 2006. \n\n[3] Bareinboim, Elias, Jin Tian, and Judea Pearl. \"Recovering from selection bias in causal and statistical inference.\" In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 2410-2416. 2014.\n\n[4] Lee, Sanghack, Juan D. Correa, and Elias Bareinboim. \"General identifiability with arbitrary surrogate experiments.\" In Uncertainty in artificial intelligence, pp. 389-398. PMLR, 2020. \n\n[5] Jaber, Amin, Adele Ribeiro, Jiji Zhang, and Elias Bareinboim. \"Causal identification under Markov equivalence: calculus, algorithm, and completeness.\" Advances in Neural Information Processing Systems 35 (2022).\n\n\n------\n\n> There are many variables and notations used throughout the paper. A table summarizing these notations and their definitions could improve readability.\n\nThank you for your suggestion. We will add a table summarizing our key notations.\n\n> How often do real-world problems satisfy the restrictive condition in Equation (8)​​?\n\nThe condition in Equation (8) specifies that following an intervention on $X$, the target variable(s) $Y$ should have the same distribution in both the sub-population and the entire population. In other words, after the intervention, $Y$ should not be impacted by the sub-population, which is a very stringent requirement.\nFor instance, in the family of graphs where $Y$ is a parent of $S$, Equation (8) does not hold. Consequently, $P_{X}(Y)$ is not identifiable from $P (\\mathbf{V} | S = 1)$, while $P_{X}(Y | S = 1)$ can still be identifiable from $P (\\mathbf{V} | S = 1)$.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2YoiO9h8MU", "reviewer": "Reviewer_4gRi", "summary": "This paper extends the sub-population causal effect identifiability (S-ID) problem to include latent variables by adapting classical graphical definitions such as connected-components and Hedges. It proposes a sound algorithm to compute causal effects in sub-populations with latent variables.", "questions": "See weaknesses section", "limitations": "Limitations are discussed", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is written well and easy to understand.\n2. Examples in each section helps to understand the underlying idea easily.\n3. All the necessary background is discussed clearly.", "weaknesses": "1. Example 1 could be a better one because socioeconomic status can cause cardiovascular disease.\n2. It would be good to include a sub section for summarising any assumptions made.\n3. It would be good to include some real-world use-cases benefiting from such setting of causal effect identification.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's suggestions and positive feedback.\n\n---\n\n> Example 1 could be a better one because socioeconomic status can cause cardiovascular disease.\n\nWe acknowledge that the causal graph might not be completely accurate - the primary goal of this example is to demonstrate the difference between ID and s-ID algorithms over a simple graph.\nFor the particular edge that the reviewer is mentioning (from socioeconomic status to cardiovascular disease), if we simplify/replace socioeconomic status with income, it might be fair to assume that this effect is mediating through $X$, which is the medication choice.\n\n----\n\n> It would be good to include a sub section for summarising any assumptions made.\n\nAs the reviewer suggested, we will add a section that summarizes the problem setup and the corresponding assumptions. We will also add a table of notations to improve the clarity. \n\n----\n\n> It would be good to include some real-world use-cases benefiting from such setting of causal effect identification.\n\nThank you for your suggestion. While our main contribution is to establish the theoretical framework for addressing the s-ID problem in the presence of latent variables, we agree that providing more real-world examples in addition to Example 1 would be valuable for readers looking to apply our proposed method. Therefore, we will include another example in the final version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Okv3HCV6Ha", "reviewer": "Reviewer_YSpJ", "summary": "This paper addresses the S-ID (Sub-population Identification) problem in causal inference, extending it to scenarios with latent variables. The S-ID problem seeks to determine if a causal effect in a specific sub-population can be uniquely computed from observational data pertaining to that sub-population. The authors introduce new graphical definitions such as S-components and S-Hedges, which are extensions of classical notions like C-components and Hedges. They present a sufficient graphical condition for determining if a causal effect is S-ID and propose a sound algorithm for solving the S-ID problem in the presence of latent variables. Additionally, they show a reduction from the S-Recoverability problem to the S-ID problem.", "questions": "1. In Remark 5.3.  the authors \"conjects that this algorithm is also complete\". Is there any example or situation that it returns a false negative?\n2. The reduction from S-Recoverability to S-ID in Section 6 seems to suggest that S-ID is a more general problem. Is there any scenario where solving S-ID would be more useful than solving the S-Recoverability problem?", "limitations": "The paper proved that both proposed algorithms are sound for s-id, but did not show any guarantee for the completeness. Nevertheless, this limitation is mentioned in conclusion.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Technical quality: The paper presents thorough theoretical analysis, including formal definitions, lemmas, examples and theorems, as well as the reduction derivation.", "weaknesses": "Empirical evaluation: there is no experiments at all besides the last section in appendix briefly describling how the authors want to conduct them. That is to say, this paper lacks experimental results or real-world case studies to demonstrate the practical applicability and performance of the proposed solution, especiallly for the two recursive algorithms\nComparison to other existing methods: The paper surely follows the id, c-id, S-Recoverability literature for related work, but still could benefit from a more extensive comparison with other latent variable models in causal inference, [1-3] to name a few\n\n[1] Liu, Yuhang, et al. \"Identifying weight-variant latent causal models.\" arXiv preprint arXiv:2208.14153 (2022).\n\n[2] Sherman, Eli, and Ilya Shpitser. \"Identification and estimation of causal effects from dependent data.\" Advances in neural information processing systems 31 (2018).\n\n[3] Kocaoglu, Murat, Karthikeyan Shanmugam, and Elias Bareinboim. \"Experimental design for learning causal graphs with latent variables.\" Advances in Neural Information Processing Systems 30 (2017).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments. We first answer the two questions and then discuss the reviewer's concerns mentioned in the weakness section. \n\n------\n\n> **Q1**: Is there any example or situation that it returns a false negative?\n\nWe have not found any examples where the algorithm produces a false negative. In fact, we have evidence to support that the s-ID algorithm always returns \"Fail\" when the target causal effect is not s-ID and have observed no false negatives. That is why we conjectured that the algorithm is complete. We have tried to prove this conjecture, and we have succeeded in proving it for some specific cases. However, proving it in general has eluded us so far (perhaps due to the complexity of s-component and s-Hedge structures and the positivity constraint).\n\n----\n\n> **Q2**: Is there any scenario where solving S-ID would be more useful than solving the S-Recoverability problem?\n\nThere are two main implications of Theorem 6.1 and Remark 6.2: \n\n- **s-ID is a more general problem**:  As the reviewer correctly points out, s-ID is more general than s-Recoverability in the sense that an algorithm for s-ID can tackle s-Recoverability problem as well.\n\n- **s-ID is a more practical setting**: Recall that given the observational distribution of a sub-population, s-Recoverability attempts to identify $P_{X}(Y)$ (the causal effect over the entire population), while s-ID attempts to identify $P_{X}(Y | S = 1)$ (the causal effect over the sub-population). Not surprisingly s-Recoverability does not succeed except under very special settings.\nThis is what Remark 6.2 argues: the condition for s-recoverability to successfully identify $P_{X}(Y | S = 1)$ is quite restrictive, whereas s-ID has a more realistic condition.\nAccordingly, there are indeed many examples where $P_{X}(Y | S = 1)$ is identifiable, but $P_{X}(Y)$ is not. For instance, please refer to Figure 4, where our algorithm returns \n\n$ P_{X_1}(Y | S = 1) = \\sum_{Z_1, Z_2} P^s (Z_1, Z_2) \\sum_{X_2} P^s (X_2 | X_1, Z_1, Z_2) \\sum_{\\tilde{X}_1} P^s(Y | \\tilde{X}_1, X_2, Z_1, Z_2) P^s (\\tilde{X}_1 | Z_1, Z_2).$\n\nWhile s-Recoverability fails to recover $P_{X_1}(Y)$ since this effect is indeed non-identifiable.\n\n------\n\n## Regarding empirical evaluation\n\nComputing a causal effect numerically typically involves several steps:\n\n- **Causal Discovery (a.k.a., Causal Identification)**: Learning the causal graph from the available data.\n\n- **Causal Effect Identification**: This step focuses on determining whether the causal effect is identifiable and, if so, deriving a formula for it.\n\n- **Numerical Estimation**: Given a finite set of samples, compute the expression derived in the identification phase numerically.\n\n- **Evaluation**: This step measures and evaluates the quality of estimators proposed in the estimation step, which often includes sensitivity analysis.\n\n\nSimilar to [1-5], this paper focuses on addressing the identification part. Still, we provided an example with real-world variables in Example 1 and also carried out a simple numerical experiment in Appendix C using a finite set of samples. However, designing an end-to-end pipeline is beyond the scope of this paper and is an interesting future research direction. This would involve collecting a real-world dataset, conducting rigorous causal discovery, designing a proper estimator based on the variable distribution, and applying various evaluations.\n\n\n[1] Tian, Jin, and Judea Pearl. \"A general identification condition for causal effects.\" In Proceedings of the Eighteenth National Conference on Artificial Intelligence (2002)\n\n[2] Shpitser, Ilya, and Judea Pearl. \"Identification of conditional interventional distributions.\" Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, 2006. \n\n[3] Bareinboim, Elias, Jin Tian, and Judea Pearl. \"Recovering from selection bias in causal and statistical inference.\" In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 2410-2416. 2014.\n\n[4] Lee, Sanghack, Juan D. Correa, and Elias Bareinboim. \"General identifiability with arbitrary surrogate experiments.\" In Uncertainty in artificial intelligence, pp. 389-398. PMLR, 2020. \n\n[5] Jaber, Amin, Adele Ribeiro, Jiji Zhang, and Elias Bareinboim. \"Causal identification under Markov equivalence: calculus, algorithm, and completeness.\" Advances in Neural Information Processing Systems 35 (2022).\n\n------\n\n> Comparison to other existing methods: The paper surely follows the id, c-id, S-Recoverability literature for related work, but still could benefit from a more extensive comparison with other latent variable models in causal inference, [1-3] to name a few...\n\nThank you for mentioning these works, especially Paper [2], which is more relevant to our problem.\nPapers [1] and [3], however, seem to be more focused on causal discovery, which is the first step of the pipeline we described earlier. Nonetheless, as the reviewer suggested, we will include more related work.\n\n-----\n\nFinally, we noticed that the reviewer is a bit concerned about the \"Soundness\" and \"Presentation\" of the paper. We would appreciate it if the reviewer could provide further details so that we can improve the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors presents a sound algorithm for checking s-identifiability of causal effects under sub-populations given the causal structure and partially observed data. The unanimous recommendation from reviewers is accept, which I concur with and recommend.
Decision: Accept (poster)