Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference

Authors: yonghan  jung
min  woo park
sanghack  lee
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 11:00:58

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.

Main paper: 
 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 (Pearl, 1995 (Pearl, , 2000 ) ✓ ✗ N/A ✗ AC (Shpitser et al., 2010) ✓ ✗ N/A ✓ SBD (Pearl and Robins, 1995) ✓ ✓ ✗ ✗ mSBD (Jung et al., 2020 )✓ ✓ ✓ ✗ SAC (Ours) ✓ ✓ ✓ ✓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:1. 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.2. We devise sequential adjustment criterion (SAC), a sound and complete graphical criterion for sequential covariate adjustment.3. 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.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. 
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 Vi } Vi∈V such that V i ← f Vi (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 }.Table 2 : Key notations Notation Definition X, X, x, x A random vector, variable, and their realized values.x f (x) A marginalization of f (x) over values in the domain of X. X(i) (X 0 , • • • , X i ) for the ordered set X = (X 0 , • • • , X n ). X ≥i (X i , • • • , X n ) for the ordered set X. De G (C), An G (C) Descendants and ancestors of C in G. pcp G (X, Y) Proper causal path set; i.e., (De G X (X) \ X) ∩ An GX (Y). dpcp G (X, Y) De G (pcp G (X, Y)).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) :EQUATIONThe 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→ 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.Z ′ i ⊊ Z min i , Z ′ i does not satisfy Eqs.To present the minimal sequential covariate adjustment set, we define a useful tool: Definition 9 (Closure (van der Zander and Liśkiewicz, 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:Algorithm 1: minSCA(X, Y, G)Input: A disjoint pair of ordered set (X, Y) and a causal graph G Output: A minimal sequential covariate adjustment set Z minSet H an 0 := Y0.for i = 1, • • • , m do Set Z ani as in Eq. ( 17).Z * i := closure G X,Y psbd,i (Y ≥i ; Xi, Z an i ∪ H an i-1 ) ∩ Z an i Z min i := closure G X,Y psbd,i (Xi; Y ≥i , Z * i ∪ H an i-1 ) ∩ Z * i end return Z min := (Z min 1 , • • • , Z min m )1. The path only contains nodes in An G (A ∪ B); and 2. 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,(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:1. There exists an ordered setZ := (Z 1 , • • • , Z m ) satisfying SAC w.r.t. (X, Y) in G.2. 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 thatZ 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 2 := {Z a , X 1 , X 2 } ∩ {Z a , Z b , Z c } = {Z a }. Therefore, Z min := (Z min 1 , Z min 2 ) = (∅, {Z a }).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. 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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 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 satisfied3 :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 (P (y | do(x)) = z m j=0 P (z j+1 , y j | h j-1 , x j , z j ).(2)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 partitionsX = (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 whereX = (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 ),whereY i := Y \ De G (X) for i = 0, (Y \ Y (i-1) ) ∩ De G (X i ) \ De G (X ≥i+1 ) for i = 1, • • • , m.(3)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).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 i) . Then, Z is said to satisfy the multi-outcome sequential back-door (mSBDH i := X (i) ∪ Y (i) ∪ Z (EQUATIONEQUATIONProposition 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:EQUATION) Za Z b Zc Z d X 1 Y 1 X 2 Y 2 Figure 1: Graph exemplify- ing mSBD in Def. 5For concreteness, consider the causal diagram G in Fig. 1 withX = (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 X1X2 derived from G by removing the edgesX 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) by Z 1 .Finally, in the graph G X2 , 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:P (y | do(x)) = z1,z2 P (y 2 | x 1 , x 2 , y 1 , z 1 , z 2 )P (z 2 , y 1 | x 1 , z 1 )P (z 1 ) = za,z b ,zc,z d P (y 2 | x 1 , x 2 , y 1 , z a , z b , z c , z d )P (z c , z d , y 1 | x 1 , z a , z b )P (z a , z b ). (7)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 toP (y 1 , y 2 | do(x 1 , x 2 )) = za,z b P (y 2 | do(x 2 ), h 1 )P (y 1 | x 1 , z a , z b )P (z a , z b ), with H 1 := X 1 ∪ Y 1 ∪ Z 1 = {X 1 , Y 1 , Z a , Z b }. Next, P (y 2 | do(x 2 ), h 1 ) = zc P (y 2 | do(x 2 ), h 1 )P (z c | h 1 ) = zc P (y 2 | do(x 2 ), h 1 , z c )P (z c | h 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, P (y 2 | do(x 2 ), h 1 , z c ) = z d P (y 2 | do(x 2 ), h 1 , z c , z d )P (z d | do(x 2 ), h 1 , z c ) = z d P (y 2 | x 2 , h 1 , z c , z d )P (z d | h 1 , z c ),where the last equality holds by Rule 2 and 3 of do-calculus, i.e., (Y2 ⊥ ⊥ 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.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 (Robins, , 2000) ) provides a condition for Z to be a sequential adjustment set:EQUATIONwhere 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.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. 
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: Equipped with G X,Y psbd,i , the sequential adjustment criterion is presented as follows:Za Z b Zc Z d X 1 Y 1 X 2 Y 2 (a) SAC Za Z b Zc Z d X 1 Y 1 X 2 Y 2 (b) G X,Y psbd,1 Za Z b Zc Z d X 1 Y 1 X 2 Y 2 (c) G X,Y psbd,2EQUATIONDefinition 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: (Y ≥i ⊥ ⊥ X i | Z i , H i-1 ) G X,Y psbd,i; and(10)EQUATIONThe 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.,EQUATIONFor instance, consider Fig. 2a , where we witnessed that the causal effectP (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,Ypsbd,1 (Fig. 2b ) and G X,Y psbd,2 (Fig. 2c ). In G X,Y psbd,1 , we observe thatZ 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 , whereZ := (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) . InX 1 Y 1 X 2 Y 2 Za Z bG, 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:EQUATIONThm. 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 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 asY 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 Z 1 meets the conditions of Eqs. (EQUATIONEqs. (13, 14) are equivalent since P (y1 | 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.Sequential adjustment criterion in Def. 7 relies on a predefined partitionZ = (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.EQUATIONEQUATIONEQUATIONThen, the following statements are equivalent:1. There exists an ordered set 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.Z := (Z 1 , • • • , Z m ) satisfying the sequential adjustment criterion w.r.t. (X, Y) in G. X 1 Z b Zc Za X 2 Y (a) G X 1 Z b Zc Za X 2 Y (b) G X,Y psbd,1 X 1 Z b Zc Za X 2 Y (c) G X,Y psbd,2To 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,Ypsbd,1 . We can see that Z an 1 satisfies Eqs. (10, 11) in the SAC becauseX 1 and {Y 1 , Y 2 } are d-separated in G X,Ypsbd,1 given Z an 1 , and they are not on the descendant of proper causal paths betweenX 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 }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.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. (10, 11) .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, 1. Z minQuestion: 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. 
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).Supplement to "Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference" In summary, under the claim, for all Z i,j ∈ Z i ,P (y ≥i | do(x ≥i ), z (j-1) i , h i-1 ) = zi,j P (y ≥i | do(x ≥i ), z (j) i , h i-1 )P (z i,j | z (j-1) i , h i-1 ).By applying this equation for all elements Z i,1 , • • • , Z i,mi in Z i , we haveP (y ≥i | do(x ≥i ), h i-1 ) = zi P (y ≥i | do(x ≥i ), z i , h i-1 )P (z i | h i-1 ). (A.1)We also note that the given condition in Eq. ( 10) implies the Rule 2 of the do-calculus (Pearl, 2000) :(Y ≥i ⊥ ⊥ X i | Z i , H i-1 ) G X,Y psbd,i =⇒ (Y ≥i ⊥ ⊥ X i | Z i , H i-1 ) G X i X ≥i+1 (A.2)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:ziP (y ≥i | do(x ≥i ), h i-1 , z i )P (z i | h i-1 ) = zi P (y ≥i | x i , do(x ≥i+1 ), h i-1 , z i )P (z i | h i-1 )by Eq. (A.2)= zi P (y ≥i+1 | do(x ≥i+1 ), h i )P (y i | do(x ≥i+1 ), h i-1 , x i , z i )P (z i | h i-1 ) = zi P (y ≥i+1 | do(x ≥i+1 ), h i )P (y i | h i-1 , x i , z i )P (z i | h i-1 ) by PT X G .Therefore,P (y | do(x)) = P (y ≥1 | do(x ≥1 ), h 0 )P (h 0 ) = z1 P (y ≥2 | do(x ≥2 ), h 1 )P (y 1 | h 1 , x 1 , z 1 )P (z 1 | h 0 )P (h 0 ) = z1,z2 P (y ≥3 | do(x ≥3 ), h 2 )P (y 2 | h 1 , x 2 , z 2 )P (z 2 | h 1 )P (y 1 | z 1 , x 1 , z 1 )P (z 1 | h 0 )P (h 0 ) . . . = z m i=0 P (y i | h i-1 , x i , z i ) m i=j P (z j | h j-1 ) = z m i=0 P (y i | h i-1 , x i , z i )P (z i+1 | h i ) = z m i=0 P (z i+1 , y i | h i-1 , x i , z i ).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.,(Y ≥i ̸ ⊥ ⊥ Z i,j | X ≥i , Z (j-1) i , H i-1 ) G X ≥i and (F Xi ̸ ⊥ ⊥ Z i,j | Z (j-1) i , H i-1 ) G . (A.3)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 Xi and Z i,j in G given Z (j-1) i ∪ H i-1 . That is, the following path exists:(F Xi → X i --Z i,j ) G ∧ (Z i,j --Y ) G X ≥i . (A.4)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 Xi and Z i,j in G is preserved in G X ≥i+1 . To witness, suppose there exists a d-connected path between F Xi 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 Xi 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:Type 1. X i --•→ X k →•← * Z i,j Type 2. X i ←•→ X k →•← * Z i,j Type 3. X i * →•← X k ←•--Z i,j Type 4. X i * →•← X k ←•→ Z i,j Type 5. X i * →•← X k * →•← * Z i,j Type 6. X i * →•← * X k →•← * Z i,jAll 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 Xi 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 :(F Xi → X i --Z i,j --Y ) 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 : Type 1. F Xi → X i --•→ Z i,j --•→ Y Type 2. F Xi → X i --•→ Z i,j ←•→ Y Type 3. F Xi → X i --•→ Z i,j * →•← * Y Type 4. F Xi → X i ←•→ Z i,j --•→ Y Type 5. F Xi → X i ←•→ Z i,j ←•→ Y Type 6. F Xi → X i ←•→ Z i,j * →•← * Y Type 7. F Xi → X i * →•← * Z i,j --•→ Y Type 8. F Xi → X i * →•← * Z i,j ←•→ Y Type 9. F Xi → X i * →•← * Z i,j * →•← * YWe 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 inH 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 formF Xi → 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,Ypsbd,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 graphG = {X i ↔ Y i , X i → Y i } where G X,Ypsbd,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. 
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 Appendix: 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 X i X ≥i+1holds. This means that there exists a d-connecting path betweenX i and Y ∈ Y ≥i given Z i ∪H i-1 in G X,Ypsbd,i while the path is blocked in G XiX ≥i+1 . The differences between G XiX ≥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 -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. 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.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 (iThis concludes the proof that the mSBD criterion implies the SAC.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(Y ≥i ̸ ⊥ ⊥ X i | Z i ∪ H i-1 ) (G X ≥i+1 ) X i ,Y ≥i pbd , for any Z i ⊆ Z \ De G (X ≥i+1 ). (A.5)Then, our goal is to show the following:Statement to be proved: For anyZ i ⊆ Z \ De G (X ≥i+1 ), (Y ≥i ̸ ⊥ ⊥ X i | Z i ∪ H i-1 ) (G X ≥i+1 ) X i ,Y ≥i pbd =⇒ (Y ≥i ̸ ⊥ ⊥ X i | Z ∪ H i-1 ) G X,Y pbd . (A.6) 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:Step 1. For anyZ i ⊆ Z \ De G (X ≥i+1 ), (Y ≥i ̸ ⊥ ⊥ X i | Z i ∪ H i-1 ) (G X ≥i+1 ) X i ,Y ≥i pbd =⇒ (Y ≥i ̸ ⊥ ⊥ X i | Z i ∪ H i-1 ) G X i ,Y ≥i pbd . (A.7)Proof of Step 1. We will prove that any d-connected path betweenX i and Y ∈ Y ≥i given Z i ∪H i-1 in (G X ≥i+1 ) Xi,Y ≥i pbd is preserved in G Xi,Y ≥i pbd . First, the d-connected path between X i and Y in (G X ≥i+1 ) Xi,Y ≥ipbd 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 Xi,Y ≥i pbd does not cut such path, the d-connection of the path remains intact in G Xi,Y ≥i pbd , which completes the proof.Second, we will witness the following:Step 2. For anyZ i ⊆ Z \ De G (X ≥i+1 ), (Y ≥i ̸ ⊥ ⊥ X i | Z i ∪ H i-1 ) G X i ,Y ≥i pbd =⇒ (Y ≥i ̸ ⊥ ⊥ X i | Z i ∪ H i-1 ∪ Z ≥i+1 ) G X i ,Y ≥i pbd . (A.8)Proof of Step 2. The left-hand side of Eq. (A.8) implies that there exists a d-connecting path betweenX i and Y ∈ Y ≥i given Z i ∪ H i-1 in G Xi,Y ≥i pbd . The path cannot be a directed path, since it is a d-connected path in G Xi,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 X i ,Y ≥i pbd holds, but (Y ≥i ⊥ ⊥ X i | Z i ∪ H i-1 ∪ Z ≥i+1 ) G X i ,Y ≥i pbd . (A.9)That is, every d-connecting path between X i and Y givenZ i ∪ H i-1 in G Xi,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-of- freedom 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 subpathZ 2 --•→ X i in G Xi,Y ≥i pbd . Case 2. The vertex Z 2 is an ancestor of H i-1 in G Xi,Y ≥i pbd . Case 3. The vertex Z 2 is an ancestor of Y in G Xi,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.Step 2, what we have shown is the following: 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, supposeZ 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 (10, 11) . However, consider the following choice of Z ′ i :{Z an 1 , • • • , Z an i-1 }, no Z ′ i ⊆ V \ F an i satisfies Eqs.Z ′ i := (Z i ∪ Z (i-1) )\{Z an 1 , • • • , Z an i-1 }. 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, (X i ⊥ ⊥ Y ≥i | Z ′ i ∪ H an i ) G X,Y psbd,i = (X i ⊥ ⊥ Y ≥i | Z i ∪ H i ) G X,Y psbd,iwhere the right-hand side independence(X i ⊥ ⊥ Y ≥i | Z i ∪ H i ) G X,Y psbd,iholds by the given assumption.This contradicts that no Z ′ i ⊆ V \ F an i satisfies Eqs. (10, 11) . The contradiction happens since we assumed thatZ an i (Z an 1 , • • • , Z an i-1) does not satisfy Eqs. (10, 11) . Therefore, we conclude that, for anyi ∈ {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 4For each i ∈ {1, 2, • • • , m}, Z min i is the minimal d-separator relative to (X i , Y ≥i ) in G X,Ypsbd,i by (van der Zander and Liśkiewicz, 2020, Proposition 5.2); i.e., excluding any vertices from Z min i violates the d-separability, andZ 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 .• 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. 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).The multi-outcome column indicates whether the outcome for each criterion is a sequence of vectors or a single vector. Therefore, BD and AC, which deal with static context, are not applicable.If an ordered set associates with an index out of range (e.g., X0, Xm+1), it is interpreted as the empty set. An interval [a, b] (e.g., those associated with or ) with b < a is also an empty set.The code is available at https://github.com/snu-causality-lab/minSAC 


Decision: The reviewers are highly consistent in their evaluation of the paper’s extension of the adjustment criterion to sequential data: This extension is valid and useful, though not particularly unexpected. The authors note that while the results are not surprising, there are also nontrivial challenges when extending the adjustment criteria to sequential data. Thus, the results are worth publishing.

Several reviewers suggest improvements in the presentation, and the authors should revise their presentation accordingly. Specifically, reviewers VqEB and aPdz note several definitions and formulas that could be made clearer and more precise. In some cases, the authors responses are defensive, but they should take the reviewer’s comments as one indication that a careful reader had questions and misunderstandings and thus some revisions are justified.

Reviews: review: paper_summary: paper_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.
strength_weakness: strength_weakness: - Simulation studies are not provided, though it is fine for a theory-focused paper.
- The computational complexity of the algorithm is not discussed.
questions: questions: What is the computational complexity of the proposed algorithm?
limitations: limitations: I did not manage to find a clear discussion of the limitations.
paper_summary: paper_summary: To 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.
strength_weakness: strength_weakness: - 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.
questions: questions: Questions:
- For an adjustment set Z where mSBD holds, SAC holds as well (corollary 1) but can SAC provide a smaller adjustment set?
- In the example of Figure 3, doesn't SAC collapse to be the same as AC?

Minor comments:
- 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.
- References for definitions in Section 3?
- H_i should be redefined in Definition 5 to make the definition complete, or refer back to where it is defined
limitations: limitations: This is clear throughout the work.
paper_summary: paper_summary: The 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.
strength_weakness: strength_weakness: The 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. 

Moreover, 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.

The authors do not provide experimental results verifying the performance of the proposed algorithms.
questions: questions: In Eq. (2) x_j and z_j are undefined for j=0.

In Eq. (2) z_{j+1} is undefined for j=m.

Write explicitly how do you define sets H_i in Definition 5. In the same way as n Definition 3? 

Why 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.

Eq. (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}.

The 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?

The 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)?

In Eq. (16) you use union of sets Z_{j}^{an} for j=1,..., i-1. For i=1 this summation makes no sense.
 
In Definition 7: (Sequential Adjustment Criterion) --> (Sequential Adjustment Criterion (SAC))
limitations: limitations: yes
paper_summary: paper_summary: The 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.
strength_weakness: strength_weakness: The 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. 

Definition 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? 

How to understand ``the causal effect is given as an adjustment''?

The conclusion in Definition 4 is not readable, please double-check. 

What type of causal graph satisfies the proposition 3? Can you provide a type to summarize this kind of causal graph?

The 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
questions: questions: See Weaknesses
limitations: limitations: Yes

Comments: title: Comment
Thank you for the rebuttal.
comment: We sincerely appreciate your encouragement of thoughtful discussion on points that other reviewers might also have been concerned about!
We are pleased that we have addressed all of your concerns. Thank you for taking the time to offer your valuable and detailed feedback.
Thank you again for your detailed comments. The discussion period is nearing its end, with just about a day remaining. Could you please confirm if our rebuttal has adequately addressed your concerns and comments?
We’re glad to have addressed all the concerns from you and other reviewers, `aPdz` and `VqEB`. Thank you for taking the time to review our work.
Thanks to the authors for all the answers that address my concerns.
Thank you for your further clarification. I am happy to increase my rating from 5 to 6.
> It would be better to add some examples for illustrating Definition 4.

Please note that Definition 4 (Partitioning Operator) has already been applied to all subsequent examples.

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> I am not confident about whether we can summary the graph that mSBD criterion fail.

What do you mean by _summary the graph that mSBD criterion fail_ ?  In this response, we presume that the reviewer is concerned with the case where the mSBD criterion fails while the proposed SAC succeeds (i.e., a type of causal graph satisfying Proposition 3).
First, we recall that SAC implies mSBD criterion as follows:
$$
(\mathbf{Y}^{\geq i+1} \perp X_i \mid  \mathbf{Z}_i, \mathbf{H}\_{i-1} ){\mathcal{G}^{\mathbf{X},\mathbf{Y}}\_{\operatorname{psbd},i}}
\implies
(\mathbf{Y}^{\geq i+1} \perp X_i \mid  \mathbf{Z}_i, \mathbf{H}\_{i-1} )\_{{\underline{X_i}}\overline{{\mathbf{X}^{\geq i+1}}}},
$$
since (1) $\mathcal{G}^{\mathbf{X},\mathbf{Y}}\_{\operatorname{psbd},i}$ cuts fewer edges than $\mathcal{G}\_{{\underline{X_i}}\overline{{\mathbf{X}^{\geq i+1}}}}$, and (2) removing more edges from $\mathcal{G}^{\mathbf{X},\mathbf{Y}}\_{\operatorname{psbd},i}$ to obtain $\mathcal{G}\_{{\underline{X_i}}\overline{{\mathbf{X}^{\geq i+1}}}}$ doesn’t decrease any d-separations. This means that it’s impossible for $\mathbf{Z}$ to satisfy SAC while violating mSBD. Equivalently, Whenever mSBD holds, SAC will also hold simultaneously.
This observation allows us to come up with a scenario in which SAC holds but mSBD fails when there exists $\mathbf{Z}_i$ that is a descendant of $\mathbf{X}_i$. We illustrated this scenario in lines 142-154. Specifically, in Fig. 2a, $\mathbf{Z}_2 := \\{Z_c,Z_d\\}$ does not satisfy the mSBD criterion since it’s a descendant of $X_2$, while it does satisfy SAC.
Thank you for the rebuttal. My concerns have been partially addressed.

It would be better to add some examples for illustrating Definition 4.

I am not confident about whether we can summary the graph that mSBD criterion fail.
Thanks for the responses. I do not have further concerns beyond these, and will retain my score until further discussion with the other reviewers.