Title: Causal Effect Identification in a Sub-Population with Latent Variables
Venue: NeurIPS 2024
Accepted: yes
Keywords: Causal Effect Identification

Selection Bias

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

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 [AMK24]. 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 [Pea95, TP02], to their new counterparts. Subsequently, we propose a sound algorithm for the S-ID problem with latent variables.

1 Introduction

Causal inference, i.e., understanding the effect of an intervention in a 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, evidence-based decisions. To perform causal inference tasks, it is crucial to differentiate between two types of data: observational and interventional [PM18].

Population

Samples from the population

Samples from a sub-population

𝑃(𝑽)

Observe

𝑽

𝑃(𝑽	|𝑆 = 1)

Observe

𝑽

(a) Observational data.

Population after an

intervention on

𝑿

Samples from the population

Samples from a sub-population

Study

𝒀

𝑃

𝑿

(𝒀)

𝑃

𝑿

(𝒀|𝑆 = 1)

Study

𝒀

(b) Interventional data.

Observational Data. Figure 1 illustrates a population that pertains to the entire sample space for a study of the causal effect of some intervention. A sampler draws samples from the population. The sampler is unbiased if it draws samples at random such that each individual in the population has

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 subpopulation . For each extracted sample, we collect data from a set of observed features denoted by V . As depicted in Figure 1a, 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 ( V | S = 1) , where S = 1 indicates that the sample belongs to a sub-population.

Interventional Data. An intervention on a subset X ⊆ 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 P X ( · ) to denote the distribution of the variables after an intervention on X . Figure 1b depicts the population after an intervention on subset X ⊆ V , where we seek to understand how changes in X would affect a set of outcome variables Y ⊆ V \ X . To analyze this causal effect across the entire population, we must compute the distribution P X ( Y ) . On the other hand, if we are merely interested in the results of the intervention on a specific sub-population, it suffices to compute the conditional distribution P X ( Y | S = 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 causal graph , a graphical representation that depicts the causal relationships between variables [Pea09, SGSH00], 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 1 lists four causal effect identification problems. The most renowned among them is the ID problem, introduced by [Pea95], which seeks to determine a causal effect for the entire population using the observational distribution pertaining to the entire population. Specifically, it aims to compute P X ( Y ) from P ( V ) . The c-ID problem, introduced by [SP06a], extends the ID problem to handle conditional causal effects, i.e., compute the conditional causal effect P X ( Y | 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-population such as P X ( Y | S = 1) can be uniquely computed from the observational distribution pertaining to that sub-population, i.e., P ( V | S = 1) . Another direction of research considers learning a causal effect from multiple datasets [LCB19, KMEK22, CLB21, KEK23, THK21, LGS24]. In all aforementioned causal inference problems, the causal graph is assumed to be known. Some recent work relax this assumption [JZB19, JRZB22] 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 are 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 a special case of ID problem, where the population is restricted to the target sub-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 ( V | S = 1) . Consequently, there are many instances where a causal effect is identifiable in the ID setting but not identifiable in the S-ID setting. In particular, when all the variables in a causal system are observable, all causal effects are identifiable in the setting of

the ID problem [Pea00]. This is not the case in the S-ID setting, and some causal effects become non-identifiable, as noted by [AMK24]. Moreover, even when a causal effect is identifiable in both the ID and S-ID settings, using the expression from the ID algorithm can lead to erroneous inference. Example 1 illustrates this case.

Example 1. Consider an example pertaining to study of the effect of a cholesterol-lowering medication on cardiovascular disease. Figure 2 depicts 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 a person. In this scenario, X and Z 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 P X ( Y ) = P ( Y | X ) .

However, we might instead be interested in a 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 routines and possibly have a higher genetic predisposition for heart disease. Let S be an indicator node for this subpopulation, Z 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 5 that the causal effect in this sub-population, P X ( Y | S = 1) , is S-ID and equals ∑ Z P ( Y | X,Z,S = 1) P ( Z | S = 1) . In

X

Z

Y

S

this example, the presence of S introduces a spurious correlation between X and Y through the path involving Z and S . Therefore, if we were to ignore the presence of S and apply the ID algorithm to the input P ( X,Y,Z | S = 1) , it would result in incorrect inference: P ( Y | X,S = 1) as opposed to the correct value P X ( Y | S = 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 to the S-ID setting in the presence of latent variables (Section 4).

We present a sufficient graphical condition to determine whether a causal effect is S-ID (Theorem 5.1). Accordingly, we propose a sound algorithm for the S-ID problem (Algorithm 1).

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 3, we cover the preliminaries and review key definitions and results for the ID problem. In Section 4, 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 to S-ID. The appendix includes proofs of our results, as well as a numerical experiment.

2 Preliminaries

Throughout the paper, we use capital letters to represent random variables and bold letters to represent sets of variables. Furthermore, to facilitate ease of reading, we have summarized the key notations in Table 2.

Graph Definitions. Acyclic directed mixed graphs (ADMGs) consist of a mix of directed and bidirected edges and have no directed cycles. Let G = ( V , E 1 , E 2 ) be an ADMG, where V is a set of variables, E 1 is a set of directed edges ( → ), and E 2 is a set of bidirected edges ( ↔ ). The set of parents of a variable X ∈ V , denoted by Pa G ( X ) , consists of the variables with a directed edge to X . Similarly, the set of ancestors of X ∈ V , denoted by Anc G ( X ) , includes all variables on a

̸

directed path to X , including X itself. For a set X ⊆ V , we define Anc G ( X ) = ⋃ X ∈ X Anc G ( X ) . In an ADMG G over V , a subset X ⊆ V is called ancestral if Anc G ( X ) = X .

A path is called bidirected if it only consists of bidirected edges. A non-endpoint vertex X i on a path ( X 1 , X 2 , . . . , X k ) is called a collider if one of the following situations arises:

Let X , Y , W be three disjoint subsets of variables in an ADMG G . A path P = ( X,Z 1 , . . . , Z k , Y ) between X ∈ X and Y ∈ Y in G is called blocked by W if there exists 1 ≤ i ≤ k such that Z i is a collider on P and Z i / ∈ Anc G ( W ) , or Z i is not a collider on P and Z i ∈ W .

Denoted by ( X ⊥ ⊥ Y | W ) G , we say W m -separates X and Y if for any X ∈ X and Y ∈ Y , W blocks all the paths in G between X and Y . Conversely, ( X ̸ ⊥ ⊥ Y | W ) G if there exists at least one path between a variable in X and a variable in Y that is not blocked by W .

For X , Z ⊆ V , G XZ denotes the edge subgraph of G obtained by removing the edges with an arrowhead to a variable in X (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 a tuple ( V , U , F , P ( U )) , where V is a set of endogenous variables, U is a set of exogenous variables independent from each other with the joint probability distribution P ( U ) , and F = { f X } X ∈ V is a set of deterministic functions such that for each X ∈ V ,

where Pa X ⊆ V \ { X } and U X ⊆ U . This SCM induces a causal graph G over V such that Pa G ( X ) = Pa X and there is a bidirected edge between two distinct variables X,Y ∈ V when U X ∩ U Y = ∅ . Henceforth, we assume the underlying SCM induces a causal graph that is ADMG, i.e., it contains no directed cycles.

An SCM M = ( V , U , F , P ( U )) with causal graph G induces a unique joint distribution over the variables V that can be factorized as

This property is known as the Markov factorization [Pea09]. Note that ∑ X denotes marginalization, i.e., 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.

Modeling a Sub-Population. We model a sub-population using an auxiliary variable S and a biased sampler from a causal environment akin to [BP12, AMK24]. Suppose M is 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 is biased, it draws samples from the conditional distribution P S ( V ) := P ( V | S = 1) , where S is an auxiliary variable defined as

S := f S ( Pa S , U S ) , where f S is a binary function, Pa S ⊆ V , and U S is the set of exogenous variables corresponding to S . Note that U S can intersect with U , but the variables in U ∪ U S are assumed to be independent. In this model, S = 1 indicates that the sample is drawn from the target sub-population. Furthermore, we define the augmented SCM M S = ( V ∪ { S } , U ∪ U S , F ∪ { f S } , P ( U ∪ U S ) ) obtained by adding S to the underlying SCM M . We denote by G S , the causal graph of M S , which is an augmented ADMG over V ∪{ S } . Note that in G S , variable S does not have any children, but it can have several parents and bidirected edges.

Intervention. An intervention on a set X ⊆ V converts M to a new SCM where the equations of the variables in X are replaced by some constants. We denote by Q [ V \ X ] := P X ( 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 P X ( Y ) , where X and Y are disjoint subsets of V . Accordingly, the causal effect of X on Y in a sub-population is denoted by P X ( Y | S = 1) . 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 a specific sub-population. In this paper, given the augmented graph G S and two arbitrary, disjoint subsets X and Y , we address the following question: Can the causal effect P S X ( Y ) be uniquely identified from the observational distribution P S ( 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 [TP02, HV06, SP06b].

Definition 3.1 (ID) . Suppose G is an ADMG over V and let X and Y be disjoint subsets of V . Causal effect P X ( Y ) is said to be identifiable (or ID for short) in G if for any two SCMs M 1 and M 2 with causal graph G for which P M 1 ( V ) = P M 2 ( V ) > 0 , then P M 1 X ( Y ) = P M 2 X ( Y ) .

Next, we review C-components, a fundamental concept to address the ID problem.

Definition 3.2 (C-component) . Suppose G 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 a 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 us 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 ⊆ V such that G [ Y ] is a single C-component. A subset H ⊆ V is called a Hedge for Y in G , if Y ⊊ H , G [ H ] is a single C-component, and H = Anc G [ H ] ( Y ) .

Example 2. Consider the ADMG G over V = { X 1 , X 2 , Y 1 , Y 2 } depicted in Figure 3a. In this case, G , G [ X 1 , X 2 ] , and G [ Y 1 , Y 2 ] are single C-components. The C-components of G [ X 1 , X 2 , Y 1 ] are { X 1 , X 2 } and { Y 1 } . Furthermore, H = { X 1 , Y 1 , Y 2 } is a Hedge for Y = { Y 1 , Y 2 } since G [ H ] and G [ Y ] are single C-components and Anc G [ H ] ( Y ) = H . Similarly, V is a Hedge for Y , but there exists no Hedge for either { Y 1 } or { Y 2 } .

The following theorem, restating the results in [SP06b] and [HV06], outlines a necessary and sufficient condition to determine the identifiability 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 of V . Causal effect P X ( Y ) is ID in G if and only if Q [ D ] is ID in G , where D = Anc G [ V \ X ] ( Y ) . Furthermore, let { D i } k i =1 be the C -components of G [ D ] , then Q [ D ] is ID in G if and only if there are no Hedge in G for any of the C -components { D i } k i =1 .

Example 3. Following Example 2, Theorem 3.4 implies that P X 1 ( Y 1 ) , P X 2 ( Y 2 ) , P { X 1 ,X 2 } ( Y 1 ) , and P { X 1 ,X 2 } ( Y 2 ) are ID since no Hedge for either { Y 1 } or { Y 2 } exists. However, P X 1 ( Y 1 , Y 2 ) is not

X

1

X

2

Y

1

Y

2

(a) ADMG G in Examples 2-3.

X

1

X

2

Z

2

Z

1

Y

2

S

Y

1

(b) Augmented ADMG G S in Examples 4-9.

ID because D = { Y 1 , Y 2 , X 2 } and the C-components of Q [ D ] are D 1 = { Y 1 , Y 2 } and D 2 = { X 2 } , and { X 1 , Y 1 , Y 2 } (or V ) is a Hedge for D 1 . Similarly, we can show that P { X 1 ,X 2 } ( Y 1 , Y 2 ) is not ID.

4 S-ID, S-component, and S-Hedge

We begin by providing a 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 S an augmented ADMGover V ∪ { S } . Furthermore, we denote by V AS and V NS the ancestors and non-ancestors of S in V , i.e.,

Definition 4.1 (S-ID) . Let X and Y be disjoint subsets of V . Conditional causal effect P X ( Y | S = 1) (or P S X ( Y ) ) is S-ID in G S if for any two augmented SCMs M S 1 and M S 2 with causal graph G S for which P M S 1 ( V | S = 1) = P M S 2 ( V | S = 1) > 0 , then P M S 1 X ( Y | S = 1) = P M S 2 X ( Y | S = 1) .

Next definition extends Q [ · ] and introduces Q S [ · ] .

Definition 4.2 ( Q S [ · ] ) . For H ⊆ V NS , we define Q S [ H ] := P V NS \ H ( H | Anc G S ( S ) \ { S } , S = 1) .

The next definition extends C-components (Definition 3.2) and introduces S-components.

Definition 4.3 (S-component) . For a subset H ⊆ V NS , let C 1 , . . . , C k denote the C-components of G S [ H ∪ Anc G S ( S )] . We define the S-components of H in G S as the subsets H i := C i ∩ H which are non-empty. Furthermore, H is called a single S-component in G S if it contains only one S-component.

Note that Q S [ · ] and S-components are only defined for the subsets of V NS . Figure 4a visualizes the structure of S-components of a subset H ⊆ V NS . In this figure, each blue subset (e.g., M 1 ) 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-components (e.g., H 1 ) of H are connected via bidirected edges in G S [ H ∪ V AS ] . Figure 4b shows the structure of a single S-component, where all the nodes of H are connected via bidirected edges in G S [ H ∪ V AS ] .

Example 4. Consider the ADMG G S in Figure 3b over V ∪ { S } , where V = { X 1 , X 2 , Y 1 , Y 2 , Z 1 , Z 2 } . Since Anc G S ( S ) = { Z 1 , Z 2 , S } , we have V AS = { Z 1 , Z 2 } and V NS = { X 1 , X 2 , Y 1 , Y 2 } . In this case, the S-components of V NS are { X 1 , X 2 } and { Y 1 , Y 2 } . Moreover, the S-components of { X 1 , Y 1 , Y 2 } are { X 1 } and { Y 1 , Y 2 } .

We now provide two crucial properties for Q S [ · ] .

Lemma 4.4. Let W , W ′ be two subsets of V NS such that W ′ ⊂ W . If W ′ is an ancestral set in G S [ W ] , then

Anc(S)

H

H 1

C 1

C 2

M2

H 2

N 2

N 1

M1

M3

(a) S-components of H are H 1 , H 2 .

Anc(S)

H

C 1

M2

N 1

N 2

M1

M3

(b) H is a single S-component.

Anc(S)

H

Y

N 1

N 2

(c) H is an S-Hedge for Y .

H

Y

(d) H is a Hedge for Y .

Lemma 4.5. Suppose H ⊆ V NS and let H 1 , . . . , H k denote the S -components of H in G S . Then,

Q S [ H ] decomposes as

Let m be the number of variables in H , and consider a topological ordering of the variables in graph G S [ H ] , denoted as V h 1 < · · · < V h m . Let H (0) = ∅ and for each 1 ≤ i ≤ m , H ( i ) denote the set of variables in H ordered before V h i (including V h i ). For every 1 ≤ j ≤ k , Q S [ H j ] can be computed from Q S [ H ] by

where Q S [ H ( i ) ] s can be computed by

The aforementioned lemmas are extensions of similar lemmas for Q [ · ] [TP03] to Q S [ · ] .

Example 5. Following Example 4, since { Y 1 } is ancestral in G S [ Y 1 , Y 2 ] , Lemma 4.4 implies that Q S [ Y 1 ] = ∑ Y 2 Q S [ Y 1 , Y 2 ] . Furthermore, since the S-components of V NS are { X 1 , X 2 } and { Y 1 , Y 2 } ,

Finally, we define S-Hedges, which extends Definition 3.3 for Hedges.

Definition 4.6 (S-Hedge) . Suppose Y ⊆ V NS is a single S-component in G S . A subset H ⊆ V NS is called an S-Hedge for Y in G S , if Y ⊊ H , H is a single S-component in G S , and H = Anc G S [ H ] ( Y ) .

When H ⊆ V NS is a single c-component, it is also a single S-component. Therefore, if H is a Hedge for Y , it will also be an S-Hedge for Y . Thus, Hedges can be seen as special cases of S-Hedges when H ⊆ V NS . Figure 4d shows the structure of H , which is a single c-component and forms a Hedge for Y . Moreover, Figure 4c 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 us to be able to determine whether a causal effect is S-ID.

Example 6. Following Examples 4 and 5, { X 1 , X 2 } is an S-Hedge for { X 2 } , because both { X 1 , X 2 } and { X 2 } are single S-components and { X 1 , X 2 } = Anc G S [ X 1 ,X 2 ] ( X 2 ) . Similarly, { Y 1 , Y 2 } is an S-Hedge for { Y 2 } .

5 Main Results

In this section, we provide a sufficient graphical condition for a causal effect to be S-ID in an ADMG. This extends the condition presented in [AMK24], which assumes that the causal graph is a DAG. Accordingly, we propose a sound algorithm for the S-ID problem in the presence of latent variables.

Recall that G S is an augmented ADMG over the set observed variables V and auxiliary variable S , and we defined V AS = V ∩ Anc G S ( S ) and V NS = V \ Anc G S ( S ) .

Theorem 5.1. For disjoint subsets X and Y of V , let X AS := X ∩ V AS , X NS := X ∩ V NS , and Y NS := Y ∩ V NS .

and P S X NS ( Y , X AS ) is S -ID in G S .

Suppose D := Anc G S [ V NS \ X NS ] ( Y NS ) and let { D i } k i =1 denote the S -components of D in G S . Conditional causal effect P S X NS ( Y , X AS ) is S -ID in G S if there are no S -Hedge in G S for any of { D i } k i =1 .

Remark 5.2. If either X NS or Y NS is an empty set, then P S X ( Y ) is S-ID in G S if and only if Equation (5) holds.

In the absence of latent variables, i.e., when G S is a directed acyclic graph (DAG), there are no S-Hedge in G S since all the edges are directed. Therefore, Theorem 5.1 states that P S X NS ( Y , X AS ) is always S-ID, and P S X ( Y ) is S-ID in G S 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 S in Figure 3b, where we want to determine whether P S { X 1 ,X 2 ,Z 1 } ( Y 1 , Y 2 ) is S-ID in G S . In this case, X AS = { Z 1 } , X NS = { X 1 , X 2 } , Y NS = { Y 1 , Y 2 } , and ( Z 1 ⊥ ⊥ { Y 1 , Y 2 }| X 1 , X 2 , S ) G S Z 1 X 1 ,X 2 . This shows that Equation (5) holds. Hence, we need to determine whether P S X 1 ,X 2 ( Y 1 , Y 2 , Z 1 ) is S-ID in G S . In this case, D = Anc G S [ Y 1 ,Y 2 ] ( Y 1 , Y 2 ) = { Y 1 , Y 2 } , which is a single S-component. Since there exists no S-Hedge for { Y 1 , Y 2 } , Theorem 5.1 implies that P S { X 1 ,X 2 ,Z 1 } ( Y 1 , Y 2 ) is S-ID in G S .

Algorithm for S-ID. So far, we have presented a graphical condition to determine whether a causal effect is S-ID. In this section, we propose a recursive algorithm that returns an expression for P S X ( Y ) in terms of P S ( V ) when the condition of Theorem 5.1 holds, and otherwise, returns FAIL.

In the proof of Theorem 5.1 presented in the appendix, we show that when Equation (5) holds, then

where W = V AS \ ( X AS ∪ Y AS ) . Thus, it suffices to find an expression for P S X NS ( Y NS | V AS ) in terms of P S ( V ) . Let D = Anc G S [ V NS \ X NS ] ( Y NS ) and { D i } k i =1 be the S-components of D in G S . From Lemmas 4.4 and 4.5 we have

Therefore, it suffices to find an expression for each D i in terms of P S ( V ) . Note that D i is a single S-component in G S . We can now propose Algorithm 1 for computing P S X ( Y ) from P S ( V ) .

Function sID takes disjoint subsets X and Y of V along with an augmented ADMG G S and conditional distribution P S ( V ) as 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 i } k i =1 in G S . For each 1 ≤ i ≤ k , it finds the corresponding S-component T i of V NS in G S that contains D i . Note that T i is well-defined as D i cannot partially intersect with the S-components of V NS in G S . Next, the algorithm seeks to compute Q S [ D i ] from Q S [ T i ] by calling Function sID-Single . If Function sID-Single succeeds in returning an expression for each i , then the algorithm uses Equations (6) and (7) to return an expression for P S X ( Y ) in terms of P S ( Y ) . Otherwise, the algorithm returns FAIL.

Function sID-Single takes two single S-components C and T in G S such that C ⊆ T and aims to drive an expression for Q S [ C ] in terms of Q S [ T ] . The procedure is recursive and uses Lemmas 4.4 and 4.5. In each recursion, the algorithm reduces T to a smaller subset T ′ such that T ′ is still a single S-component in G S and C ⊆ T ′ (lines 7-10). Eventually, the function either returns FAIL or an expression for Q S [ C ] .

Algorithm 1 Computing P S X ( Y ) from P S ( V )

1: Function sID ( X , Y , G S , P S ( V ) )

2: Output: Expression for P S X ( Y ) in terms of P S or FAIL

3: V AS ← V ∩ Anc G S ( S ) , V NS ← V \ Anc G S ( S )

4: X AS ← X ∩ V AS , X NS ← X ∩ V NS

5: Y AS ← Y ∩ V AS , Y NS ← Y ∩ V NS

6: if ( X AS ̸ ⊥ ⊥ Y | X NS , S ) G S X AS X NS then

7: Return FAIL

8: D ← Anc G S [ V NS \ X NS ] ( Y NS )

9: { D 1 , . . . , D k } ← S-components of D in G S

10: for i in [1 : k ] do

11: T i ← The S-component of V NS that contains D i

1: Function sID-Single ( C , T , Q S [ T ] )

2: Input : Two single S-components C and T in G S such that C ⊆ T

3: Output: Expression for Q S [ C ] in terms of Q S [ T ] or FAIL

4: A ← Anc G S [ T ] ( C )

5: if A = C : Return ∑ T \ C Q S [ T ]

6: if A = T : Return FAIL

7: if C ⊊ A ⊊ T , then

8: T ′ ← The S-component of A in G S that contains C

9: Compute Q S [ T ′ ] from Q S [ T ] using Lemma 4.5

10: Return sID-Single ( C , T ′ , Q S [ T ′ ] )

12: Compute Q S [ T i ] using Lemma 4.5

13: Q S [ D i ] ← sID-Single ( D i , T i , Q S [ T i ])

14: if Q S [ D i ] = FAIL then

15: Return FAIL

16: W ← V AS \ ( X AS ∪ Y AS )

17: Return ∑ W P S ( Y AS , W | X AS ) ∑ D \ Y NS ∏ i Q S [ D i

]

Example 8. Consider again ADMG G S depicted in Figure 3b, where we want to apply Algorithm 1 for causal effect P S X 2 ( Y 2 ) . Herein, X NS = { X 2 } , Y NS = { Y 2 } , and X AS = Y AS = ∅ . Function S-ID passes the condition in line 6 and defines D = { X 1 , Y 1 , Y 2 } , leading to D 1 = { X 1 } and D 2 = { Y 1 , Y 2 } . It then defines T i 's in line 12 as T 1 = { X 1 , X 2 } and T 2 = { Y 1 , Y 2 } . In line 13, it uses Lemma 4.5 to compute Q S [ T 1 ] = ∑ Y 1 ,Y 2 Q S [ V NS ] and Q S [ T 2 ] = Q S [ V NS ] ∑ Y 1 ,Y 2 Q S [ V NS ] (See Example 5). It then calls Function sID-Single , which returns Q S [ D 1 ] = ∑ X 2 Q S [ T 1 ] and Q S [ D 2 ] = Q S [ T 2 ]

. Finally, in line 20, the function returns

where Q S [ X 1 ] = P S ( X 1 | Z 1 , Z 2 ) and Q S [ Y 1 , Y 2 ] = P S ( Y 1 , Y 2 | X 1 , X 2 , Z 1 , Z 2 ) .

Example 9. Following the previous example, suppose we want to apply Algorithm 1 for computing causal effect P S X 1 ( Y 1 , Y 2 ) . In this case, the algorithm needs to compute Q S [ Y 1 , Y 2 ] and Q S [ X 2 ] . However, when the algorithm calls sID-Single ( X 2 , { X 1 , X 2 } , Q S [ X 1 , X 2 ]) , Function sID-Single returns FAIL. Accordingly, Function sID returns FAIL for P S X 1 ( Y 1 , Y 2 ) .

Remark 5.3. Algorithm 1 is sound for the S-ID problem in the presence of latent variables. We conjecture that this algorithm is also complete , meaning that whenever it returns FAIL, the corresponding causal effect is not S-ID.

6 Reduction from S-Recoverability to S-ID

Recall that the objective in S-Recoverability is to compute P X ( Y ) from P S ( V ) [BTP14], while S-ID aims to compute P S X ( Y ) from P S ( V ) . [BT15] proposed RC, a sound algorithm for the SRecoverability problem. Subsequently, [CTB19] proved that RC is complete. In this section, we present a reduction from the S-Recoverability problem to the S-ID problem. This indicates that solving S-ID can solve the S-Recoverability problem (but not the other way around).

Theorem 6.1. For disjoint subsets X and Y of V , P X ( Y ) can be uniquely computed from P S ( V ) in the augmented ADMG G S if and only if

and P S X ( Y ) is S -ID in G S .

Algorithm 2

Reduction from S-Recoverability to S-ID

1: Input: X , Y , G S , P S

2: Output: Expression for P X ( Y ) in terms of P S or FAIL

3: if ( Y ̸ ⊥ ⊥ S | X ) G S X then

4: Return FAIL

5: else

6: Return sID ( X , Y , G S , P S )

X

1

X

2

Z

2

Z

1

Y

S

Remark 6.2. Equation (8) is a very restrictive condition, and when it holds, Rule 1 of do-calculus implies that P X ( Y ) = P S X ( Y ) .

Theorem 6.1 implies that when Equation (8) does not hold, then P X ( Y ) is not S-Recoverable. However, this causal effect might be identifiable in the target sub-population, i.e., P S X ( Y ) is S-ID.

As a consequence of Theorem 6.1, we propose Algorithm 2 for computing P X ( Y ) from P S ( V ) . The algorithm takes as input two disjoint subsets X and Y of V along with an augmented ADMG over V ∪ { S } and the conditional distribution P S ( V ) . It first checks Equation (8) in line 3, and then calls Algorithm 1 as a subroutine to compute P S X ( Y ) from P S ( V ) when it is S-ID in G S .

Example 10. Consider the augmented ADMG G S in Figure 5. In this graph, ( Y ̸ ⊥ ⊥ S | X 1 ) G S X 1 , thus, Theorem 6.1 implies that P X 1 ( Y ) cannot be uniquely computed from P S ( V ) . On the other hand, P X 2 ( Y ) can be identified from P S ( V ) since ( Y ⊥ ⊥ S | X 2 ) G S X 2 and due to Theorem 5.1, P S X 2 ( Y ) is S-ID in G S . In this case, Algorithm 2 returns the following expression for P X 2 ( Y ) in terms of P S

7 Conclusion

The S-ID problem, introduced by [AMK24], asks whether, given the causal graph, a causal effect in a sub-population can be identified from the observational distribution pertaining to the same sub-population. [AMK24] addressed this problem when all the variables in the causal graph are observable. In this paper, we studied the S-ID problem in the presence of latent variables and provided a sufficient graphical condition to determine whether a causal effect is S-ID. Consequently, we proposed a sound algorithm for S-ID. While this paper proves the soundness of our proposed method, we also conjecture that our approach is not only sound but also complete. Finally, by presenting an appropriate reduction, we showed that solving S-ID can solve the S-Recoverability problem.

Acknowledgments

We thank the anonymous reviewers for their feedback. This research was in part supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF40_180545 and Swiss SNF project 200021_204355 /1.

References

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[BMS20] Rohit Bhattacharya, Daniel Malinsky, and Ilya Shpitser. Causal inference under interference and network uncertainty. In Uncertainty in Artificial Intelligence , pages 1028-1038. PMLR, 2020.

[BP12] Elias Bareinboim and Judea Pearl. Controlling selection bias in causal inference. In Neil D. Lawrence and Mark Girolami, editors, Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics , volume 22 of Proceedings of Machine Learning Research , pages 100-108, La Palma, Canary Islands, 21-23 Apr 2012. PMLR.

[BT15] Elias Bareinboim and Jin Tian. Recovering causal effects from selection bias. Proceedings of the AAAI Conference on Artificial Intelligence , 29(1), Mar. 2015.

[BTP14] Elias Bareinboim, Jin Tian, and Judea Pearl. Recovering from selection bias in causal and statistical inference. Proceedings of the AAAI Conference on Artificial Intelligence , 28(1), Jun. 2014.

[CLB21] Juan Correa, Sanghack Lee, and Elias Bareinboim. Nested counterfactual identification from arbitrary surrogate experiments. Advances in Neural Information Processing Systems , 34:6856-6867, 2021.

[CTB19] Juan D. Correa, Jin Tian, and Elias Bareinboim. Identification of causal effects in the presence of selection bias. Proceedings of the AAAI Conference on Artificial Intelligence , 33(01):2744-2751, Jul. 2019.

[HV06] Yimin Huang and Marco Valtorta. Identifiability in causal bayesian networks: A sound and complete algorithm. In AAAI , pages 1149-1154, 2006.

[JAK23] Fateme Jamshidi, Sina Akbari, and Negar Kiyavash. Causal imitability under contextspecific independence relations. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors, Advances in Neural Information Processing Systems , volume 36, pages 26810-26830. Curran Associates, Inc., 2023.

[JRZB22] Amin Jaber, Adele Ribeiro, Jiji Zhang, and Elias Bareinboim. Causal identification under markov equivalence: Calculus, algorithm, and completeness. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems , volume 35, pages 3679-3690. Curran Associates, Inc., 2022.

[JZB19] Amin Jaber, Jiji Zhang, and Elias Bareinboim. Causal identification under Markov equivalence: Completeness results. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning , volume 97 of Proceedings of Machine Learning Research , pages 2981-2989. PMLR, 09-15 Jun 2019.

[KEK23] Yaroslav Kivva, Jalal Etesami, and Negar Kiyavash. On identifiability of conditional causal effects. The 39th Conference on Uncertainty in Artificial Intelligence , 2023.

[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.

[LGS24] Jaron J.R. Lee, AmirEmad Ghassami, and Ilya Shpitser. A general identification algorithm for data fusion problems under systematic selection. In The 40th Conference on Uncertainty in Artificial Intelligence , 2024.

[MJEK22] Ehsan Mokhtarian, Fateme Jamshidi, Jalal Etesami, and Negar Kiyavash. Causal effect identification with context-specific independence relations of control variables. In International Conference on Artificial Intelligence and Statistics , pages 11237-11246. PMLR, 2022.

[Pea95] Judea Pearl. Causal diagrams for empirical research. Biometrika , 82(4):669-688, 1995.

[Pea00] Judea Pearl. Causality: Models, reasoning and inference. Cambridge, UK: CambridgeUniversityPress , 19(2):3, 2000.

[Pea09] Judea Pearl. Causality . Cambridge university press, 2009.

[PM18] Judea Pearl and Dana Mackenzie. The book of why: the new science of cause and effect . Basic books, 2018.

[Rub74] Donald B Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology , 66(5):688, 1974.

[SGSH00] Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search . MIT press, 2000.

[SP06a] Ilya Shpitser and Judea Pearl. Identification of conditional interventional distributions. Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence , 2006.

[SP06b] Ilya Shpitser and Judea Pearl. Identification of joint interventional distributions in recursive semi-markovian causal models. In Proceedings of the National Conference on Artificial Intelligence , volume 21, page 1219. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999, 2006.

[SS18] Eli Sherman and Ilya Shpitser. Identification and estimation of causal effects from dependent data. Advances in neural information processing systems , 31, 2018.

[THK19] Santtu Tikka, Antti Hyttinen, and Juha Karvanen. Identifying causal effects via contextspecific independence relations. Advances in neural information processing systems , 32, 2019.

[THK21] Santtu Tikka, Antti Hyttinen, and Juha Karvanen. Causal effect identification from multiple incomplete data sources: A general search-based approach. Journal of Statistical Software , 99(5), 2021.

[TP02] Jin Tian and Judea Pearl. A general identification condition for causal effects. In Proceedings of the Eighteenth National Conference on Artificial Intelligence (AAAI 2002) , pages 567-573, Menlo Park, CA, 2002. AAAI Press/The MIT Press.

[TP03] Jin Tian and Judea Pearl. On the identification of causal effects. Technical report, Department of Computer Science, University of California, 2003.

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Appendix

The structure of the appendix is 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, 4.5, and Theorems 5.1, 6.1 are presented in Appendix C. In Appendix D, we will conduct an experiment to compare the outputs of S-ID algorithm and the classic ID algorithm.

A Additional Example

U

1

U

2

S

Y

X

In this section, we provide an example where ignoring S results in an identifiable effect, whereas the target causal effect is, in fact, not S-ID.

Consider the following two SCMs.

SCM M 1 :

where ⊕ denotes the XOR operator, and Bern ( p ) denotes a Bernoulli random variable with parameter p .

SCM M 2 :

According to the above equations, we have

Similarly for M 2 we have

Thus, P M 1 ( X,Y | S = 1) = P M 2 ( X,Y | S = 1) > 0 . Furthermore, we have

Similarly, for SCM M 2 :

̸

This shows that P S X ( Y ) is not S-ID in this causal graph, as P M 1 ( X,Y | S = 1) = P M 2 ( X,Y | S = 1) > 0 , but P M 1 x =0 ( Y = 1 | S = 1) = P M 2 x =0 ( Y = 1 | S = 1) .

Note that ignoring the sub-population, the causal effect P X ( 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 X , Y , Z , W be four disjoint subsets of V . The following three rules, commonly referred to as Pearl's do-calculus rules [Pea00], provide a tool for calculating interventional distributions using the causal graph.

· Rule 1 : If ( Y ⊥ ⊥ Z | X , W ) G X , then P X ( Y | Z , W ) = P X ( Y | W ) . · Rule 2 : If ( Y ⊥ ⊥ Z | X , W ) G XZ , then P X , Z ( Y | W ) = P X ( Y | Z , W ) . · Rule 3 : If ( Y ⊥ ⊥ Z | X , W ) G XZ ( W ) , where Z ( W ) := Z \ Anc G X ( W ) , then P X , Z ( Y | W ) = P X ( Y | W ) . Lemma B.1 (TP03) . For two sets W ′ ⊂ W , if W ′ is an ancestral set in G [ W ] , then

Lemma B.2 (TP03) . Let C ⊆ V , and assume that C is partitioned into C -components C 1 , . . . , C m in the subgraph G [ C ] . Then we have

Q [ C ] decomposes as

Let k denote the number of variables in H , and let us assume a topological order of variables in C as V c 1 < V c 2 < · · · < V c k in G C . Let C i be the set of variables in C ordered before V c i (including V c i ), for i = 1 , 2 , . . . , k , where C 0 is an empty set. Then each Q [ C j ] , j = 1 , 2 , . . . , m , is computable from Q [ C ] and is given by

where each Q [ C i ] is given by

Corollary B.3. Let H 1 ⊔ H 2 ⊔ · · · ⊔ H m be a partition of set C , where for each C i there exists 1 ≤ j ≤ m such that C i ⊆ H j . Then, we have

Lemma B.4. Let H be a subset of V NS , we have the following equation

Proof. According to definition of Q S , we have

Note that Anc ( S ) is an ancestral set in G ; Hence, for any H , we have

This implies Equation (13).

Lemma B.5. Let X NS and Y NS be disjoint subsets of V NS = V \ Anc G S ( S ) . We have

where D = Anc G [ V NS \ X NS ] ( Y NS ) , and D i 's are S -components of D .

Proof. Using marginalization over V NS \ ( X NS ∪ Y NS ) , we have

Since D is an ancestral set in G [ V NS \ X NS ] , according to Lemma 4.4, we have

Therefore, the first property of Lemma 4.5 implies that

Lemma B.6. If Function sID-Single returns FAIL for the inputs C and T , then T is an S -Hedge for C .

Proof. When the algorithm returns Fail,

C and T are both single S-components since the inputs of the algorithm have to be Scomponents.

If A := Anc G [ T ] ( C ) , then T = A .

According to the definition of s-Hedge 4.6, T is an s-Hedge for C .

C Proofs of Main Results

Lemma 4.4. Let W , W ′ be two subsets of V NS such that W ′ ⊂ W . If W ′ is an ancestral set in G [ W ] , then we have

Proof. According to Lemma B.4, we have the following equations

Therefore, by replacing the above equations in Equation (1), it is sufficient to show that

Since P ( Anc ( S )) > 0 , this is equivalent to

Note that W ′ ∪ Anc ( S ) in an ancestral set in G [ W ∪ Anc ( S )] , and W \ W ′ = ( W ∪ Anc ( S )) \ ( W ′ ∪ Anc ( S )) . Hence, Lemma B.1 concludes the proof.

Suppose H ⊆ V NS and let H 1 , . . . , H k denote the S-components of H in G S . Then,

Q S [ H ] decomposes as

Let m be the number of variables in H , and consider a topological ordering of the variables in graph G S [ H ] , denoted as V h 1 < · · · < V h m . Let H (0) = ∅ and for each 1 ≤ i ≤ m , H ( i ) denote the set of variables in H ordered before V h i (including V h i ). For every 1 ≤ j ≤ k , Q S [ H j ] can be computed from Q S [ H ] by

where Q S [ H ( i ) ] s can be computed by

Proof. First part: according to the definition of Q S [ . ] , we have

Note the last equality holds because Anc ( S ) is an ancestral set in G . Now, we have

Similarly, for each i ∈ [1 : k ] , we have

Therefore, the above equations imply that

Let C i 's be the corresponding C-component of H i (i.e., H i ⊆ C i and C i is a C-component of H i ∪ Anc ( S ) ). According to the definition of C i and Corollary B.3 we have

Moreover, since C i is a C-component of Anc ( S ) ∪ H i , then C i \ H i should be union of some C-components of Anc ( S ) . Therefore, for each i , Corollary B.3 implies

Putting the above equations together, we have

Hence,

Consequently, we have

Note that C i \ H i are disjoint subsets of Anc ( S ) , where each of them is the union of some Ccomponents of Anc ( S ) . If C k +1 := Anc ( S ) \ ⋃ i C i , then Corollary B.3 implies the following.

Finally, applying Corollary B.3 for H ∪ Anc ( S ) , we have

This concludes the first part.

Second part: the proof is similar to proof of Lemma B.2. Define C := H ∪ Anc ( S ) . Let C 1 , . . . , C m be C-components of C (w.l.o.g. assume that H i ⊆ C i for i ∈ [1 : k ] ). Consider a topological order of V h 1 < V h 2 < · · · < V h n for H , and V s 1 < · · · < V s n ′ for Anc ( S ) . Since any V h i is not an ancestor of Anc ( S ) , V s 1 < · · · < V s n ′ < V h 1 < V h 2 < · · · < V h n is a topological order for H ∪ Anc ( S ) . According to Lemma B.2, we have

where ( c 1 , c 2 , . . . , c n + n ′ ) = ( s 1 , . . . , s n ′ , h 1 , . . . , h n ) . For each i ∈ [1 : n ] , let H i := { V h 1 , V h 2 , . . . , V h i } and H 0 = ∅ . For each i > 0 , we have

where the last equality holds according to Lemma B.4. It suffices to show that

Hence, according to this equation and Lemma B.2, we have

Equation 18 implies that

Based on the definition C j and H j , C j \ H j is the union of some C-components of Anc ( S ) . Denote the c-components of C j \ H j by A 1 , . . . , A t . Applying Lemma B.2 to C = Anc ( S ) with topological order V s 1 < · · · < V s n ′ , we have

.

.

.

By multiplying all Q [ A i ] , we have

By substituting this in Equation 19, we obtain

To complete the proof, it suffices to show that Q S [ H ( i ) ] are computable from Q S [ H ] . Since we have used the topological order, H i is an ancestral set in G [ H ] . Therefore, Lemma 4.4 implies that

This shows that Q S [ H i ] is uniquely computable from Q S [ H ] .

Theorem 5.1. For disjoint subsets X and Y of V , let

Conditional causal effect P S ( Y ) is S -ID in G S if and only if

and P S X NS ( Y , X AS ) is S -ID in G S .

Suppose D := Anc G S [ V NS \ X NS ] ( Y NS ) and let { D i } k i =1 denote the S -components of D in G S . Conditional causal effect P S X NS ( Y , X AS ) is S -ID in G S if there are no S -Hedge in G S for any of { D i } k i =1 .

Proof. We first show that ( X AS ⊥ ⊥ Y | X NS , S ) G S X AS X NS is a necessary condition. Suppose ( X AS ̸⊥ ⊥ Y | X NS , S ) G S X AS X NS . Denote by G ′ the equivalent DAG of G S , obtained by adding the unobserved variables. Theorem 2 in [AMK24] shows that there are two SEMs M 1 and M 2 compatible with G ′ such that

̸

We use these SEMs to construct SCMs M ′ 1 and M ′ 2 compatible with G S , in which we have

Note that all causal effects in both models are the same for G S and G . Hence, when ( X AS ̸⊥ ⊥ Y | X NS , S ) G S X AS X NS holds, then P S X ( Y ) is not S-ID. It shows that this condition is a necessary condition.

Now suppose that ( X AS ⊥ ⊥ Y | X NS , S ) G X AS X NS holds. According to Rule 2 of do-calculus, we have

The above equation shows that P X ( Y | S = 1) is s-ID in G S if and only if P X NS ( Y | X AS , S = 1) is s-ID in G S . Moreover,

Since X NS ∩ V AS = ∅ , acccording to Rule 3 of do-calculus, we have P X NS ( X AS | S = 1) = P ( X AS | S = 1) . Hence,

This shows that S-Identifiability of P X NS ( Y | X AS , S = 1) and P X NS ( Y , X AS | S = 1) are equivalent (note that P ( X AS | S = 1) > 0 due to the positivity assumption in Definition 4.1).

Let W := V AS \ ( X AS ∪ Y AS ) , then

The above equalities have been obtained by a marginalization over W , chain rule, and replacing the definition of W , respectively. According to Rule 3 of do-calculus, since X NS ∩ V AS = ∅ , we have

Moreover, according to Lemma B.5, we have

Combining the aforementioned equations with Equation (21), we get

Now, note that according to Lemma B.6, if there is not any S-Hedge for D i s, then Function sID-Single will compute Q S [ D i ] s. Therefore, we can compute P X NS ( Y , X AS | S = 1) using the above equation, which concludes the proof. As a result, if Equation (5) holds, we have

Theorem 6.1. For disjoint subsets X and Y of V , P X ( Y ) can be uniquely computed from P S ( V ) in the augmented ADMG G S if and only if

and P S X ( Y ) is S -ID in G S .

Proof. If ( Y ̸ ⊥ ⊥ S | X ) G X , then [BT15, Theorem 2] implies that P X ( Y ) is not S-recoverable. If ( Y ⊥ ⊥ S | X ) G X , then according to Rule 1 of do-calculus, we have

Therefore, the identifiability of P X ( Y ) is equivalent to P X ( Y | S = 1) , which concludes the proof.

U

1

X

Z

U

2

S

Y

D Numerical Experiment

We conduct a numerical experiment to demonstrate the significance of the s-ID problem and assess the output of Algorithm 1. 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 7.

Herein, ⊕ denotes the XOR operation, all the variables { U 1 , U 2 , ε x , ε z , ε s 1 , ε s 2 , ε s 3 } are independent, and Bern ( p ) denotes a Bernoulli random variable with parameter p . Now, consider the sub-population with the following mechanism:

We now consider the problem of estimating the causal effect of X on Y in this sub-population. Particularly, our goal is to calculate P X =0 ( Y = 1 | S = 1) .

D.1 Theoretical Analysis

To analyze and compare the S-ID and ID algorithms, we first determine the exact values of P X =0 ( Y = 1 | S = 1) and P X =0 ( Y = 1) . According to the equation of Y in the SCM, we have

Since U 2 is a Bernoulli random variable with parameter 0.5, the above probability always equals 0.5. For P X ( Y | S = 1) , we have

Therefore, for X = 0 and Y = 1 , we need to compute P ( U 2 = 1 | S = 1) . By using the equation of S in the model,

Note that P ( U 2 = 0) = P ( U 2 = 1) = 0 . 5 , hence, we have

Since ε s 1 and ε s 3 and Z are independent variables, let W be Z × ε s 1 ⊕ ε s 3 ; then, we have

Note that W and U 2 × ε s 2 are independent, and S = W + U 2 × ε s 2 ; thus,

Hence,

D.2 Empirical Analysis

The ID algorithm returns the following simple expression for P X ( Y )

On the other hand, the s-ID algorithm returns

Next, we 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 sub-population. Recall that our goal was to estimate P X =0 ( Y = 1 | S = 1) . Consider the following two approaches.

If we 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

Suppose we ignore the presence of sub-population and apply the ID algorithm. In this scenario, we have to estimate P X ( Y ) using the empirical distribution of variables, i.e., ˆ P ( V ) . If we use the ID algorithm, we need to estimate the quantity mentioned in equation (26). The empirical estimate for this case is

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 in Equation (25) shows that our approach accurately computes the target causal effect. On the other hand, ignoring the subtleties related to sub-population can lead to erroneous estimation.

NeurIPS Paper Checklist

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Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: All steps of the conducted experiment are detailed in Appendix D. We utilize synthetic data corresponding to a causal model defined in Appendix D. Hence, the data generation process is straightforward. After that, we only need some estimation of certain distributions from this data to reproduce the results.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

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(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

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5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [No]

Justification: As discussed in the previous question, our numerical experiment consists of a few simple steps. One can generate data according to the defined causal model in our experiment and reproduce the results. Hence, we have not provided the code.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

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6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We thoroughly elaborated on all the steps of our experiment in Appendix D.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: Our experiment aims to compare the outcomes of our algorithm and the ID algorithm, which provide estimands for causal effects. We focus on scenarios with an infinite sample size, eliminating the need to report error rates. Further details can be found in Appendix D.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

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If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: In Appendix D, we note that our numerical experiment does not need specific computational power.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

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Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [No]

Justification: We think there is no societal impact as our work primarily focuses on theoretical analysis of a problem in causal effect identification.

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The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

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Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The focus of our paper is the theoretical understanding of a specific problem in the area of causal inference. Therefore, this question is not applicable.

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The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

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Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [NA]

Justification: The focus of our paper is the theoretical understanding of a specific problem in the area of causal inference. Therefore, this question is not applicable.

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The answer NA means that the paper does not use existing assets.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nTechnical quality: The paper presents thorough theoretical analysis, including formal definitions, lemmas, examples and theorems, as well as the reduction derivation.\n\nWEAKNESSES:\nEmpirical 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).\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-27"}

comments: 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.

STRENGTHS:
1. The paper is written well and easy to understand.
2. Examples in each section helps to understand the underlying idea easily.
3. All the necessary background is discussed clearly.

WEAKNESSES:
1. Example 1 could be a better one because socioeconomic status can cause cardiovascular disease.
2. It would be good to include a sub section for summarising any assumptions made.
3. It would be good to include some real-world use-cases benefiting from such setting of causal effect identification.

QUESTIONS:
See weaknesses section

LIMITATIONS:
Limitations are discussed
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-11

comments: 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.

STRENGTHS:
- The paper addresses an important problem in causal inference.
- The paper is very well-written and covers the prerequisites very well.

WEAKNESSES:
See the Questions section below.

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.
- There are many variables and notations used throughout the paper. A table summarizing these notations and their definitions could improve readability.
- How often do real-world problems satisfy the restrictive condition in Equation (8)​​?

LIMITATIONS:
The limitations of the proposed approach have not been discussed.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n- 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 $Q[D]$.\n\nQUESTIONS:\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- 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?\n\nLIMITATIONS:\nOK.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-17"}