Unified Covariate Adjustment for Causal Inference
Authors: Yonghan Jung
Jin Tian
Elias Bareinboim
Venue: NeurIPS 2024 poster (2024)
Keywords: causal effect estimation
causal effect identification
debiased machine learning
semiparametric inference
front-door adjustment
Primary area: causal_inference
TL;DR: We develop an estimation framework that can estimate broad causal effect identification efficiently, achieving doubly robustness.

Abstract: Causal effect identification and estimation are two crucial tasks in causal inference. Although causal effect identification has been theoretically resolved, many existing estimators only address a subset of scenarios, known as the sequential back-door adjustment (SBD) (Pearl and Robins, 1995) or g-formula (Robins, 1986). Recent efforts for developing general-purpose estimators with broader coverage, incorporating the front-door adjustment (FD) (Pearl, 2000) and more, lack scalability due to the high computational cost of summing over high-dimensional variables. In this paper, we introduce a novel approach that achieves broad coverage of causal estimands beyond the SBD, incorporating various sum-product functionals like the FD, while maintaining scalability -- estimated in polynomial time relative to the number of variables and samples. Specifically, we present the class of UCA for which a scalable and doubly robust estimator is developed. 
In particular, we illustrate the expressiveness of UCA for a wide spectrum of causal estimands (e.g., SBD, FD, and more) in causal inference. We then develop an estimator that exhibits computational efficiency and doubly robustness. The scalability and robustness of the proposed framework are verified through simulations.

Unified Covariate Adjustment for Causal Inference
Yonghan Jung', .Jin Tian?, and Elias Bareinboim
Purdue University
Mohamed bin Zayed University of Artificial Intelligence
CColumbia University
'jung222ppuruue.. edu, jjin. tian@mbzuai. ac. ae, eeb0cs cowumbia. edu
Abstract
Causal effect identification and estimation are fundamental tasks found throughout the data sciences. Although causal effect identification has been solved in theory, many existing estimators only address a subset of scenarios, known as the sequential back-door adjustment (SBD) (Pearl and Robins] 1995a) or g- g-formula (Robins) 1986). Recent efforts for developing general-purposs estimators with broader coverage, incorporating the front-door adjustment (FD) (Pearl) 2000) and others, are not scalable due to the high computational cost of summing over high- dimensional set of variables. In this paper, we introduce a novel approach that achieves broad coverage of causal estimands beyond the SBD, incorporating various sum-prod sum-product functionals like the FD, while achieving scalability - estimated polynomial time relative to the number of variables and samples in the problem. Specifically, we present the class of unified covariate adjustment (UCA) for which illustrate develop scalable and doubly robust estimator. In particular, the expressiveness UCA for a wide spectrum of causal estimands (e.g.. SBD, FD, and others) in causal inference. We then develop an estimator that exhibits computational efficiency and double robustness. Experiments corroborate the scalability and robustness of the proposed framework.
Introduction
Causal inference a crucial aspect of scientific research, with broad applications ranging from social sciences to economics, and from biology to medicine. Two significant tasks in causal inference are causal effect identification and estimation. Causal effect identification concerns determining the conditions under which the causal effect can be inferred from a combination of available data distributions and causal graph depicting the data bata-gennr process. Causal effect estimation, on the other hand, develops an estimator for the identified causal effect expression using finite samples.
Causal effect identification theories have been well-established across various scenarios. These include cases where the input distribution is purely observational (Tian and Pearl 2003 Shpitser and Pearl 2006 Huang and Valtorta 2006) (known as obxervational identification obslD combi- nation of observational and interventional (Bareinboim and Pearl 2012aa Lee et al. 2019) (referred to
as generalized identification gID); scenarios where the target query input distributions originate from different populations (Bareinboim and Pearl 2012b Bareinboim et al. [0144] Bareinboim and
Pearl | Correa et al.) 2018 Lee et al.) al.]]| (2200) (known recoverabiliry or transportaiility; or cases where the target query is counterfactual (Rung 3) (Correa et al.) [2021] (referred to as Ctf-ID) beyond interventional (Rung 2) the Ladder of Causation (Pearl and Mackenzie, 2018, [Bareinboim]
et al., 2020). In these situations, algorithmic solutions have been devised that take input distributions along with specified target queries and formulate identification functionals as arithmetic operations (sums/nntegration, products, ratios) conditional distributions induced from input distributions.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Despite all the progress, existing estimators cover only subset of all identification scenarios. Specifically, well-established estimators for the back-door (BD) adjustment (Pear 1995), represented
as B|y| x. P2). and sequential back-door adjustment (SBD) (Robins] 1986, Pearl and Robins
1995bp and off-policy evaluations (OPE) (Murphy (2003), which is an SBD with policy interventions.
are known for their robustness to the bias (Bang and Robins 2005 Robins et al. 2009 van der Laan
and Gruber 2012, Murphy, 2003, Rotnitzky et al. 2017; Luedtke et al. 2017 Uehara et al. [2022] Díaz et al. 2023). These estimators are also scalable; i... evaluable in polynomial time relative to the number of covariates (IZ)) and capable in the presence of mixed discrete and continuous covariates.
However, SBDs only address fraction of the broader spectrum of identification scenarios.
Beyond SBD, recent efforts have expanded developing estimators for the front-door (FD) ad- justment Σ Z_.e' B[y] 22 (xx') (Pearl 1995). At first glance, this adjustment appears similar to SBD, as both involve the sum-product of conditional probabilities. However, FD involves treatments variables in dual roles- one being summed (x' in En E x', ((xx)) and the other being fixed (x in P(z r²)). While FD estimators achieving doubly robustness have been developed (Fulcher et al. 2019 Guo et al.] 2023). they lack scalability due to the necessity of summing over the values of Z (i.e., Σe,, thereby limiting its practicality when Z is figh-dimensionall or continuous.
Similar challenges arise in more general identification scenarios beyond SBD and FD. Recent efforts have focused on developing estimators for broad causal estimands, such as Tian's 's adjustment TTian and Pearl 2002a)), which incorporates FD) and other cases where causal effects are represented as
sum-product functionals (Bhattacharya et al. 2022). These efforts also include work on covering any identification functional (Jung et al. 202 Xia et al.) 2021 2022. Bhattacharya et al., [022 Jung et al. 2 While these estimators are designed to achieve a wide coverage of functionals, they lack scalability due to the necessity of summing over high- nigh-dimensional variables.
Thus far, we have assessed the pair (functional class, estimator) based two criteria: (1) cov- erage of the functional class, and (2) scalability of the corresponding estimators. Scalable esti- mators achieving doubly robustness have been established predominantly for BD/SBD classes. While recent studies have developed estimators with a strong emphasis on coverage (e.g., any identification functional), less attention has been given to achieving scalability.
Coverage
Scalability
Function class
Prior
UCA
Prior
UCA
BD/SBD/OPE
V
/
/
FD
V
/
x
/
Tian's
V
/
x
/
obsID/gID
V
x
/
Ctf-ID
? /
In this paper, we establish a novel pair of a functional class and its corresponding estima- tion frameworks designed to ensure scalability while covering broad spectrum of identifica- tion functionals. Our work aims to maximize coverage, enabling the effective development of scalable estimators with the doubly robust prop- erty. This functional class, termed unified covariate adjustment (UCA), integrates a sum-product of conditional distributions appearing in many causal inference scenarios such as BD/FD, Tian's s adjustment, S S-admissibillity in treatment-one cottmeenttootthee-ee cnn-hhe-treated (ETT) (Heckman) 1992), and nested counterfactuals (Correa et al. 2021)
Transportability
? /
Table 1: Scope. / denotes the addressed area (by UCA or prior works). X denotes the unaddressed area. A denotes the partially addressed area. ? indicates areas where no known results are present.
(Bareinboim and Pearl 2016). of
The coverage of the proposed class is further demonstrated through the application to a novel estimand for the counterfactual directed effect (Ctf-DF) derived from fairness analysis (Plečko and Bareinboim, 2024). For the proposed UCA class, we develop scalable and doubly robust estimator evaluable computationally efficiently relative to the number of samples. Table visualizes the scope of our framework. The contributions of this paper are as follows:
1. We introduce unified covariate adjustment (UCA), a comprehensive framework that encompasses a broad class of sum-product causal estimands. This framework's 's expressiveness is demonstrated across various scenarios beyond SBD, including Tian's 's adjustment that incorporates FD and others as well novel counterfactual scenarios in fairnsss analysis.
2. We develop corresponding estimator that is computationally efficient and doubly robust and provide its finite sample guarantee. We demonstrate scalability and robustness to bias both theoretically and empirically through simulations.
2

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X1
X
Z
R X
B
Y
X
W
(a)
(b)
(c)
(d)
(e)
Figure 1: (a) Front-door in Example (b) Verma in Example 2 (c) Napkin, (d) Standard faimess model in Example and (e) Example graph from (Jung al.) 2021a Fig. lb)
Notations. We use (X, X, 2) to denote a random vector, variable, and their realized values, respectively. For a function f(z;) for i = 1,2,... we use E, f(z:) = f(zi) + ((Z2).... Also, for function f(z), we use Σx f(z) to denote the oummation/integratio of discrete/continuous random variables Z. For example, we write the back-door adjustment as when Z is mixture of discrete/continuous variables. Given ordered set X = {X,,--., ,nn we denote X
over mixture
Eg Ep|Y x. (z))
{Xi,... Xi} and X = {X++1.'... Xm for m X|. For a discrete X, we use (X) as function such that xx(X = 1 if X = x; (x) 0 otherwise. P(V) denotes a distribution over V and P(v) a probability
V v, We use Ep[f(V)] and VP[f(V)] to denote the mean and variance of f(V) relative to P(V). We use L.2norm of with P. If function f is a consistent estimator of having a rate Tn. we will use f We will use f - (1( if - is bounded in probability. Also, f f is said to be bounded in probability at rate r, if f f Or(rn)- [n] {1,... ,n. is collection of index. D = {V (1)
||f[|r v (E)((((V)
or(rn). We will say ff is Le-consistent if
fll. = o1()).
i [n]} denotes sample set, where V (i) denote the ith sample in D. The empirical average of f(V) with samples D is ED[[(V) = (1/1D)) Env.,ED (V)
Structural causal models. We use structural causal models (SCMs) (Pearl 2000; Bareinboim eet al., 2020) as our framework. An SCM M is a quadruple M = (U. P(U). ) where U set of exogenous (latent) variables following a joint distribution P(U). and V is set of endogenous (observable) variables whose values are determined by functions F = {fv hvvvv such that Vú + Sv. (pa,, u;) where PA; C V and U; U. Each SCM M induces distribution P(V) and a causal graph G = (M)) over V in which directed edges exists from every variable in PA; to V; and biddree-bdd arrows encode common latent variables. Performing 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.. (pa,. Uz) )) by the constant x for all X X and induces an interventional distribution P(V do(x)). For any Y C V, the potential response Y (u) is defined as the solution of Y in the submodel Mx given U u, which induces a counterfactual variable Yx-
Related work. Our work is an extension of existing sequential back-door adjustment (SBD) estima- Bang and Robins 2005 Robins et al. (2009] van der Laan and
tors (Mises) 1947| Bickel et al. 1993
Gruber Romitzzy et al. 2017 Luedtke et 2017; Diaz et al. 2023) to a broader class of sum-product functionals, such the front-door adjustment (FD) and Tian's adjustment (Tian and Pearl 2002a) which generalizes FD and more, and nested counterfactuals, which will be detailed in later sections. Our work is aligned with recent works of Chemoohhukov et al. (2022); Li and Luedtke (2023); Quintas-Martinez et al. (2024), which examined SBD derived from various joint distributions. Specifically, [i and Luedtke (2023) considered the SBD setting where conditional distributions are induced from different sources. In contrast, we study a broader class of sum-product functionals from multiple populations. Also, Quintas-Marttnnez Martinez et al. (024) considered the Markovian model II PV(V. PA,) where each Pt can be distinct. In contrast, we study a broader class of estimands that are not confined conditioning on PA, On the other hands, (Chernozhukov considered the case where covariate distributions are allowed to be changed, and demonstrated that FD can be captured through this technique. Our work expands on these findings by covering broader class, such as the Tian's adjustment and nested counterfactual in fairness literature, and by providing a more formal theory that includes finite sample guarantees and asymptotic analysis.
al.) (022)
Unifted Covariate Adjustment
A class of causal estimands termed unified covariate adjustment (UCA) is defined as follows:
3

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Definition 1 (Unified Covariate Adjustment (UCA)), Let V (PP denote the following probability
measure over an ordered set V := [C,,RI,.. Cm, Rm: Y == 1m+1): y P == pm (Y
Sm) [== ===1 (C S [R; [S, RR), where
P (V) E [m [ is a set of distributions in the form of PV()) = QV(V - = s where Q' is a distribution, 1 is (potentially empty) set. Each pairs Po(V) and pó (V) can be the same (V) ()())) or distinct (P²(V) ))++ + p (V)).
For i E [m + 11. S == (CC--)RR S
Each R, is controlled by a pre-specified / known probability measure dR. == OR. (r: ss \ri)
where Ex, oR. (r: 8i \ r.) and S oi, I almost surely (e.g. dr.. = Lr. (R:)).
Then, the expectation of Y over T oP is called Unified Covariate Adjustment (UCA):
m
40 = Evip: M = Σ Epm+i (Y| sm
II Pi(c, s0 (ra 8t \ rs)..
(I)
cur
We will exemplify that UCA encompasses many well-known causal estimands, including the se- quential back-door adjustment (SBD) (Robins 1986; Pearl and Robins 1995b), front-door adjust- ment (Pearl 1995), Tian's ' adjustment (Tin and Pearl (002a)). S-admissibility in transportabil- ityrrecoverability (Bareinboim and Pearl 2016), effecc-of-trratmeent-- -rretteentoon------ee settimeentto--th-tteeeee -hetheetreeted (ETT) (Heckman)
1992). nested counterfactuals (Correa et al. [002]) treatment- treammmet interaction (Jung et al. 2023b).
and off-policy evaluation (Murphy] 2003). This section particularly focuses recently developed and lesser-known estimands, for which scalable estimators have been rarely explored. Appendix B provides additional examples, demonstrating how UCA can represent well-known estimands such as off-policy evaluation and S-admissibility.
At first glance. UCA closely resembles the sequential back-door adjustment (SBD) (Robins 1986
Pearl and Robins 1995b). Indeed, UCA is reduced to SBD in the special case where P [(V)
for all i= 1.... m and oR. == Ar, (R.); i.c., 1/0 = Σ. Ep(Y c(m) U (m))) I1mh P(c. c(i-1) U ((-)))) However, UCA provides flexibility to represent target estimands beyond SBD
by allowing p to be any distribution that aligns with the target estimand, permitting arbitrary
conditional distributions beyond the observational distribution P. To demonstrate, consider the
front-door adjustment (FD) scenario (Pearl 1995) depicted in Fig. la
E|y | do(x)) = Σ IELY I (,,2,, P( P(Gx)'))
(2)
c,z,a'
Even though FD cannot be expressed using SBD because the treatment variable X is being fixed (in P(= c)) and summed (with Σn) simultaneously, it can be represented through UCA as follows:
Example (FD as UCA). FD can be written as the expectation Y over P(Y Z, X, C(Z
x, P(X, C). We set C₁ {X, C}, C2 {2), R 0, P (C1) = P(X, C), (² S,
P(Z x, C) with S = (X), S₁ (C), and Pa S3) = P(Y Z,X,C) with S C..
Next, consider Verma's 's equation (Verma and Pearl 1990 Tian and Pearl 2002b) with Fig. lb
B|y do(x))= = Σ E|Y b,, bb,,,)](() a, a,,'')
x))
b,a.x'
where X is fixed to x in LE| to FD, due to the dual role of X, the existing SBD framework is not suitable to express Verma's equation, which can be represented through UCA as follows:
x. a, b] and P(a | xx) while summed in P(b | a, x') and P(x'). Similar
Example 2 (Verma as UCA). Verma's 's equation is expressible as the expectation Y over P(Y B, A. P(B A, (AA x) P(X). We set C [X]. C2 (A). C3 (B), and R = 0. We
map Pl (C,) = P(X). (2, S1) = P(A 2) with S = 0, S = [X). p3 (C, S₂) = P(B
A, X) with S2 = (A, X). and Pt(Y S²)= = P(Y B, x) with S3 = (B, A}, S = (X).
In both examples, the variable S = X is bifurcated, being fixed in some conditional distributions (c.g., P(x Iz c) in the front-door criterion (FD)) and summed over Σ x' in others (c.g..PPy 2, 2, 2,'4,4 .c)

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Algorithm 1: Tian-to-UCA(G V = (Vi,... Vk, Y))
Input: graph G and set of topologically ordered variables V == (Vi.... VK, V). Set C1 = (V1,-.. Vk-1, 1 V3 = X, Vi+1, Ve+t, ) as an ordered sequence, where (V1, Vk-1) are predecessors of X, and (V++ (V++1,' ... Ve+i,) are successors of X within Sx.
Set P'(C)) P(C1), = and R V. while V \ ([Y) C * 0 do
if 1 _ Sx set C, as the next sequence of vertices V (DXJU that are not in Sx:
S.-1 C((-1) { and PC, S.-1) P(C. 2)) with S {X).
else, set C; the sequence of vertices in V \ ((Y) u ((-1) that are in Sx: S C((-1);
and (C, 1 = P(C, (S-1) with 8 = i i
end
Sct m + i. I E Sx. set Sm = C(m), Sh = 0, and (Y Sm) = PPY Sm). Otherwise, set Sm = ((m) \ {X}, Shh [X}, and pm+ (Y Sm) = P(Y Sm; x).
return B YY] where & [P] = pm+1 (Y (.)) II P'(C, S,-1).
in FD). Both FD and Verma's equations special cases of Tian's 's adjustment (Tian and Pearl 2002aa, which states that E[Y do(x)) is identifiable under certain conditions. Specifically, when and its children ch ((X) in the graph G are not connected by bidirected edges, it can be expressed as:
K
E[Y do(x)] = ΣΣ E DY
v (K)
P / ((i-1)), =
viay
where V == (V1, ..... VK, Y) is a topologically ordered set with V₁ == X for some k being the treatment variable X, ("(u; v(i-1)) = P(u; | v(k-1), x, Uk+1,-.. v(-1) (i.e., X is fixed to x)) if V Sx where Sx is the set of vertices connected with X through bidirected edges, and P'(x; ((-)) P(u; v(+-1), ,,', 0k+1, 1111) (i.e., X is summed with Ex;) if V. Sx. In Tian's adjustment, X is bifurcated into summed through Σx, and fixed to x. We exhibit the expressiveness of UCA for Tian's 's adjustment:
Proposition 1. Tian's 's adjustment in Eq. is UCA-expressibble through Algo.
Next, exhibit the coverage of the UCA for counterfactual quantity in the fairness literature. Specifically, we focus on the counterfactual directed effect (Ctf-DE) in the Standard fairness model (SFM) (Plečko and Bareinboim, 2024), illustrated in Fig. ld This model includes several key components: the protected (discrete) attribute (X), such as race; the baseline covariates (2), like age; the mediator variables (W) affected by X, for example, educational level; and the outcome variable (Y), such as salary. Consider a scenario where we investigate the the query, "What would be the expected salary for someone who is Black, but hypothetically of Asian race and had been educated as White person typically would be?". . The query is represented as Ctf-DE: E[Yx Wx-21 X = x2). where rp. 31, and 22 correspond to the races Asian, White, and Black, respectively. This query can be identified through the algorithm in (Correa et al. 2021) under the SFM in Fig. [d]
E|yx E[Yx=ma -xX X
x2]
Σ E| X
To, w, 2)P(W X = P({z X = x2). (5)
This identification functional is UCA excreexxressible:
Example (Ctf-DE as UCA), The Cif-DE is expressible through the expectation Y over P(Y
X = 2o, W, P(W X,Z)P(Z X = x2) Ml, (X). Sei R₁ := [X}, oh, == LI, (X), (C, P(z X z2) with C₁ { and Sh = [X}. p2 (C2 S₁) P(W X, with C2 (W}
and S [), (Y S²) P(Y X = Fu, W. Z) with S (W, 2) and S (X).
Despite the broad expressiveness of UCA, as illustrated in this section and ppendix B not all causal estimand functionals are UCA-expressible. To witness, consider the 'napkin' estimand described
in (Pearl and Mackenzie 2018 Jung et al. 2021a) with G in Fig. Ic] defined as P(y do(x)) =
E. P(aln,w) P(w) Here, the functional for B|Y do(x)) is represented not the expectation of a product of conditional distributions, but rather as quotient of sums of conditional distributions. The napkin estimand is not UCA- UCA-expressible. Intuitively, if target functional is expressed as
Σ. (()))))
5

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an expectation of probability measure that is represented as a product of multiple conditional distributions, it can be captured through UCA. A formal criterion the following:
Theorem 1 (Expressiveness) Suppose functional vn is expressed as the mean of the following
measure, pm++1( S')) I Pi(C, i (R, S R.), where S - (C U (()) \ S
for each i = 1....,,0 m and (V(V) j = 1,..., m + 1 are distributions of the form PV(V)- = Q' (V S = 1)-1). Then, the functional do can be expressed through UCA in Eq.
3 Scalable Estimator for Unified Covariate Adjustment
So far, we discussed the coverage of UCA. In this section, we construct a scalable estimator for UCA that achieves doubly robustness property and provides its finite sample guarantee. We define the estimator with two sets of nuisance parameters μ and π. is a collection of regression parameters, and π is a collection of ratio parameters.
We introduce sets to define regression nuisances. Define B == S,nck- for i = 2, m as bifurcated set, which is a subset of S; in pi+ (S:) that is fixed s at p (C; S while marginalized out over p (C, Sy-1) for some j < i (e.g.. X in FD). Set B,, 0. We use B to denote an independent copy of B; (variables following the same distribution as B but independent of B and V). With B and B we define S = ((S; BB) \ B, U and S; == S \ R, for i 2, m. Define the regression nuisance parameters follows:
Mo (Sm) = Bpm+1 [Y S.m]] and pn (Sm) == Er... Rmn. (Im Sm Rm) (Im. Sm). For
i m- 1,
(6,5, B3) = Epara 11b (SH+1) S, B.) Ab Σ i (r. S, (R) S)). r;
(6)
(7)
Equipped with the regression nuisances, UCA can be computed as follows: Proposition 2. UCA in Eq. (T) can be parameterized as 1/u = Ep 1ac (S:)).
Whenever no variables are being summed and fixed simultaneously (i.e., B;-1 = 0 for all i 2.... .m) in the UCA functional, as in Eq. (5) in Ctf-DE, the standard SBD adjustment or examples in Appendix B we can estimate ja through nested regression methods with off-the-shelf regression models and compute UCA in Eq. (1) as 1i0 = Ep [i](() 1]] This approach aligns with existing
SBD estimators (Bang and Robins 2005 Robins et al. 2009; van der Laan and Gruber 2012
Rotnitzky et al. 2017 Luedtke et al. 2017
Diaz et al. 2023). For instance, in Ctf-DE in Example 3
"/ 2) = pr W, Z, vo). 16 (W, 2)
1 W Z). (X Z) = l (W, Z) X, 2T.
ZZ( = p1(11, 2). and th = Ep
||2] These nuisances can be estimated efficiently with
regression models run in polynomial time relative to the number of variables and samples (e.g.. neural
networks (LeCun et al. 2015) or XGBoost (Chen and Guestrin 2016)).
Beyond the SBD framework, the regression nuisances are capable of representing functionals in the presence of variables being summed and fixed simultancously (c.g., FD in Eq. (2) or Verma Eq. (3). As an example, consider FD in Eq. ( with its UCA representation in Ex- ample First, define (2, X, C) = Ep(Y | Z, X, CJ with S2 = 12 C}. Next, we have B + i nC = {X} and, S, = (Z,X' C} where X' is an independent copy of X. Con- sequently, AZ,
X', C) = (Z, C), where (Z,X', C') is plugged into a function p.. Next,
define M (C, = (2, expectation, Ep Ep [x](C Ee, P(c, x')((, correctly specifies FD in Eq. (2) as follows:
x. C.X'). Finally, we have (C, X) = (C, The
Σ
Σ P(c, (udlea)) = Σ P(c,,')) (2 x' ar, [x,,,"]
() E Plax)))( (),0(((),,)00(0555
Σ P(c,))))(z | x,,) ,, 0,
where the equation = holds since X' is an independent copy of X, so it's independent of Z.
Empirically, generating B' involves permuting copied samples of B,, an used in recent works in (Chermozhukov et al. 2022 Xu and Gretton 2022). We name this approach empirical bifurcation:

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Algorithm 2: DML-UCA((D').)
(Sample splitting) For each E [m + 1], randomly split D' 14 p into L-folds. Let Dj denote the E-th partition, define D'=; = D Di- We use (D))) refer to the samples of VW in Dj
for E [L] do for m...... 1
Learn (S, B3) by regressing p" (B-))(D))))) onto ,,((" (where 1m0 +1 = Y).
2. Evaluate i² (S, )) using empirical bifurcation under the policy dR,
Compute म (S. by evaluating já=, using samples D).
4. For nuisance parameter n. satisfying Eq. (0). learn #_e using samples (D' :j [i+ 1]}
5. Evaluate #े = #' ((()) jj E li 1]}).
end end
Return the DML-UCA estimator d
m
B Di (#2(
Ab) EDy
(8)
Definition (Empirical bifurcation). An empirical bifurcation for B following distribution P the procedure of copying samples of B ~ P and randomly permuling to obtain new samples B'.
In general, the regression nuisances can be estimated from data by employing empirical bifurcation and off-the offtthe-shelf regression models.
Next, we define the ratio nuisance parameters π. Define To" (Sm) as the solution functional satisfying Epo+1 [40" (S... >ry"" d/o- Recursively, for m 1.... define T B,) as functional satisfying the following equation, for any j²* E (2(())))
Ep a [ (S BB +1 (S,+1, B +1 Epan a (S, B ( (S+11) S, B[],,
(9)
where the closed form solution is given as follows:
II (S(C; 1S,) (R2 |S, R,)) PL+I(S, B()
(10)
For the example FD, म = P(Z[x.C) and T/J = P()(C)
P(z)
P(2|X.C)
Equipped with the ratio nuisances, UCA can be computed as follows: Proposition 3. UCA in Eq. can be parameterized 4/u Pm Y..
Estimating the ratio nuisances may challenging due the distribution ratio of continuous/high- dimensional variables. To address the challenge, we use Bayes' rule to transform the distribution ratio more tractable form. For example, FD, the treatment X is a singleton binary, instead of estimating F = P(Z CJ an equivalent estimand = = P C(X can be estimated. This approach allows to use off-the- the-shelf probabilistic classification methods for estimating distribution ratios, allowing scalable computation. detailed procedure for ratio estimation in Appendix C.2
P(Z]x,C)
P(X
Combining regression and
we present a double/debiased machine learning (DML)
Chernozhukov et al., 2018 -based estimator & for the UCA, titled 'DML-UCA' in Algo. 2 We
provide detailed nuisance specification for various examples in Appendix and B
DML-UCA provides a scalable estimator for functionals expressible through UCA. When the target
query is BD/SBD, DML-UCA aligns with existing doubly robust SBD estimators (Bang and Robins
2005
Robins et al, 2009 van der Laan and Gruber 2012 Rotnitzky et al.] Luedi et al.
2017 Díaz et al. (023). Beyond SBD, DML-UCA can be estimated in polynomial time relative to the number of variables and samples, ensuring its scalability:
Theorem 2 (Scalability). Algo. 2 runs in O(Knmax T(m, Dmax K)), where K is the number of distinct in P. Amax max | E [K]] and T(m, Tmaaxs K) is the time complexity for
7

---

Estimand
Estimator
Complexity
Plug-in
0(n2n)
BD/SBD
IPW [Rosenbaum and Rubin 1983
OM [Robins 1986
(m + +T(m,, n))
AIPW
(Rotnitzky et al.
1998
HD Tian's '
Fulcher et al. 2019 Guo ct al. 2023
0(n2 +T(m,, m))
Bhaltacharya et al.
2022
DML-UCA (BD, FD and Tian's)
On T(m, n))
UCA
DML-UCA (general)
O(K T(m, Ammoey K))
obsID
DML-ID Jung et al.
2021a
O(n22m + Tm,,
gID DML-gID (Jung et al. 2023a
P + T(m, Rmaxx K))
Table 2: Comparison of time complexities of existing estimators for estimands: Temax == max (DD1) is the number of samples, m is the number of variables, and T(m, Tmax, K) (or T(m, n) = T(m, 7max = n, K = 1)) is the time complexity for learning nuisance parameters for the tar- get functional. The plug-in estimator for BD is one where Ep x. z) and P(z) are estimated from data, and Eg EY x, z) P(z) is evaluated. Details are in Sec. C.4
learning nuisances üig and tt.. Specifically, (m maxx K)) O(K L x (T₁, + +T))), where Tu = max(T;; i E [m],e [L]} Tx == mTx : [m], l [4]] and T ki and Tx denote the time complexity for learning and evaluating #t and Pby respectively.
An an example, for XGBoost (Chen and Guestrin 2016), T = Tu = Onumaes x depth mmax log Rmaxx, where numaae and depth are the number and depth of trees in XGBoost.
tree x
summarizes the comparison of time complexities for existing estimators. As shown in the table, scalable estimators with polynomial time complexity have only been developed for BD/SBD estimands. Existing estimators beyond SBD often lack scalability. For instance, existing estimators for FD (Fulcher et al., 2219)] Guo et al., 2023) or Tian's adjustment (Bhattacharya et al., (022) face exponential time complexity in the dimension of mediators. In contrast, DML-UCA'S i polynomial time complexity positions it uniquely scalable solution within the UCA functional class, which includes FD and Tian's is adjustment as special cases. For general ObsID/gID estimands beyond the UCA class, scalable estimators have yet to be developed.
3.1 Error analysis
In this section, we show that DML-UCA exhibits doubly robustness, in addition scalability. Since UCA is composed of multiple (possibly distinct) distributions, we provide tool to distinguish them.
Definition 3 (Index set). The index sets It .......... IIK partition {1.-.., .m + 1} such that indices i and j are in the same set I& if and only if Pd (V) = (V).
We will use pk for k = 1,... K to denote the distribution pi for i E Ig. Then, the functional P: a in Eq. (1) can be written as follows:
V (P:a| [(P(" k 1, K):
(11)
Since multiple distributions are involved in UCA, deriving an influence function for each distribution Pe becomes necessary. A standard influence function is typically defined for a single distribution P. and thus, does not suffice for studying multi-distribution setting. To address the issue, we employ a
partial influence function (PIF) (Pires and Branco. 2002). an influence function defined relative to each p formal definition is in Appendix C For UCA, PIFs are given as follows:
Theorem 3 (PIF for UCA). Assume that pb oc and 0 < क 11-1) for 1, 1, m + 1, and
oc almost surely for 1, ..... m.
Define Th = {ub} and n == {mf-1 .Pb. 16
4 ($ vo) = m (06 1 fi>1
(12)
j
4/0
fi
8

---

Let V = UiEz St and ni = Usez, To- Then, the k-th PIF for UCA is % = 6² (V n vo)
=
Eiez, p'(S': No: vn).
Equipped with PIFs, we provide a finite-sample guarantee for DML-UCA, extending Chernozhukoy al. (023) which analyzed DMI. estimators for BDs.
Theorem 4 (Finite sample guarantee). Suppose p fiž and < mbs # < oc almost surely for i 1, m. Suppose the third moment of 6b for k 1,... K exist. Let Φb = & nos v/o
and 8 == = (V.; vil v0o). Let R2 = (1/L) Et= ( (Epg 1001 Ep (0t1). Then,
The error & vho is decomposed as follows:
K
m
1 4/
R
Bp [(#] - T6) - pb)).
(13)
2. Let p2.0 = Vp loo).
With probability (W.P) greater than 1 - E,
K
12
K
L
K
Σ
R S
(14)
P
3. Let 3 Kk,0
= -Bpal 1pp 1100 Let NORMAL(T) denote the standard normal CDE W.P greater than 1
[VIDAt
1 D
0.4748x
pk
R1 <
- NORMAL(x))
(15)
PR.O
2T
PROV DF
t=1
This is novel finite sample guarantee of DML-based estimators for functionals beyond SBD. Finite sample analyses for functionals beyond SBD have been studied only for the non-doubly robust estimators (Bhattacharyya et al.] (002)). For doubly robust estimators, only asymptotic analyses were provided for FD (Fulcher al. [2019] Guo et al. 2023). Tian's adjustment (Bhattacharya et al.) 2022). and obsll) Gung et al. 102 Thm. Jeluciaaes that the error can be decomposed into two terms R2 and Rt The term 12 closely approximates a standard normal distribution variable, and R2, comprises the error of (ft, #t #1-1) and ji', exhibiting doubly-robustnes, behavior. Specifically, the nuisance parameters pig 77. and # converge at a rate of n 11/ (where n represents the size of the smallest sample set), then DML-UCA converges at a faster rate of n 11/2, This point becomes evident in the corresponding asymptotic analysis:
Corollary 4 (Asymptotic error). Assume phy Phy co and 0 < T #i oo almost surely. Suppose the map of ot uniformly differentiable with respect to ni and the derivative of J w.r.l. o is bounded by some constants. Suppose jig and #t are La-consistent. Then,
m EE
-
v/0
Σ
R
T.
Opti (llb -6lI(1133 ((()))
1=1
and VIDK [Rt converges in distribution to normal(0, pi.))-
Experiments
In this section, demonstrate the scalability and doubly robustness of the DML-UCA estimator, where nuisances are learned through XGBoost (Chen and Guestrin 2016). We specify an SCM M for FD (Fig. [aj, Verma (Fig. lb). and the example graph in (Jung al. 2021a) (Fig. [e]. and generate datasets x ~ pk from the SCM. The target estimand is denoted as Vn. Details are in Appendix
Further simulations provided in Appendix F
Scalability. demonstrate scalability DML--UCA, we compare the running time with existing
estimators of (Fulcher al. 2019 (FD) and (Jung al. 2021a) (Verma's equation and the estimand

---

300
300
230
250
200
200
150
150
100
100
50
100
2657
303
365220
(8)
(b)
(c)
0.05
PM
0.8
.225
DPL
0.04
0.7
.200
.175
0.03
-125
0.02
.100
.05
0.3
050
1500000
1070
21000
0.2 200000
1i00 1000
200200 2000
1000
(d)
(c)
(f)
Figure 2: Comparison of DML-UCA ('DML') with existing estimators using (Top) running-time-
plots (x-axis: the dimension of summed variables, y-axis: running time); and (Bottom) AAE-plots (x-axis: the sample size, y-axis: errors). DML-UCA is compared with Fulcher et al. (2019) for
FD; (b.e) (Jung al. 2021a) for Verma's equation; and (c,f) Jung al. (2021a) for Jung's equation.
with Fig. le E do(x1, =2)=,, Ep(y T2, 22, 21, x1, P(2, which we call 'Jung' S equation"). For each example, we increment the dimension of the summed variables, run 100 simulations, take the average of running times, and compare this average. We label this plot as 'run-ting 'run-time-plo" 2019) for FD Fig. [aa we fix |C| = and increment 2 {2, 4, 6, 8, 12. 20, 30, 50, 100). When comparing with (Jung et al. 2021a), for Verma's equations in Figs. 2b we fix |A| = 2 and increment B
presented in the top side of Fig. 2 In the comparison with (Fulcher et al.
{2, 4, 6, 8, 12, 20, 30, 50, 100). For Jung's equation
Fig. 2c we fix |Z|
and R {2, 4, 6, 8, 12, 20.30, 30, 100}. The timeout for the run-time is set to 300 seconds. For all scenarios, the run- run-time of existing estimators increases rapidly over dimensions due to the summation operation while DML-UCA scales well for high-dimensional covariates.
Doubly robustness. To demonstrate doubly robustness, compare the error of DML-UCA with existing estimators for FD of Fulcher et al. (2019) and for Verma's and Jung's equations of Jung et al. 22221a) We use Jest for est {DML, Fulcher, Jung] to denote each estimator. We use the average absolute error (AAE), which is an average of the error of the estimated versus true causal effect of X X:
v>)( To witness the fast convergence of DML-UCA, we enforce the convergence rate of nuisance estimates to be no faster than the decaying rate n 1/4 by adding the noise term é ~ normal(n 1/4 -1/4) to nuisances, inspired by the experimental design in (Kennedy) (023).. We ran 100 simulations for each number of samples n = (2500, 5000, 10000, 20000). We label the plot as 'AAE-plot', presented in the bottom side of Fig. For each example, DML-UCA outperforms other estimators, exhibiting fast convergence.
domain(X) X) ExEdomain(x) 1
5 Conclusions
We introduce a framework that encompasses broad class of sum- sum-product causal estimands, called UCA class, for which scalable estimators were previously unavailable. We demonstrate the expres- siveness of the UCA class, which includes not only BD/SBD but also broader classes such as Tian's adjustment incorporating FD and Verma, and Ctf-DE, for which the existing SBD-based framework not applicable. We develop an estimator for UCA called DML-UCA that can estimate the target estimand in polynomial time relative to the number of samples and variables, ensuring scalability. We provide sinite-sample guarantees and corresponding asymptotic error analysis for DML-UCA, demonstrating its fast convergence. These scalability and fast convergence properties are empirically verified through simulations. Our results pave the way toward developing an estimation framework maximizing both coverage and scalability in Table I
10

---

Acknowledgments
We thank anonymous reviewers for constructive comments improve the manuscript. This research is supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation.
References
Bang, H. and Robins, J. M. (2005). Doubly robust estimation missing data and causal inference models. Biometrics, 61(4):962-973
Bareinboim, E., Correa, J. D., lbeling, D., and Icard, T. (2020). On pearl's hierarchy and the foundations of causal inference. In Probabilistic and Causal Inference: The Works of Judea Pearl, Technical Report R-60. Causal Artificial Intelligence Laboratory, Columbia University.
Bareinboim, E. and Pearl, J. (2012a). Causal inference by surrogate experiments: z-identifiability. In In Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence, pages 113-120. AUAI Press.
Bareinboim, E. and Pearl, J. (2012b). Transportability of causal effects: Completeness results. In Proceedings of the 26th AAAI Conference on Artificial Intelligence, pages 698-704.
Bareinboim, E. and Pearl, J. (2016). Causal inference and the data-fusion problem. Proceedings of the National Academy of Sciences, 113(27):7355-7352
Bareinboim, E., Tian, J., and Pearl, J. (2014). Recovering from selection bias in causal and statistical inference. Proceedings of the 28th AAAI Conference on Artificial Intelligence, pages 2410-2416.
Berry, A. C. (1941). The accuracy of the gaussian approximation to the sum of independent variates. Transactions of the american mathematical sociery, 49(1):122-136.
Bhattacharya, R., Nabi, R., and Shpitser, I. (2022). Semiparametric inference for causal effects in graphical models with hidden variables. Journal of Machine Learning Research, 23:1-76.
Bhattacharyya, A., Gayen, S., Kandasamy, S., Raval, V., and Variyam, V. N. (2022). Efficient interventional distribution learning the pac framework. In International Conference Artificial Intelligence and Statistics, pages 7531-7549. PMLR.
Bickel, P. J., Klaassen, C. A., Bickel, P. J., Ritov, Y., Klaassen, J., Wellner, J. A., and Ritov, Y. (1993). Efficient and adaptive estimation for semiparametric models, volume 4. Johns Hopkins University Press Baltimore.
Chen, T. and Guestrin, C. (2016). Xgboost: A scalable tree boosting system, In Proceedings of the 22nd ACM SIGKDD International Conference Knowledge Discovery and Data Mining, pages 785-794.
Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E.,, Hansen, C., Newey, W., and Robins, (2018). Double/debiased machine learning for treatment and structural parameters: Dou- ble/debiased machine learning. The Econometrics Journal, 21(1).
Chernozhukov, V., Newey, W., Singh, R., and Syrgkanis, V. (2022). Automatic debiased ma- chine learning for dynamic treatment effects and general nested functionals. arXiv preprint arXiv:2203. 13887.
Chernozhukov, V., Newey, W. K., and Singh, R. (2023). A simple and general debiased machine learning theorem with finite-sample guarantees. Biometrika, 110(1):57-2664
Corrca, J., Lee, S., and Bareinboim, (2021). Nested counterfactual identification from arbitrary surrogate experiments. Advances Neural Information Processing Systems, 34.
Correa, J. D., Tian, J., and Bareinboim, E, (2018). Generalized adjustment under confounding and selection biases. In Proceedings of the 32nd AAAI Conference Artificial Intelligence.

---

Díaz, I,, Williams, N., Hoffmnnn K. L., and Schenck, E. J. (2023). Nonparametric causal effects based on longitudinal modified treatment policies. Journal of the American Statistical Association, 118(542):846-8555)
Esseen, C.-G. (1942). On the liapunov limit enror in the theory of probability. Ark. Mat. Astr, Fys., 28:1-19.
Fulcher, I. R., Shpitser, I., Marealle, S., and Tchetgen Tchetgen, E. J. (2019). Robust inference on population indirect causal effects: the generalized front door criterion. Journal of the Royal Stalistical Society: Series B (Statistical Methodology).
Guo, A., Benkeser, D., and Nabi, R. (2023). Targeted machine learning for average causal effect estimation using the front-door functional. arXiv preprint arXiv:2312.1022
Heckman, J. (1992). Randomization and Social Policy Evaluation. In Manski, C. and Garfinkle, I., editors, Evaluations: Welfare and Training Programs, pages 201-230. Harvard University Press, Cambridge, MA.
Huang, Y. and Valtorta, M. (2006). Pearl's calculus of intervention is complete. In Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, pages 217-224. AUAI Press.
Jung, Y., Díaz, I., Tian, J., and Bareinboim, E. (2023a). Estimating causal effects identifiable from combination observations and experiments. In Proceedings of the 37th Neural Information Processing Systems.
Jung, Y., Tian, J., and Bareinboim, E. (2021a). Estimating identifiable causal effects markov equivalence class through double machine learning. In Proceedings of the 38th International Conference Machine Learning.
Jung. Y., Tian, J., and Bareinboim, E. (2021b). Estimating identifiable causal effects through double machine learning. In Proceedings of the 35th AAAI Conference on Artificial Intelligence.
Jung, Y., Tian, J., and Bareinboim, E. (2023b). Estimating joint treatment effects by combining multiple experiments. In Proceedings of the 40th International Conference on Machine Learning.
Kemnedy, E. H. (2023). Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics, 17(2):3008-33 -049..
Kennedy, E. H., Balakrishnan, S., G' Sell, M., et al. (2020). Sharp instruments for classifying compliers and generalizing causal effects. Annals of Statistics, 48(4):2008-200 2030.
LeCun, Y., Bengio, Y. and Hinton, G. (2015). Deep learning. nature, 521(75533:436-444
Lec, S. and Bareinboim, E. (2020). Causal effect identifiability under cartial-bbservabili In Proceedings of the 37th International Conference on Machine Learning.
Lee, S., Correa, J., and Bareinboim, E. (2020). General transportabilttyyyyttessiii observations and experiments from heterogemeous domains. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 10210-10217.
Lee, S., Correa, J. D., and Bareinboim, E. (2019). General identifiability with arbitrary surrogate experiments. In Proceedings of the 35th Conference Uncertainty in Artificial Intelligence. AUAI Press.
Li, S. and Luedtke, A. (2023). Efficient estimation under data fusion. Biometrika, 110(4):0041-1054
Luedtke, A. R., Sofrygin, O., van der Laan, M. J., and Carone, M. (2017). Sequential double robustness in right-censored longitudinal models. arXiv preprint arXiv: 1705.0245..
Mises, R. v. (1947). On the asymptotic distribution of differentiable statistical functions. The annals of mathematical statistics, 1803)330--348
Murphy, S. A. (2003). Optimal dynamic treatment regimes. Journal of the Royal Statistical Society Series B: Statistical Methodology, 65(2):331-355.
12

---

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4):669-710.
Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press, New York. 2nd edition, 2009.
Pearl, J. and Mackenzie, D. (2018). The book of why: the new science of cause and effect. Basic Books.
Pearl, and Robins, J. (1995a). Probabilistic evaluation of sequential plans from causal models with hidden variables. In Proceedings of the 1 1Ith Conference Uncertainty in Artificial Intelligence, Morgan Kaufmann Publishers Inc.
pages 144-453
Pearl, J. and Robins. J. (1995b). Probabilistic evaluation of sequential plans from causal models with hidden variables. In Proceedings of the Eleventh conference on Uncertainty in artificial intelligence, pages 444-453.
Pires, A. M. and Branco, J. A. (2002). Partial influence functions. Journal of Multivariate Analysis, 83(2):451 -468.
Plečko, D. and Bareinboim, E. (2024), Causal faimess analysis: A causal toolkit for fair machine learning. Foundations and Trends@ in Machine Learning, 17(3):304-588
Quintas-Martinez, V., Bahadori, M. T., Santiago, E., Mu, J., Janzing, D., and Heckerman, D. (2024). Multiply-robust causal change attribution. arXiv preprint arXiv:2404.08839.
Robins, J. (1986). new approach to causal inference in mortality studies with a sustained exposure appliicaionn to control of the healthy worker survivor effect. Mathematical modelling,
Robins, J., Li, L., Tehetgen, E., and van der Vaart, A. W. (2009). Quadratic semiparametric von mises calculus. Metrika, 69:227-247.
Robins, J. M. and Rotnitzky, A. (1995). Semiparametric efficiency in multivariate regression models with missing data. Journal of the American Statistical Association, 90(429):
Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studics for causal effects. Biometrika, 70(1):41555.
Rotnitzky, A., Robins, J., and Babino, L. (2017). On the multiply robust estimation of the mean of the g-functional. arXiv preprint a75.v.170..0858..
Rotnitzky, A., Robins, M., and Scharfstein, D. O. (1998). Semiparametric regression for re- peated outcomes with nonignorable nonresponse. Journal of the umerican statistical association, 93(444):
Shevtsova, I. (2014). On the absolute constants in the berry-esseen-type inequalities. In Doklady Mathematics, volume 89, pages 378-381. Springer.
Shpitser, I. and Pearl, J. (2006). Identification of joint interventional distributions in recursive semi- markovian causal models. In Proceedings of the 21st AAAI Conference on Artificial Intelligence, page 1219. Menlo Park, CA: Cambridge, MA; London; AAAI Press; MIT Press; 1999.
Tian, J. and Pearl, J. (2002a). A general identification condition for causal effects. In Proceedings of the 18th National Conference on Artificial Intelligence, pages 567-573.
Tian, J. and Pearl, (2002h). On the testable implications of causal models with hidden variables. In Proceedings of the 18th conference Uncertainty in artificial intelligence, pages 519-527. Morgan Kaufmann Publishers Inc.
Tian, J. and Pearl, J. (2003). On the identification of causal effects. Technical Report R-290-I., UCLA.
Uehara, M., Shi, C., and Kallus, (2022). A review off-policy evaluation reinforcement learning, arXiv preprint arXiv:2212.06355.
13

---

van der Laan, M. J. and Gruber, S. (2012). Targeted minimum loss based estimation of causal effects of multiple time point interventions. The international journal of biostatistics, 8(1).
Verma, T. and Pearl, (1990). Causal networks: Semantics and expressiveness. Machine intelligence and pattern recognition, volume 9, pages 69-76. Elsevier.
Xia, K., Lee, K.-Z.,, Bengio, Y., and Bareinboim, E. (2021). The causal-neural connection: Ex- pressiveness, learnability, and inference. Advances in Neural Information Processing Systems, 34.
Xia, K. M., Pan, Y., and Bareinboim, E. (2022). Neural causal models for counterfactual identification and estimation. In The Eleventh International Conference on Learning Representations.
Xu, L. and Gretton, A. (2022). A neural mean embedding approach for back-door and front-door adjustment. In The Eleventh International Conference on Learning Representations.
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Supplement to "Unified Covariate Adjustment for Causal Inference"
Contents
Introduction
2
Unified Covariate Adjustment
3 Scalable Estimator for Unified Covariate Adjustment
3.1
Error analysis
Experiments
Conclusions
10
Nuisance Specification
17
A.I
Front-door adjustment in Example
17
A.2
Verma's s equation in Example
17
A.3
Counterfactual directed effect in Example
18
A.4
Example Estimand for Fig. le
19
More UCA Examples
20
B.1
Effect of the treatment on the treated (ETT)
20
B.2
Transportability (S-admissibility)
20
B.3
Off-policy evaluation
21
B.4 4
Treatment-treatment interactions
21
More Results
22
C.I
Formal definition of Partial influence function (PIF)
................................
22
Density Ratio Estimation
22
................................
Analysis of non- UCA functionals
23
C33.1 On Case
23
C,3,2 On Case
24
Time Complexity
................................
24
C.4.1
BD/SBD
................................
25
C.4.2
Front-door adjustment (FD)
26
C.4.3
Tian's s adjustment
26
C.4.4
DML-UCA (BD, FD, and Tian s).
27
C.4.5
DML-UCA (general).
27
C.4.6
DML-ID (obslD).
27
C.4.7
DML-gID (gID).
27
Proofs
27
D.I
Proof of Proposition
27
15

---

D.2
Proof for Proposition
28
D.3
Proof for Proposition
28
D .
Proof for Theorem
28
D.5
Proof for Theorem
28
D.6
Proof for Theorem
29
D.7
Proof for Theorem
29
D.7.1
Helper lemmas
30
D.7.2
Preliminary Results
33
D.7.3
Proof of Theorem
33
D.7.4
Proof of Theorem
33
D.7.5
Proof of Theorem
34
D.8
Proof for Corollary
35
More Experiments
35
Details in Experiments
36
F.1 FD (Fig.
la for Simulation in Fig. 2a
36
F2
Verma (Fig.
lb) for Simulation in Fig. 2b
37
F3
Example estimand (Fig. le for Simulation in Fig. 2c
38
F.4
ETT in Sec. B for Simulation in Fig. E.4a
40
F5
Transportability in Sec. B for Simulation in Fig. E.4b
41
F6
FD with continuous mediators for Simulation in Fig.
42
F7
Verma's equation with continuous mediators for Simulation in Fig. E.4d
43
F88
Cuf-DE in Example for Simulation in Fig. E.4c
44
16

---

A Nuisance Specification
A.1 Front-door adjustment in Example
We first note that the front-door adjustment is an expectation of the following product:
P)C,X)P(Z xz C)Y C,,,
which implies that
C X): C2 [Z); C3 (Y) (i.e., m 2).
* S {C} and S, = [C
S = {X}
B1 S2nC ns (X}.
$2 [C,X', and $ X.
p2 = P(- ())
The regression nuisances are the followings:
()))) == 16(2, x, C) 2,X,C]
(S2) 16(
(S,, BB) = (6(X''))= =)pp=)/p2,, 22 C,
(),) = C))
The ratio nuisances are the followings:
P(Z x,
17, C)
(A.1)
P(Z X,C)
P(x)
TC(C) P(# (C)
(A.2)
The representation for DML-UCA is
Ep (,, ,,(,,X,C)] Ep (1)) (', X'C) - -v(('' [x]
Er (X),
A.2 Verma's equation in Example 2
We first note that the Verma's equation in Eq. 3 is expectation of the following product:
P(X) PBB A,X) B,A,)),
which implies that
C, (X): C2 (A); C3 (B): and C4 (Y) (i.e., m = 3).
S1 0; S2 (4, and S3 (B,
.
st = {X} and s = (X}.
B, = S, C, ns: = {X}.
$3 [B, A]: S2 = X') and $ (X}.
p2 = pl P(: z).
17

---

The regression nuisances are the followings:
(S,) = PY A) = Ep(Y B, A,
[Sa) = P²(B, A) = Ep B,
(S2) Mã(A, X) Ep (p A) A,
p ($²) - MO(A,
(S Bi) = (A x,
($.) = Px)
The ratio nuisances are the followings:
(B))) Exe P(B| A,2))
(A.3)
P(B A,
P(A | x)
(A, X) = P(A
(A.4)
(A.5)
The representation for DML-UCA
+ Ep -,(B,AA))) a]]
(, - ()( + Ep 1 (A X')]
A.3
Counterfactual directed effect in Example 3
From the fact that Ctf-DE in Eq. (5) is represented the expectation of Y over P(Y X W, PWW
X, ZZ X = x2) 2, (X). Set
C (Z]: C2 (:): 3; and C2 {Y}.
.
81 (X, Z}, S2 = (
R1 (X).
.
S = (x2): 8 (X): = (X).
B, = 0 for all i.
S, S. R. for all i
The regression nuisances are the followings:
(S2) = rê(W, Z) = Y W, xo, Z]
(S2)) = r6(W, 2) Er(Y W. Z]
(S,) = vb(X,, Z) = Ep lu (W, Z) |
(Si)) - Mb( Z).
The ratio nuisances are the followings:
P(W x1, P(Z x2)
7 (W, Z)
(A.6)
P(W, Z zo)
1x, (X) |22)
To 2X = P(X ZZ)(Z))
(A.7)
18

---

The representation for DML-UCA is
Ep|ng(W W. 2) (i( X = 20)
1 (,) Z) 16(X, Z)
Eplug 2) [z].
A.4
Example Estimand for Fig. le
Given Fig. le
the causal effect is given as
E|Y do(x),, x2)]
Σ
Ep(y
2, p())))
r,2,a
which is the expectation of Y over the probability measure
P(Y X2, 2, xı, 2) (X2).
Set
C = (X1, C2 (R): C3 {Y}
R, and R₂ = (X2) with of Lzg (X2).
. 8 (2). S2 = (R, X2, Z,XI)
8 (Xi).
. B, S, -s (
$, = (R, Xi, Z}. S S..
p2 = P( æ1).
The regression nuisances are the followings:
(S,) = P6(
2. = Ep|y R, X1]
r = (,(R, = Eply R, Ø2, Z
A (S,, BB) = =1(2,x;)= E> I2,
X,) |21, 2,
(S:) ri(Z,
The ratio nuisances are the followings:
(X2) P(R| x1,2)
2, 2) P(X2 R, R2X,, Z) P(R) X1,Z)'
(A.8)
P(Z) P(x)
P(x)
=1)=)= P(Z x)) "((x Z)P(Z)
(A.9)
P(x Z)
The representation for DML-UCA
Ep [ R,X,, 2) - 2IHI
+ (Z 2, - AZ(2X))) =x= = x11
+ E,,,)(
19

---

B More UCA Examples
B.1 Effect of the treatment on the treated (ETT)
Let V (Z, YJ be of variables where Z is covariate, X is treatment and Y is an outcome. The target estimand is
(
Σ EplY x 222 x').
(B.1)
The ETT estimand can be written as an expectation of Y over the probability measure
= P(Y
(X).
This factorization implies that C₁ = { R = (r]
= (X}. where R, = {X}, and
R 1 (X)). Also, S [X] U Z. Finally,
(C(Ci) P(Z x')
P² (S1) P(Y X,Z))
The regression nuisances are the followings:
(S1) = (x, Eply X
(S R1) (Z) (z, Z).
The ratio nuisances are the followings:
P(Z (X)
P(a' P(Z
1x() P(X Z)P(Z)
P(x' Z) (X)
To (x(
P(X,Z)
P(x))
P(X Z) P(x')
The representation for DML-UCA is
(X () (X, 2)H1 + EEppi a.]
B.2 Transportability (S-adnissibility)
Let V (Z, X.Y) be of variables where Z is covariate, X is treatment and Y is outcome. Let S denote the domain indicator such that 5 = means the target domain, and S = 1 denotes the source. The S-admissibilityy estimand appeared transportability scenario is
Y do(=))
Σ
EplY x. z. 1]P(z =).
(B.2)
The estimand can be written expectation of Y over the probability measure
V P(Y X,Z,S (z S (X),
From this factorization, we have C₁ : Z and R == X. Also, set ('((i) == P(Z 5 = 0) with
S = S. Set (Y (,) = P(Y X, Z | 1) with S S and S [X] UZ.
The regression nuisances are the followings:
( = n6((X, Z) = Ep(Y X,2,S =
/ = (Z) (2, Z).
The ratio nuisances are the followings:
1x(X)
P(Z S
(X Z) P(X Z,S = P(Z S
20

---

The representation for DML-UCA is
( () (5(X,)) S = 1] - (Z) S 0].
B.3 Off-poolccy evaluation
Let V {Z. X,YY be of variables where Z covariate, is treatment and is an outcome. Let (X Z) denote the behavioral policy that agent observed; ic.,
(Z, XY) P(Y X, 2) (X P())
(B.3)
Let oXX Z) denote policy be evaluated. Then, the effect of the policy of is given
y
Σ
Ep(y x, z (æ P())
(B.4)
The policy treatment effect in Eq.
B. can be represented as UCA as follow.
C =
R1 (X)
d o'(X Z)
S (X] UZ.
Set (C) + P(Z), o (R) 11) + (X Z). and P² R,) + P(Y X) Then,
o(P:o) Σ
Epl Y CI, R. (ci)
c,R
> Eply x z|o0 zz)(
E o]
(Bq. (B.4)
The regression nuisances are the followings:
(C (R = M)(X = Ep(Y X
(C(U)) = (Z) = Σ (z. (x. Z)o² (s Z)
The ratio nuisances are the followings:
(X, (X Z
P(X Z)
The representation for DML-UCA is
Ex, Z) NX(X, (1(X,20 2)H + Erldd())
B.4 Treatment-freatmment interactions
Let V (Z, X, be set of variables where Z covariate, X is a treatment and is an outcome. The estimand for treatment-treatment interaction discussed in Jung al. (2023b) is
E do(x1, x2)] = Σ E do(x))) z. 2, do(z1)).
(B.5)
which is an expectation of Y over product of probability measure
P(Y Z, do
P(Z do 11) (X1),
21

---

which satisfies an additivity. Therefore, B[Y do(21, is UCA-expressible, Such reduction can be done since the probability measure satisfies additivity w.r.t all conditional distributions and the policy 1s, (X1). Specifically, set
C₁ Z
R {Xi}
S = {Xi) UZ.
Also, set
(c) = P(Z | do(x1))
P2 [ = P(Y |1,, Z, (o(2))
R 1z, (X1).
The regression nuisances are the followings:
vO(CC)) ( = px, Z) Ep(Y X1, 2,
Ab(C = Eply 21. Z, do(2))]
The ratio nuisances are the followings:
1,, (XI) do(ri))
क (X, Z = P(X, Z, do(2)) do(x2))
which can estimated through density estimation approach using the probabilistic classification Sec. 5.4).
method described in (Diaz al. 2023
The representation for DML-UCA
Ep (X. 2) (6(X2))) do(x22) + -EE (z)() do(x))) )]
C More Results
C.1 Formal definition of Partial influence function (PIF)
Let g(p! ,P) denote K-multi-dif distribution functional. For the k-th component, let P. = p + t(Q - pt)
Definition C.1 (Partial influence function (PIF) (Pires and Branco 2002
for t E [0, 11. where q is an arbitrary distribution absolutely continuous w.r.. pa The k-th
partial inftuence function a function (V; (P*), such that Epi 10t (V:1t(('*)) go)] Vps f (V:n² (p*), go)l < and (p) .... .P..... ,Pk 1=- Eo- ( },
C.2 Density Ratio Estimation
Two available approaches for estimating the density ratio are the followings. The first approach is apply the Bayes rule for rewriting the density ratio into more tractable form. For example, consider the problem of estimating Th for FD, which is given as
P(Z x, C) = P(Z X,C)
Suppose Z, C. are high- sigh-dimensional random vectors, and X a binary singleton variable. Then, P(X C') or P(X Z, C) are tractable to estimate compared P(Z X, C), since estimating P(X can be done using off-the-shelf probabilistic classification method. Here, Th can be written
22

---

as tractable form as follows:
x,)
m X(Z)X,C)
P(Z,X,C) P(z
P(X C)P(C) P(Z,z,C)
P(C) P(Z,C) P( | P(X (2C)
P(C) P(Z, C) P(X [] P( Z,C)
P( () P(X 2,C)
P(X о) P(æ Z,C)
The second approach is to recast the density ratio into the classification problem (Diaz al. 2023 Sec. 5.4). For example, consider the ratio nuisance appeared in Treatment-treatmeett interactions:
1a, PX do(x)))
T (X, Z) P(X1 do(r2)))P(2 do(x2))
P(Z be estimated following procedure. Let D1 P(Z do(x1) and D2 P(Z | do(x2) denote samples. Let Da == D1 UD2. Let À denote an indicator such that = means samples from D and À means they are from D2- Without loss of generality, [D] ['2]]-
Here, P(Z))o(zz)
Then,
P do(x)))
P(Z x= P(A= 1) P(A = 2)
P() 0 Z)
P(Z (o()))) P(Z 1- P( = 0) P(A = P)
P(2 1 Z)
Then, instead of estimating the density ratio explicitly as P(Z]do(2.)) estimand P()=0/Z)
p(Z)))(22) we can estimate the equivalent using any off-the-shelf probabilistic classification method.
P(A=1 Z
C.3 Analysis of non-UCA functionals
We consider two cases where target estimand cannot be expressed through UCA:
1. Case 1. The target estimand is not in form of the product (e.g., the target estimand is the quotient of sum-products of two conditional distributions ).
2. Case 2. For a target estimand that is represented as the expectation of Y over the measure
II m
V'PPo] = pm+1 (Y (
| (R, \R)),
where (*(V) Q'(V S = 5) for some distribution Q', 3i E {2, .., + 1} such that 8 # (C(-1) UR ( St S
In this section, we will provide example functionals that cannot be expressed through UCA.
C.3.1
On Case 1
Consider Fig. 1c where the causal effect P(y) | do(x)) is identifiable and given
PΦy do(x)) Ene Ply, r. w) P(w)
(C.1)
Σn P(x r, P(w)
Here, the functional for E[Y do(x)] represented not as the expectation of a product of conditional distributions, but rather as a quotient of sums of conditional distributions. The napkin estimand UCA-expressible.
23

---

C
C
14
X
7
R
7
X
R
7
(a)
(b)
(c)
Figure C.3: (a-c) Example for Case 2 (Generalized identification under partial observability (Lee and
Bareinboim 2020 Fig. 1))
C.3.2 On Case 2
Consider Figs. C.3a C.3c A goal is to identify P(y do(x:)) from Fig. C.3c from input distri-
butions: (1) an interventional distribution P(c,= do(x)) with Fig, C.3a and (2) an observational
distribution P(r, z, y) with Fig. C.3b This problem is entitled as generalized identification under
partial observability (Lee and Bareinboim 2020).
Here, the causal effect is identifiable and given as (Lee and Bareinboim, 2020)
Y do(x)] = Σ Er(Y r, P(z (P()) do(()))
(C.2)
C,T.2
This functional is an expectation of the probability measure Y over P(Y R. Z)
() do(x)). Based on this probability measure, apply the following setting:
.
C [C}, C₂ (R} and C3 {Z}.
R = 0.
P'(C,) = P(C do(x)) with S 01
P² 8/) = P(R) with S = and S = 0.
(C 8/2) = P(Z do(œ), C) with s = 0 and S = (C).
P'(Y 8/)) = P(Y) R, Z) with S = 0 and S = (R, Z).
Here, S = 0 + (C(1) U R S [C). Also, S = [C} + (C (R())
s {C, R}.
Finally, 5 = (R, )) * (CO) UCA-class.
R (3) S = (C 2}. Therefore, Eq. C.2 is not within
Now, we will witness that the target estimand cannot be correctly represented through the nested regression and empirical bifurcation. Applying the nested regression, we have
( = r 2) = Ep(Y R, /
(3())) 1RR 2) = Ep(Y R, R,
Then.
(S()) (R Z) C.do(x)) Σ Ep(Y |r,=) (c,
This doesn't correctly represent the target estimand in Eq. C.2 because P(r, do(x)) is not decomposed into P(r) and P(z r, c, do(x)).
C.4 Time Complexity
In this section, we provide detailed analysis on the time complexity in Table 2 Here, n is the sample size. When there are multiple sample sets (i.e., K > 1), we will use 7mmax to denote the size of the largest sample set. K the number of sample sets Eq. (11). m the number of variables in causal graph.
24

---

C.4.1 BD/SBD
Here, we focus on the back-door adjustment, since the SBD can be analyized similarly, The back-door (BD) adjustment estimand is given as
Σ
Plug-in. The plug-in estimator composed of two stage learn the conditional probability table and and evaluate it for each samples. For a detailed description, we define some notations. Let D (V6) j 1,.... n.. where V ()) denote the j'th sample. For any W C V, we will use w to denote the sub-sample of D that W is fixed to w; i... Dw = ()) E D such that W ()) w).
We will use I(Dw) to denote the index set for Dw Finally, we will use fow = (Dwl-
For the BD adjustment, the plug-in estimator is
Σ FlY xx,
(C.3)
where
E x, z|
Σ Y()
(C.4)
7xx
jEI()xxx)
P(z) = f2
(C.5)
For the fixed (x, z), learning B[Y x, z] and P(z) take O(n). Such learning needs to be done for all possible realizations (x, z). where the cardinality of the realization is ((2)). As result, the computational complexity is )n22)).
We first consider the inverse probability-weighing (IPW) estimator (Rosenbaum) and Rubin 1983). The IPW estimator is
IPW, OM, AIPW
1x(X (1))
Y())
(C.6)
n (((() Z(a))
where 7 (XX| Z) is the evaluated function for P(X Z). Learning the nuisance parameter ग takes T(n, m) and evaluating the IPW estimator takes (n)) As a result, the time complexity for the IPW estimator is O(n + T (nn, m)). Next, we consider the outcome-model (OM) estimator
Robins 1986). The OM estimator is
Σ ((X ()) ()). n
(C.7)
where (XX, Z) is the evaluated function for [Y X, Z]. Learning the nuisance parameter र takes T(n, m) and evaluating the OM estimator takes (n). As result, the time complexity for the OM estimator is Onn + T(n,m))
Finally, the AIPW estimator (Robins and Rotnitzky
1995
1x 1x (x) ((0)
1x (6)
Yo +
A X 6) Z (6)
fixX (0) (())
(C.8)
(X (1) (()))
Learning the nuisance parameters and takes T(n, m) and evaluating the OM/IPW estimator takes Onn). As result, the time complexity for the OM estimator is O(n + m(n.
25

---

C.4.2 Front-door adjustment (FD)
The front-door adjustment (Pearl 2000) is
Σ
P(z z) Er c/ c()).
x,C
The FD estimators of (Fulcher al. 2019
Guo et al. 2023 is
(() 2, (())
(1() - (() (1)) (6))
(0): (()) C(t))
1x (x)
(a) Z (())
Σ x,)(2 (()))x(x, C(m)
(x)) (1), (()
X.2
(C) X n x),(2, ((n)),
(C.9)
where D (,,, D), E(Z,X, X, C) and #(x, C) are the evaluated functions for Ep|Y C, Z].
P(Z X, C) and P(X C). Learning these nuisances takes T(n, m) time. Equipped with ju, &, f. evaluating the FD estimator takes )(n2m),, since the evaluation over a samples is repeated for 0(2m) since realization of every (x, z). Therefore, the overall time complexity is O + T(n, m)).
C.4.3 Tian's adjustment
The Tian's adjustment (Tian and Pearl 2002a) is
EΣ
Ep. Y ((K) II P'(u v(-1)).
vlay
The estimator for Tian's adjstment proposed where (V: is given
(Bhattacharya et al.
2022 is Di=1 s(V (V()))).
(X)
Σ
II
Σ y
II
P(u, (b-1)) V3
P(V,
V(-1))
WEVE
2242
Lx(X
Σ nEVENiiSS
II #' Fiii V
Σ y
P(ey ((1-1)))
22 |1 /
P(V, Vū-I)
nvivv P(V, VU-D))
Σ
Σ V
II
P(u, ((-1)) x==o' if Vi
II IV P(V, VG-1)
WEWEVSKiing
V2tt1
VEEE24
IV P(V, V(i-1)
Σ
Σv
II
P(uj v iVVeES
II P(V, VG-1))
Vekevekiinsy
VjEV2
Σ
II V
P(u vD-1))
II
P(e / v(r-1)).
v2k11
V.EVeklnns,
Learning these nuisances takes T(n, m) time. Equipped with nuisances corresponding to P(o; (ii))) evaluating the estimator takes O(n2m), since the evaluation over n samples is repeated for O(2") realization of every Therefore, the overall time complexity is O(n2" + T(n, m)).
26

---

C.4.4 DML-UCA (BD, FD, and Tian's).
For BD, FD, and Tian's, s, the time complexity can be derived by specializing Thm. 2 with K = and max = n, and T(m, n) = I. x (T + Tw). Then, the complexity in Thm. reduces to O(n + n)m,
C.4.5 DML-UCA (general),
Set ('m, Amaax K) = Ix K (Tμ +Tx). Then, the complexity in Thm. reduces O T'max
Tmm, Rmax:: K)).
C.4.6 DML-ID (obsID).
The DML-ID estimator of (Jung et al. (021a) writes the identification functional of causal effect as am arithmetic function of multiple sequential back-door adjustments, where the arithmetic function an arbitrary combination of marginalization, product, and division. Let f([A* k 1,... _K}) denote the DML-ID, where each A is the sequential back-door adjustment, and f denotes the arithmetic function.
To evaluate the DML-ID functional, the first step is learn all nuisances composing each A This takes ( n)) time. The second step is to evaluate f({ k 1, 1,.... K}). Whenever contains marginalization over some random vector, the time complexity for evaluating is O(n2").. In DML-ID, such marginalization
happen (22)) times in worst case. Therefore, evaluating J([A* k 1,... K}) can take O(n2m 2 O(n22m). As a result, the total time complexity is (m222m T(n, m)).
C.4.7 DML-gID (gID).
The DML-gID estimator of (Jung et al. 2023a) writes the identification functional of causal effect as am arithmetic function multiple generalized sequential back-door adjustments called g-mSBD (Jung al. 2023a), where the arithmetic function is arbitrary combination of marginalization, product, and division. Let f({ g-mSBD, and f denotes the arithmetic function.
1, ..... JJ) .)) denote the DML-ID, where each A is the
To evaluate the DML-gID functional, the first step is to learn all nuisances composing each A This takes O(T(m, Amax, K)) time. The second step is to evaluate f({A) : jj= ,,.".,,}}) Whenever contains marginalization over some random vector, the time complexity for evaluating it 2"") In DML-gID, such marginalization can happen 0 K) times in worst Therefore, evaluating f(fA', j 1,..., ,J) can take ( 2m K) = 0(K 2mm)
As result, the total time complexity is O(Knmax 22m T(m, Mmaxx K)).
D Proofs
D.1 Proof of Proposition
The algorithm provides a product of probabilities in form of
V
P
II
P(V: v(i)
II VeeVisx
P(V, V(0 (x,)).
V.ESx
Then, by taking an expectation over Y, it gives
ΣΣ Ep (K) II Pl ((i1))).
vlay
27

---

D.2
Proof for Proposition 2
First, define
set = ((S U (B+1) R \ B,)
(((((++1) U U (B+1) \ (1++)) B;) UB;
and
set, == S; U (CO (S))BB UB
Then, claim that set, \ set, C.+1- This holds when B,+1 B; S. 0. To prove this with contradiction, suppose B,+1 S; * 0. This holds when B,+1 C S.. Recall that B,+1 is subset of fixed variables in S in 2it Then, Bi+1 S S means that this variable will be fixed in pi+1_ However, for this variable to be bifurcated some C, this variable should be within B; However, this contradiction of the definition of B,+1 and B, Therefore, B,+1 B, S; and set; \ set, C++1.
Then.
(S. B:) = E1t+ a 116 (S:+1) S, B
E lin
1
(((S,+1 U BB41) R441 UB) S, B[]
Σ
(c++1 S; B,) R.4s (1441)46* ( (c-1),
4+1
(C-11, 141,, B))
Σ
(1+ (S:) i+1
(ra1))n ++1
(CaF1, S, B)).
01-1,1141
By recursion.
Ep1 (a)))
E ((m))
I[ s (ry s) \ry (ry)EE (Y sm).
D.3 Proof for Proposition 3
By definition of mz"
D.4 Proof for Theorem
conditions in Theorem met, the estimand reduces to the UCA by definition.
D.5 Proof for Theorem 2
1. The samplc-splittin takes O((m + 1)mmaxx.
2. For the fixed f, learning jig for i = nuisances takes (Tμ X m L).
takes ( x m). Therefore, learning all regression-
3. For the fixed learning #f for 1,... m takes O m). Therefore, learning all ratio- nuisances takes ( m L).
4. Evaluating the DML estimator in Eq. (8) takes O(mm + 1)mmax).
In total, the time complexity is
O((m + 1)mmax) + OTT, X m x L) OT x m x L) + O((m + 1)mmax) L (T T>)))
=0 X {Neuax
28

---

D.6 Proof for Theorem 3
Define V = IT pk (CR Sk-1)0*(BA, Ss RR). Define
i-1
P/= (L-1)o
S; R,) II
PC 1S=1 (Rs Sp Rs).
(Rs
k=1
Define
(S)) = ppt+ lat 1 S/].
For any Pt, we choose the following parametric submodel:
:: Pi 1(Q' PP).
For any m....
2/ = Ev, (nd
Fix i (1, m... Then, consider the differention with respect to pi+1
0
Bug [u((s))]
0t p ())
Ot
0
0t E M (s)
Ot E
pp (5)(
8
Øt p
(ij (St+1)) )]
Øt. Pp ( (S))
E pt
| (At +1 (S++1) 1b (
Equ io { 1 (S:+1) ) (S
Also, consider the differentiation with respect pl:
ot Ep ( = Eq: (S1) - vho)-
Then, define sp as the differentiation with respect to p as
8 (S, 16. 1ho) =
m [i
{/b
10 111
ifi>1
v00
=fi=1.
Then,
i Øt ((1 ... ....... pk)
dP: ô
V(Pl, ... ............ .t.....
pm+1 σ)
Öt ÖP;
1=0
VEIx
(=0
Σ
Egi (p'(( mh v/))].
TETx
which completes the proof.
D.7 Proof for Theorem
Structure the proof. Theorem will proven based on Lemma D.2 Lemma D.3 and Lemma D.4 Specifically, we proceed the proof follows:
1. We will prove Lemma D.2 Lemma D.3 and Lemma
D.4
2. Berry-Essen'' inequality (Berry 1941 will be stated as a preliminary in Prop. D.I
3. Theorem 4 will be proven based on the main lemmas and Berry-Essent 's inequality.
29

---

D.7.1 Helper lemmas
We first state and prove helper lemmas.
Lemma D.1.
V0
1++1
-b33 + Eml(Aa))
(D.I)
i=1
Proof of Lemma D.1 D.1
By the total expectation law, suffices to show that
(P) = Ep1 (ab(()))
This holds from Prop.
Lemma D.2 (Decomposition).
Define the following
m
+ (fa #)
E Pit
[+ (1t
fie)] + Ep [724]
(D.2)
T (Mo: To)
E pi+a 11
31 Ep [M
(D.3)
The following decomposition holds:
( ft (140 nO) =
E3 pp
0 6r
{u6 - f0" H -
(D.4)
1=1
Proof of Lemma D.2
First,
(f,, - 1/10 $( ft 4 To)-
Also,
E pm 11 [ (M00) f1 }1 + 1m [TT'" mm
EE pm
[T] m 4u
Vi
Epm+i ((=)" Am (iath _ 16
For 1,... 1, define
- Ep S. B.].
Then.
E Epita [+ ( [ + Ep [= Ep (Trngla
= Ep (T - # H'H Hhlp IH..
Also, for any + and corresponding + and for all 1,... 1, by the definition of the T/ nuisance,
E Ep [m/ i+1 [11 = E (x
30

---

Then.
Epm+1 [ (100 |] + E pm (TT ( - pm (=T" Mo
Epan ff (1b(lttt))) (²") Epria [= Epeia (monala't]
Epiti [= 15 [Mbl -2 2² Epa [T - 40
Epan Hmm - H'L - 131
Note that Epa p ] Ep [itt), since
Ep jū (S1 U B1)
Ep RR (R1 |S] (() Ri)]
('((S) (UIS)UB))) R1)
R. (Rı Si RR) UBi)) Ri)
= p2
P2(S) B1)
(C((i) RR. (R. S1 (R)) UB,) \ R.)
p2
(2 BB)
pa mdp
Therefore,
( f)
D(Ho, TO)
Epir 1p - 2 Mit IH
iid Lemma D.3 (Stochastic Equicontinuity), Let P. Let D Do DD, where n = (Dol. Let
be function estimated from D1. Then, in probubility greater than e for any e (0, 1).
BDo-P r|]
upl-c 1l flle
(D.5)
Vnc
which implies that
[lf fllp
Ep.-Plff fl] Op
n
Proof of Lemma D.3 This proof is from (Kennedy et al. 2020 Lemma 2). Since f is function of Dj, we will denote fr. Define following random variable of interest:
X = E-DE-rlfe f]-
Then, the conditional expectation of X given D, is zero, since
EP
Σ (v.)))
D
Splfo,(V.)) Dil = - (Plfo,()) Pi] Epllo, (i),
31

---

where the third equality holds by the independence of Do and D1. Therefore,
Ep(X] Di] - - Di]
= =EplED, IfD. fl Pi] Ep(BplfD, f] Di] EE[EEPID, fI Pi] - Ep[EplfD -f] Di] 0.
Also,
p D1] = r [0D fl Dil
Vr /] Di)
Wplfo, f D1)
n I fo. f11?.
n
By applying the (conditional-) Chevyshev's inequality,
1 P(IX Ep(X Dil 2 t 1 S
1
Vplx
Dil
llfo. - -f1p.
1
n12
Then,
P([ 2 1) P(IX [x Di]] 2 1)
-P(DD) [P X - Ep|X Dill 2 t D2)]
S nt2 So. - -11
In other words, probability greater than 1 Ilo. f13
If
I1,o, -flle vne
then
I1fo,-fll -flle in the probability greater than e for any e e (0.
X
V/c
Lemma D.4 (Combining concentration inequalities). Suppose P(A > t) I bx/t² for k
L.... K. Then,
K
As S
iK
21-
bk
Proof. The event ΣK Ag S tK includes the case where Aş < t for 1, K. Therefore,
P
Σ A S iK
2 P(1 < and and Ax
1 P(1 > oo or Ak > t)
K
2 Σ P(A >
K by
2 21 Σ
1
1=1
32

---

D.7.2 Preliminary Results
Proposition D.1 (Berry-Esseen's inequality (Berry) 1941 Esseen 1942 Shevtsova 2014)). Sup- pose D [X,, . are independent and identically distributed random variables with [x[X]] = 0, E,([X]] = x² and [r[[X,]] |1 = =k². Then, for all x and n,
P
Ep [X] <x) =(x)) v σ Vnn
0.4748k
00
D.7.3 Proof of Theorem (1)
By Lemma
D.2
we decompose the error as follow:
K
$ - 0
Ep
[o0l
(D.6)
k=1
Ep ph 10t - -b11
(D.7)
I
{=1 k=1
7
Epiti H46 - 1 mi
(D.8)
T
Define
R = P
[0o]
1
Σ
E
D -px o8)
L
t=1
Then it completes the proof.
D.7.4 Proof of Theorem 4- (2)
We first study the term Epr (ot]. By Chebyshev's inequality,
P
EDi ED [001
>
Pk.0 VIDA
22
Equivalently,
pk.u
P
P(E lobil >)
By Lemma D.4
K
K
Pk.0
P
E D pa
|00| < 11
>
DE
By Lemma D.3
pk ([Bpy-pu
>
112
1
11
(D.9)
12
D
By Lemma D.4
K
p
E B pa [3] -61|
s Kk2
2
1
(D.10)
12
DD
Fm Am
km
33

---

p D
and l2 = /
110
- pk.
Choose / =
===1
(D)l
Then, with probability
greater than -, 6
K
PEO
2
7
K
1
& 1
R
K
DE
Df
k=1
2
K
Pk.0
K
KKy
V
+
IDjl
N=
k=
D.7.5 Proof of Theorem 4- (3)
By Lemma D.3
100
18112
p* (Eezz_ 108
>t)
- t2
(D.11)
DDA
By Lemma D.4
p&
Pp. -p4 10b
ob1 21
2
1
1
1
2 pk
(D.12)
T.
7
[D/
Equivalently, by choosing
1 E 1==1
1108 -
13
Di1
wpl-c
1
|
of
Edp ph 100 - 0b1
(D.13)
L
|D
Define
4
EDe [00]
(D.14)
L
B*
Epy pa (0)
-b1
(D.15)
I.
C*
Epp ph 18 08.1
(D.16)
T,
t=1
|| o lp
A
(D.17)
DD
Here,
12th = A + B²
(D.18)
Then,
p (R² < x)
(D.19)
pt (Ax Bg < x))
(D.20)
pd (As < x - Bx)
(D.21)
S pd (As < x + Ck)
(D.22)
w
< p (Ax < + Ab).
(D.23)
34

---

Then.
|p (AA < AA) +(x)
(D.24)
|p (Ax < A+) (0(++An)) + As) +(x)
(D.25)
$ pd (As < + Ax) ((x+AA))) + 10(x + Ax) - +(x)
(D.26)
0.4748K0
+ (x + AR) (z))
(Prop. D.1
(D.27)
phoo VID*
0.4748K0
+ (x))aa
"Mean-value theorem) (D.28)
P V DD
0.4748K0
2m A
(D.29)
PR.OVIAA
This completes the proof.
D.8 Proof for Corollary
By Cauchy-Schwarty inequality,
1 Σ
Bpati H10 p&Н#
_ O pi-t (11e
pblll #!|l
(D.30)
Given assumption, the upper bound in Eq. 15 converges at Opt (D/)) Therefore, we conclude
that Rtth converges in distribution to normal (0, pi.)).
E More Experiments
In this section, demonstrate the DML-UCA estimator through examples for the ETT, S- admissibility, FD, Verma's equation, and Ctf-DE described in Sec. For each example, the proposed estimator constructed using a dataset D² following a distribution p Our goal to provide em- pirical evidence of the fast convergence behavior of the proposed estimator compared competing baseline estimators. We consider two standard baselines in the literature: the 'regression-bacte estimator (reg)' only uses the regression nuisance parameters and the 'ratio-based estimator (ratio)" that only uses the ratio nuisance parameters म, while our DML-UCA estimator ('dml') uses both. Details of the regression-basee ('reg') ) and the ratio-based ("ratio") estimators provided in Sec. A Details of experimental setting is provided in Sec. In this experiments, we set all variables other than the treatment variable X as continuous.
We compare DML-UCA estimator to the regression-based estimator ('reg") and the ratio-based jest for est E (reg, (reg,,w, dml} to denote the regression-
estimator ("ratio"). In particular, based, probability and DML-UCA estimators. We assess the quality of the estimators by computing the average absolute error AAE which defined as follow. For the ETT and Ctf-DE, AAE = jeess Vol. where v/ = E{ X = 1] for the ETT and vp = E( | X = 2] for the Ctf-DE. For the other examples, AAE == Jomqin(X] Exeedomain( 1n (x) (x)
where tu (x: = E|y do(x)), (x)) is an estimator for v'o(xr) and dom (X) is a cardinality of the domain of X. Nuisance functions are estimated using XGBoost (Chen and Guestrin 2016). We ran 100 simulations for each number of samples n (2500, 5000, 10000, 20000) and drew the AAE plot. We evaluate the AAESE in the presence of the "converging noise c' as in Sec.
Statistical Robustness. The AAE plots for all scenarios are presented Fig. E.4 For all examples, all the estimators ('reg', 'pw", 'dml') converge as the sample size grows. Furthermore, the proposed DML-UCA estimator outperforms the other two estimators by achieving fast convergence. This result
35

---

0.40
0.20
0.45
0.4
0.40
0.15
0.25
0.20
0.35
0.10
0.15
0.20
0.10
0.2
0.29
0.30
0.05
0.20
0.00
1000
2000
2009
1990
2070
Trer
3000
19991
795
(a)
(b)
(c)
(d)
(c)
Figure E,4: (a) ETT in Sec. B (b) Transportability (S-admissibility) in Sec. B
(C) Front-door in
Example (d) Verma in Example 2
(e) Ctf-DE
Example
corroborates the robustness property in Thm. which implies that DML-UCA converges faster than the other counterparts.
F Details in Experiments
model for estimating nuisances. We implemented the model using Python. In modeling nuisance using the XGBoost, we used the command xgboost... XCBCC @CBClassifies (eval careallmmerii=eeee cloglosss Y to use the XGBoost. We tuned the parameters for each examples empirically guarantee the convergence of the regression and ratio nuisances. For each examples, the same parameters are used globally for implementing DML-UCA, regression-bed based estimator, ratio-based estimator, or other competing estimators (Fulcher
As described in Sec. used the XGBoost (Chen and Guestrin 2016
al. 2019 Jung al. 2021a).
Now, we present the structural causal models (SCMs) utilized for generating the dataset. Furthermore, we include a segment of the code employed to generate the dataset.
F1
FD Fig.
aa for Simulation in Fig. 2a
We define the following structural causal models:
U ~ normal (0.5, 5),
Uz. normal (0, 1), for 1, ......,,, dz
C. fc, where C {C, i 1,... .dc}
X = UC,C,U),
Z: = (2, where {Z: i= = 1,... dzp
Y = Z,C,
where
fu,(U) =
+ exp (0.25U + 20 -
(C, = Binary
1 exp(2CTT U)
fz, (C,x) = Binary
1+ e+exp(2X - + 0,5CT1+ Uz,)
1
S,CC,,UU cx + (1/d2)
2U)
The parameterization for XGBoost used in called (mu_params) and called (pi_params) is the following:
Detailed parametrization of parameters including leamning rates, maximum depth of the trees, etc. are explained in www.thedos.rrrrcccccoooooooooo. http:///ww....e//////a////ttto/a module-xgboodd segbbost.trraiing
html#
36

---

mu-params =
'booster;; 'gbtree', 'eta': 0.3,
"gammaa: 0,
mmax_eephh': 10,
chin_child_ 1,
subsample' 1.0,
colssample_bytre'': 1, 'lambda': 0.0,
"alpha': 0.0,
'objective:: 'reg: squarederoor'', 'rmse',
'n_jobs': }
pi_params 'booster'; 'gbtree', 'eta': 0.3,
"gamma': 0, max__epth': 10,
minn chincciid____htt" 1,
subsample' 0.0,
colsammle__ytree" 1,
objjecive': 'binary:logistcc', : 'logloss',
lragdlamdda's 0.0,
alg_alpha': 0.0,
'nthread': 4
F.2 Verma (Fig, lb) for Simulation in Fig. 2b
We define the following structural causal models:
UXb nwrmalomlal.mmomo (1, 5),
UAY normal (00,00l-11
UA ~ normal (0,
UB normal (0, 1)
X = /x (Ux B)
A = fA. (X, UAY), for = 1, ..... da
B. fB. (X,Ux), for 1.... dp
Y = (B.UAv),
where
+ exxp(2Uxx (1)xxxx8)1) (1
fx Binary
1
(X, Binary
1+exp(2X + U UAY) 1
(x,(X,UXB) -Binary
oxp 1 Up + 0.5UxB)
(B, = 1 exp(p211 0.50Ar)
37

---

The parameterization for XGBoost used in called (mu_params) and π called (pi-params) is the following:
mu mu_params
booster': : 'gbtree',
eta': 0.35,
"gammaa 0,
"max_ max__epph': 6,
min_chiil__eiihht 1,
'subaample': 1.0,
ccossample.bttree 1,
'lambda': 0.0,
'alpha': 0.0,
'objective': 'reg:squarederror',
'eval_mee :ell__etric': 'rmse',
'n_jobs': 4 # Assuming you have 4 cores
pi_params
'booster': 'ghtree',
'eta': 0.1, ggamaaa 0, mmx mexx_hepth: 10, min_child__wiihht ssubsample': 0.0,
colsample_bytree'. 1,
"objective:: 'binary:logistcc,, Change as per your objective 'eval_metric' 'logloss', Change per your needs
'reg_lambda" 0.0,
'reg_aphha: 0.0,
'nthread': }
F33
Example estimand (Fig. Te) for Simulation in Fig. 2c
We define the following structural causal models:
Ux, ~ (1,
Ux,,Y ~ normal (-1, 5),
Uz,y normal (0.5, 0.5)
UR ~ normal (0, 0.5)
Uz ~ normal (0, 0.5)
U X, normal (0,00,)
X1 = (U,,Ux,,)) /x,(Ux,,,Uxx,x)
Z, = fz, (X1. Ux1.2. for 1, 1.... .dz
R. (Xi,,),, for 1, dR
Y = fy (B.UAY),
38

---

where
(x,(Ux,,; Binary
+ expe(UUUxx UxxY
1
fn1) Binary
1 + exp((2X - 1 + UR)
1
Ux1,2,, Binary
e 1 + Uz + Ux,,Z + Uzz
1
(Z, Binary expx 1)2 Ux.))
UxX,,
1 exp((1ddRRRT 2 2(U x Uzy))
The parameterization for XGBoost used in called (mu_params) and π called (pi_params) is the following:
mu mu_params =
'booster: 'gbtree',
'eta:: 0.3,
gamma': 0,
max__epth': 8,
mii_chiild_weihht 1,
ssusample'' 0.8,
colsample_bytree" 0.8, 'lambda': 0.0,
'alpha': 0.0,
"objective': 'reg:squarederror',
goval_metric'. 'rmse',
'n_jobs': 4 # Assuming you have 4 cores
pi_params
'booster': 'gbtree',
leta': 0.1, "gamma': 0, maxx max__tpth': 10,
wain_child_weight's 1,
suubamppe' 0.75,
colsample_bytree" 0.75,
objective'; 'binary:lgiisic', # Change as per your objective "eval_metric'' 'logloss', Change as per your needs
'ambdaa 0.0,
'reg_alpha'; 0.0,
'nthread':
39

---

F4
ETT in Sec.Bfor Simulation in Fig. E.4a
We define the following structural causal models:
Ux normal(0, 1)
Uy ~ 0.5 textttnormal(0.) 1)
Z 0.25normal(O, (,
X fx(Z)
Y = (x
where
(x Binary
1 Ux)
(Z.
+
The parameterization for XGBoost used in le called (mu_params)) and called (pi_params) the following:
mu mu_params =
boosstr': 'gbtree',
eeta: 0.5,
gamma': 0,
max__epth': 15,
min_child_weiihht 1,
ssubaampl'' 0.8,
colsample_bytree'. 1,
"lambda':
'alph': 0,
"objectiv'': 'reg:squarederror',
geval_metric' 'rmse',
'n_jobs': 4 # Assuming you have 4 cores
pi_params
'booster': 'gbtree',
'eta': 0.3, "gamma': 0, maax max__tpth': 10,
miin_child_weight's 1,
subsample' 1,
colsample_bytree".
objective'; 'binar:logistic', # Change as per your objective
'eval_metric'' 'logloss', Change as per your needs
'ambdaa 1,
'reg_alpha'; 0, 'nthread':
40

---

E,5
Transportability in Sec. for Simulation in Fig. E.4b
We define the following structural causal models:
Ux normal(O,)
Uy 0.5 textttnormml(0.1 1)
Z 0.25normal(O, 0.5, d2) + Snormal 0.5, d2)
X (x(
Y = fy(,
where
[x( = Binary
(T+cp((1/22(22111))))
fy (Z. X)
1+
- -)
The parameterization for XGBoost used in y called (mu_paramss) and called (pi_params) the following:
mu mu_params =
'oosstr': 'gbtree',
'eta: 0.3,
gamma': 0,
max__epph': 15,
min_child_weiihtt 1,
ssubaampl'' 0.8,
colsample_bytree'. 1,
'lambda':
'alph': 0,
"objective': 'reg:squarederror',
geval_metric' 'rmse',
'n_jobs': 4 # Assuming you have 4 cores
pi_params
'booster': 'gbtree',
leta': 0.1, "gamma': 0, maxx max__hept': 10,
min_child_weight! 1,
subsample' 1,
colsample_bytree":
objective'; 'binar:logistic', # Change as per your objective 'eval_metric"" 'logloss', Change as per your needs
lambbaa 1,
'reg_alpha'; 0, 'nthread':
41

---

F.6
FD with continuous mediators for Simulation in Fig.
E.4c
We define the following structural causal models:
Uc normal (0, 1, de))
U ~ normal (0,
C = fc(U)
X [x(U, C)
Z = C2
Y = Cr(U,
where
Jc(U) = 0.25U + 2U -1
(U C) == Binary
(x
Cz + exp(0.1CT1(2X - 1 XX
(2, = 1 exp CT1 (2Z 1) U))
The parametcrization for XGBoost used in i called (mu_params)) and π called (pi_params) the following:
mu paraasrmmm 'ooster': 'gbtree',
etaa:: 0.01, 'gaamaa: 0,
mmax max__epph': 10,
cain_chhll_weight 1,
subsample' 1.0,
colsample_bytree"" 1,
"lambda': 0.0,
'alphaa: 0.0,
'objective:: 'reg:squarederror",
wvrl_metrich 'rmse', 'n_jobs': 4
pi_params {
'booster:: 'gbtree', 'eta': 0.3,
'gamma': 0,
maxx_ death'' 20,
min_chhild_wiight's 1,
subsample': 0.0,
colsample_bytree" 1,
objective'; 'binary: 'eval_metrice 'logloss',
'reg_lambda:: 0.0,
'reg_alpha: 0.0, 'nthread':
42

---

E.7
Verma's equation with continuous mediators for Simulation in Fig. E.4d
We define the following structural causal models:
Uxb normal (1, 1),,),,
UAY ~ normal -1,
X = fx(UxB)
A TA(X,UAN
B = fa(X,
= (Y(B,UAr):
where
fx(UxB) = Binary
+exp(2Uxx== xB
1
TA(X,UAY) = Binary
+ - 0.5UAY)
1 1 + 0.50x
Te(X,UXB) = Binary
fy = 1 + exp - -
The parameterization for XGBoost used in 4 called (mu_params) and called (pi_params) the following:
mu_ params
booster': 'gbtree',
leta': 0.35, "gammaa: 0,
mmax_ maaxh' 6,
wein_child_weight': 1,
subsample': 1.0,
colssmmple_bttre'' 1, 'lamdda' 0.0,
"alphaa 0.0,
'objective:: rrg::squarderror""
: 'rmse',
'n_jobs': # Assuming you have cores
pi_params {
'booster': 'gbtree', 'eta': 0.1,
"gammaa: 0,
"max_ max__epph': 10,
min_child_weightt 1,
subsample'; 0.0,
colsample_bytree" 1,
'objectiv':: 'binary:logistic', # Change per your objective deval_metrice 'logloss', Change as per your needs
'reg_lambda" 0.0,
"reg_alpha': 0.0,
'nthread'; 4)
43

---

F88 Cif-DE in Example for for Simulation in Fig. E.4e
We define the following structural causal models:
U normal (0, 2),
X = fx(U)
Z (z(
W = fw(X, Z)
Y = fr 1x(.2,WW,,
where
0 [+exp(2xxx
< 0.5
[x(U)=
1f 0.5 [+exp(2Ux=-] 0.8
2 0.8 +exp((UUxB-1)
f2(U)= 1 exp( -U +1
Sw (x, = + exp(X - -1 2)
fr (,) 1 (3X - 1 0.1Z + 0.1W + W(X - 1))
The parameterization for XGBoost used in called (mu_params) and π called (pi_params) is the following:
mu params {
booster': 'gbtree',
'eta: 0.3, # vab
gamma': 0.0,
max__epth' 6, #vb (same as va)
min_chiild_weight!..
subsample' 1.0,
colsammle_bytree' 1,
'lambda': 0.0,
'alpha': 0.0,
"bbjectiv'': 'reg:squarederror",
goval_metric' 'rmse',
'n_jobs': 4 # Assuming you have 4 cores
pi_params
'booster': 'gbtree',
'eta': 0.05, "gammaa: 0, mmax_ max__hpph': 10,
whin_chhiddweighh'' 1,
ssubsampl'' 1.0, colsample_bytree""
objective': 'multi:
# Change as per your objective
'num_class'; 3, 'eval_metric" 'mlogloss', # Change per your needs
'reg_lambda': 0.0,
'reg_alpha: 0.0,
'nthread':
44

---

45

---

NeurIPS Paper Checklist
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integral part of your paper submission. They visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will published with the paper.
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1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We clearly state our main claim in the abstract. The contributions are described in the introduction and are clearly reflected throughout the entire paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made 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 other settings.
is fine to include aspirational goals as motivation as long is clear that these goals not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
46

---

Answer: [NA]
Justification: We explicitly state the assumptions that theories are depending on. This assumption serving as limitations of the work.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers penalize honesty concerning limitations.
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3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: Detailed proofs are provided in the supplemental material. Additionally, we have provided the full set of assumptions for each theorems.
Guidelines:
The answer NA means that the paper does include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced 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 provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
47

---

Answer: [Yes]
Justification: We provide the detailed recipe for running the simulation and the guideline for implementing theories. Also, we provide a detailed method on how the synthetic data are generated.
Guidelines:
The answer NA means that the paper does not include experiments.
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Reviews:
[{"review_id": "EvaY5oJ97I", "reviewer": "Reviewer_Q5Gy", "summary": "The paper introduces a novel framework, unified covariate adjustment (UCA), which covers a broad class of sum-product causal estimands and additionally develops a scalable estimator (via DML-UCA) that ensures double robustness.", "questions": "See Weaknesses", "limitations": "yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The paper presents a well-developed theoretical framework with clear assumptions and derivations. The motivation to extend existing estimands is well-articulated, and the comparisons with prior work are thoroughly examined.\n\n* The paper is well-written and clearly presents the motivation, methodology, and contributions.\n\n* It is interesting to revisit the coverage and scalability of previous studies and provide comprehensive evaluations.", "weaknesses": "* The proposed method is based on structural causal models, which have been extensively studied.  UCA-class is an extension of the sequential back-door adjustment (SBD), and there are already existing studies that address similar questions \n\n* The authors mention in Example 1 that the Front-Door adjustment (FD) can be represented using the UCA framework. Are there any assumptions required to validate this representation? Similarly, to represent the Verma constraints as UCA in Example 2, are there any criteria that could be followed to verify the representation in practice? What are the requirements and limitations for implementing these representations to ensure the reliability and validity of the estimation in general?", "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!\n\n> UCA-class is an extension of the sequential back-door adjustment (SBD), and there are already existing studies that address similar questions\n\nIndeed, UCA is an extension of the SBD, and we have appreciated and cited papers regarding estimating the SBD estimand. However, naively applying the SBD estimators to the UCA-class (e.g., FD, Verma) may lead to biased estimators whenever the value of SBD estimand and the UCA estimand don't match. Also, modifying the existing SBD estimators to the UCA-class is non-trivial since variables that are fixed and marginalized at the same time (e.g., ‘X’ in FD) are not properly treated in the SBD estimators. A special method (such as the empirical bifurcation in Def. 6) is required to develop an estimator. Furthermore, for Tian’s adjustment, the weighting nuisances $\\pi^{i}_0$ do not have the same form as those in SBD. In summary, developing doubly robust estimators for the UCA class is a novel and non-trivial task.\n\n---\n\n> Are there any assumptions required to validate this representation? Similarly, to represent the Verma constraints as UCA in Example 2, are there any criteria that could be followed to verify the representation in practice? What are the requirements and limitations for implementing these representations to ensure the reliability and validity of the estimation in general?\n\nSome causal queries that satisfy known graphical criteria (e.g. BD/SBD, FD, or Tian’s adjustment) can be represented as an UCA. Recall that representing the Tian's adjustment through the UCA estimand is demonstrated in the paper. \n\nOn the other hand, we have further developed the sufficient criterion to determine if a given estimand can be represented as a valid UCA expression. The idea behind the criterion is as follows: If\n\n1. The target estimand is expressed as the mean of the product of conditional distributions over $(\\mathbf{C}_1, \\mathbf{R}_1, \\cdots, \\mathbf{C}_m, \\mathbf{R}_m)$; and \n\n2. The variables that are marginalized and fixed simultaneously (e.g., $X$ in the front-door adjustment) only appear in $\\mathbf{C}_1$ (that is,  $\\mathbf{S}^{b}\\_{i-1} \\cap \\mathbf{C}\\^{\\geq 2} = \\emptyset$), \n\nthen the proposed methods can be applied. These conditions are sufficient for applying the empirical bifurcation technique (Def 2) that allows scalable estimation.", "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": "i7zSwHPOGJ", "reviewer": "Reviewer_Z7av", "summary": "The paper presents a class of adjustment formulas called unified covariate adjustment (UCA) which is shown to be able to express many classes of adjustments known in the existing literature. A scalable and doubly robust estimator for UCA is also presented along with some experimental results.", "questions": "- How are the symbols in Table 1 determined? \"scalable\" is defined to be \"evaluable in polynomial time relative to number of covariates and capable in the presence of mixed discrete and continuous covariates\", but about \"coverage\"?\n- Consider rephrasing the awkward-sounding sentence \"Our work strives maximizing coverage...\" on Line 73-76?\n- $S^b_{i-1}$ first appears in Line 141. How is this defined? How does it differ from $S_{i-1}$?\n- In Line 144, $S_0, C_0, R_0$ are referenced but how are they defined? Also, how do the $S_i$s relate to the variable set $V$? Is $S_i$ the c-component of the $i^{th}$ vertex?\n- Equation (4): The notation $v^{(i−1)}$ is undefined. Is $v^{(i−1)} = \\\\{v_1, \\ldots, v_i\\\\}$ as in the Tian-Pearl 2002 paper?\n- Line 312: What is $O_{P^{i+1}}$? How does it differ from just the usual big-O notation? Also, how do the errors in the estimation scale with the number of samples? Lines 307-309 only say \"**if** the terms converge at a rate of $n^{-1/4}$, then DML-UCA coverges at a rate of $n^{-1/2}$\". Why do those terms converge at a rate of $n^{-1/4}$?\n- Given a general causal graph and causal query, how does one find a suitable and valid UCA expression? What is the procedure?\n\nPotential typos:\n- Double comma on Line 94\n- Extra ) on Line 153", "limitations": "Nil", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "The proposed UCA estimator seems to be very expressive and is able to model many existing estimators in the literature. Examples were given to show how existing estimators can be expressed as a UCA estimator.\n\nThe paper also proposes a doubly robust method to obtain UCA estimates.", "weaknesses": "- It is of my *personal opinion* that the paper lacks polish. There are many instances of technical notations being used without first properly defining them, making it hard to understand and follow the discussion (see the Questions section for some of them). The reader should not be expected to guess the meaning of certain notations by cross-referencing across subsequent pages (at best check the preliminaries/notation section) or even across other paper references.\n- The paper claims on Lines 57-58 that \"while these estimators are designed to achieve a wide coverage of functionals, they lack scalability due to the necessity of summing over high-dimensional variables\" but the general definition of UCA in equation (1) also involves summing over potentially many variable values in $c \\cup r$. Please explain clearly why UCA avoids scalability issues.\n- Line 302-304: It is not true that only asymptotic analyses were known for all these estimators. For example, [1] gives non-asymptotic finite sample guarantees for the Tian-Pearl adjustment. The paper would benefit from a comparison against such prior works and illustrate how DML-UCA adjustments are indeed more sample efficient as compared to existing estimators.\n- Why is there no experiments against the Tian-Pearl adjustment?\n- No code was released (though some parameters were given the appendix).\n\n[1] Arnab Bhattacharyya, Sutanu Gayen, Saravanan Kandasamy, Vedant Raval, Vinodchandran N. Variyam Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:7531-7549, 2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback and for the opportunity to provide further elaboration. \n\n---\n\n> technical notations being used without first properly defining them\n\nWe will further proofread the paper and the preliminaries. \n\n---\n\n> Please explain clearly why UCA avoids scalability issues.\n\nExisting BD/SBD estimators avoid scalability issues by replacing marginalization with nested expectation. However, when a variable is both marginalized and fixed simultaneously (e.g., FD: $\\sum_{x’,c}E[ Y | z,x’,c] P(z | x, c) P(x’,c)$, where $X$ is fixed to $x$ in $P(z | x,c)$ and marginalized by $\\sum_{x’}$ in other components), representing the marginalization operator as a nested expectation is non-trivial. These challenges lead to potential scalability issues for previous FD estimators (Fulcher et al., 2019; Guo et al., 2023), which have a complexity of $O(n2^m + T(n,m))$ ($m$: the dimension of variables, $T(n,m)$: time complexity of learning nuisances). In contrast, the newly proposed UCA estimator uses empirical bifurcation to replace marginalization with nested expectation, achieving $O(n + T(n,m))$ by leveraging the empirical bifurcation method.\n\n---\n\n> [1] gives non-asymptotic finite sample guarantees for the Tian-Pearl adjustment.\n\nWe will cite all the referred papers. Please note that lines 302-304 discuss DML-style estimators, while the mentioned paper provides finite sample guarantees for a basic plug-in estimator under the discrete random variable setting.\n\n---\n\n> Why is there no experiments against the Tian-Pearl adjustment?\n\nFigure 2(b,e) shows the experimental results for Verma's graph (Figure 1b), which is an example instance of the Tian-Pearl adjustment.\n\n---\n\n> No code was released (though some parameters were given the appendix).\n\nWe will make the code available after the revision. Thank you. \n\n---\n\n> How are the symbols in Table 1 determined? \"scalable\" is defined to be \"evaluable in polynomial time relative to number of covariates and capable in the presence of mixed discrete and continuous covariates\", but about \"coverage\"?\n\nCoverage indicates whether established estimators exist. For obsID/gID classes, estimators like those by Jung et al. (2021a, 2023a), Xia et al. (2021, 2022), and Bhattacharya et al. (2022) are marked in the \"Prior\" column. The UCA, covering Tian's adjustment, is checked in its respective column. \n\n---\n\n> Consider rephrasing the awkward-sounding sentence \"Our work strives maximizing coverage...\" on Line 73-76?\n\nThank you. We will rephrase as follows: “Our work aims to maximize coverage, enabling the effective development of scalable estimators with the doubly robust property.”\n\n---\n\n> $S^b_{i-1}$ first appears in Line 141. How is this defined? How does it differ from $S_{i-1}$? \n\n$\\mathbf{S}^b_{i-1}$ represents a set of variables fixed to $\\mathbf{s}^b_{i-1}$ in a conditional distribution $P^i(\\mathbf{V}) = Q^i(\\mathbf{V} \\mid \\mathbf{S}^b_{i-1} = \\mathbf{s}^b_{i-1})$ (e.g., $\\mathbf{S}^b_1 = \\{X\\}$ with $\\mathbf{s}^b_1 = \\{x\\}$ in FD, Example 1). Meanwhile, $\\mathbf{S}_{i-1}$ is a subset of the union of $\\mathbf{C}^{(i-1)}$ and $\\mathbf{R}^{(i-1)}$, excluding the fixed set. \n\n---\n\n> $S_0, C_0, R_0$  are referenced but how are they defined? Also, how do the $S_i$s relate to the variable set $V$? Is $S_i$  the c-component of the $i$th  vertex?\n\n1. $S_0$, $C_0$, and $R_0$ are all defined as the empty set. We will add this in the preliminaries.\n2. We define $\\mathbf{S}\\_{i-1} := (\\mathbf{C}\\^{(i-1)} \\cup \\mathbf{R\\}^{(i-1)} ) \\setminus \\mathbf{S}\\^{b}\\_{i-1}$ in line 144. This is a subset of the variable set $\\mathbf{V}$.\n3. Thank you for the good question. $\\mathbf{S}_i$ is not a c-component. We only used $\\mathbf{S}\\_{X}$ to denote the c-component containing $\\{X\\}$. We will improve the notation to distinguish them more explicitly. \n\n---\n\n> Equation (4): The notation $v^{(i-1)}$ is undefined. Is $v^{(i-1)} = \\{v_1,\\cdots,v_i\\}$  as in the Tian-Pearl 2002 paper?\n\nYes, it's defined in line 98.\n\n---\n\n> Line 312: What is $O_{P^{i+1}}$?  How does it differ from just the usual big-O notation? Also, how do the errors in the estimation scale with the number of samples? \n\n> Why do those terms converge at a rate of $n^{-1/4}$?\n\n1. As written in line 104, $O_P$ is a stochastic boundedness called the big-O in probability [van der Vaart, \"Asymptotic Statistics.\"] (1998). The expression $f(\\mathbf{V}) = O_{P^{i+1}}(n^{-1/4})$ means that $n^{1/4} \\times f(\\mathbf{V})$ will be bounded even when $n$ increases to infinity. This indicates that $f(\\mathbf{V})$ decreases at least as fast as $n^{1/4}$. If the error term is $O_{P^{i+1}}(n^{-1/4})$, then it decreases at the rate of $n^{1/4}$. \n2. Theorem 3 and Corollary 3 show that when nuisances converge at the rate of $n^{-\\alpha}$ ($\\forall \\alpha \\in (0,1)$), the estimator can converge at the double rate: $n^{-2\\alpha}$. We demonstrate this with $\\alpha = 1/4$, since $n^{-1/4}$ is the fastest convergence rate for modern ML models like neural networks [1].\n\n[1] Györfi, László, et al. “A distribution-free theory of nonparametric regression” (2002)\n\n---\n\n> how does one find a suitable and valid UCA expression? \n\nSome causal queries that satisfy known graphical criteria (e.g. BD/SBD, FD, or Tian’s adjustment) can be represented as an UCA. On the other hand, we have further developed the sufficient criterion to determine if a given estimand can be represented as a valid UCA expression. The idea behind the criterion is as follows: If\n\n1. The target estimand is expressed as the mean of the product of conditional distributions over $(\\mathbf{C}\\_1, \\mathbf{R}\\_1, \\cdots, \\mathbf{C}\\_m, \\mathbf{R}\\_m)$; and \n\n2. The variables that are marginalized and fixed simultaneously (e.g., $X$ in the front-door adjustment) only appear in $\\mathbf{C}_1$,\n\nthe proposed methods can be applied. This criterion will be added in the revised version of the paper. \n\n---\n\n> Double comma on Line 94, Extra ) on Line 153\n\nWe will fix the typos, thank you.", "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": "IOrpDbF66u", "reviewer": "Reviewer_ZBfF", "summary": "This paper describes the estimand framework \"unified covariate adjustment (UCA)\" and discusses its coverage with multiple examples (Front-door, Verma's equation, Counterfactual directed effect and most importantly Tian's adjustment). Then it develops an estimator for this function class and shows that it is scalable. Lastly the authors run experiments on simulated data to empirically demonstrate the robustness and the scalability of their estimator.", "questions": "* Could you give intuition concerning the meaning of the $\\bm{C}_i$ and $\\bm{R}_i$ and $\\bm{S}_i$ in Def 1 as well as $\\bm{\\mu}$ and $\\bm{\\pi}$ for non experts.\n\n* Could you discuss the limits of DML, ie. when the dimension grows and when the sample size shrinks.\n\n* Concerning the experiments, each point corresponds to the average of 100 simulations. Could you provide the variances as well?", "limitations": "* The authors do not discuss which function classe are not covered by UCA. Moreover, they do not discuss the limits of their estimator DML when the dimension grows and when the sample size shrinks.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "* This work is original as it introduces - to the best of my knowledge - the only scalable estimator for many functions classes. It makes a lot of novel causal inference results applicable in real applications.\n\n* The overall quality of this work is very good, the results are provided as theoretically sound theorems and empirically confirmed through simulated data experiments.\n\n* This paper is clear, the contributions are clearly stated and the paper is well structured with some useful examples.", "weaknesses": "* All the proofs are provided in the supplementary material and the intuition for the proofs are not provided in the main paper.\n\n* Some further experiments could be interesting even they are not necessary. For example: how does DML compare to prior scalable estimators on BD/SBD? In figure 2a,b,c how much can the dimenion of the summed variables grow before the running time of DML reaches unreasonable values (eg. 2000)? Similar question for figure2d,f how little can thee sample size be before the errors of DML reaches unreasonable values?\n\n* UCA could be more thoroughly delimited. It is well defined and the authors give examples of scenarios that are included, how they do not clearly discuss which type Ctf-ID are covered and which are not (This is also true for obsID/gID and transportability). Furthermore, there is no discussions concerning non-covered function classes and what difficulties prevent estimation in those cases.\n\n* Some intuition regarding the meaning of the mathematical objects would have been appreciated (eg. what are the sets $\\bm{C}_i$ and $\\bm{R}_i$ in Def 1)? \n\n* While pseudo-code is provided, giving access to the code is always appreciated.", "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, and appreciate the positive assessment of our work.\n\n---\n\n> Some further experiments could be interesting even they are not necessary. For example: how does DML compare to prior scalable estimators on BD/SBD? In figure 2a,b,c how much can the dimenion of the summed variables grow before the running time of DML reaches unreasonable values (eg. 2000)? Similar question for figure2d,f how little can thee sample size be before the errors of DML reaches unreasonable values?\n\nWe have provided a set of experimental results in the attached pdf file. \n\n---\n\n> UCA could be more thoroughly delimited. It is well defined and the authors give examples of scenarios that are included, how they do not clearly discuss which type Ctf-ID are covered and which are not (This is also true for obsID/gID and transportability). Furthermore, there is no discussions concerning non-covered function classes and what difficulties prevent estimation in those cases.\n\nSome cases where the target estimand cannot be expressed through UCA are discussed in Appendix C.3. A summary of this discussion will be provided in the main paper.\n\n---\n\n> Some intuition regarding the meaning of the mathematical objects would have been appreciated (eg. what are the sets $C_i$ and $R_i$ in Def 1)?\n> Could you give intuition concerning the meaning of the $C_i$ and $R_i$ and $S_i$  in Def 1 as well as $\\mu$ and $\\pi$ for non experts.\n\nThe meaning of $C_i$, $R_i$, and $S_i$ depends on specific cases. In all examples, we specified what $C_i$, $R_i$, and $S_i$ meant. For the BD/SBD, we can view $C_i$ as a set of covariates, $R_i$ as a treatment, and $S_i$ as predecessors of $C_{i+1}$. $\\mu$ is a (nested-) expectation functional representing the UCA estimand, and $\\pi$ is the probability-weighting-based functional representing the UCA estimand. We will provide more explanation to give an intuition about these mathematical objects in the paper. Thank you.\n\n---\n\n> While pseudo-code is provided, giving access to the code is always appreciated.\n\nWe will make the code available after the revision. Thank you. \n\n---\n\n> Could you discuss the limits of DML, ie. when the dimension grows and when the sample size shrinks.\n\nAs shown in the experiment in the PDF of the global response, the proposed DML estimator remains scalable when the dimension is high.\n\nWhen the sample size shrinks, it’s possible that the error of the DML estimator is amplified because its error decomposes into the product of the errors of nuisances; i.e., $\\text{[error of DML]} = \\sum_{i=1}^{m} \\text{[error of } \\mu^i \\text{]} \\times \\text{[error of } \\pi^i \\text{]}$. If the sample is small so that the errors of nuisances become large, the resulting DML estimator may have a larger error since the error is multiplied. However, as the sample size grows, the DML estimator is guaranteed to converge faster whenever nuisances are converging to the truth.\n\n---\n\n> Concerning the experiments, each point corresponds to the average of 100 simulations. Could you provide the variances as well?\n\nIn all plots in Figure 2, the confidence intervals of the error with $\\alpha = 0.05$ are shown as error bars.\n\n---\n\n> The authors do not discuss which function classe are not covered by UCA. \n\nSome function classes where the target estimand cannot be expressed through UCA are discussed in Appendix C.3. \n\nWe will add a sufficient criterion to determine which estimands can be represented as a UCA estimand in the revision of the paper. The idea behind the criterion is as follows: If \n\n1. The target estimand is expressed as the mean of the product of conditional distributions over $(\\mathbf{C}_1, \\mathbf{R}_1, \\cdots, \\mathbf{C}_m, \\mathbf{R}_m)$; and \n\n2. The variables that are marginalized and fixed simultaneously (e.g., $X$ in the front-door adjustment) only appear in $\\mathbf{C}_1$, \n\nthe proposed methods can be applied. These conditions are sufficient for applying the empirical bifurcation technique (Def 2) that allows scalable estimation.", "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": "RID93wiTGq", "reviewer": "Reviewer_cBfm", "summary": "The paper introduces a new framework for identifying causal estimands, referred to as Unified Covariate Adjustment (UCA). \n\nIt demonstrates that the UCA-expressible class (a class of causal estimands identifiable by UCA) is extensive, encompassing estimands identified by the (sequential) back-door adjustment, front-door adjustment, and Tian’s adjustment. \n\nFurthermore, the paper proposes an estimation strategy for the UCA-expressible class using machine learning methods and analyzes the error of such estimators. \n\nNotably, the proposed estimator is scalable and achieves the double robustness property.", "questions": "page 2, Table 1: Can the authors clarify or provide examples of why UCA does not cover the functionals identified by obsID/gID? \n\npage 3, line 94: There are double commas.\n\npage 4, line 145: Given that the probability is at most 1, does this mean that $r_i$ is discrete? If so, can the authors explain why $r_i$ has to be discrete?\n\nSection 3.1: \n\n- Estimation of $\\mu$: \nI think the error $\\| \\widehat{\\mu}^i - \\mu_0^i \\|$ is not only affected by the estimation errors from the $i$th stage, but also by the estimation errors from the previous stages. In other words, the estimation error of $\\mu$ accumulates as the index $i$ increases. Can the authors provide some comments on this error accumulation?\n\n- Bias structure: \nThe bias structure $R_2$ in Theorem 3 consists of [error of $\\mu^i$]$\\times$[error of $\\pi^i$] and [error of $\\mu^i$]$\\times$[error of $\\pi^{i-1}$]. The second cross-$i$ product term seems non-trivial. Can the authors provide an example of causal estimands that only has the bias structure of [error of $\\mu^i$]$\\times$[error of $\\pi^i$] without cross-$i$ product terms?\n\nSections F.1.4 and F.1.5: Typo: textttnormal", "limitations": "Limitations are not explicitly discussed in the paper. \n\nThe authors might consider including the weaknesses mentioned above as limitations.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1. The paper provides good examples to illustrate the relationship between the UCA-expressible class and other classes of estimands identified by different adjustments.\n\n2. It offers comprehensive identification and estimation strategies, thereby presenting a complete causal inference methodology.\n\n3. The theoretical analysis of the estimator's scalability and error characterization is solid.", "weaknesses": "1. While the paper provides sufficient conditions for an estimand being not UCA-expressible (i.e., necessary conditions for an estimand being UCA-expressible), it lacks necessary conditions for an estimand being not UCA-expressible (i.e., sufficient conditions for an estimand being UCA-expressible). However, addressing this issue might be beyond the scope of this paper.\n\n2. In the presence of unmeasured confounders, causal estimands (e.g., average treatment effects) may not be identifiable by UCA since identification requires bridge functions and a specifically designed ID algorithm (e.g., Shpitser et al. 2023, JMLR, The Proximal ID Algorithm). It may be worth mentioning these points at the end of Section 2, along with the \"napkin\" estimand.\n\n3. The authors claimed the estimator is doubly robust (page 8, line 307) under certain conditions, which includes $n^{-1/4}$ convergence rate for $\\widehat{\\mu}^i$. However, I believe this condition may not be achievable, as the error tends to accumulate for large $i$; see Question below.", "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, and appreciate the positive assessment of our work.\n\n---\n\n> While the paper provides sufficient conditions for an estimand being not UCA-expressible (i.e., necessary conditions for an estimand being UCA-expressible), it lacks necessary conditions for an estimand being not UCA-expressible (i.e., sufficient conditions for an estimand being UCA-expressible). However, addressing this issue might be beyond the scope of this paper.\n\nAs mentioned, developing necessary conditions is challenging and beyond the scope of this paper. We believe this question has the potential to open a new direction for enhancing the proposed method. Thank you for the insightful question.\n\n---\n\n> In the presence of unmeasured confounders, causal estimands (e.g., average treatment effects) may not be identifiable by UCA since identification requires bridge functions and a specifically designed ID algorithm (e.g., Shpitser et al. 2023, JMLR, The Proximal ID Algorithm). It may be worth mentioning these points at the end of Section 2, along with the \"napkin\" estimand.\n\nIn Table 1, we provide the coverage of the UCA, which states that not all obsID/gID functions are identified. We will further mention this point as suggested at the end of Section 2.\n\n\n---\n\n> The authors claimed the estimator is doubly robust (page 8, line 307) under certain conditions, which includes $n^{-1/4}$  convergence rate for $\\hat{\\mu}^i$. However, I believe this condition may not be achievable, as the error tends to accumulate for large $i$; see Question below.\n> Estimation of $\\mu$: I think the error $\\|\\hat{\\mu}^i - \\mu^i_0 \\|$  is not only affected by the estimation errors from the $i$th stage, but also by the estimation errors from the previous stages. In other words, the estimation error of $\\mu$ accumulates as the index $i$ increases. Can the authors provide some comments on this error accumulation?\n\nGood point. Due to the nature of nested expectation (and nested regression), the error of the nuisances can accumulate as $i$ decreases from $m$ to $1$. The term $\\|\\hat{\\mu}^i - \\mu^i_0 \\|$ indeed represents the accumulated error of $\\hat{\\mu}^i$. Even with this accumulation, the error decomposes into the product of the errors of the nuisances; i.e., $\\text{[error of DML]} = \\sum_{i=1}^{m} \\text{[error of $\\mu^i$]} \\times \\text{[error of $\\pi^i$]}$ still holds. As long as $\\hat{\\mu}^i$ converges to $\\mu^i_0$ (which is likely in practice with flexible ML models), even if errors accumulate, the rate of convergence of the DML estimator outperforms that of competing estimators (OM, PW).\n\nOn the other hand, we note that $n^{-1/4}$ is used to exemplify the debiasedness property because it’s the fastest rate that a neural network can achieve [1] . \n\n[1] Györfi, László, et al. “A distribution-free theory of nonparametric regression” (2002)\n\n---\n\n> page 2, Table 1: Can the authors clarify or provide examples of why UCA does not cover the functionals identified by obsID/gID?\n\nUCA does not cover all the functionals identified by obsID/gID. We provide an example called Napkin estimand in lines 202-204, with a causal diagram in Figure 1c. The identification estimand is given as $\\frac{ \\sum_{w}P(x,y \\mid r,w)P(w) }{ \\sum_{w}P(x \\mid r,w)P(w) }$. The UCA cannot handle cases where the functional is given as a ratio of two functions. A detailed discussion on the coverage of the UCA is provided in Section C.3. \n\n\n> page 4, line 145: Given that the probability is at most 1, does this mean that $r_i$ is discrete? If so, can the authors explain why $r_i$  has to be discrete?\n\nNo, it means that the value of $R_i$ is governed by the probabilistic measure $\\sigma^i$. Whether $R_i$ is continuous or discrete depends on the choice of $\\sigma^i$. For example, if $\\sigma^i$ is a uniform distribution over $[a,b]$, then $R_i$ can be any real number within $[a,b]$. However, if $\\sigma^i$ is a Bernoulli distribution, then $R_i$ is a $\\{0,1\\}$ binary variable.\n\n---\n\n> Bias structure: The bias structure $R_2$  in Theorem 3 consists of [error of $\\mu^i$] $\\times$ [error of $\\pi^{i}$] and [error of $\\mu^i$] $\\times$ [error of $\\pi^{i-1}$]. The second cross-$i$ product term seems non-trivial. Can the authors provide an example of causal estimands that only has the bias structure of [error of $\\mu^i$] $\\times$ [error of $\\pi^i$] without cross-$i$ product terms?\n\nConsider the back-door adjustment where $i=1$. Since the error term only contains [error of $\\mu\\^i$] $\\times$ [error of $\\pi\\^i$], the second cross-term doesn’t exist when $i=1$.\n\n---\n\n> Sections F.1.4 and F.1.5: Typo: textttnormal\n> page 3, line 94: There are double commas.\n\nThank you for catching the typo. These will be fixed. \n\n---\n\n> The authors might consider including the weaknesses mentioned above as limitations.\n\nWe will discuss more about the limitations based on provided feedback. 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": {}}]

Author rebuttal:
We attached a PDF to report the experimental results in respond to the following questions from Reviewer ZBfF: 

1. > In figure 2a,b,c how much can the dimenion of the summed variables grow before the running time of DML reaches unreasonable values (eg. 2000)?

2. > how little can thee sample size be before the errors of DML reaches unreasonable values?

In summary, the proposed DML-UCA estimator can be evaluated under a high-dimensional setting where $d = 50000$. The estimator can also be evaluated under a small sample size setting, with samples varying from 10 to 100.

Meta-review: The paper introduces a new causal identification framework which covers a wide range of estimands. The authors provided detailed (and ultimately satisfactory) responses to the initial reviews and all reviewers have recommended acceptance.
Decision: Accept (poster)