Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference
Authors: Yonghan Jung
Min Woo Park
Sanghack Lee
Venue: NeurIPS 2024 poster (2024)
Keywords: adjustment criterion
sequential back-door criterion
g-formula
Primary area: causal_inference
TL;DR: We develop a sound and complete criterion for sequential back-door adjustment (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
Yonghan Jung
Min Woo Park2 Sanghack Lee
Purdue University SSouul National University
jung2228purdue edu
(alsdn0110, sanghack) @snu ac. kr
Abstract
Covariate adjustment, also known as back-door adjustment, is a fundamental tool in causal inference. Although 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 back- door adjustment (Jung al., (020). are sound but incomplete; i.c., there are graphical scenarios where the causal effect is expressible via covariate adjustment, yet these criteria do not cover. this paper, we exemplify this incompleteness and then present the sequential adjustment criterion, sound and complete criterion for sequential covariate adjustment. We provide 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 algorithm for identifying minimal sequential covariate adjustment set, which optimizes efficiency by ensuring that no unnecessary vertices are included.
Introduction
Covariate adjustment (also known the back-door adjustment (Pearl 1995 where Z is a set of covariates, X is a treatment, and an outcome -is one 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) sound graphical condition determine if the causal effect is expressible as a covariate adjustment. [Shpitser et al.) (20T0) extended the BD and developed a sound and complete graphical criterion. called adjustment criterion (AC). van der Zander et al. (2014) and Maathhus 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. (0118) addressed by developing a sound and complete adjustment criterion.
Σg P(y x, P())
Beyond such static adjustment criterion, sequential covariate adjustment (also known as the 8-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 single outcome, Jung al. (2020) extended it to accommodate multiple outcome variables called multi-oue multi-outcome sequential dock-door (mSBD) criterion. However, as detailed in Sec. 3 this criterion not complete; i.e., there exists an example where the mSBD criterion is not satished, but the causal effect is identified as sequential covariate adjustment. While it is known that the sequential ignorability condition (Richardson and Robins 2013) is sound and complete for
*Coresponniigg authors
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Table Comparison of graphical criteria for (sequential) covariate adjustments
Criterion
Static
Sequential
Multi-outcome
Completeness
BD Pearl 1995
2000
N/A
AC Shpitser al.
2010
x
N/A
SBD
Pearl and Robins
1995
/
X
mSBD
Jung et al.
2020
/
X
SAC (Ours)
/
the g-formula estimand, it does not provide a graphical criterion that can be tested based on graph compatible with the distribution.
To address this challenge, we devise sound and complete graphical criterion for the sequential covariate adjustment. Table [ visualizes our contribution against previous graphical criteria for covariate adjustments Specifically, our contributions are as follows:
1. We demonstrate the incompleteness of mSBD criterion by exemplifying graph such that the causal effect is given as sequential covariate adjustment where the mSBD criterion is not satisfied. Moreover, discuss the impracticality of other existing criteria.
2. We devise sequential adjustment criterion (SAC), sound and complete graphical criterion for sequential covariate adjustment.
We develop a constructive sequential adjustment criterion that yields a set that satisfies the SAC 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), P). 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 = = [fv: Iviv such that V, + fv, (pa,, u,) where PA, C V [Vi} and U, S U. Each SCM M induces distribution P(V) and a causal graph g = G(M) over V in which there exists arrows encode correlated
a directed edge from every variable in PA, to V3 and latent variables. Performing an intervention fixing X x is represented through the do-operator, do(X X = x), which encodes the operation of replacing the original equations of X (i.e., fx (paz, uz)) by the constant r for all X E X and induces an interventional distribution P(V do(x)))
Graphs. We consider graph G having vertices V and edges E composed of directed (V: - Vj) and bidirected edges (V/2 ++ Vj). An ordered sequence of distinct nodes in G is called a path between V; and V; in G if (1) the start node is Vi and the end node is Vj. and (2) there is an edge (directed or bidirected) between any two subsequent variables in the sequence. A variable V,, is called a collider on the path if the path contains two edges having arrow heads toward V;; i.c., *- V2 <- 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 C V are d-connected w.r.t. Z C V (A U B) in g if there exists a path between Vi E A and V, E B where every non-colliders on the path is not contained in Z and all colliders are in Ang (Z). We say that Z d-separates (A, B) in G if they not d-connected w.r.t. Z in G. We denote G by G with removing those edges toward and from
B. For disjoint sets of vertices AJ. Am C A, we use (Ai,... Am) to represent an ordered set, where the order needs to be highlighted. For the unordered set, we use braces as (A,, Am )-
"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, not applicable.

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Table 2: Key notations
Notation
Definition
X,X,x,a
A random vector, variable, and their realized values.
Ex f(x)
marginalization of (x) over values in the domain of X.
X
(Xa..'. X.,) for the ordered set X (Xo.... ,X,)-
X2
(X, x,.) for the ordered X.
(e An)(C)
Descendants and ancestors
G.
pepg( (X, Y)
Proper causal path set: ie., (Deax (X) X) n Angx
(X Y)
Deg Dpc(pcpy(X) Y)).
Adjustment Criterion. We revisit the problem of characterizing criterion such that the causal effect is given adjustment. Formally, the adjustment set relative to (X. Y) in G is defined as set Z such that, for any P compatible with G, the following is satisfied (Shpitser et al. 2010):
P(y do(x))
Σ P(y x. P(z).
(1)
The adjustment criterion (Shiitser et al. 2010) provides a sound and complete criterion for the causal effect P(y do(x)) to be expressed as covariate adjustment in Eq. I To present the adjustment criterion, we define a proper causal path between X and Y as a directed path from any of X E X any of Y Y on which the path does contain node in X \ {X}. A proper causal path set pcpg (X, Y) is collection of nodes excluding X on all proper causal paths from X E X to Y e Y. defined in Table 2 dpcpg (X, Y) descendant of the proper causal path set defined in Table 2
The adjustment criterion utilizes this subgraph called the proper back-door graph:
Definition (Proper Back Door Graph (van der Zander et al. 2014)). Let (X, Y) denote a disjoint pair in V. The proper back-door graph Gold is a graph obtained from G by removing an edge X + for each X E X and D E pcpg(XX Y).
Then, the adjustment criterion and its characterization given follows.
Definition 2 (Adjustment Criterion (Shpitser et al., 2010 van der Zander et al. 2014 Ler (X, Y) denote disjoint pair in V. A set variables Z CVV. (XUY ) is said satisfy the adjustment
criterion relative to X, Y) in G if(1) (Y 1L X Z) oxx and (2) Z n dpepa X, =)
Proposition (Shpitser al. (2010) van der Zander et al. (0014). Z satisfies the adjustment criterion relative to (X, Y) in G if and only if ZZ is an adjustment set relative to (X. Y) in G.
For concreteness, in Fig. 3 Z
LZ. Z,} satisfies the adjustment criterion relative to X See (van der Zander et al. 2014 Sec. 4.2) for details. In the later
[2) and Y == (Yi. } 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 follows:
Definition (Sequential Covariate Adjustment). Let (X, Y) denote pair of ordered sets such that X (X,.... Xm) and Y (Yo.... mm). Let Z V (X Y) denote vertices ordered as (Z....., Zm). Define H, := x UY UZ The set Z is said to be a sequential adjustment set relative to (X, Y) in G for any P compatible with G, the following is satisfied))
P(y do(x))
II[ P(zj+1,85 hy-1,Xj,2,)) ==0
(2)
The right-hand side of Eq. is called the sequential covariate adjustment of Z w.r.t. (X, Y) in G.
'If an ordered set associates with an index out of range (e.g., Xo. X 1), it is interpreted as the empty set. An interval (o, b] (e.g.. those associated with Σ or U) with < is also empty
3

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In this definition, sequential covariate adjustment relies on specific partitions X = (X,. .m)
and oY..... Y m. To standardize these partitions, we employ a fixed structure based on topological order of (X,Y) in G so to ensure that the partition aligns with the causal structure.
Definition 4 (Partitioning Operator), Let X and denote disjoint vertices where X (X1,.... .mm)
is topologically ordered along with g. Then, the operator PT partitions Y relative to X and 9; i.e, PT (Y) (Yo,... Y. where
Y, (Y) Deg (X)
for 0, for i 1, ,
(3)
((Y Y(-1)) (eg())) \ (
Hereafter, will denote (X, Y) pair of vertices where X (Xi, Xm) is topologically ordered with respect to & and Y as the ordered PT (Y).
3.1 Limitation of mSBD Criterion
The nulti-outcomee sequential back-door (mSBD) criterion (Jung et al. 2020) has been developed to determine whether ordered set Z (Z1.... Zm) constitutes sequential adjustment set:
Definition 5 (Multi-outcome Sequential Back-Door Criterion (Jung et al. 2020 Lei (X, Y) denote a pair of ordered sets in G. Let Z
(X Y) be ordered as (Zi..... Zm). and let
H. = ( U (i) ( (Z))) Then, is said satisfy the multi-ouse outcome sequential back-door (mSBD) criterion w.r.t. (X, Y) in G if. for = l..... m,
Zg is non-descendant of 22 in 9; and
(4)
( 11 X, H; 2.) X
(5)
Proposition (mSBD adjustment (Jung et al., 2020 ) If (Zi,.... Zm) satisfies the mSBD criterion w.r.l. (X in G. the causal effect Ply do(x)) is identifiable and expressible as sequential covariate adjustment of Z w.r.l. (X, Y) G:
m P(y do(x)) = Σ II z 1=0
P( 1,,1,
(6)
For concreteness, consider the causal diagram 9 in Fig. with X X2) and Y (Y,, Y2). Applying the partitioning operator PTT. we divide Y into two subsets: Y; {Yi} and Y. (Y2). Next, we define Z == (Z,, Z2). where Z, == (Za, [b} and Ze == (Ze, Za). This ordered set Z satisfies the mSBD criterion relative to [X, Y) in G. The first condition of the mSBD criterion holds since Zi does not include any descendants of X1 in G, and similarly, Zz excludes any descendants of X2. To check the second condition of the mSBD criterion, consider the graph 9 Xx,X, derived from by
Figure 1: Graph exemplify- ing mSBD in Def. 5
removing the edges X1 + Yı and [Zg + +2,, Z. + X2, X2). In this graph, (Y,, Y2) are conditionally independent of X1 given Z₁ and X², because any path from X2 to (Y1 Y2} is blocked (d-separated) by Z1. Finally, in the graph Gx2+ where the edge X₂ Y22 d-separated when conditioned Z, U Zz. This analysis confirms that Z satisfies the mSBI) criterion, and therefore, the causal effect is given as follows:
is removed, any paths between X2 and
P(y do(x))=
Σ P(yz X1,X2, V1, Z1, 22) P(zz, X1,z))P(2)
Σ
P(yz 01, M/, Zas Tb, Zer Za) Pizc, Zdo 91 Za; 2b) P(za, (Za)2)). (7)
Rd
Despite its soundness (Prop. D. 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 = (Zi..... Zm) does not satisfy the mSBD criterion relative to (X, Y) in G.

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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 9 in Fig. 2a with X (X1, and Y = (Y,, Y2) (i.e., Yo =0) 0) with Y1 = {{1} and Y2 = {Y2) }. Set Z 1 Z2), where Z [Ze Zo} } and Zg == [Ze, Za). Then, while Z₁ satisfies both Eqs. 4 5 of the mSBD criterion, Z2 does not satisfy the mSBD condition in Eq. 4 since Z₂ includes the descendant of X2- 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 Z₁ satisfies the mSBI)
criterion in Eqs. (4 leading to P(y 1/2 do(
Eenits P(y2 do(x2),hi P(y1 F1, Sas 2b) P(za (2 26). with H1 == X1 Y2 UZ = {X1, Y1, Za. Zo}. Next,
P(y2 do( h1) Σ P(y2 do(x2). h1 P((c h1)
P P(ye do(x2), ao) hi);
where the last equality holds by Rule of do-calculus Further,
Pearl 1995:: (Y2 IL Ze X2, Hi)ox
P(y2 do(x2), h1, = Σ P(yz do(x2), h1, tcr za) P(za do(x)),
Σ P(ya x2, hi, zo, P(zd h),
where the last equality holds by Rule 2 and 3 of do-caleulus, i.e., (Y/ IL X2 H1, Zo, Za)exa and (Za X2 H1, Ze)sxx respectively. Combining these derivations, P(y 1/2 do(x x2)) is expressible as the same as in Eq. 77. 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 effeet can be represented as sequential covariate adjustment. For instance, the no-ummeauured- confounding assumption (Rosenbaum and Rubin 1983) is sufficient verification tool. However, as illustrated Fig. 2a the causal effect is expressible through sequential covariate adjustment even though an unmeasured confounder between the treatment X2 and the outcome Y1 presents.
On the other hand, the sequential ignorability criterion (Robins Z to be sequential adjustment set:
1986 2000 provides condition for
Y2 11 X, H,
(8)
where Y²*(x) is counterfactual variable induced from Y2' 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 Zi, Ze as sequential adjustment set in Fig 2a because they fail to accommodate both Yx()) and Zy in a SWIG. This limitation arises since the counterfactual variable (1zz), instead of the observable Zz, will appear the SWIG. In contrast, twin-networks are capable of exploring the relationship between (22) and Zz. Yet, their practicality decreases expands exponentially, leading to substantial
with larger cardinality (m), as the size of computational demands. These observations motivate the necessity of computationally efficient and
effective criterion to determine if causal effect aligns with sequential covariate adjustment.
4 Sequential Adjustment Criterion
In this section, we devise a sequential adjustment criterion (SAC), which provides a sound and com- plete method for characterizing the causal effect P(y do(x)) as a sequential covariate adjustment of Z = (Z,.... Zmm with respect to (X, Y). To begin with, we define a special subgraph called proper sequential back-door graph:
5

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Z
(a) SAC
(b) GXY
(c) psbd,
Figure (a) Graph exemplifying SAC in Def. and (b,c) its proper sequential back graphs
Definition 6 (Proper Sequential Back- Graph), For a disjoint pair (x, Y) and the index i, the X proper sequential back-door graph pabd,s is proper back-door graph induced from 9 9x244 relative (X,, Y2:); i.e..
pabdd. = (Gx24T)) X2
(9)
As constructing the proper back-door graph from G takes linear time O( + [E]) (van der Zander et al. 2014), the sequential back-door graph can also be constructed in linear time O[IV+ [E].
Equipped with 5 the sequential adjustment criterion is presented as follows:
Definition 7 (Sequential Adjustment Criterion (SAC)). Let (X.Y) denote a disjoint pair. Let
Z = (Z,,... 2m) denote topologically ordered set of vertices disjoint to (X, where each Z; non-descendant X Then, said to satisfy sequential adjustment criterion (SAC) w.r.f.
(X, Y) in G if the following conditions are satisfied for = 1,.... m
(Y X₂ Zi, H
11 ox pabd."
and
(10)
Z: n dpcpg (X,, Y 0.
(11)
The sequential adjustment criterion characterizes the sequential covariate adjustment:
Theorem 1 (Soundness and Completeness). Let (X, Y) denote a disjoint pair, and let Z =
(Zz.... Zm) denote an ordered set of vertices disjoint to (X,Y). Then, the following are equivalent:
1. Z satisfies SAC relative to (X, () in G.
2. Z is sequential adjustment set relative to (X, Y) in G; ie.,
P(y do(x)) = Σ II m
P(z b1,1,
(12)
==
For instance, consider Fig. Za where we witnessed that the causal effect P(y1, 3/2 do( x2)) is given as a sequential covariate adjustment of Z = [Z, [Za: 7. Zg (Ze, Za)). ). To confirm
that this Z satisfies SAC relative to (X, Y) for X (X1 X2) and Y (Y,, Y2) with Y, (Yi
X.XYY and Y2 = {Y2), we examine pabd,, (Fig. 2b and XxYY (Fig. 2c). In psbd, we observe that Z, = [Za, ] d-separates X1 and (Yi, Y2f. Moreover, Z, does not contain any descendant of c.YY proper causal paths from X1 to (Yr, Y22 within this subgraph. Similarly, in G ppbed.2- the path between X₂ and Y2 is d-separated conditioned on Z₂ [Ze, Za} and H1 [Za: Zo, X1, Y1). Additionally, Ze is not on any descendant of proper causal paths between X₂ and Yz. Consequently. fulfills SAC relative to (X, Y) in the graph G.
Gx.Y
The soundness and completeness of SAC provide extensive coverage, incorporating existing covariate adjustment criterion such BD, AC, SBD, and mSBD in Table Clearly, SAC covers the mSBD) criterion due to completeness. In other words, the mSBD criterion sufficient for the sequential adjustment criterion.
Corollary (mSBD = SAC), Z satisfies the mSBD criterion relative to (X, Y) in G, then Z satisfies the SAC relative (X, Y) in G.
To demonstrate, consider G Fig. I where Z (Z,, Zz) with Z, == (Za; Zb) and Zg = (2a, 7d) This Z satisfies the mSBD criterion relative to (X, Y) for X X2) and Y (Yi, Y2). To

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ensure that Z also satisfies the SAC relative to (X, Y) in G. we verify that (Z1, Zz) meet the conditions in Eqs. [T) respectively. Since Z, and Ze are both non-descendants of X, and X2. respectively, we only need to verify that they meet the conditions of Eq. (11) for (X1, (Yi, Y2)) and (X2, Y2) respectively. First, consider ppbdd. which removes edges {X1 - Y1, Zb X2, Ze X2) from G. In this graph, X1 is d-separated from (Y1, Y2) given Zz. Next, consider G. pshd, which removes the edge {X2 + Y2) from G. In this graph, X2 is d-separated from Y/2 given (Z1 X1, Y1) and Zg. 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 Here, Z == {Za, Zo} satisfies AC relative to (X, Y) for X = (X, Y) and Y (Y,, Y2). However, Z does not satisfy the mSBD criterion since it descendant set of X == (Xi}. In contrast, AC is sufficient criterion for SAC.
Fig. 3
Theorem 2 (AC = of Z and topological order of X such that Z satisfies the SAC relative to (X, Y) in G.
SAC). Z satisfies the AC relative to (X, Y) in G. then there exists a partition
For concreteness, consider G in Fig. van der Zander et al. G. 7 = Za. } satisfies the adjustment criterion relative to
2014 In X, for
X = (X1, X2) and Y = (Yi, Y2) in G, and the causal effect given as an adjustment as follows:
P(y1, V2 do(x1, = Σ P(y x(z).
(13)
Thm. iimpliss that Z meets the SAC relative to (X, Y) in graph G. To demonstrate this, for X = (X1, X2), we apply the partitioning operator (Y1), and Y2 0. We set Zi = Za. } Since Y, is empty, it is sufficient to demonstrate that Z meets the conditions of Eqs. (10) and (IT). First, Z, meets the condition of Eq. 11 it contains
Figure 3: (van der Zander et al. 2014 Fig. 2)
in Def. 4 and obtain Yo (Y2). Y,
no descendants on the causal paths from X1 to Y1. Consider the modified graph G
ppbdd. obtained
by removing the edges (X, - Y1. Zn + X2). this reduced graph, X, and Y, are d-separated given Z and Yo. confirming the conditions of SAC. Therefore, Z = (Z1, Z2) satisfies SAC relative
to (X, Y) in G. and the causal effect described as
P(y1, U2 dz (2)))))))
= Σ P(y I1, Ras 2b: 1/2 P(zas 2b 122 P(V2).
(14)
Eqs. 113 14h are equivalent since P(y I1, T2, Zas 2h, y2) Plyl 31, Za, 2b, 82) and P(ya
Ii, .22; Roz 2b. P(g2 | Ra; #b) by the conditional independence implied by G.
4.1
Constructive Sequential Adjustment Criterion
Sequential adjustment criterion in Def. 7 relies a predefined partition Z (Z,,... Zmm). ). 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, Define a == (Z Z and corresponding (FTm, FFA) and (H .... Han H alterna-
Y) denote a disjoint pair in G.
tively as follows: For i = 1, 1,...,,. m,
H 1 X D UZZ
(15)
F = XUYUHEU U (Xp, Y Y²) De (X2+11)
(16)
z.m = Augx.
subdd. (XX:) UH.) F.n.
(17)
Then, the following statements equivalent:
1. There exists an ordered set Z = (Zi,'" 2m) satisfying the sequential adjustment criterion w.r.t. (X, Y) in G.
7

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Z satisfies Eqs. 10 since X2 and Y are d-separated in pabdd. given Z {X1]. Therefore, [Za: Zn, Zz)) satisfies SAC relative (X, Y) g. (a) g
(b) 9 X.Y pabd,1
X.Y (c) g pabd,2
Figure Example for minSCA and corresponding proper sequential back-door graphs
2. The ordered set Z satisfies SAC relative (X, Y) in g.
The set Zan be constructed in (m(([V) + El)) since the procedures for constructing dpcpg (X, Y and G 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. consider Fig. Set Yo = 0, Y1 = ]Y and Y2 == {Y2) by the mechanism of the partitioning operator PT GxXYY 1 and xx pehd,2 depicted in Figs. (2b 2cp. Then, = = ZZ, Zb, Zee Za} since these
2a
ancestors of (X1, Y1, Y2) in psbdd, We can see that Z satisfies Eqs. (10 11 in the SAC because X1 and {Y1, }2} are d-separated in G.YYY pabd,1 given - and they are not on the descendant of proper causal paths between X1 and (Yi, Y2 Next, Z = 0. It satisfies Eqs. (10 since X2 and Y2 are d-separated in Spobd.2 given (X1
D U Z Combining with our previous observation that
Z₁ ZZa [ and Z2 {Ze, Za} satisfy Eqs. (10 I1]. this example demonstrates that indeed
7 = IZ Z2n) satisfies the SAC relative to (X. Y in G.
As example that the second statement implies the first one, consider Fig. 4a where Yo
Y1 0 and Y2 == {Y} by the mechanism of the partitioning operator PT Next, G and GX are depicted in Figs. 44b 4c). Then, z
0, since none of [za Zz, Ze} are the ancestors of (X1, YJ. Z satisfies Eqs. (10 11 in the SAC, since the Xi and Y are d-separated in @paba,1 given Zqd. Next, Z = [Za: Zo, Zz) since they are ancestors of X₂ in 9pshe, This
a
am
4.2 Minimal Sequential Adjustment Criterion
We have demonstrated the effectiveness of constructing sequential adjustment sets. However, when dealing with large graphs, a may include a large number of vertices, leading to high computational costs when evaluating sequential covariate adjustments. This situation highlights the need for more parsimonious adjustment set. To address this, we introduce the minimal sequential covariate adjustment set, which is the smallest subset of a without sacrificing the validity of the adjustment.
Definition 8 (Minimal Sequential Covariate Adjustment Set). Let (X, Y) denote disjoint pair in G. An ordered set of vertices min == (Zmin 2mmnn where each Zmnn are non-descendant of 2 is said to be a minimal sequential covariate adjustment set if. for each i 1,...,,0 m,
1. Zmmin satisfies Eqs. 11.1
171.
2. For any Z S Zmmin Z does not satisfy Eqs.
10
11
To present the minimal sequential covariate adjustment set. we define useful tool:
Definition 9 (Closure (van der Zander and I Liskiewicz, 2020)). For disjoint vertices A, B, and C in 9, the closure of A with respect to B and C in G, denoted as closureg A; B, C), is the union of A and the collection of V E Ang BB that are connected A via path satisfying the following:
8

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Algorithm 1: minSCA(X, Y,9
Input: disjoint pair of ordered set (x, Y) and causal graph G
Output: A minimal sequential covariate adjustment set mii
Set H Yo.
1.... do Set z as in Eq.
Z: = closure X.Y pubd.
X1
2, - Z
Z = closure XxYY
(X;;
2 Z
U H
niz -1 n
pubdd
return Z (Z
Zala
1. The path only contains nodes in An U); 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, closureg( A; B, C) is set of vertices in Ang Ang(A) BB) that is d-connected to A in conditioned on C, where the paths are through Ang B). Finding the closure can efficiently
done in ((V EE) time (van der Zander and Liskiewicz 2020), where [V] and EE are the number
of vertices and edges of the graph y. For example, consider closure (X2; (Za, Zo: Zo X1))
pobdd2
with G pabd,2 in Fig. 4c First, the ancestor of (X2, Y are [Ze: Zn, X1). Since Zo and X₃ adjacent to X2, they contained in the closure. However, {Z,, Ze) not in the closure, since all the non-collider vertices on the path between X₂ and {7,, Ze} are contained {Zu. 7,, Ze, x1). (X2; Y, (Za, Zo, Ze: X1)).
Therefore, X1,
closure ppubd,
Equipped with the closure, we propose procedure to construct the minimal sequential covariate adjustment in Algo. [ The minimal sequential covariate adjustment set, Zmin is derived directly from the ordered set Znn, which characterizes the sequential covariate adjustment. Therefore, Z naturally retains this characterizing property.
Theorem (Construction of Minimal Sequential Covariate Adjustment Set). Let (X, Y) denote disjoint pair in G. Then, the following statements are equivalent:
1. There exists an ordered Z (Zi,... Zmm) satisfying SAC w.r.t. (X, Y) in G.
2. The ordered Zuin minSCA(X X, Y, 9) from Algo. is minimal sequential covariate adjustment set.
Constructing Zin MinSCA(X, Y) can be achieved in (m VI + EE)), the construction of each 0 in Eq. 17 and the required closures each take O(VV + El) and are repeated times.
We demonstrate Thm. 4 with causal diagram in Fig. 4a
We have shown that Zan = ( Zgn) with Z = and Z = Za, Zn. Ze} satisfie SAC relative to (X, in G. Therefore,
only demonstrate Algo. for Zan = {Za, Zo Ze}. Run line Zž :=
closure pobda X2, Zay Zo, Ze, X1)) n (Za, Zo, Zoj. First, {Za, Ze} is included in the closure
since they are adjacent to Y. Next, Z, is included the closure, since the path Y + Z, < Zb does not include non-collider vertex. Therefore, Z5 Zan (2., Zo, Zo}. Next, run line 5. We already witnessed that (Za, X2) = closure @xY. (X2; (Za; Zo, Zos X1)), which leads to Zmni = Za, X2) n (Za, Zo, Za} = [.a). Therefore, Z = (Zmin Zmin) = (0, {2a)).
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

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proposed criterion includes and extends the mSBD and standard adjustment criterion, confirming its comprehensive nature since they imply the sequential adjustment criterion [ and Thm. (2) Further, we develop a procedure for constructing the sequential adjustment set that captures the sequential covariate adjustment (Thm. B). Finally, we devise an algorithm identify the minimal sequential covariate adjustment set (Algo. and Thm.
Acknowledgments and Disclosure of Funding
We thank anonymous reviewers for constructive comments to improve the manuscript. This work was partly supported by the ITTP IRS-2022-1122295332099 NRF (RS-2023-0021 000211904/0 1904/20%, RS-2023- 00222663/20%) and MFDS (23112MFDS202220%) grant funded by the Korean government.
References
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Perkovic, E., Textor, J., Kalisch, M., Maathuis, M. H., et al. (2018). Complete graphical characteriza- tion and construction of adjustment in markov equivalence classes of ancestral graphs. Journal of Machine Learning Research, 18(220):166..
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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. 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). the validity of covariate adjustment for estimating causal effects. In Proceedings of the Twenty-Sixth Conference Uncertainty in Artificial Intelligence, pages 527-536. AUAI Press.
van der Zander, B. and Liskiewicz, M. (2020). Finding minimal d-separators in linear time and applications. In Uncertainty in Artificial Intelligence, pages 637-647. PMLR.
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10

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Supplement to "Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference"
Contents
1
Introduction
2
Preliminaries
3
Limitations of Existing Criterion for Sequential Covariate Adjustment
Limitation of mSBD Criterion
3.2
Limitation of Other Existing Criteria
Sequential Adjustment Criterion
Constructive Sequential Adjustment Criterion
4.2
Minimal Sequential Adjustment Criterion
5
Conclusion
Proofs
12
A. Proof of Theorem
12
A.2
Proof of Corollary
16
A.3 Proof of Theorem
16
Proof of Theorem
18
A.5
Proof of Theorem
19

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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 A-B. B. Furthermore, we symbolize the type of paths, that are not mutually exclusive, as follows:
1. A - B means that there is directed path from A to B.
2. A f-o+ B means that there is a non-directed path with no colliders and all non-colliders on the path are not conditioned.
3. A *- B means that there is non-directed path containing at least one collider, and either the collider one of its descendants is conditioned.
A.1 Proof of Theorem
In order to prove both the soundness and completeness of SAC, we structure the proof in two parts. First, will demonstrate the soundness of SAC through Lemma S.T. followed by proving its completeness in Lemma S.2
Lemma S.1 (Soundness of SAC). Let (X, Y) denote disjoint pair of ordered sets where Y is partitioned with PTg Let Z (Z),.. ....... Zmm) denote an ordered set of vertices disjoint to (X, Y) in G. Z satisfies the SAC relative (X, Y) in G, then Z is sequential adjustment set relative to (X. Y) in G.
Proof of Lemma S.I We first assume that the following claim holds, and then demonstrate that the claim valid. For any fixed Z; (Z,,.... Zm), suppose the following claim is true.
Claim. If satisfies SAC relative to (X, Y) in G, then there exists an ordered set Z; (Z,,1 mm) (where m; [Z,]) such that, for each Z,,j in the ordered set, and with
z = (2,,1, Z1z-1),, at least one of the following two cases holds:
Case 1. (Y IL Zij X
(1
H. H.-1)o x2X
Case (Fx; 1L Zr. 2 H,-1)e
where Fx; a new parent of X,, called the regime, augmented in the graph G.
For the fixed partition of Z; that makes the claim true, suppose Case holds for all Z Z, Then,
Pl do(x () 2 (j-1) 1,-1)
Σ P(y |d((22)) 20 (1-1) (h-1)) I 22 (-1) h--1) 2
Aid
Σ P(y do(x2)) 2 20) h 1 2 .1-1)
by Case 1.
Now, suppose Case 2 holds for all Zig. Then,
2l do(x 21) ) zz h1-1) (j-1)
Σ P(yzi |do(x2)), 2 2 h hi-1) do((2))) 2, (1-1) h.-1)
Ai
Σ P(y2i do(x2)) ) (0 h P(zs) do(xi)) 2, (-11) h.-1)
by the assumption Z;
Ai
Σ P(y do(x2)) ) (00 ( 12 (2(--1) .h-1)
by Case 2.
Ad
12

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In summary, under the claim, for all Zi, E Z,.
P(y²* do(x z (i-1) h;-1)= = Σ P(y do(x=)), 2 (1-1)) |2 121-111 h(-1).
By applying this equation for all elements " Zim Z, we have
P(y do h1-1) Σ P(y do(x2)) Z; h, 11- Pz) h.-1).
(A.1)
2.
We also note that the given condition in Eq. (10) implies the Rule 2 of the do-calculus (Pearl
2000p:
(
X; Z; H.
(Y X; Zr H xx22441
(A.2)
pubd.a
since the d-separation is always preserved when cutting the edge from the graph. The right-hand side of Eq. (A. can be written further as follows:
Σ P(y 2' do(x2 21), 2,)
2:) P(z: h;-1)
21
Σ P(y Ty do(x2 b; P(z; h:-1)
by Eq. A.2
Σ P(y2itz |o(xx 21122 Pyy: do(x2it1), hi Tig Zi) Pi) h
Σ Ply2itl |do(2 P(y: h;-1, P(zz h;-1)
by PT
Therefore,
P(y do(x))
P(y21 [o(xx2)), ho) P(ho)
P(y22 do(x hn) 21, P(bo)
Σ P ), h1, 22, 22, P(ze P1) 21, 21) ho) P(ho)
22
Ply, h, F4 2;) II
P(z 1
1=0
1=
Σ
Plys b;,,, 2:) ha)
1=0
ΣI 1=
P(z.r,y b;,1,
In conclusion, the sequential adjustment criterion 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 satisfies the SAC relative to (X, Y) in 9 but for all orders of elements Z, (denoting an order as (Z.)), there always exists some Zr, E Z, with 2 Z (..j-1) (which is predecessors of Zi.; under the order π (Z.)) such that Zi, does not satisfy either Case or Case 2, i...
2 L Lin 2 z, - H H1-1)g and (Fx, 1 Zn, 1 HH
(A.3)
13

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This means that there exists a d-connecting path between Z and Y E Y in g xze given X2 U
z U - and a d-connecting path between Fx, and Zrj in 9 given z 1 U H That is, the following path exists:
(F - Zisle A (Zr, Y0zEr
(A.4)
We further state the following. In this setting, there exists a d-connecting path between Zz,j and Y in IxIniI, which is evident since adding edge (from 9xar to (xa.1t) does not block any d-connecting paths. Moreover, the d-connecting path between Fx, and Zi.j in G is preserved in Xx2+-T. To witness, suppose there exists d-connected path between Fx, and Zn,j in G, while the path is blocked/cut in Gx=tEr This implies that the path contains edge + X5 for some Xx E X2 By the topological order and the assumption that Z, non-descendant of X the directed path from X to X, and the directed path from Xx to Zi are both impossible. Thus, we only consider non-directed path from Fx, to Zi.j- First, X5 on the path connecting X; and Z cannot be collider, as the path would be d-separated if Xg were not conditioned. Additionally, Xx cannot be an ancestor of z(--1) U H due to the topological order of X. the mechanism of the partitioning operator PTş and the assumption that Z, is non-descendant of X 22+11 Therefore, we consider the path between X; and Z through Xx, where neither X nor its descendant forms a collider. The possible types of paths in G are as follows:
Type 1. X; - Xx + Zi,j
Type 2. X; a Xxx , Zzj
Type 3. X; * X 60- Zio
Type 4. X; x> X k 7,.
Type 5. X, * X * Ztd
Type 6. X; *>0*** X - Zie
All cases imply the existence of a directed path from X₁ to any vertex in zi an ancestor of ~ (1) UH This contradicts our setting, just as in the case where Xx a collider. It confirms that the d-connecting path between Fx, and Zz,5 in g is preserved in 9x As result, Eq.
UH making X
A.4 means the presence of the following path in GxaFT:
(Fx, > X; 7 Y) Y)s X224T
Then, there are nine types of d-connected path between X, and Y in 9 Exeil given z U UH
Type 1, Fx, + X; - Z,, -
Type Fx. + X; - Zij x Y
Type 3. Fx, + X; - Zij +1o+* Y
Type 4. Fx, + X₁ +o+ Lij - Y
Type 5. Fx, X; +o- Zij 60) Y
Type 6. Fx, + X; +o Zij **i+ Y
Type Fx, + X; * Zij -
Type 8. Fx X; * Zij t Y
Type 9. Fx, + X; * Zij *tot* Y
We will show that each type of path belongs to one of the following three cases:
Case 1. There always exists another order n' for Z; such that the path is blocked, even if the path is d connected under the oreder æ.
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.
14

---

The paths in Type are directed from X, to Y, which contradicts the second condition of SAC in Eq. (II..
Consider Type 2. Since Z is a collider, and ( ()-1) U H is conditioned on, it forms a d-separated path between X; and Y. which is preserved in Gpsib. psbddii' First, consider the where this path is not blocked by further conditioning Z; Z 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 Z z In other words, there exists another order x' for Z, such that this path is blocked whenever { (Zr,g) U.Z ( U H;-1 is conditioned on. Therefore, without loss of generality, we can ignore Type 2 by considering any other order x' in which the colliding path is blocked.
Consider Type 3. Since the case where Zi,j forms collider is considered in Type 2, we only need to consider the case where Zz,5 - lies on the path. In this case, Zz,j has a directed path to any vertex in H;-1 U - However, it is impossible for the vertex to be involved in H; this would contradict the valid topological order of SAC. Therefore, we can conclude that there exists a directed path from X; to Z Z through Zi.. Then, the following two cases are possible: First, there exists directed path from Y to Z. This case contradicts the condition of SAC in Eq. (II), Z would be a descendant of Y. Second, there exists a divergent path between Y and Z, meaning that Z forms collider on the path between X; and Y. Note that we can ignore Type for the same reason discussed in the case of Type 2.
The path for Type 4-6 be open when X; is ancestor of the conditioning set Z H;-1- Due the valid topologiaal order of SAC, we only consider the case where the conditioning set is z (1-1)
This implies the existence of some path of the form Fx, + X; + Z (E z ((1-)) that does not belong to Type 4-6, allowing to reduce this types to another type.
We now consider the path for Type 7-9. As we witnessed that X, cannot a collider opening the path, it should contain X, >> which implies that X, is an ancestor of some collider and its descendant
some vertices in z ( U H, Hence, the collider must be a vertex in z z(1-1), as - 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 Zrjj Z. satisfies Case or Case 2, meaning the claim is Irue.
Lemma S.2 (Completeness SAC). Let (X, Y) denote a disjoint pair of ordered sets where Y partitioned with PTş. Let Z == (Z,,-.., Zmm denote an ordered set of vertices disjoint to (X, Y). There exists graph such that Z is not sequential adjustment set relative to (X, Y) whenever Z does not satisfy SAC relative to (X, Y) in G.
Proof of Lemma S.2. If does not satisfy SAC in G, then there exists Z, Z not satisfying Eq. 100 or Eq. in SAC.
First, suppose Z, does not satisfy Eq. TO Then, there must be path between X; and Y Y2 such that all non-colliders on the path are not in Z, U UHH1 and all colliders on the path are in
(Z, U H Since the path is defined in g Y ppbbd, and by the mechanism of the partitioning
Angxx pubb..
operator PTX. the path cannot be directed path.
Now, suppose the path is divergent path that does not contain any colliders; i.e., X; < Y. In this case, consider simple graph 9 {X, +> Y, X, + Yj} where gx contains subgraph (X, Y.). Then, P(y do(x)) cannot be given identifiable. Suppose. instead, the path is colliding path; i.e., X * Y where the collider is an ancestor of the conditioned vertices Z, HHHH Then, we can consider a simple graph 9 that contains subgraph {X, + Z, X₁ + Y,,X, ++ Z, ++ Yi). Here again, P(y do(x)) cannot be given sequential adjustment since it is not even identifiable.
sequential adjustment since it is not even
15

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Next, suppose Z, does not satisfy Eq. TTI. Then, we may consider graph a simple graph Ç that contains a subgraph X; - Z, + Y. Then, there exists a SCM M such that P(y do(x)) is not given an adjustment over Z, (Shpitser al. (010))
This concludes the proof that the sequential adjustment criterion is complete.
A.2 Proof of Corollary
We will prove that if there exists Z, E Z that does not satisfy Eq. (10) or Eq. does not satisfy Eq. Eq. 5 in mSBD criterion.
11 SAC, then Z;
For the sake of contradiction, suppose Z, does not satisfy Eq.
10 but (X, IL Y
1 Z,)g X:X2FF1 holds. This means that there exists d-connecting path between X2 and Y Y2
given - in pobad, while the path is blocked 9 X, x,X2++T The differences between X, x,x
and psid.: imply that the d-connecting path should contain X; -. Since the path cannot be di- rected path, it must be a colliding path where the collider an ancestor of any vertices in Z, U H This means that there exists a directed path either X; - Z for some Z Z; or X; - H for some H E H; If X, - Z, such Z; fails satisfy the mSBD criterion because Z, contains descendant of X, If X; - H, it implies that there exists a directed path from X; to X₁ E ( Ys E ( or 2x 7 (1-1, In any case, contradicts the topologiaal order X, the mechanism of partition operator PT or the assumption that Z is non-descendant of X 21 in G.
Now, suppose Z, does not satisfy Eq. (TT). Then, such Z; does not satisfy Eq. 4 since failure to satisfy Eq. (TT) means that Z; is descendant of X; in G.
This concludes the proof that the mSBD criterion implies the SAC.
A.3 Proof of Theorem
We will prove that the SAC fails for any partition of Z across all topological ordering X, then also fails to satisfy the adjustment criterion. The failure of SAC Z, can be stated as
1 X, Z, U UH HH-1)
x for any Z, S Z Dog (X
(A.5)
Then, goal is to show the following:
Statement to be proved: For any Z; C Z (X2(1))
(Y 1 X; Z SUH HH-1)
X v.
Y A X; Z H
(A.6)
Note that the right-hand side of Eq. A.6 implies (Y A X Z) "pdd
meaning that Z does not satisfy
the adjustment criterion relative to X, Y) in G. To this end, we first witness the following:
Step For any Z, C Z Deg (X 2(t1),
(Y 11 X₂ Z Z.HH,1)
X.Y
XXIIT) pbu
(Y
2Í H X; Z H.HH))) x21.
(A.7)
pbd
Proof of Step I. We will prove that any d-connected path between X, and Y E 2 given Z-HH; Y2 is preserved in G.X. Y First, the d-connected path between X, and Y in Y2 must be non-directed path in G, since the path cannot contain incoming edges to any vertices in 22++1, otherwise, would cut. As the operation applicd to G. does not cut such path, the d-connection of the path remains intact in pbd X2 which completes the proof. D
in (G xx2111) pbd
(Gx phd
16

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Second, we will witness the following:
Step For any Z. C Deg X 21t1),
(Y A X; Zi H 22
pbd
(Y 2: A X;
Z 4UH,-1
1
(A.8)
X2
pbd
Proof of Step 2. The left-hand side of Eq.
A.8
implies that there exists a d-connecting path between The path cannot be a directed path, since it is a
X; and Y E Y given Z; J H in g G...
"bd
d-connected path in G.b.a Y and by the mechanism of PT. Therefore, the path is either divergent path X; + y, or a colliding path X; **o+* Y where the collider is an ancestor of some vertices Z; H (indeed, the collider an ancestor Z, only, since cannot be an ancestor H;-1 because it would create directed path from X, any vertex in H;-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 2: M Xi Z; H
1) x2 holds, but phd
IL
Z AAUHHIUZZH
v2i. pbd
(A.9)
That is, every d-connecting path between X, and given Z, H in G.X., Y blocked by additionally conditioning on Z
Suppose the above path a divergent path X; +o+ Y. Then can consider two cases; the path
contains subpath Ze - X; or Zn - Y where Z2 Z Since we have full degree-of- freedom of choosing Z = (Z; i 1,.... m) where each Z, is non-descendant of X2++1 in G. only reason that the vertex Zg would in Zy Z and not in Z; is that it is descendant of some Xę x Therefore, Ze -o- X, means that there is directed path Xp - X,. However, contradicts the valid topological order of X, since Xp E X X On the one hand, suppose the above path be a divergent path X; +o- Y that contains a subpath Z₂ - Y. This means that there exists directed path from Xp Y Y This case also contradicts SAC condition in Eq. (II) because Ze lies on the proper causal path. Therefore, the path cannot be divergent path.
Suppose the path is colliding path X; *>><<* Y where the collider an ancestor of some vertices in H,-1. In order for the path to be blocked by conditioning Z there must be Z2 E Z on the path that falls into one of the following cases:
There exists subpath Zg _ X; in G X; r2: vé:
Case 1.
pbd
The vertex Z2 is an ancestor of H;-1 in G Nx Y2
Case 2.
phd
Case 3. The vertex Zz is an ancestor of Y in G X
The existence of Z2 even with the full freedom of choosing any partition of Z means that there exists a directed path from Xp to Zz as discussed. Therefore, Case 1 implies existence of any directed path from Xp to X,, which contradicts the valid topological order of X.
Case 2 is impossible because implics the existence of directed path from Zg to any vertex in H (again, directed path from X, the vertex in H,-1), which would results in an invalid topological order.
Finally, Case 3 also impossible implies directed path from X, to Y, indicating that Zg 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:
17

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Step Step 2: For any Z, S Deg X (2
(Y 2 A X Z, H
x. pdd
x2
Y x P X; Z UH U 22+11
(A.10)
Y2 pbd
We will now show that Eq. (A.6) holds. Let the left-hand side of Eq. (A.6) holds; i.e., there exists
d-connected path between X; and Y E Y given Z; U - in (9 (XxerII X2, We have
established that such d-connected path is preserved in G.X. Y conditioning on Z, U uz as shown by Eq. (A.10). If this path does not contain any vertices in X (X:). then the path not affected by the operation applied to G which completes the proof. Otherwise, even though the path contains vertices in X \\ {X,,, if the subpath between Xa X \\ac {X+} and any vertex in Y X
not proper causal path to Y 2 then the path is not affected by the operation applied Spd which completes the proof.
Suppose the path contains a subpath that a proper causal path from Xa E X (X;) to Y. Then such X, should be included X by the mechanism of PTş. Then, as the path between X, and X; cannot be directed in either side, the path between Xa and X; is either divergent path Xa <-- X; or colliding path X, *>0<<+ X;. Furthermore, since there is no incoming edge to Xa, the path should be Xa -- X, Additionally, the path d-connected when conditioning on Z, L H This implies that there exists directed path from X, e X to any vertex in Z, U H which contradicts our partioning mechanism PTT Therefore, there must be proper causal path from X, Y.
By proving that Eq. A.6 have completed the proof that the AC implies the SAC.
A.4 Proof of Theorem
Lemma A.1 (fvan der Zander al. 2014 Lemma 3.4)). Let X, Y, I, R. be sets of vertices with
I C R, R n [X Y = 0. If there exists d-separator Zo with I C Zo R in 9 then = Ang (XUXUUY R is a d-separator.
Lemma A.2 (Construction). Let (X,Y) denote a disjoint pair of ordered sets. For any i E (1,2, ,m), suppose each Zj E (Zz,... ZZ-1) is non-descendant of X2i+1 and disjoint to (X, Y). Lei Z be an ordered set in Eq. (17) defined with respect to (+) which is denoted as
Z (Z (I)) Then, the following statements are equivalent.
1. Z, (1-11) is non-descendant of X 2++1 satisfying Eqs.
10
11
2. There exists set Z. S VVXXU UH, H DDg X
2441
satisfying Eqs. 10
11
3. There exists set Z. S V
Dog X
x2++1)) satissyyng the following:
P(y²i | do(x2)), ha-1)= Σ 2(y do(x2it!), h:) P(z: h; h::1) wiz Zi, h;
Proof of Lemma 4.2
By Theorem I (2) + (3) holds. Additionally, (1) (3) by Lemma S.I
which is equivalent to (1) = (2). Therefore, we only need to prove (2) (1).
We apply Lemma A. by replacing G in Lemma A.I as pabd, in Lemma as (X.). Y Lemma
X.YY
as 2 In Lemma H Zo in Lemma an union of H and certain Z, satisfying Eqs. 10
11 and R as an union H and V (XUYUHL
U d Y²1) UDg (x) Then, Lemma A. states that, under the statement (2), there exists a d-separator between X; and Y2', which
Z H = (Angxy ([X,)U UXX,UY221 Y HHH,)) F.) U H This completes the proof.
We emphasize that Lemma A.2 stated for a fixed ( 1-11),, ie.. Z in Lemma A.2 dependent on some specific choice of (=1) This dependency is highlighted by the notation Z (Z)).
18

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Then, the remaining part of the proof is to show that, for any given Z if Z, satisfies Eqs. 10
11.. then Z,n (Zi" , _ satisfies Eqs. (10 To show through contradiction, suppose
7 p 2, does not satisfy Eqs. 110 even if Z; satisfies them given Z
By
Lemma A.2 this implies that, given {Z]".
no Z C CVV F," satisfies Eqs. 10
However, consider the following choice of Z
-
Then, z dpopg (X,, Y since Z: n dpopg X1, Y2 0 and every set in ( is non-descendant of X,. Also,
(X, IL Z
(X; 11 Y
Z H
X
where the right- side independence (X; Y
Z HHH
holds by the given assumption.
This contradicts that no Z V \ F satisfies Eqs. (10) assumed that ne (Z
11
The contradiction happens since we
_ ) does not satisfy Eqs.
10
11
Therefore, we conclude that, for any E (1,2, m). Z,
(Zi",...
2, satisfies Eqs.
10
whenever Z, (Z))l)) satisfies Eqs. 10 11b This completes the proof.
A.5 Proof of Theorem 4
For each E (1,2, ... .m). Z is the minimal d-separator relative to (X,, Y21) in 9 X.Y
ppbdd.i by
(van der Zander and Liskiewicz 2020 Proposition 5.2); i.c., excluding any vertices from Zmmin violates the d-separability, and Z, is valid d-separator whenever Z is a valid d-separator relative to (X,, 22 in psbd.
"The code is available at att.::////ttttoo.ooooccccuaaatad
19

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Reviews:
[{"review_id": "D6vs740CfC", "reviewer": "Reviewer_UrnJ", "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.", "questions": "What is the computational complexity of the proposed algorithm?", "limitations": "I did not manage to find a clear discussion of the limitations.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The problem considered is important in causal inference. \n- The discussion of the limitations of existing criterion is insightful, often illustrated with example.\n- 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.\n- The computational complexity of the algorithm is not discussed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for sharing your thoughts and feedback with us.\n\n> Simulation studies are not provided, though it is fine for a theory-focused paper.\n\nAs you noted, our work is theory-oriented, presenting a complete criterion for sequential covariate adjustment and introducing an algorithm for constructing a (minimal) sequential adjustment set. Our focus has been on clearly elucidating the theories and algorithms through appropriate examples. We believe that simulation studies or experimental validations would not enhance our theoretical work. However, to demonstrate the practical benefits of our theories, we will provide implementation code during the revision process.\n\n---\n\n> The computational complexity \n\n1. The proper sequential back-door graph (in Def. 6) can be constructed in $O(|\\mathbf{V}|+|\\mathbf{E}|)$ time [1].\n\n2. Provided the partition of $\\mathbf{Z}= (\\mathbf{Z}_1,\\cdots,\\mathbf{Z}_m)$, checking if such $\\mathbf{Z}$  satisfies the SAC in Def. 7 takes $O(m|\\mathbf{V}| + m|\\mathbf{E}|)$, since checking each d-separation takes $O(|\\mathbf{V}| + |\\mathbf{E}|)$. \n\n3. Constructing $\\mathbf{Z}^{an}_i$ in Theorem 3 takes $O(|\\mathbf{V}| + |\\mathbf{E}|)$ since finding the set dpcp, descendent and ancestor sets take $O(|\\mathbf{V}| + |\\mathbf{E}|)$. \n\n4. As mentioned in line 286, finding closure can be done in $O(|\\mathbf{V}| + |\\mathbf{E}|)$. \n\n5. Combining all of these analyses, the computational complexity of the proposed method is $O(m|\\mathbf{V}| + m|\\mathbf{E}|)$.\n\n---\n\n<Reference>\n\n[1] 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Q7EBuwCXks", "reviewer": "Reviewer_sfA3", "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.", "questions": "Questions:\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", "limitations": "This is clear throughout the work.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- 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.", "weaknesses": "- 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and valuable feedback! \n\n---\n\n> The work does seem to be a simple extension of AC to the sequential case. \n\nOur proposed method is an extension of the AC, similar to how the sequential back-door adjustment extends the back-door adjustment. The principle of our method is to open the proper causal paths and block the proper non-causal paths. However, we disagree that the extension procedure is simple. \n\nAs the saying goes “The devil is in the details”, there are nontrivial challenges when extending the AC to the SAC: \n\n1. Unlike the adjustment case, partitioning the variables is essential. Choosing a proper partition that preserves causal interpretation (e.g., $\\mathbf{X}_i$ causes $\\mathbf{Y}^{\\geq i}$) is nontrivial. \n\n2. There is a significant gap between conjecturing and actually witnessing the soundness and completeness of SAC. Specifically, the formal approach for witnessing the soundness and completeness of AC proposed by [1] is hardly generalizable to SAC cases. The original proof of AC by [1] uses the counterfactual network [2] graph to show the soundness and completeness of AC. Such a proof technique is hardly generalizable for sequential cases, since, by the nature of the counterfactual network, the number of treatment variables to be considered in the proof increases exponentially with the number of treatments. To circumvent such challenges, we devised a new proof technique that can be commonly applied to both AC and SAC.\n\n3. One naive approach is to condition the previous set $\\mathbf{H}_{i-1}$, and apply the AC with respect to $(X_i, \\mathbf{Y}^{\\geq i})$ in the graph where the previous set is conditioned on. However, the graph induced by conditioning the previous set may not be a DAG (with bidirected edges), even if the original causal graph is a DAG. This happens when the previous set contains colliders [3].\n\nCircumventing such challenges while adhering to the principle—to open the proper causal paths and block the proper non-causal paths—is a nontrivial problem requiring formal treatments, as we have done in this paper.\n\n--- \n\n> For an adjustment set Z where mSBD holds, SAC holds as well (corollary 1) but can SAC provide a smaller adjustment set?\n\nJust to clarify, please note that mSBD and SAC determine whether a causal effect can be expressed as Sequential Covariate Adjustment (SCA) when $\\mathbf{Z}$ is given, rather than constructing $\\mathbf{Z}$ itself. Algorithm 1 (minSCA) is designed to provide a smaller (or equal) adjustment set such that no strict subset of it is a valid adjustment set.\n\nFor example, consider Figure 4a. Here, $\\mathbf{Z} = (\\emptyset, \\{Z_a, Z_b, Z_c\\})$ satisfies the mSBD criterion (and SAC) relative to $((X_1, X_2), (\\emptyset, Y))$. However, as discussed in lines 301-308, Algorithm 1 (minSCA) yields $\\mathbf{Z}^{min} = (\\emptyset, \\{Z_a\\})$ and this set also satisfies the SAC. This example illustrates that the minSCA procedure can yield a smaller adjustment set.\n\n--- \n\n> In the example of Figure 3, doesn't SAC collapse to be the same as AC?\n\nIn the example, the set $\\mathbf{Z} = \\\\{ Z_a,Z_b \\\\}$ satisfies AC, and $\\mathbf{Z} := (\\\\{Z_a,Z_b\\\\},\\\\{\\\\})$ satisfies SAC. In other words, the set $\\mathbf{Z} = \\\\{ Z_a,Z_b \\\\}$ is a valid admissible \\& sequentially admissible set. However, this observation doesn't hold in general. For example, $\\mathbf{Z} := (\\\\{\\\\},\\\\{\\\\})$ also satisfies the SAC in the same graph. However, the AC is not satisfied, because $\\mathbf{Z}$ fails to block the path $Y_2 \\to Z_b \\to X_2$ in the proper back-door graph $\\mathcal{G}_{\\text{pbd}}^{\\mathbf{X},\\mathbf{Y}}$.  \n\n---\n\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\nOur definition is comparable with the definitions in the previous papers as follow. First, a causal path (equivalently, a _directed path_) is said to be proper if it does not intersect $\\mathbf{X}$ except at the end point [1]. Equivalently, the path is _proper_ if only its start node is in $\\mathbf{X}$ [4]. These definitions match with our definition in line 78, which states that the path does not contain a node in $\\mathbf{X} \\setminus \\\\{ X \\\\}$. In other words, the path starting from $X$ is not overlapped with $\\mathbf{X}$ if and only if the path doesn't contain a node in $\\mathbf{X} \\setminus \\\\{ X \\\\}$. Therefore, our definition is equivalent to the definitions in the previous papers. \n\n---\n\n> References for definitions in Section 3?\n\nWe presume that you are referring to Definition 3 (SCA). We will refer to it just before Definition 3, which we already referenced sequential covariate adjustment in lines 28-29 of the Introduction.\n\n--- \n\n> H_i should be redefined in Definition 5 to make the definition complete, or refer back to where it is defined\n\nThank you for the suggestion. We first note that $\\mathbf{H}_{i-1}$ has been defined in Def. 3. We will invoke this definition in Def. 5.\n\n---\n\n<Reference>\n\n[1] Shpitser, I., VanderWeele, T., and Robins, J. M. (2010). On the validity of covariate adjustment for estimating causal effects. \n\n[2] Shpitser, I. and Pearl, J. (2008). Complete identification methods for the causal hierarchy. *Journal of Machine Learning Research*, 9:1941–1979.\n\n[3] Richardson, T., & Spirtes, P. (2002). Ancestral graph Markov models. *The Annals of Statistics*, 30(4), 962-1030.\n\n[4] 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "biOjScqRK6", "reviewer": "Reviewer_VqEB", "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.", "questions": "In 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))", "limitations": "yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The 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).", "weaknesses": "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. \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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the valuable feedback and detailed comments. \n\n---\n\n> 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\nThank you for your feedback. We implicitly defined a set whose indices are outside their original range as an empty set. For example, \n\n1. In Eq. (2), all the subscripted values whose indices are out of range (e.g., $\\mathbf{x}\\_0$, $\\mathbf{z}\\_0$ and $\\mathbf{z}\\_{m+1}$ are considered as an empty set. \n\n\n2. In Eq. (16), it implies that we go over index j such that $1 \\leq j \\leq i-1$. Hence, if $i=1$ implies that there is no union, i.e., an empty set will be returned for the term. \n\nWe will explicitly state this in the revised version of the paper. \n\n---\n\n> 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?\n\nHere is the proof of the equivalence, as derived in line 393 in the Appendix: \n\n$\\underbrace{P_\\mathbf{x}(\\mathbf{y}) = \\sum_\\mathbf{z} \\prod\\_{j=0}^n P(\\mathbf{y}_j \\mid \\mathbf{x}^{(j)},\\mathbf{z}^{(j)},\\mathbf{y}^{(j-1)})\\prod\\_{k=1}^n P(\\mathbf{z}_k \\mid \\mathbf{x}^{(k-1)},\\mathbf{z}^{(k-1)},\\mathbf{y}^{(k-1)})}\\_{ \\text{Jung et al.,2020, Thm1}}$\n\n$\\qquad \\quad = \\sum\\_\\mathbf{z} \\prod\\_{j=0}^n P(\\mathbf{y}_j \\mid \\mathbf{x}_j,\\mathbf{z}_j,\\mathbf{h}\\_{j-1})\\prod\\_{k=1}^n P(\\mathbf{z}_k \\mid \\mathbf{h}\\_{k-1})$\n\n$\\qquad \\quad =\\sum_\\mathbf{z} \\prod\\_{j=0}^n P(\\mathbf{y}_j \\mid \\mathbf{x}_j,\\mathbf{z}_j,\\mathbf{h}\\_{j-1})P(\\mathbf{z}\\_{j+1} \\mid \\mathbf{h}_j)$\n\n$\\qquad \\quad =\\underbrace{\\sum\\_\\mathbf{z} \\prod\\_{j=0}^n P(\\mathbf{z}\\_{j+1},\\mathbf{y}_j \\mid \\mathbf{h}\\_{j-1},\\mathbf{x}_j,\\mathbf{z}_j,) = P(\\mathbf{y} \\mid do(\\mathbf{x}))}\\_{\\text{Ours Eq. (2)}}$\n\n---\n\n> Write explicitly how do you define sets H_i in Definition 5. In the same way as n Definition 3? \n\nYes. We will invoke the definition of $\\mathbf{H}_i$ in Def. 3 to Def. 5.\n\n\n\n---\n\n> 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.\n\nThis is because a set serves as a basic unit for graphical operators like \\(Pa\\), \\(An\\), and so on. Furthermore, by treating \\(X_i\\) as a set, we can maintain consistency in handling \\(X_i\\), \\(\\mathbf{Y}_i\\), and \\(\\mathbf{Z}_i\\) simultaneously.\n\n---\n\n> 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}.\n\nThank you for catching these typos. We will make the necessary corrections.\n\n---\n\n>  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\nWe use $\\mathbf{H}\\_{i-1}$ in (15) to main comprehensibility. Specifically, Equation (15) is read as follow -- the forbidden set is an union of (1) treatments, (2) outcomes, (3) predecessors ($\\mathbf{H}\\_{i-1}$), (4) dpcp, and (5) a descendent set. \n\n\n---\n\n> In Definition 7: (Sequential Adjustment Criterion) --> (Sequential Adjustment Criterion (SAC))\n\nWe will revise as suggested. Thanks.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QRLoNYA6Eu", "reviewer": "Reviewer_aPdz", "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.", "questions": "See Weaknesses", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The 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.", "weaknesses": "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. \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", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the valuable feedback.\n\n---\n\n> 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 \nIn Eq. (2) included in Definition 3, $\\mathbf{X}$ and $\\mathbf{Y}$ are already defined as $(\\mathbf{X}_1,\\cdots ,\\mathbf{X}_m)$ and $(\\mathbf{Y}_0,\\cdots ,\\mathbf{Y}_m)$, respectively.\n\n---\n\n> __(1)__ 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. __(2)__ How to determine a sequential adjustment set from G?\n\n \n1. That is a standard and established way to express it. Similar expressions can be found in existing works [1, 2].\n2. Section 4.1 and Section 4.2 address methods for constructing $\\mathbf{Z}$, focusing on how to determine the sequential adjustment set. \n\n---\n\n> How to understand “the causal effect is given as an adjustment”? \n\nIt means that the causal effect (left-hand side of Eq. (13)) is expressed as covariate adjustment (right-hand side of Eq. (13)). For further explanation of the adjustment criterion, please refer to lines 73-75 of our paper or the existing works referenced in lines 17-27.\n\n---\n\n> The conclusion in Definition 4 is not readable \n\nIt would be greatly appreciated if you could specify which part you find \"not readable,\" so that we can better address your comment.\n\n---\n\n> What type of causal graph satisfies the proposition 3? Can you provide a type to summarize this kind of causal graph?\n> \n\nIn lines 142-154, we illustrate such an example immediately following Proposition 3, where mSBD is not satisfied, but the causal effect can still be expressed by SCA.\n\n---\n\n> 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\nOur paper falls into the causal effect identification category in which the key task is to express the causal effect as a function of observational distribution using assumptions encoded in a causal graph. Therefore, we disagree that the presence of causal graphs is a limitation of our work. \n\nAs you noted, our work is theory-oriented, presenting a complete criterion for sequential covariate adjustment and introducing an algorithm for constructing a (minimal) sequential adjustment set. Our focus has been on clearly elucidating the theories and algorithms through appropriate examples. We believe that simulation studies or experimental validations would not enhance our theoretical work. However, to demonstrate the practical benefits of our theories, we will provide implementation code during the revision process.\n\n\n---\n\n<Reference>\n\n[1] Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4):669–710.\n\n[2] Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press, New York. 2nd edition, 2009.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The 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.
Decision: Accept (poster)