Title: Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference
Venue: NeurIPS 2024
Accepted: yes
Keywords: adjustment criterion

sequential back-door criterion

g-formula
Abstract: Covariate adjustment, also known as back-door adjustment, is a fundamental tool in causal inference. Although a sound and complete graphical identification criterion, known as the adjustment criterion (Shpitser, 2010), exists for static contexts, sequential contexts present challenges. Current practices, such as the sequential back-door adjustment (Pearl, 1995) or multi-outcome sequential back-door adjustment (Jung, 2020), are sound but incomplete; i.e., there are graphical scenarios where the causal effect is expressible via covariate adjustment, yet these criteria do not cover. In this paper, we exemplify this incompleteness and then present the *sequential adjustment criterion*, a sound and complete criterion for sequential covariate adjustment. We provide a constructive sequential adjustment criterion that identifies a set that satisfies the sequential adjustment criterion if and only if the causal effect can be expressed as a sequential covariate adjustment. Finally, we present an algorithm for identifying a *minimal* sequential covariate adjustment set, which optimizes efficiency by ensuring that no unnecessary vertices are included.

Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference

Abstract

Covariate adjustment, also known as back-door adjustment, is a fundamental tool in causal inference. Although a sound and complete graphical identification criterion, known as adjustment criterion (Shpitser et al., 2010), exists for static contexts, sequential contexts present challenges. Current practices, such as the sequential back-door adjustment (Pearl and Robins, 1995) or multi-outcome sequential backdoor adjustment (Jung et al., 2020), are sound but incomplete; i.e., there are graphical scenarios where the causal effect is expressible via covariate adjustment, yet these criteria do not cover. In this paper, we exemplify this incompleteness and then present the sequential adjustment criterion , a sound and complete criterion for sequential covariate adjustment. We provide a constructive sequential adjustment criterion that identifies a set that satisfies the sequential adjustment criterion if and only if the causal effect can be expressed as a sequential covariate adjustment. Finally, we present an algorithm for identifying a minimal sequential covariate adjustment set, which optimizes efficiency by ensuring that no unnecessary vertices are included.

1 Introduction

Covariate adjustment (also known as the back-door adjustment (Pearl, 1995)) -∑ z P ( y | x , z ) P ( z ) where Z is a set of covariates, X is a treatment, and Y is an outcome-is one of the most prevalent method for estimating causal effects from observational data. Sufficient conditions have been established to determine whether causal effects can be expressed through covariate adjustment. For example, back-door criterion (BD) (Pearl, 1995) is a sound graphical condition to determine if the causal effect is expressible as a covariate adjustment. Shpitser et al. (2010) extended the BD and developed a sound and complete graphical criterion, called adjustment criterion (AC). van der Zander et al. (2014) and Maathuis and Colombo (2015) extended the adjustment criterion to general graph classes including directed acyclic graphs, maximal ancestral graphs, or partial ancestral graphs. The criterion from Maathuis and Colombo (2015) is sound but incomplete, which Perkovic et al. (2018) addressed by developing a sound and complete adjustment criterion.

Beyond such static adjustment criterion, sequential covariate adjustment (also known as the g-formula (Robins, 1986)) has been studied. For example, sequential back-door (SBD) criterion (Pearl and Robins, 1995) extended the back-door criterion to the sequential setting. Since the SBD criterion only considered a single outcome, Jung et al. (2020) extended it to accommodate multiple outcome variables called multi-outcome sequential back-door (mSBD) criterion. However, as detailed in Sec. 3, this criterion is not complete; i.e., there exists an example where the mSBD criterion is not satisfied, but the causal effect is identified as a sequential covariate adjustment. While it is known that the sequential ignorability condition (Richardson and Robins, 2013) is sound and complete for

the g-formula estimand, it does not provide a graphical criterion that can be tested based on a graph compatible with the distribution.

To address this challenge, we devise a sound and complete graphical criterion for the sequential covariate adjustment. Table 1 visualizes our contribution against previous graphical criteria for covariate adjustments. 2 Specifically, our contributions are as follows:

We demonstrate the incompleteness of mSBD criterion by exemplifying a graph such that the causal effect is given as sequential covariate adjustment where the mSBD criterion is not satisfied. Moreover, we discuss the impracticality of other existing criteria.

We devise sequential adjustment criterion (SAC), a sound and complete graphical criterion for sequential covariate adjustment.

We develop a constructive sequential adjustment criterion that yields a set that satisfies the SAC if and only if the causal effect can be expressed as a sequential covariate adjustment. Furthermore, we present an algorithm to identify a minimal sequential covariate adjustment set.

The proofs for all results are provided in Sec. A in the supplementary document.

2 Preliminaries

We introduce the necessary notation and background. Key notations are summarized in Table 2.

Structural Causal Models. We use Structural Causal Models (SCMs) (Pearl, 2000). An SCM M is a quadruple M := ⟨ U , V , P ( U ), F ⟩ . U is a set of exogenous (latent) variables following a joint distribution P ( U ) . V is a set of endogenous (observable) variables whose values are determined by functions F = { f V i } V i ∈ V such that V i ← f V i ( pa i , u i ) where PA i ⊆ V \ { V i } and U i ⊆ U . Each SCM M induces a distribution P ( V ) and a causal graph G = G ( M ) over V in which there exists a directed edge from every variable in PA i to V i and dashed-bidirected arrows encode correlated latent variables. Performing an intervention fixing X = x is represented through the do-operator, do( X = x ) , which encodes the operation of replacing the original equations of X (i.e., f X ( pa x , u x ) ) by the constant x for all X ∈ X and induces an interventional distribution P ( V | do( x )) .

Graphs. We consider a graph G having vertices V and edges E composed of directed ( V i → V j ) and bidirected edges ( V i ↔ V j ). An ordered sequence of distinct nodes in G is called a path between V i and V j in G if (1) the start node is V i and the end node is V j , and (2) there is an edge (directed or bidirected) between any two subsequent variables in the sequence. A variable V i is called a collider on the path if the path contains two edges having arrow heads toward V i ; i.e., ∗→ V i ←∗ where ∗→∈ {→ , ↔} . A path is directed (or causal), if it contains only directed edges, all pointing in the same direction. We say that two sets of vertices A , B ⊆ V are d-connected w.r.t. Z ⊆ V \ ( A ∪ B ) in G if there exists a path between V i ∈ A and V j ∈ B where every non-colliders on the path is not contained in Z and all colliders are in An G ( Z ) . We say that Z d-separates ( A , B ) in G if they are not d-connected w.r.t. Z in G . We denote G AB by G with removing those edges toward A and from B . For disjoint sets of vertices A 1 , · · · , A m ⊆ A , we use ( A 1 , · · · , A m ) to represent an ordered set, where the order needs to be highlighted. For the unordered set, we use braces as { A 1 , · · · , A m } .

Adjustment Criterion. We revisit the problem of characterizing a criterion such that the causal effect is given as an adjustment. Formally, the adjustment set relative to ( X , Y ) in G is defined as a set Z such that, for any P compatible with G , the following is satisfied (Shpitser et al., 2010):

The adjustment criterion (Shpitser et al., 2010) provides a sound and complete criterion for the causal effect P ( y | do( x )) to be expressed as a covariate adjustment in Eq. (1). To present the adjustment criterion, we define a proper causal path between X and Y as a directed path from any of X ∈ X to any of Y ∈ Y on which the path does not contain a node in X \ { X } . A proper causal path set pcp G ( X , Y ) is a collection of nodes excluding X on all proper causal paths from X ∈ X to Y ∈ Y , defined in Table 2. dpcp G ( X , Y ) is a descendant of the proper causal path set defined in Table 2. The adjustment criterion utilizes this subgraph called the proper back-door graph :

Definition 1 (Proper Back-Door Graph (van der Zander et al., 2014)) . Let ( X , Y ) denote a disjoint pair in V . The proper back-door graph G X , Y pbd is a graph obtained from G by removing an edge X → D for each X ∈ X and D ∈ pcp G ( X , Y ) .

Then, the adjustment criterion and its characterization are given as follows.

Definition 2 (Adjustment Criterion (Shpitser et al., 2010; van der Zander et al., 2014)) . Let ( X , Y ) denote a disjoint pair in V . A set of variables Z ⊆ V \ ( X ∪ Y ) is said to satisfy the adjustment criterion relative to ( X , Y ) in G if (1) ( Y ⊥ ⊥ X | Z ) G X , Y pbd and (2) Z ∩ dpcp G ( X , Y ) = ∅ .

Proposition 1 (Shpitser et al. (2010); van der Zander et al. (2014)) . Z satisfies the adjustment criterion relative to ( X , Y ) in G if and only if Z is an adjustment set relative to ( X , Y ) in G .

For concreteness, in Fig. 3, Z := { Z a , Z b } satisfies the adjustment criterion relative to X := { X 1 , X 2 } and Y := { Y 1 , Y 2 } . See (van der Zander et al., 2014, Sec. 4.2) for details. In the later sections, we will extend this result to the sequential context.

3 Limitations of Existing Criterion for Sequential Covariate Adjustment

In this section, we demonstrate the limitations of existing criteria for sequential covariate adjustment, which is defined as follows:

Definition 3 (Sequential Covariate Adjustment) . Let ( X , Y ) denote a pair of ordered sets such that X = ( X 1 , . . . , X m ) and Y = ( Y 0 , . . . , Y m ) . Let Z ⊆ V \ ( X ∪ Y ) denote vertices ordered as Z = ( Z 1 , . . . , Z m ) . Define H i := X ( i ) ∪ Y ( i ) ∪ Z ( i ) . The set Z is said to be a sequential adjustment set relative to ( X , Y ) in G if for any P compatible with G , the following is satisfied 3 :

The right-hand side of Eq. (2) is called the sequential covariate adjustment of Z w.r.t. ( X , Y ) in G .

In this definition, sequential covariate adjustment relies on specific partitions X = ( X 1 , · · · , X m ) and Y = ( Y 0 , · · · , Y m ) . To standardize these partitions, we employ a fixed structure based on a topological order of ( X , Y ) in G so as to ensure that the partition aligns with the causal structure.

Definition 4 (Partitioning Operator) . Let X and Y denote disjoint vertices where X = ( X 1 , · · · , X m ) is topologically ordered along with G . Then, the operator PT X G partitions Y relative to X and G ; i.e., PT X G ( Y ) = ( Y 0 , · · · , Y m ) , where

Hereafter, we will denote ( X , Y ) as a pair of vertices where X = ( X 1 , · · · , X m ) is topologically ordered with respect to G and Y as the ordered set PT X G ( Y ) .

3.1 Limitation of mSBD Criterion

The multi-outcome sequential back-door (mSBD) criterion (Jung et al., 2020) has been developed to determine whether an ordered set Z = ( Z 1 , · · · , Z m ) constitutes a sequential adjustment set:

Definition 5 (Multi-outcome Sequential Back-Door Criterion (Jung et al., 2020)) . Let ( X , Y ) denote a pair of ordered sets in G . Let Z ⊆ V \ ( X ∪ Y ) be ordered as Z = ( Z 1 , . . . , Z m ) , and let H i := X ( i ) ∪ Y ( i ) ∪ Z ( i ) . Then, Z is said to satisfy the multi-outcome sequential back-door (mSBD) criterion w.r.t. ( X , Y ) in G if, for i = 1, . . . , m ,

Proposition 2 (mSBD adjustment (Jung et al., 2020)) . If Z = ( Z 1 , . . . , Z m ) satisfies the mSBD criterion w.r.t. ( X , Y ) in G , the causal effect P ( y | do( x )) is identifiable and expressible as sequential covariate adjustment of Z w.r.t. ( X , Y ) in G :

For concreteness, consider the causal diagram G in Fig. 1 with X = ( X 1 , X 2 ) and Y = ( Y 1 , Y 2 ) . Applying the partitioning operator PT X G , we divide Y into two subsets: Y 1 = { Y 1 } and Y 2 = { Y 2 } . Next, we define Z := ( Z 1 , Z 2 ) , where Z 1 := { Z a , Z b } and Z 2 := { Z c , Z d } . This ordered set Z satisfies the mSBD criterion relative to ( X , Y ) in G . The first condition of the mSBD criterion holds since Z 1 does not include any descendants of X 1 in G , and similarly, Z 2 excludes any descendants of X 2 . To check the second condition of the mSBD criterion, consider the graph G X 1 X 2 derived from G by removing the edges X 1 → Y 1 and { Z b → X 2 , Z c → X 2 , Y 1 → X 2 } . In this graph, ( Y 1 , Y 2 ) are conditionally independent of X 1 given Z 1 and X 2 , because any path from X 1 to { Y 1 , Y 2 } is blocked (d-separated)

Za

Zc

Zd

X

1

Y

1

X

2

Y

2

by Z 1 . Finally, in the graph G X 2 , where the edge X 2 → Y 2 is removed, any paths between X 2 and Y 2 are d-separated when conditioned on Z 1 ∪ Z 2 . This analysis confirms that Z satisfies the mSBD criterion, and therefore, the causal effect is given as follows:

Despite its soundness (Prop. 2), the mSBD criterion is not complete. In other words, there are causal graphs such that P ( y | do( x )) is expressible through sequential covariate adjustment, even though Z := ( Z 1 , . . . , Z m ) does not satisfy the mSBD criterion relative to ( X , Y ) in G .

Proposition 3 (Incompleteness of mSBD criterion) . There exists a causal graph G where the causal effect P ( y | do( x )) can be represented as a sequential covariate adjustment with Z such that Z does not satisfy the mSBD criterion relative to ( X , Y ) in G .

To illustrate the incompleteness of the mSBD criterion, consider the graph G in Fig. 2a with X = ( X 1 , X 2 ) and Y = ( Y 1 , Y 2 ) (i.e., Y 0 := ∅ ) with Y 1 = { Y 1 } and Y 2 = { Y 2 } . Set Z := ( Z 1 , Z 2 ) , where Z 1 := { Z a , Z b } and Z 2 := { Z c , Z d } . Then, while Z 1 satisfies both Eqs. (4, 5) of the mSBD criterion, Z 2 does not satisfy the mSBD condition in Eq. (4) since Z 2 includes the descendant of X 2 . Nevertheless, the causal effect P ( y | do( x )) is still expressible through sequential covariate adjustment as in Eq. (7). To demonstrate, we begin with the observation that Z 1 satisfies the mSBD criterion in Eqs. (4, 5), leading to P ( y 1 , y 2 | do( x 1 , x 2 )) = ∑ z a , z b P ( y 2 | do( x 2 ), h 1 ) P ( y 1 |

where the last equality holds by Rule 1 of do-calculus (Pearl, 1995): ( Y 2 ⊥ ⊥ Z c | X 2 , H 1 ) G X 2 . Further,

where the last equality holds by Rule 2 and 3 of do-calculus, i.e., ( Y 2 ⊥ ⊥ X 2 | H 1 , Z c , Z d ) G X 2 and ( Z d ⊥ ⊥ X 2 | H 1 , Z c ) G X 2 , respectively. Combining these derivations, P ( y 1 , y 2 | do( x 1 , x 2 )) is expressible as the same as in Eq. (7). This exemplifies the incompleteness of mSBD criterion.

3.2 Limitation of Other Existing Criteria

Besides mSBD, there are other methods attempting to establish conditions for determining if the causal effect can be represented as sequential covariate adjustment. For instance, the no-unmeasuredconfounding assumption (Rosenbaum and Rubin, 1983) is a sufficient verification tool. However, as illustrated in Fig. 2a, the causal effect is expressible through sequential covariate adjustment even though an unmeasured confounder between the treatment X 2 and the outcome Y 1 presents.

On the other hand, the sequential ignorability criterion (Robins, 1986, 2000) provides a condition for Z to be a sequential adjustment set:

where Y ≥ i ( x ) is a counterfactual variable induced from Y ≥ i by fixing X = x in the SCM. Graphical methods such as single world intervention graphs (SWIGs) (Richardson and Robins, 2013) and twin-networks (Pearl, 2000; Shpitser and Pearl, 2008) help in analyzing such counterfactual variables. However, each has its limitations in efficiently verifying sequential adjustment sets. SWIGs, for example, face challenges in capturing Z 1 , Z 2 as a sequential adjustment set in Fig 2a, because they fail to accommodate both Y 2 ( x 2 ) and Z 2 in a SWIG. This limitation arises since the counterfactual variable Z 2 ( x 2 ) , instead of the observable Z 2 , will appear on the SWIG. In contrast, twin-networks are capable of exploring the relationship between Y 2 ( x 2 ) and Z 2 . Yet, their practicality decreases with larger cardinality ( m ), as the size of twin-networks expands exponentially, leading to substantial computational demands. These observations motivate the necessity of computationally efficient and effective criterion to determine if a causal effect aligns with a sequential covariate adjustment.

4 Sequential Adjustment Criterion

In this section, we devise a sequential adjustment criterion (SAC), which provides a sound and complete method for characterizing the causal effect P ( y | do( x )) as a sequential covariate adjustment of Z := ( Z 1 , · · · , Z m ) with respect to ( X , Y ) . To begin with, we define a special subgraph called proper sequential back-door graph :

Za

Zc

Zd

X

1

Y

1

X

2

Y

2

(a) SAC

Za

Zc

Zd

X

1

Y

1

X

2

Y

2

(b)

G

X

,

Y

psbd,1

Za

Zc

Zd

X

1

Y

1

X

2

Y

2

(c)

G

X

,

Y

psbd,2

Definition 6 (Proper Sequential Back-Door Graph) . For a disjoint pair ( X , Y ) and the index i , the proper sequential back-door graph G X , Y psbd, i is a proper back-door graph induced from G X ≥ i +1 relative to ( X i , Y ≥ i ) ; i.e.,

As constructing the proper back-door graph from G takes linear time O ( | V | + | E | ) (van der Zander et al., 2014), the sequential back-door graph can also be constructed in linear time O ( | V | + | E | ) . Equipped with G X , Y psbd, i , the sequential adjustment criterion is presented as follows:

Definition 7 (Sequential Adjustment Criterion (SAC)) . Let ( X , Y ) denote a disjoint pair. Let Z := ( Z 1 , · · · , Z m ) denote a topologically ordered set of vertices disjoint to ( X , Y ) where each Z i is non-descendant of X ≥ i +1 . Then, Z is said to satisfy sequential adjustment criterion (SAC) w.r.t. ( X , Y ) in G if the following conditions are satisfied for i = 1, · · · , m :

The sequential adjustment criterion characterizes the sequential covariate adjustment:

Theorem 1 (Soundness and Completeness) . Let ( X , Y ) denote a disjoint pair, and let Z := ( Z 1 , · · · , Z m ) denote an ordered set of vertices disjoint to ( X , Y ) . Then, the following are equivalent:

1. Z satisfies SAC relative to ( X , Y ) in G .

2. Z is a sequential adjustment set relative to ( X , Y ) in G ; i.e.,

For instance, consider Fig. 2a, where we witnessed that the causal effect P ( y 1 , y 2 | do( x 1 , x 2 )) is given as a sequential covariate adjustment of Z = ( Z 1 = { Z a , Z b } , Z 2 = { Z c , Z d } ) . To confirm that this Z satisfies SAC relative to ( X , Y ) for X = ( X 1 , X 2 ) and Y = ( Y 1 , Y 2 ) with Y 1 = { Y 1 } and Y 2 = { Y 2 } , we examine G X , Y psbd,1 (Fig. 2b) and G X , Y psbd,2 (Fig. 2c). In G X , Y psbd,1 , we observe that Z 1 = { Z a , Z b } d-separates X 1 and { Y 1 , Y 2 } . Moreover, Z 1 does not contain any descendant of proper causal paths from X 1 to { Y 1 , Y 2 } within this subgraph. Similarly, in G X , Y psbd,2 , the path between X 2 and Y 2 is d-separated conditioned on Z 2 = { Z c , Z d } and H 1 = { Z a , Z b , X 1 , Y 1 } . Additionally, Z 2 is not on any descendant of proper causal paths between X 2 and Y 2 . Consequently, Z fulfills SAC relative to ( X , Y ) in the graph G .

The soundness and completeness of SAC provide extensive coverage, incorporating existing covariate adjustment criterion such as BD, AC, SBD, and mSBD in Table 1. Clearly, SAC covers the mSBD criterion due to its completeness. In other words, the mSBD criterion is sufficient for the sequential adjustment criterion.

Corollary 1 (mSBD = ⇒ SAC) . If Z satisfies the mSBD criterion relative to ( X , Y ) in G , then Z satisfies the SAC relative to ( X , Y ) in G .

To demonstrate, consider G in Fig. 1, where Z := ( Z 1 , Z 2 ) with Z 1 := { Z a , Z b } and Z 2 := { Z c , Z d } . This Z satisfies the mSBD criterion relative to ( X , Y ) for X = ( X 1 , X 2 ) and Y = ( Y 1 , Y 2 ) . To

ensure that Z also satisfies the SAC relative to ( X , Y ) in G , we verify that ( Z 1 , Z 2 ) meet the conditions in Eqs. (10, 11), respectively. Since Z 1 and Z 2 are both non-descendants of X 1 and X 2 , respectively, we only need to verify that they meet the conditions of Eq. (11) for ( X 1 , ( Y 1 , Y 2 )) and ( X 2 , Y 2 ) respectively. First, consider G X , Y psbd,1 , which removes edges { X 1 → Y 1 , Z b → X 2 , Z c → X 2 } from G . In this graph, X 1 is d-separated from ( Y 1 , Y 2 ) given Z 1 . Next, consider G X , Y psbd,2 , which removes the edge { X 2 → Y 2 } from G . In this graph, X 2 is d-separated from Y 2 given ( Z 1 , X 1 , Y 1 ) and Z 2 . Therefore, Z also satisfies the SAC relative to ( X , Y ) in G .

The broader coverage of SAC compared to the mSBD criterion is evident when compared to AC. First, the mSBD criterion is not extensive enough to cover the AC criterion. To witness, consider Fig. 3. Here, Z := { Z a , Z b } satisfies AC relative to ( X , Y ) for X = ( X , Y ) and Y = ( Y 1 , Y 2 ) . However, Z does not satisfy the mSBD criterion since it is a descendant set of X 1 := { X 1 } . In contrast, AC is a sufficient criterion for SAC.

Theorem 2 (AC = ⇒ SAC) . If Z satisfies the AC relative to ( X , Y ) in G , then there exists a partition of Z and a topological order of X such that Z satisfies the SAC relative to ( X , Y ) in G .

For concreteness, consider G in Fig. 3 (van der Zander et al., 2014). In G , Z := { Z a , Z b } satisfies the adjustment criterion relative to ( X , Y ) for X = ( X 1 , X 2 ) and Y = ( Y 1 , Y 2 ) in G , and the causal effect is given as an adjustment as follows:

Thm. 2 implies that Z meets the SAC relative to ( X , Y ) in graph G . To demonstrate this, for X = ( X 1 , X 2 ) , we apply the partitioning operator in Def. 4 and obtain Y 0 = { Y 2 } , Y 1 := { Y 1 } , and Y 2 = ∅ . We set Z 1 := { Z a , Z b } . Since Y 2 is empty, it is sufficient to demonstrate that

X

1

Y

1

X

2

Y

2

Za

Z 1 meets the conditions of Eqs. (10) and (11). First, Z 1 meets the condition of Eq. (11) as it contains no descendants on the causal paths from X 1 to Y 1 . Consider the modified graph G X , Y psbd,1 , obtained by removing the edges { X 1 → Y 1 , Z b → X 2 } . In this reduced graph, X 1 and Y 1 are d-separated given Z 1 and Y 0 , confirming the conditions of SAC. Therefore, Z := ( Z 1 , Z 2 ) satisfies SAC relative to ( X , Y ) in G , and the causal effect is described as

Eqs. (13, 14) are equivalent since P ( y 1 | x 1 , x 2 , z a , z b , y 2 ) = P ( y 1 | x 1 , z a , z b , y 2 ) and P ( y 2 | x 1 , x 2 , z a , z b ) = P ( y 2 | z a , z b ) by the conditional independence implied by G .

4.1 Constructive Sequential Adjustment Criterion

Sequential adjustment criterion in Def. 7 relies on a predefined partition Z = ( Z 1 , · · · , Z m ) . A natural question is how to construct such ordered partition of the covariates. We propose a method to construct a sequential adjustment set, which characterizes sequential covariate adjustment in that the ordered set meets the SAC if and only if the causal effect can be expressed through a sequential covariate adjustment.

Theorem 3 (Construction of Sequential Adjustment Set) . Let ( X , Y ) denote a disjoint pair in G . Define Z an := ( Z an 1 , · · · , Z an m ) and corresponding ( F an 1 , · · · , F an m ) and ( H an 0 , · · · , H an m -1 ) alternatively as follows: For i = 1, · · · , m ,

Then, the following statements are equivalent:

There exists an ordered set Z := ( Z 1 , · · · , Z m ) satisfying the sequential adjustment criterion w.r.t. ( X , Y ) in G .

X

1

Zc

Za

X

2

Y

(a)

G

X

1

Zc

Za

X

2

Y

(b)

G

X

,

Y

psbd,1

X

1

Zc

Za

X

2

Y

(c)

G

X

,

Y

psbd,2

2. The ordered set Z an satisfies SAC relative to ( X , Y ) in G .

The set Z an can be constructed in O ( m ( | V | + | E | )) since the procedures for constructing dpcp G ( X i , Y ≥ i ) and G X , Y psbd, i in Eq. (16), as well as finding ancestors and descendants in Eqs. (16, 17), each require O ( | V | + | E | ) and are repeated m times.

To concretely exemplify that the first statement implies the second one in Thm. 3, consider Fig. 2a. Set Y 0 := ∅ , Y 1 := { Y 1 } , and Y 2 := { Y 2 } by the mechanism of the partitioning operator PT X G . G X , Y psbd,1 and G X , Y psbd,2 are depicted in Figs. (2b, 2c). Then, Z an 1 := { Z a , Z b , Z c , Z d } , since these are ancestors of { X 1 , Y 1 , Y 2 } in G X , Y psbd,1 . We can see that Z an 1 satisfies Eqs. (10, 11) in the SAC because X 1 and { Y 1 , Y 2 } are d-separated in G X , Y psbd,1 given Z an 1 , and they are not on the descendant of proper causal paths between X 1 and { Y 1 , Y 2 } . Next, Z an 2 := ∅ . It satisfies Eqs. (10, 11) since X 2 and Y 2 are d-separated in G X , Y psbd,2 given { X 1 , Y 1 } ∪ Z an 1 ∪ Z an 2 . Combining with our previous observation that Z 1 := { Z a , Z b } and Z 2 := { Z c , Z d } satisfy Eqs. (10, 11), this example demonstrates that indeed Z an := ( Z an 1 , Z an 2 ) satisfies the SAC relative to ( X , Y ) in G .

As an example that the second statement implies the first one, consider Fig. 4a, where Y 0 := ∅ , Y 1 := ∅ and Y 2 := { Y } by the mechanism of the partitioning operator PT X G . Next, G X , Y psbd,1 and G X , Y psbd,2 are depicted in Figs. (4b, 4c). Then, Z an 1 := ∅ , since none of { Z a , Z b , Z c } are the ancestors of { X 1 , Y } . Z an 1 satisfies Eqs. (10, 11) in the SAC, since the X 1 and Y are d-separated in G X , Y psbd,1 given Z an 1 . Next, Z an 2 := { Z a , Z b , Z c } since they are ancestors of X 2 in G X , Y psbd,2 . This Z an 2 satisfies Eqs. (10, 11), since X 2 and Y are d-separated in G X , Y psbd,2 given Z an 2 ∪ { X 1 } . Therefore, Z an := ( Z an 1 , Z an 2 ) = ( ∅ , { Z a , Z b , Z c } ) satisfies SAC relative to ( X , Y ) in G .

4.2 Minimal Sequential Adjustment Criterion

We have demonstrated the effectiveness of constructing sequential adjustment sets. However, when dealing with large graphs, Z an may include a large number of vertices, leading to high computational costs when evaluating sequential covariate adjustments. This situation highlights the need for a more parsimonious adjustment set. To address this, we introduce the minimal sequential covariate adjustment set, which is the smallest subset of Z an without sacrificing the validity of the adjustment.

Definition 8 (Minimal Sequential Covariate Adjustment Set) . Let ( X , Y ) denote a disjoint pair in G . An ordered set of vertices Z min := ( Z min 1 , · · · , Z min m ) where each Z min i are non-descendant of X ≥ i +1 is said to be a minimal sequential covariate adjustment set if, for each i = 1, · · · , m ,

Z min i satisfies Eqs. (10, 11).

For any Z ′ i ⊊ Z min i , Z ′ i does not satisfy Eqs. (10, 11).

To present the minimal sequential covariate adjustment set, we define a useful tool:

Definition 9 (Closure (van der Zander and Li´ skiewicz, 2020)) . For a disjoint vertices A , B , and C in G , the closure of A with respect to B and C in G , denoted as closure G ( A ; B , C ) , is the union of A and the collection of V ∈ An G ( A ∪ B ) that are connected to A via a path satisfying the following:

The path only contains nodes in An G ( A ∪ B ) ; and

All non-collider vertices on the path are not in C ; Equivalently, any vertex on the path that is in C is a collider on the path.

Verbally speaking, closure G ( A ; B , C ) is a set of vertices in An G ( A ∪ B ) that is d-connected to A in G conditioned on C , where the paths are through An G ( A ∪ B ) . Finding the closure can be efficiently done in O ( | V | + | E | ) time (van der Zander and Li´ skiewicz, 2020), where | V | and | E | are the number of vertices and edges of the graph G . For example, consider closure G X , Y psbd,2 ( X 2 ; Y , { Z a , Z b , Z c , X 1 } )

with G X , Y psbd,2 in Fig. 4c. First, the ancestor of { X 2 , Y } are { Z a , Z b , Z c , X 1 } . Since Z a and X 1 are adjacent to X 2 , they are contained in the closure. However, { Z b , Z c } are not in the closure, since all the non-collider vertices on the path between X 2 and { Z b , Z c } are contained in { Z a , Z b , Z c , X 1 } . Therefore, { Z a , X 1 , X 2 } = closure G X , Y psbd,2 ( X 2 ; Y , { Z a , Z b , Z c , X 1 } ) .

Equipped with the closure, we propose a procedure to construct the minimal sequential covariate adjustment in Algo. 1. The minimal sequential covariate adjustment set, Z min , is derived directly from the ordered set Z an , which characterizes the sequential covariate adjustment. Therefore, Z min naturally retains this characterizing property.

Theorem 4 (Construction of Minimal Sequential Covariate Adjustment Set) . Let ( X , Y ) denote a disjoint pair in G . Then, the following statements are equivalent:

There exists an ordered set Z := ( Z 1 , · · · , Z m ) satisfying SAC w.r.t. ( X , Y ) in G .

The ordered set Z min = minSCA ( X , Y , G ) from Algo. 1 is a minimal sequential covariate adjustment set.

Constructing Z min = minSCA ( X , Y , G ) can be achieved in O ( m ( | V | + | E | )) , as the construction of each Z an i in Eq. (17) and the required closures each take O ( | V | + | E | ) and are repeated m times.

We now demonstrate Thm. 4 with a causal diagram G in Fig. 4a. We have shown that Z an := ( Z an 1 , Z an 2 ) with Z an 1 := ∅ and Z an 2 := { Z a , Z b , Z c } satisfies SAC relative to ( X , Y ) in G . Therefore, we only demonstrate Algo. 1 for Z an 2 := { Z a , Z b , Z c } . Run line 4: Z ∗ 2 := closure G X , Y psbd,2 ( Y ; X 2 , { Z a , Z b , Z c , X 1 } ) ∩ { Z a , Z b , Z c } . First, { Z a , Z c } is included in the closure since they are adjacent to Y . Next, Z b is included in the closure, since the path Y ↔ Z c ← Z b does not include non-collider vertex. Therefore, Z ∗ 2 = Z an 2 = { Z a , Z b , Z c } . Next, run line 5. We already witnessed that { Z a , X 1 , X 2 } = closure G X , Y psbd,2 ( X 2 ; Y , { Z a , Z b , Z c , X 1 } ) , which leads to Z min := { Z , X , X } ∩ { Z , Z , Z } = { Z } . Therefore, Z min := ( Z min , Z min ) = ( ∅ , { Z } ) .

5 Conclusion

We develop a sound and complete graphical criterion for sequential covariate adjustment. We start by highlighting the limitations of the existing mSBD criterion, acknowledging its soundness but noting its lack of completeness. (Prop. 3). Against this background, we introduce the sequential adjustment criterion (Def. 7), which characterizes sequential covariate adjustment (Thm. 1). The

proposed criterion includes and extends the mSBD and standard adjustment criterion, confirming its comprehensive nature since they imply the sequential adjustment criterion (Prop. 1 and Thm. 2). Further, we develop a procedure for constructing the sequential adjustment set that captures the sequential covariate adjustment (Thm. 3). Finally, we devise an algorithm to identify the minimal sequential covariate adjustment set (Algo. 1 and Thm. 4).

Acknowledgments and Disclosure of Funding

We thank anonymous reviewers for constructive comments to improve the manuscript. This work was partly supported by the IITP (RS-2022-II220953/20%), NRF (RS-2023-00211904/20%, RS-202300222663/20%), and MFDS (23212MFDS202/20%) grant funded by the Korean government.

References

Jung, Y., Tian, J., and Bareinboim, E. (2020). Estimating causal effects using weighting-based estimators. In Proceedings of the 34th AAAI Conference on Artificial Intelligence .

Maathuis, M. H. and Colombo, D. (2015). A generalized back-door criterion. The Annals of Statistics , 43(3):1060-1088.

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika , 82(4):669-710.

Pearl, J. (2000). Causality: Models, Reasoning, and Inference . Cambridge University Press, New York. 2nd edition, 2009.

Pearl, J. and Robins, J. (1995). Probabilistic evaluation of sequential plans from causal models with hidden variables. In Proceedings of the Eleventh conference on Uncertainty in artificial intelligence , pages 444-453.

Perkovic, E., Textor, J., Kalisch, M., Maathuis, M. H., et al. (2018). Complete graphical characterization and construction of adjustment sets in markov equivalence classes of ancestral graphs. Journal of Machine Learning Research , 18(220):1-62.

Richardson, T. S. and Robins, J. M. (2013). Single world intervention graphs (SWIGs): A unification of the counterfactual and graphical approaches to causality. Center for the Statistics and the Social Sciences, University of Washington Series. Working Paper , 128(30):2013.

Robins, J. (1986). A new approach to causal inference in mortality studies with a sustained exposure period-application to control of the healthy worker survivor effect. Mathematical modelling , 7(9-12):1393-1512.

Robins, J. M. (2000). Robust estimation in sequentially ignorable missing data and causal inference models. In Proceedings of the American Statistical Association , volume 1999, pages 6-10. Indianapolis, IN.

Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika , 70(1):41-55.

Shpitser, I. and Pearl, J. (2008). Complete identification methods for the causal hierarchy. Journal of Machine Learning Research , 9:1941-1979.

Shpitser, I., VanderWeele, T., and Robins, J. M. (2010). On the validity of covariate adjustment for estimating causal effects. In Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence , pages 527-536. AUAI Press.

van der Zander, B. and Li´ skiewicz, M. (2020). Finding minimal d-separators in linear time and applications. In Uncertainty in Artificial Intelligence , pages 637-647. PMLR.

van der Zander, B., Li´ skiewicz, M., and Textor, J. (2014). Constructing separators and adjustment sets in ancestral graphs. In Proceedings of the UAI 2014 Conference on Causal Inference: Learning and Prediction-Volume 1274 , pages 11-24.

Supplement to 'Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference'

Contents

A Proofs

In this appendix, we provide detailed proof of Theorems presented in the main paper.

Notations. We symbolize the d-connection between two vertices A and B as A - -B . Furthermore, we symbolize the type of paths, that are not mutually exclusive, as follows:

A - - ◦→ B means that there is a directed path from A to B .

A ←◦→ B means that there is a non-directed path with no colliders and all non-colliders on the path are not conditioned.

A ∗ →·← ∗ B means that there is a non-directed path containing at least one collider, and either the collider or one of its descendants is conditioned.

A.1 Proof of Theorem 1

In order to prove both the soundness and completeness of SAC, we structure the proof in two parts. First, we will demonstrate the soundness of SAC through Lemma S.1, followed by proving its completeness in Lemma S.2.

Lemma S.1 (Soundness of SAC) . Let ( X , Y ) denote a disjoint pair of ordered sets where Y is partitioned with PT X G . Let Z := ( Z 1 , · · · , Z m ) denote an ordered set of vertices disjoint to ( X , Y ) in G . If Z satisfies the SAC relative to ( X , Y ) in G , then Z is a sequential adjustment set relative to ( X , Y ) in G .

Proof of Lemma S.1 . We first assume that the following claim holds, and then demonstrate that the claim is valid. For any fixed Z i ∈ ( Z 1 , · · · , Z m ) , suppose the following claim is true .

Claim. If Z satisfies SAC relative to ( X , Y ) in G , then there exists an ordered set Z i = ( Z i ,1 , · · · , Z i , m i ) (where m i = | Z i | ) such that, for each Z i , j in the ordered set, and with Z ( j -1) i := ( Z i ,1 , · · · , Z i , j -1 ) , at least one of the following two cases holds:

where F X i is a new parent of X i , called the regime , augmented in the graph G .

For the fixed partition of Z i that makes the claim true, suppose Case 1 holds for all Z i , j ∈ Z i . Then,

Now, suppose Case 2 holds for all Z i , j . Then,

In summary, under the claim, for all Z i , j ∈ Z i ,

By applying this equation for all elements Z i ,1 , · · · , Z i , m i in Z i , we have

We also note that the given condition in Eq. (10) implies the Rule 2 of the do-calculus (Pearl, 2000):

since the d-separation is always preserved when cutting the edge from the graph. The right-hand side of Eq. (A.1) can be written further as follows:

Therefore,

In conclusion, the sequential adjustment criterion is sound under the claim. It now remains to prove that the claim is true.

Proof of the claim . We will prove the claim by contradiction. Suppose that Z satisfies the SAC relative to ( X , Y ) in G but for all orders of elements in Z i (denoting an order as π ( Z i ) ), there always exists some Z i , j ∈ Z i with Z ( j -1) i := Z ( π , j -1) i (which is predecessors of Z i , j under the order π ( Z i ) ) such that Z i , j does not satisfy either Case 1 or Case 2, i.e.,

This means that there exists a d-connecting path between Z i , j and Y ∈ Y ≥ i in G X ≥ i given X ≥ i ∪ Z ( j -1) i ∪ H i -1 , and a d-connecting path between F X i and Z i , j in G given Z ( j -1) i ∪ H i -1 . That is, the following path exists:

We further state the following. In this setting, there exists a d-connecting path between Z i , j and Y in G X ≥ i +1 , which is evident since adding an edge (from G X ≥ i to G X ≥ i +1 ) does not block any d-connecting paths. Moreover, the d-connecting path between F X i and Z i , j in G is preserved in G X ≥ i +1 . To witness, suppose there exists a d-connected path between F X i and Z i , j in G , while the path is blocked/cut in G X ≥ i +1 . This implies that the path contains an edge → X k for some X k ∈ X ≥ i +1 . By the topological order and the assumption that Z i is non-descendant of X ≥ i +1 , the directed path from X k to X i and the directed path from X k to Z i , j are both impossible. Thus, we only consider non-directed path from F X i to Z i , j . First, X k on the path connecting X i and Z i , j cannot be a collider, as the path would be d-separated if X k were not conditioned. Additionally, X k cannot be an ancestor of Z ( j -1) i ∪ H i -1 due to the topological order of X , the mechanism of the partitioning operator PT X G , and the assumption that Z j is non-descendant of X ≥ j +1 . Therefore, we consider the path between X i and Z i , j through X k , where neither X k nor its descendant forms a collider. The possible types of paths in G are as follows:

All cases imply the existence of a directed path from X k to any vertex in Z ( j -1) i ∪ H i -1 , making X k an ancestor of Z ( j -1) i ∪ H i -1 . This contradicts our setting, just as in the case where X k is a collider. It confirms that the d-connecting path between F X i and Z i , j in G is preserved in G X ≥ i +1 . As a result, Eq. (A.4) means the presence of the following path in G X ≥ i +1 :

Then, there are nine types of d-connected path between X i and Y in G X ≥ i +1 given Z ( j -1) i ∪ H i -1 :

We will show that each type of path belongs to one of the following three cases:

Case 1. There always exists another order π ′ for Z i such that the path is blocked, even if the path is d-connected under the oreder π .

Case 2. The path contradicts the first condition of SAC in Eq. (10).

Case 3. The path contradicts the second condition of SAC in Eq. (11).

The paths in Type 1 are directed from X i to Y , which contradicts the second condition of SAC in Eq. (11).

Consider Type 2. Since Z i , j is a collider, and Z ( j -1) i ∪ H i -1 is conditioned on, it forms a d-separated path between X i and Y , which is preserved in G X , Y psbd,i . First, consider the case where this path is not blocked by further conditioning on Z i \ Z ( j -1) i . This implies that Eq. (10) is not satisfied, thereby, contradicting the initial assumption that Z satisfies SAC. Therefore, the path must be blocked by further conditioning on Z i \ Z ( j -1) i . In other words, there exists another order π ′ for Z i such that this path is blocked whenever { Z i , j } ∪ Z ( π ′ , j -1) i ∪ H i -1 is conditioned on. Therefore, without loss of generality, we can ignore Type 2 by considering any other order π ′ in which the colliding path is blocked.

Consider Type 3. Since the case where Z i , j forms a collider is considered in Type 2, we only need to consider the case where Z i , j → lies on the path. In this case, Z i , j has a directed path to any vertex in H i -1 ∪ Z ( j -1) i . However, it is impossible for the vertex to be involved in H i -1 , as this would contradict the valid topological order of SAC. Therefore, we can conclude that there exists a directed path from X i to Z ∈ Z ( j -1) i through Z i , j . Then, the following two cases are possible: First, there exists a directed path from Y to Z . This case contradicts the condition of SAC in Eq. (11), as Z would be a descendant of Y . Second, there exists a divergent path between Y and Z , meaning that Z forms a collider on the path between X i and Y . Note that we can ignore Type 3 for the same reason discussed in the case of Type 2.

The path for Type 4-6 be open when X i is an ancestor of the conditioning set Z ( j -1) i ∪ H i -1 . Due to the valid topological order of SAC, we only consider the case where the conditioning set is Z ( j -1) i . This implies the existence of some path of the form F X i → X i → Z ( ∈ Z ( j -1) i ) that does not belong to Type 4-6, allowing us to reduce this types to another type.

We now consider the path for Type 7-9. As we witnessed that X i cannot be a collider opening the path, it should contain X i → , which implies that X i is an ancestor of some collider and its descendant is some vertices in Z ( j -1) i ∪ H i -1 . Hence, the collider must be a vertex in Z ( j -1) i , as H i -1 being the collider would contradicts the valid topological order of SAC. Therefore, we can ignore such type of paths.

Therefore, any type of path leads to a contradiction, which validates the claim. In conclusion, there exists an order such that every Z i , j ∈ Z i satisfies Case 1 or Case 2, meaning the claim is true .

Lemma S.2 (Completeness of SAC) . Let ( X , Y ) denote a disjoint pair of ordered sets where Y is partitioned with PT X G . Let Z := ( Z 1 , · · · , Z m ) denote an ordered set of vertices disjoint to ( X , Y ) . There exists a graph G such that Z is not a sequential adjustment set relative to ( X , Y ) in G whenever Z does not satisfy SAC relative to ( X , Y ) in G .

Proof of Lemma S.2. If Z does not satisfy SAC in G , then there exists Z i ∈ Z not satisfying Eq. (10) or Eq. (11) in SAC.

First, suppose Z i does not satisfy Eq. (10). Then, there must be a path between X i and Y ∈ Y ≥ i such that all non-colliders on the path are not in Z i ∪ H i -1 and all colliders on the path are in An G X , Y psbd, i ( Z i ∪ H i -1 ) . Since the path is defined in G X , Y psbd, i and by the mechanism of the partitioning operator PT X G , the path cannot be a directed path.

Now, suppose the path is a divergent path that does not contain any colliders; i.e., X i ←◦ → Y . In this case, we can consider a simple graph G = { X i ↔ Y i , X i → Y i } where G X , Y psbd, i contains a subgraph { X i ↔ Y i } . Then, P ( y | do( x )) cannot be given as a sequential adjustment since it is not even identifiable. Suppose, instead, the path is a colliding path; i.e., X ∗ →·← ∗ Y where the collider is an ancestor of the conditioned vertices Z i ∪ H i -1 . Then, we can consider a simple graph G that contains a subgraph { X i ← Z i → Y i , X i → Y i , X i ↔ Z i ↔ Y i } . Here again, P ( y | do( x )) cannot be given as a sequential adjustment since it is not even identifiable.

Next, suppose Z i does not satisfy Eq. (11). Then, we may consider a graph a simple graph G that contains a subgraph X i → Z i → Y i . Then, there exists a SCM M such that P ( y | do( x )) is not given as an adjustment over Z i (Shpitser et al., 2010).

This concludes the proof that the sequential adjustment criterion is complete .

A.2 Proof of Corollary 1

We will prove that if there exists Z i ∈ Z that does not satisfy Eq. (10) or Eq. (11) in SAC, then Z i does not satisfy Eq. (4) or Eq. (5) in mSBD criterion.

For the sake of contradiction, suppose Z i does not satisfy Eq. (10), but ( X i ⊥ ⊥ Y ≥ i H i -1 , Z i ) G Xi X ≥ i +1 holds. This means that there exists a d-connecting path between X i and Y ∈ Y ≥ i given Z i ∪ H i -1 in G psbd, i while the path is blocked in G X i X ≥ i +1 . The differences between G X i X ≥ i +1 and G X , Y psbd, i imply that the d-connecting path should contain X i → . Since the path cannot be a directed path, it must be a colliding path where the collider is an ancestor of any vertices in Z i ∪ H i -1 . This means that there exists a directed path either X i - - ◦→ Z for some Z ∈ Z i or X i - - ◦→ H for some H ∈ H i -1 . If X i - - ◦→ Z , such Z i fails to satisfy the mSBD criterion because Z i contains a descendant of X i . If X i - - ◦→ H , it implies that there exists a directed path from X i to X k ∈ X ( i -1) , Y k ∈ Y ( i -1) or Z k ∈ Z ( i -1) . In any case, it contradicts the topological order of X , the mechanism of partition operator PT X G , or the assumption that Z i -1 is non-descendant of X ≥ i in G .

| X , Y

Now, suppose Z i does not satisfy Eq. (11). Then, such Z i does not satisfy Eq. (4) since failure to satisfy Eq. (11) means that Z i is a descendant of X i in G .

This concludes the proof that the mSBD criterion implies the SAC.

A.3 Proof of Theorem 2

We will prove that if the SAC fails for any partition of Z across all topological ordering of X , then Z also fails to satisfy the adjustment criterion. The failure of SAC at Z i can be stated as

Then, our goal is to show the following:

Note that the right-hand side of Eq. (A.6) implies ( Y ̸⊥ ⊥ X | Z ) G X , Y pbd , meaning that Z does not satisfy the adjustment criterion relative to ( X , Y ) in G . To this end, we first witness the following:

Proof of Step 1. We will prove that any d-connected path between X i and Y ∈ Y ≥ i given Z i ∪ H i -1 in ( G X ≥ i +1 ) X i , Y ≥ i pbd is preserved in G X i , Y ≥ i pbd . First, the d-connected path between X i and Y in ( G X ≥ i +1 ) X i , Y ≥ i pbd must be a non-directed path in G , since the path cannot contain incoming edges to any vertices in X ≥ i +1 ; otherwise, it would be cut. As the operation applied to G X i , Y ≥ i pbd does not cut such path, the d-connection of the path remains intact in G X i , Y ≥ i pbd , which completes the proof.

Second, we will witness the following:

Proof of Step 2. The left-hand side of Eq. (A.8) implies that there exists a d-connecting path between X i and Y ∈ Y ≥ i given Z i ∪ H i -1 in G X i , Y ≥ i pbd . The path cannot be a directed path, since it is a d-connected path in G X i , Y ≥ i pbd , and by the mechanism of PT X G . Therefore, the path is either a divergent path X i ←◦→ Y , or a colliding path X i ∗ →·← ∗ Y where the collider is an ancestor of some vertices in Z i ∪ H i -1 (indeed, the collider is an ancestor of Z i only, since it cannot be an ancestor of H i -1 because it would create a directed path from X i to any vertex in H i -1 , which contradicts our setting). For the sake of contradiction, suppose there exists a partition of Z and an topological order of X such that ( Y ≥ i ̸⊥ ⊥ X i | Z i ∪ H i -1 ) G Xi , Y ≥ i pbd holds, but

That is, every d-connecting path between X i and Y given Z i ∪ H i -1 in G X i , Y ≥ i pbd is blocked by additionally conditioning on Z ≥ i +1 .

Suppose the above path is a divergent path X i ←◦→ Y . Then we can consider two cases; the path contains subpath Z 2 - - ◦ → X i or Z 2 - - ◦ → Y where Z 2 ∈ Z ≥ i +1 . Since we have a full degree-offreedom of choosing Z := ( Z i : i = 1, · · · , m ) where each Z i is non-descendant of X ≥ i +1 in G , only reason that the vertex Z 2 would be in Z k ∈ Z ≥ i +1 and not in Z i is that it is a descendant of some X p ∈ X ≥ i +1 . Therefore, Z 2 - - ◦→ X i means that there is a directed path X p - - ◦→ X i . However, it contradicts the valid topological order of X , since X p ∈ X ≥ i +1 . On the one hand, suppose the above path be a divergent path X i ←◦→ Y that contains a subpath Z 2 - - ◦→ Y . This means that there exists a directed path from X p to Y ∈ Y ≥ p . This case also contradicts SAC condition in Eq. (11) because Z 2 lies on the proper causal path. Therefore, the path cannot be a divergent path.

Suppose the path is a colliding path X i ∗ →·← ∗ Y where the collider is an ancestor of some vertices in H i -1 . In order for the path to be blocked by conditioning on Z ≥ i +1 , there must be Z 2 ∈ Z ≥ i +1 on the path that falls into one of the following cases:

Case 1. There exists a subpath Z 2 - - ◦→ X i in G X i , Y ≥ i pbd .

Case 2. The vertex Z 2 is an ancestor of H i -1 in G X i , Y ≥ i pbd .

Case 3. The vertex Z 2 is an ancestor of Y in G X i , Y ≥ i pbd .

The existence of Z 2 even with the full freedom of choosing any partition of Z means that there exists a directed path from X p to Z 2 as discussed. Therefore, Case 1 implies existence of any directed path from X p to X i , which contradicts the valid topological order of X .

Case 2 is impossible because it implies the existence of a directed path from Z 2 to any vertex in H i -1 (again, a directed path from X p to the vertex in H i -1 ), which would results in an invalid topological order.

Finally, Case 3 is also impossible as it implies a directed path from X p to Y , indicating that Z 2 lies on the proper causal path.

Therefore, every case leads to a contradiction, demonstrating Eq. (A.8) holds.

Combining Step 1 and Step 2, what we have shown is the following:

We will now show that Eq. (A.6) holds. Let the left-hand side of Eq. (A.6) holds; i.e., there exists a d-connected path between X i and Y ∈ Y ≥ i given Z i ∪ H i -1 in ( G X ≥ i +1 ) X i , Y ≥ i pbd . We have established that such d-connected path is preserved in G X i , Y ≥ i pbd conditioning on Z i ∪ H i -1 ∪ Z ≥ i +1 , as shown by Eq. (A.10). If this path does not contain any vertices in X \ { X i } , then the path is not affected by the operation applied to G X , Y pbd , which completes the proof. Otherwise, even though the path contains vertices in X \ { X i } , if the subpath between X a ∈ X \ { X i } and any vertex in Y ≥ i is not a proper causal path to Y ≥ i , then the path is not affected by the operation applied to G X , Y pbd , which completes the proof.

Suppose the path contains a subpath that is a proper causal path from X a ∈ X \ { X i } to Y . Then such X a should be included in X ≥ i +1 by the mechanism of PT X G . Then, as the path between X a and X i cannot be directed in either side, the path between X a and X i is either a divergent path X a ←◦→ X i or a colliding path X a ∗ →·← ∗ X i . Furthermore, since there is no incoming edge to X a , the path should be X a →·←∗ X i . Additionally, the path is d-connected when conditioning on Z i ∪ H i -1 . This implies that there exists a directed path from X a ∈ X ≥ i +1 to any vertex in Z i ∪ H i -1 , which contradicts our partioning mechanism PT X G . Therefore, there must be no proper causal path from X a to Y .

By proving that Eq. A.6 holds, we have completed the proof that the AC implies the SAC.

A.4 Proof of Theorem 3

Lemma A.1 ((van der Zander et al., 2014, Lemma 3.4)) . Let X , Y , I , R be sets of vertices with I ⊆ R , R ∩ ( X ∪ Y ) = ∅ . If there exists a d-separator Z 0 with I ⊆ Z 0 ⊆ R in G then Z = An G ( X ∪ Y ∪ I ) ∩ R is a d-separator.

Lemma A.2 (Construction) . Let ( X , Y ) denote a disjoint pair of ordered sets. For any i ∈ { 1, 2, · · · , m } , suppose each Z j ∈ ( Z 1 , · · · , Z i -1 ) is non-descendant of X ≥ j +1 and disjoint to ( X , Y ) . Let Z an i be an ordered set in Eq. (17) defined with respect to Z ( i -1) , which is denoted as Z an i ( Z ( i -1) ) . Then, the following statements are equivalent.

Z an i ( Z ( i -1) ) is non-descendant of X ≥ i +1 satisfying Eqs. (10, 11).

There exists a set Z i ⊆ V \ ( X ∪ Y ∪ H i -1 ∪ De G ( X ≥ i +1 )) satisfying Eqs. (10, 11).

There exists a set Z i ⊆ V \ ( X ∪ Y ∪ H i -1 ∪ De G ( X ≥ i +1 )) satisfying the following:

Proof of Lemma A.2 . By Theorem 1, (2) ⇔ (3) holds. Additionally, (1) = ⇒ (3) by Lemma S.1, which is equivalent to (1) = ⇒ (2) . Therefore, we only need to prove (2) = ⇒ (1) .

We apply Lemma A.1 by replacing G in Lemma A.1 as G X , Y psbd, i , X in Lemma as { X i } , Y in Lemma as Y ≥ i , I in Lemma as H i -1 , Z 0 in Lemma as an union of H i -1 and a certain Z i satisfying Eqs. (10, 11),and R as an union of H i -1 and V \ ( X ∪ Y ∪ H i -1 ∪ dpcp G ( X i , Y ≥ i ) ∪ De G ( X ≥ i +1 )) . Then, Lemma A.1 states that, under the statement (2) , there exists a d-separator between X i and Y ≥ i , which is Z an i ∪ H i -1 := ( An G X , Y psbd, i ( { X i } ∪ Y ≥ i ∪ H i -1 ) \ F i ) ∪ H i -1 . This completes the proof.

We emphasize that Lemma A.2 is stated for a fixed Z ( i -1) ; i.e., Z an i in Lemma A.2 is dependent on some specific choice of Z ( i -1) . This dependency is highlighted by the notation Z an i ( Z ( i -1) ) .

Then, the remaining part of the proof is to show that, for any given Z ( i -1) , if Z i satisfies Eqs. (10, 11), then Z an i ( Z an 1 , · · · , Z an i -1 ) satisfies Eqs. (10, 11). To show through contradiction, suppose Z an i ( Z an 1 , · · · , Z an i -1 ) does not satisfy Eqs. (10, 11), even if Z i satisfies them given Z ( i -1) . By Lemma A.2, this implies that, given { Z an 1 , · · · , Z an i -1 } , no Z ′ i ⊆ V \ F an i satisfies Eqs. (10, 11).

However, consider the following choice of Z ′ i :

Then, Z ′ i ∩ dpcp G ( X i , Y ≥ i ) = ∅ , since Z i ∩ dpcp G ( X i , Y ≥ i ) = ∅ and every set in Z ( i -1) is non-descendant of X i . Also,

where the right-hand side independence ( X i ⊥ ⊥ Y ≥ i | Z i ∪ H i ) G X , Y psbd, i holds by the given assumption. This contradicts that no Z ′ i ⊆ V \ F an i satisfies Eqs. (10, 11). The contradiction happens since we assumed that Z an i ( Z an 1 , · · · , Z an i -1 ) does not satisfy Eqs. (10, 11).

Therefore, we conclude that, for any i ∈ { 1, 2, · · · , m } , Z an i ( Z an 1 , · · · , Z an i -1 ) satisfies Eqs. (10, 11) whenever Z i ( Z ( i -1) ) satisfies Eqs. (10, 11). This completes the proof.

A.5 Proof of Theorem 4

For each i ∈ { 1, 2, · · · , m } , Z min i is the minimal d-separator relative to ( X i , Y ≥ i ) in G X , Y psbd, i by (van der Zander and Li´ skiewicz, 2020, Proposition 5.2); i.e., excluding any vertices from Z min i violates the d-separability, and Z min i is a valid d-separator whenever Z an i is a valid d-separator relative to ( X i , Y ≥ i ) in G X , Y psbd, i 4 .

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The main claims are clearly stated in the abstract. The contributions are explicitly stated/visualized in the introduction.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: Every assumptions and settings that theories are built upon is explicitly stated. These may be viewed as the limitations.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: Detailed proofs are provided in the supplemental material. Every assumption is clearly stated.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental Result Reproducibility

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: [NA]

Justification: Our work does not contain experimental results. Examples are served as empirical evidence of our theories, and our paper contains full information to follow examples.

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.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

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

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

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: [NA]

Justification: This paper does not include any data analysis.

Guidelines:

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

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

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: [NA]

Justification: This paper does not include any data analysis.

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: [NA]

Justification: This paper does not include any data analysis.

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.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

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: [NA]

Justification: This paper does not include any data analysis.

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.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: This work conforms the NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts

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

Answer: [NA]

Justification: This paper deals with identification of causal effect through sequential covariate adjustment, which has been a classic causal inference problem. Given the nature of theoretical causal inference work, it is not likely that this work causes notable societal impacts.

Guidelines:

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.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

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: This paper mainly deals with theoretical results that are unlikely misused.

Guidelines:

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.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

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: [Yes]

Justification: We have credited the original owners of the code and mentioned its license.

Guidelines:

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

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: The code for our work includes detailed explanations.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: Our work does not involve human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: Our work does not involve crowdsourcing or research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses the incompleteness of current sequential covariate adjustment criteria in causal inference. It introduces a sound and complete graphical criterion for sequential covariate adjustment, termed Sequential Adjustment Criterion (SAC), and provides an algorithm for identifying a minimal sequential covariate adjustment set. This work demonstrates the limitations of existing criteria, proposes the SAC, and develops a method to construct and optimize adjustment sets for efficient causal effect estimation.\n\nSTRENGTHS:\nThe paper demonstrates the limitations of the multi-outcome Sequential Back-Door (mSBD) criterion by presenting examples where mSBD fails to identify causal effects that can be identified using sequential covariate adjustment.\n\nThe paper presents a new criterion, SAC, which is both sound and complete for sequential covariate adjustment. \n\nDevelops an algorithm to find the minimal sequential covariate adjustment set, reducing unnecessary computations.\n\nWEAKNESSES:\nThe presentation of the manuscript is narrow and difficult to follow. For example, in Eq. 2, the subscript should be used to distinguish the orders of variables, specially, in the left-hand side of the equation. \n\nDefinition 3 seems incorrect since the presentation 'Z is said to be a sequential adjustment set relative to (X,Y) in G if…' is difficult to understand. How to determine a sequential adjustment set from G? \n\nHow to understand ``the causal effect is given as an adjustment''?\n\nThe conclusion in Definition 4 is not readable, please double-check. \n\nWhat type of causal graph satisfies the proposition 3? Can you provide a type to summarize this kind of causal graph?\n\nThe paper primarily provides theoretical examples and proofs to demonstrate the effectiveness of SAC. Further, the causal graph should be given and the lack of extensive validation with real-world data sets might be seen as a limitation.Y\n\nQUESTIONS:\nSee Weaknesses\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates the problem of identifying total causal effects via sequential covariate adjustments, which generalizes the standard, well-studied covariate adjustment. Unlike the standard static case, where there exists sound and complete graphical identification criterion, for the sequential counterpart only sound but incomplete criterion is known. The incompleteness is demonstrated in the submitted work. The main achievement of the paper is the sequential (graphical) adjustment criterion which is sound and complete for sequential covariate adjustment that is much more involved, but more powerful than in the standard static adjustment.\n\nSTRENGTHS:\nThe research of this work is well motivated and concerns an important problem in causality. The task to identify the total causal effects via covariate adjustments in well studied and the authors improve the previous results in this area presenting the first sound and complete graphical criterion for the sequential covariate adjustment which nicely extends the sound and complete criterion for the standard case.\n\nTo prove the main result, the authors extend in an elegant way the constructive adjustment criterion by (van der Zander et al., 2014) based on the adjustment criterion proposed in (Shpitser et al., 2010). A nice technical result is a construction of sequential adjustment set (in Theorem 3).\n\nWEAKNESSES:\nThe presentation of the paper needs improvement. A number of definitions and formulas require clarification and improving of mathematical precision. For example, in Definition 3 the authors consider H_i and in Eq. (2) they use in the formula h_{j-1}. So, what is the definition of h_{-1} and of h_{0}? Note that according to the definition of H_i, for H_{-1}, the sets X^{(-1)} and Y^{(-1)} are undefined, and for H_{0}, the set X^{(0)} is undefined. It seems that the authors mean, in such cases H_i are empty. Also, some sets Y^{(i)} can be empty. These issues should be clarified. \n\nMoreover, it is not clear what is the relationship between the sequential covariate adjustment formula presented in Definition 3 and the corresponding formula in (Jung et al., 2020, Theorem 1) used to identify causal effect by mSBD adjustment. In the submitted paper the authors present in Proposition 2 (mSBD adjustment (Jung et al., 2020)) formula (2) that differs from the original mSBD adjustment formula given in (Jung et al., 2020) in Theorem 1 as Eq. (4). So, to what extent is the incompleteness of mSBD criterion presented in Proposition 3 justified?  For more remarks, see paragraph \"question\" below for details.\n\nThe authors do not provide experimental results verifying the performance of the proposed algorithms.\n\nQUESTIONS:\nIn Eq. (2) x_j and z_j are undefined for j=0.\n\nIn Eq. (2) z_{j+1} is undefined for j=m.\n\nWrite explicitly how do you define sets H_i in Definition 5. In the same way as n Definition 3? \n\nWhy do you consider in Definition 3, 4, etc. in the sequence X = (X_1, . . . ,X_m) the elements X_i as sets? In fact every X_i denotes here a single variable.\n\nEq. (7) is not correct, resp., it does not correspond to the definition given in Eq. (2) in Definition 3 as well as formula used in Eq. (6). Specifically, in the factor P(y2 | x1, x2, z1, z2) in Eq. (7) variable y1 in the conditioning set is missing. Note that in this example we have: Y1 = {Y1}.\n\nThe formula presented in Eq. (2) for sequential covariate adjustment is different than the formula in (Jung et al., 2020, Theorem 1) used to identify causal effect by mSBD adjustment. Could you show that they are equivalent?\n\nThe next issue concerns Theorem 3. The authors define there in Eq. (15) the forbidden sets F_i using in the right-hand  side the sets H_{i-1}. Again, the sets H_{i} should be defined. E.g. if H_i := X^{(i)} ∪ Y^{(i)} ∪ Z^{(i)} -- as assumed in Definition 3 --  then X ∪ Y ∪ H_{i−1} in (15) is equal to X ∪ Y ∪ Z^{(i)}. So why do you use  H_{i−1} in (15)?\n\nIn Eq. (16) you use union of sets Z_{j}^{an} for j=1,..., i-1. For i=1 this summation makes no sense.\n \nIn Definition 7: (Sequential Adjustment Criterion) --> (Sequential Adjustment Criterion (SAC))\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-08"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nTo estimate causal effects given observational data, previous works provide graphical criterion that is not complete in the sequential and multi-outcome cases. The following work extends the complete adjustment criterion to these cases. The non-completeness of previous sequential and multi-outcome criterion is highlighted and then a complete graphical criterion is proposed. The authors then propose an algorithm that finds the minimum adjustment set that satisfies SAC relative to some treatment and outcome in the graph.\n\nSTRENGTHS:\n- The paper is clearly written, the problem being solved is clear as well as the limitations of previous work. The examples also help a lot in this.\n- Despite the low novelty (in weakness) this has not been done before and having access to a complete graphical criterion for sequential adjustment is of practical importance.\n\nWEAKNESSES:\n- The work does seem to be a simple extension of AC to the sequential case. I'm not sure what the main new insight here is, as the concepts required for SAC seem very simple extensions of AC. The adjustment in AC opens all causal paths and blocks all others, following this logic onto the sequential case seems to imply the criterion given in SAC.\n\nQUESTIONS:\nQuestions:\n- For an adjustment set Z where mSBD holds, SAC holds as well (corollary 1) but can SAC provide a smaller adjustment set?\n- In the example of Figure 3, doesn't SAC collapse to be the same as AC?\n\nMinor comments:\n- Definition of proper causal paths is different from previous papers (eg Shipster 2010), would be good to give intuition here. The definition of proper causal path may not be strictly true.\n- References for definitions in Section 3?\n- H_i should be redefined in Definition 5 to make the definition complete, or refer back to where it is defined\n\nLIMITATIONS:\nThis is clear throughout the work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}

comments: SUMMARY:
This paper develops a complete and constructive criterion for sequential covariate adjustment. An algorithm is provided to identify a minimal sequential covariate adjustment set, which is efficient by ensuring that no unnecessary vertices are included.

STRENGTHS:
- The problem considered is important in causal inference. 
- The discussion of the limitations of existing criterion is insightful, often illustrated with example.
- The theoretical results and algorithm appear to be coherent and sound, though I am not familiar with the details of covariate adjustment to verify them and go through the proofs.

WEAKNESSES:
- Simulation studies are not provided, though it is fine for a theory-focused paper.
- The computational complexity of the algorithm is not discussed.

QUESTIONS:
What is the computational complexity of the proposed algorithm?

LIMITATIONS:
I did not manage to find a clear discussion of the limitations.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12