Bounds on Representation-Induced Confounding Bias for Treatment Effect Estimation
Authors: Valentyn Melnychuk
Dennis Frauen
Stefan Feuerriegel
Venue: ICLR 2024 spotlight (2024)
Keywords: causal inference
representation learning
individualized treatment effect estimation
Primary area: causal reasoning
TL;DR: Bounding bias of representations for estimating conditional average treatment effect

Abstract: State-of-the-art methods for conditional average treatment effect (CATE) estimation make widespread use of representation learning. Here, the idea is to reduce the variance of the low-sample CATE estimation by a (potentially constrained) low-dimensional representation. However, low-dimensional representations can lose information about the observed confounders and thus lead to bias, because of which the validity of representation learning for CATE estimation is typically violated. In this paper, we propose a new, representation-agnostic refutation framework for estimating bounds on the representation-induced confounding bias that comes from dimensionality reduction (or other constraints on the representations) in CATE estimation. First, we establish theoretically under which conditions CATE is non-identifiable given low-dimensional (constrained) representations. Second, as our remedy, we propose a neural refutation framework which performs partial identification of CATE or, equivalently, aims at estimating lower and upper bounds of the representation-induced confounding bias. We demonstrate the effectiveness of our bounds in a series of experiments. In sum, our refutation framework is of direct relevance in practice where the validity of CATE estimation is of importance.

Published as a conference paper at ICLR 2024
BOUNDS ON REPRESENTATIIN-INDUUEE CONFOUND- ING BIAS FOR TREATMENT EFFECT ESTIMATION
Valentyn Melnychuk, Dennis Frauen & Stefan Feuerriegel
LMU Munich & Munich Center for Machine Learning
Munich, Germany
me meInychuk@ lmu.
ABSTRACT
State-of-the-at methods for conditional average treatment effect (CATE) estimation make widespread use of representation learning. Here, the idea is to reduce the variance of the low-sample CATE estimation by a (potentially constrained) low- dimensional representation. However, low-dimensions. representations can lose information about the observed confounders and thus lead to bias, because of which the validity of representation learning for CATE estimation is typically violated. wewpenntationngnosott refutation framework for
In this paper, we propose a new, estimating bounds on the representation iinduced confounding bias that comes from dimensionality reduction (or other constraints on the representations) in CATE estimation. First, we establish theoretically under which conditions CATE is non-identifiable given low-dimensional (constrained) representations. Second, our remedy, we propose neural refutation framework which performs partial identification of CATE or, equivalently, aims at estimating lower and upper bounds of the representation -nneeutttioo-nnueeedd induced confounding bias. We demonstrate the effectiveness of our bounds in a series of experiments. In sum, our refutation framework is of direct relevance in practice where the validity of CATE estimation is of importance.
INTRODUCTION
Estimating conditional average treatment effect (CATE) from observational data is important for many applications medicine (Kraus et al. 2223 Feuerriegel et al. [2024), marketing / and economics (Imbens & Angrist 1994). For example, medical professionals use electronic health records to personalize care based the estimated CATE.
Different machine learning methods have been developed for CATE estimation (see Sec. 2 for an overview). In this paper, focus on representation learning methods (e.g., Johansson al. 2016
Shalit et al. 2017 Hassanpour & Greiner 2019b a. Zhang et al. 2020 Assaad et al. [022 1 Johansson
al. 2022). Representation learning methods have several benefits: (1) They often achieve state-of- the-art performance, especially in low-sample regimens. (2) They provide generalization bounds for best-in-class estimation, regardless of whether ground-truth CATE belongs to a specified model class,
(3) They manage to reduce the variance of the low-sample CATE estimation by using a (potentially constrained) wiw-dimesiionll representation. Often, constraints are imposed on the representations such as balancing with an empirical probability metric and invertibility.
While representation learning methods benefit from reducing variance, they also have shortcoming: low-dimensional (potentially constrained) representations can lose information about covariates, including information about ground-truth confounders. representations thus lead to bias, because of which the validity of representation learning methods may be violated. To this end, we introduce the notion representation intrrseennatio--ndceedd confounding bias (RICB). As result of the RICB, the validity of representation learning for CATE estimation is typically violated, and we thus offer remedies in our paper.
we show later, such low-dimensional
In this paper, we propose new, pnnressenaationaagnssiss refutation framework for estimating bounds on the RICB that comes from dimensionality reduction (or other constraints on the representation). First, we show in which settings CATE is non-identifiable due the RICB. We further discuss how

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Table 1: Overview of key representation learning methods for CATE estimation.
Methed
Invabbiit
Balowing witl
expiiial pobbabiity reciass
los to weigllii
TARNNi Stalket al
Jobanioc et
BNN
Divasoon at
2016
Stalioral
N017 Motmooo stal
N022
ESCTR Wangoral N0M
IPM (MMD WM)
KL18
Jotossoon dial
IIM (MBII, w.M
Learnatle weight
Beadt BOST ARCE
Dzet 202
Mctrotaak
looning)
Lood sistlarie
NSW
1022
Rescepoor Ceviner
200ki
DR.CFR
Resarporr Croleer
2009
DeR-CFR
Wh
IPM (MMI, WM)
Reprosmratio propanayy
DRLITE
Phang al br37
Recmsttnctio loss
Cooseefarral vararne
IIM (MMD, w35
Connrati prrpesaty
PM
Upsasplis vi saaxlng
integral pubckili netik XMD
diorppagy WM. Waseratio JSD Junce Statto dieggose
different constraints imposed on the representation impact the RICB. Then, remedy, perform partial identificatonn of CATE or, equivalently, estimate lower and upper bounds on the RICB.
We empirically demonstrate the effectiveness of our refutation framework on top of many state-of- the-art representation learning methods for CATE estimation. We thereby show that the established representation learning methods for CATE estimation such BNN (Johansson et al. (0016); TARNet, CFR, RCFR (Shalit et al., 2017; Dohansson et al. 2018; 2022h; CFR-ISW (Hassanpour & Greiner 2019a]; and BWCFR (Assaad al. ) be more reliable for decision-making when combined with our refutation framework. We thus evaluate decision-making based on CATE, finding that the policies with deferral that account for our bounds on the RICB achieve lower error rates than the decisions based on CATE estimates from the original representation learning method. As such, our refutation framework offers a tool for practitioners to check the validity of CATE estimates from representation learning methods and further improve the reliability and safety of representation learning in CATE estimation.
In sum, our contributions are following methods can be non-identifiable due to a representaiion-induued confounding bias. To the best of our knowledge, we are the first to formalize such bias. (2) We propose
! (1) We show that the CATE from representation learning
pepresentation-agnosse refutation framework to perform partial identification of CATE, so that we estimate lower and upper bounds of the representation- induueed confounding bias. (3) We demonstrate the effectiveness of our bounds together with a wide range of state-of-the-an CATE methods.
2 RELATED WORK
An extensive body of literature has focused on CATE estimation. We provide an extended overview in Appendix [ while, in the following, we focus on two important streams relevant to our framework.
Representation learning for CATE estimation. Seminal works Johanssoon et al. 2016); ; Shalit ct al. (2017): and Johansson et al. (022)) provided generalization bounds for representation learning methods aimed at CATE estimation under the assumption of invertibility of the representations. Therein, the authors demonstrated, that generalization bounds depend on the imbalance of the invertible representations between treated and untreated subgroups, and some form of balancing is beneficial for reducing the variance. However, although invertibility of representations is still required theoretically, it is usually not enforced in practice (see Table [T.. Namely, due to bias-variance trade- off, cow-dimensional (non- -on-innettible) representations can still generalize better, although potentially containing confounding bias (Ding al. 2017).
Numerous works offer a variety of tailored deep representation learning methods for CATE estimation by extending TARNet (Shalit al.) 2017 Johansson et al. 2022) (the simplest representation leaning CATE estimator) with different ways to enforce (i) invertibility and (ii) balancing. For example, (i) invertibility was enforced via local similarity in SITE (Yao et al. 2018). and via reconstruction loss in DKLITE (Zhang et al., (2020). (ii) Balancing was implemented as both (1) constraint on the representation with some empirical probability metric and (2) loss re-weighting. Examples of (1) include balancing with integral probability metrics in, g.. BNN (Johansson et al. 2016). and CFR 2022); Shanen-Shannon divergence in, c. g.. DACPOL (Atan)
(Shalit et al.] [017:Johansson et al.
et al. 2018), CT (Melnychuk et al. 2022). and MitNet (Guo et al. 2023): ) middle point distance
available htapaaa.e//tcee////h/bo.oo//////
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Original clustered cousal
Representation (() x ((8)
Transforedd causal diagrams
Vallay of ( for CATE
diagram
x X (1))
(+6e))
(0)
9:X.
((ta)) Lso
4: x (9())
cuefburder
Bapduciee of Molka
Figure 1: The validity of CATE estimation is influenced by the different constraints imposed on representations ((-). In red: different violations of valid CATE estimation.
in SITE (Yao et al. 2018); and counterfactual variance in DKLITE (Zhang et al., 2020). (2) Loss
re-weighting was implemented with learnable weights in RCFR (Johansson et al. 2018 2022);
inverse propensity weights in, e. e.g.. CFR-ISW (Hassanpour & Greiner 2019a) and BWCFR (Assaad
et al. 2021); and upsampling in PM (Schwab et al., 2018) and StableCFR (Wu et al. 2023). We
provide the full overview of key representation learning methods for CATE in Table [
Handling unobserved confounding. Sensitivity models are used to estimate the bias in treatment effect estimation due to the hidden confounding by limiting the strength hidden confounding through a sensitivity parameter. There are two main classes of sensitivity models: outcome sensitivity
models (OSMs) (Robins et al. [2000] Blackwell 2014 Bonvini et al. 2022) and propensity (marginal) sensitivity models (MSMs)
et al. 202 Bonvini et al. 2022; Dorn & Guo 2022
Frauen et al. 2023 Oprescu et al. (202)) We later adopt ideas from the MSMs to derive our refutation framework, because they only require knowledge of propensity scores wrt. covariates and representation, but not the actual expected potential outcomes as required by OSMs.
Of note, our method uses MSMs but in an unconventional way. Importantly, we do not MSMs for sensitive analysis but for partial identification. As such, we do not face the common limitation of MSMs in that the sensitivity parameter (which guides the amount of hidden confounding) must be chosen through expert knowledge. In contrast, our application allows us to estimate the sensitivity parameters in the MSM from data.
Research gap: To the best of our knowledge, no work has studied the confounding bias in low- dimensional (constrained) representations for CATE estimation. Our novelty is formalize the representation-inducedd confounding bias and to propose a neural refutation framework for estimating bounds.
VALIDITY OF REPRESENTATION LEARNING FOR CATE
In the following, we first formalize representation learning for CATE estimation (Sec. 3.1 We then define the concept of valid representations and give two conditions when this is violated (Sec. 3.2) Finally, we lay out the implications of invalid representations (Sec. 3.3
Notation. Let X be a random variable with a realization x, distribution P(X), density/probability function P(X = z) and domain X, respectively. Furthermore, let P(Y X x, = a) be the conditional distribution of the outcome Y. Let x (x) = P(A
X x) denote the covariate propensity score for treatment A and covariates X, and n (x) = E(Y = a, x)) an expected covariate-conditiona. outcome. Analogously, Th (b)
(xX) /)) and (²())) E(Y A= = a, (X) = o) are the representation propensity score and the expected representation- conditional outcome, respectively, for treatment A and representation (X)). Importantly, in the definitions of Th μe -h and μº upper indices serve as indicators that the nuisance functions relate either to the covariates or the representations and, therefore, not arguments. Let Y Y[a) denote a potential outcome after intervening on the treatment do(A = a). dist is some distributional distance.
PP(A a
3.1 REPRESENTATION LEARNING FOR CATE
Problem setup. Assume we have observational data D with binary treatment A (0, 1). high- dimensional covariates E X C Rd= and a continuous outcome Y E e C R. In medicine, A could be ventilation, X the patient's 's health history, and Y respiratory rate.
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Without a loss of generality, we assume that covariates X implicit cluster (Anand al. 2023
of four sub-covariates, X = [X",X", xx,xx,, namely, (1) noise, (2) instruments, (3) outcome- predictive covariates, and (4) confounders (Cinelli et al. 2022) (see clustered causal diagram in Fig. D. The noise, in turn, can be partitioned onto (1.1) an independent noise, (1.2) descendants of instruments, (1.3) descendants of outcome-pree ouediieepredicctvv covariates, and (1.4) M-bias-inducing covariates, namely, X = {-X 8. X=,,, XX X X xe."), j. Importandy, some sub-covariates could be empty, and the partitioning of X usually unknown in practice, and, thus, it will be only used in this paper to provide better intuition (all experiments later consider the partitioning unknown). The observational data are sampled i.i.d. from a joint distribution, i. e., is the sample size. Furthermore, potential outcomes are only observed for factual treatments, i. i.e., Y = A AY[1] + (1 - A) ( which referred as the fundamental problem causal inference.
{Iz, Qi Y/ } P(X, A, Y), where
The conditional average treatment effect (CATH) is then defined as
r (x) = [1] - 0 X = x),
(I)
where upper index r indicates that CATE is with respect to the covariates x. Identification of CATE
from observational data relies on Neyman-Rubin potential outcomes framework (Rubin 1974). As
such, we assume (i) consistency: if A = then Y = Y[a]; (ii) overlap: PPO < TH (X)
1: and
(iii) exchangeability: A IL (Y) ((]) X. Under assumptions (i)-(ii), the CATE is identifiable
from observational data, P(X, A, Y), i. from expected covariate-conditional outcomes, 7 (=x)
u (1) 1 (z).
Representation learning for CATE estimation. Representations learning methods CATE estimators (Johansson et al. 2016; (Salit et al. 2017 Johansson al. 2022) do not assume a specific partitioning of covariates X and generally consist of two main components: (1) the representation subnetwork
( and (2) the potential outcomes predicting subnetwork(s), e.. fi Both components are as follows. (1) The representation subnetwork maps all the covariates X
(XX to low- or equal-
dimensional representation space with some measurable function, x(-) Rd + R& do S d Additional constraints can be imposed on ((.). For example, balancing with empirical probability
metric, dist (X( | A = 0), (x) A = 1)] ~ 0, and invertibility, (-)( xs X. (2) The potential outcomes predicting subnetwork(s) then aim at estimating CATE with respect to the representations, namely
(Ø) E(Y - 0 (XX)=
=), +),
where ø = (z). Yet, fo and fi can only access wttprsseennatiocccnnniiiiiin outcomes, and, therefore. instead, estimate Aa (Ø) and μ (Ø), respectively.
3.2 VALID REPRESENTATIONS
In the following, we discuss under what conditions representations o(-) are valid for CATE estimation, namely, do not introduce an intinite-data bias.
Definition (Valid representations). We call a representation (.) valid for CATE if satisfies the following two equalities:
(bxx)) and
(
(d),
where (-) and (-) are CATES wrt. covariates and representations from Eq. and ②). respec- tively.
If equality (i) is violated, then we say that the representation suffers from loss in heterogeneity. If (ii) violated, it suffers from a representation-innneee confounding bias. Hence, if any of the two violated, we have invalid representations (see Fig. left).
Characterization of valid representations. We now give two examples of valid representations based on whether information about some sub-covariates {X" X", X", XA} is fully preserved in (X).
'The latter can have either one subnetwork with outputs (SNet) as in, e. g.. BNN Johansson al. 2016); or two subnetworks (TNet) as in, e.g., TARNet (Shalit et al.) (017). 'The transformation (b.) can also be seen mapping between micro-level covariates X to low-dimensional
macro-level representations (Ruhenstein et al. 2017).
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Invertible representations. A trivial example of a valid representation is invertible representation Shalit et al. 2017: Zhang et al. 2020; Johansson et al. 2022):
(T)
B[([) 100 x 4 ()(()) E Y ((x)
((()) ( - (*())) B (Y A 0, a 1=====0000)0 (*(=)) = (0(=)) - (+()))
where equality (ti) follows from setting 4 (2) to Φ. Notably, (-) is an invertible transformation, then
X (X), X (X), X X (XX), X I x (X),
(4)
e.e. there no loss of information on each of the sub-covariates. Eq. 4 holds, as conditioning
on ((X) renders each sub-covariate deterministic, thus implying independence. In Lemma D of
Appendix B we also show that the opposite statement true: when all the statements hold in Eq.
then Φ is an invertible transformation.
Removal of noise and instruments. Another class of valid representations are those, which lose any amount of information about the noise, X or the instruments, X
X J x ((X) or X X (X)..
(5)
The validity follows from the d-separation in the clustered causal diagram (Fig. and invertibility of (.)) wrt. X and X (see Lemma 2 in Appendix
(x) [[ - 1 - = - [0 2 ++)) B( [0] (x)) = 7 ()(()),
(x) Y A 11
A = 0, م x²*)
E A AA (Y 0, (x)) (((((=)))
where equality (fi) follows from setting (xx) to d.
In actual implementations, representation learning methods for CATE achieve (1) the loss of informa- tion about the instruments through balancing with an empirical probability metric, and (2) the loss of information about the noise by lowering the representation size (which is enforced with the factual outcome loss).
3.3 IMPLICATIONS OF INVALID REPRESENTATIONS
In the following, discuss the implications for CATE estimation from (1) loss of heterogeneity and (2) RICB for CATE estimation, and in what scenarios they appear in low-dimensional or balanced representations.
(i) Loss of heterogeneity. The loss of heterogeneity happens whenever r"(x) 7 (T)()) It means that the treatment effect at the covariate (individual) level is different from the treatment effect the representation (aggregated) level. As an example, such discrepancy can occur due to aggregation such one-dimensional representation (e.g., in the covariate propensity score, $(X) = of (X)). In this case, the treatment effect (Ø) denotes a propensity conditional average treatment effect, which is used in propensity matching.
The loss of heterogeneity happens whenever some information about X or X is lost in the representation, i....
X x ((X X X" ((X).
(7)
This could be the case due too low dimensionality of the representation so that the information on X or X is lost, or due to too large balancing with an empirical probability metric so that we lose X A Then, the equality (i) does not any longer hold in Eq. i.e.
7 (2(x)) BY [0] (x)))
B[ - [o] ,x")((( = x" x" (xx)) da dx
XAXXY
(() = A x²= (T))) dz dx ++
MAXXY
Although the CATE wrt. representations, (+)), and the CATE wrt. covariates, 7" 7*(x), are different, the CATE wrt. representations is identifiable from observational data P(O(X), ), A, Y) due to the exchangeability wrt. representations, A IL (Y[0), Y(1) ((X). This is seen in the following:
(41) (0) B[Y[I] - o | Φ) E [1] A 1, B(Y A nK(²) (o))). (8)
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(it) Representation-indueed confounding bias (RICB). This situation happens when information about X A is lost or when M-bias is introduced, e.e some information is lost about both X and X but not X 8.m Yet, M-bias is rather a theoretic concept and rarely appears in real-world studies (Ding) & Miratrix 2015); therefore, we further concentrate on the loss of confounder information in repre- sentations. As described previously, the information on X could be lost due an incorrectly chosen dimensionality of the representation or due to too large balancing with an empirical probability metric. In this case, in addition to the loss of heterogeneity, we have pnnrreeenattoon-inucceee confounding bias (RICB). That is, the CATE wrt. representations is non-identifiable from observational data, (()(X), Y). This follows from
(+)
(14)
B[ - [[o I () ((((I) 1, - B( | A A= = (6( 10( (9)
Technically, both the CATE wrt. representations, (o). and the RICB, (d) (4f (Ø) My (o)). are still identifiable from P(X, A, Y). Yet, the identification formula has intractable integration in Eq. (8) and the original CATE wrt. covariates, which makes the whole inference senseless.
Motivated by this, we shift our focus in the following section to partial identification of the CATE wrt. to representations (or, equivalently, the RICB), as partial identification turns out to be tractable. As result, we can provide bounds for both quantities. With a slight abuse of formulations, we use 'the bounds on the RICB and 'the bounds the representation CATE' interchangeably, as one can be inferred from the other.
Takeaways. (1) The minimal sufficient and valid representation would aim to remove only the
information about noise and instruments (Ding et al. 2017] Johansson et al. 2022). In low-sample settings, we cannot guarantee that any information is preserved a low-dimensional representation, so we apriori assume that some information is lost about the cub-covariates. (2) The loss of heterogeneity does not introduce bias but can only make CATE less individualized, namely, suitable only for subgroups. For many applications, like medicine and policy making, subgroup-level CATE is sufficient. (3) The RICB automatically implies a loss of heterogeneity. Therefore, we consider the RICB to be the main problem in representation learning methods for CATE, and, in this paper, thus aim at providing bounds on the RICB, as the exact value is intractable.
4 PARTIAL IDENTIFICATION OF CATE UNDER RICB
In the following, we use the marginal sensitivity model to derive bounds on the RICB (Sec. 4.1. Then, we present neural refutation framework for estimating the bounds, which can be used with any representation learning method for CATE (Sec. {.2). Here, we do not assume the specific partitioning of X (which we did before in Sec. 3.1 only for the purpose of providing better intuition).
4.1 BOUNDS ON THE RICB
We now aim to derive lower and upper bounds on the RICB given by (0) and (0)), respectively For this, we adopt the marginal sensitivity model (MSM) for CATE with binary treatment (Kallus
ct al. 2019 Jesson et al. 2021 Dorn & Guo 2022 Oprescu et al. 2023 Frauen et al.) 2023)
The MSM assumes that the odds ratio between the covariate (complete) propensity scores and the representation (nominal) propensity scores can be bounded. Applied our setting, the MSM assumes
r(6)-1 < (Tā(0)/ o (o)) (#()))/=((+) < (T) for E s.t. ((x) = のb
(10)
where [())= 2 is ceresentatiin-deeeedee sensitivity parameter. In our setting, there are no undeserved confounders, and, therefore, the sensitivity parameters can be directly estimated from combined data P(X, ((XX),,YY
We make the following observations. For ((Ø) = 1 for all di then (1) all the information about the propensity score n (a:) is preserved in the representation (xx), and (2) the representation does not contain hidden confounding. If (()) > 1, we lose the treatment assignment information and
"The interval (+)) contains intractable 7 (r) and can be inferred from P(O(X), Y) and tractable sensitivity parameters.
'Note that this is crucial difference from other applications of the MSM aimed at settings with unobserved confounding, where, instead, the sensitivity parameter is assumed chosen from background knowledge.

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Legend
6:30
Louas
Hale
FC
der deroccon
Gradin Cooknng
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4FG
FC.
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ATAX]
Aqu
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Slage Filing representation network
Stage 1. Estimaling [( of MSM F(Y A (Xx]
Stage 2. Essimeling upper and lower bounds RICS
Figure 2: Our neural refutation framework for estimating bounds on the RICB. In Stage O, we fit some representation learning method for CATE, possibly with different constraints like balancing with an empirical probability metric and invertibility, or loss re- weiiggtting. In Stage 1, we estimate the sensitivity parameters of the MSM, [()), and the representation-contee conoeiatttiooooontiinnnllll conditiooal outcome distribution, P(Y| | A a, (xx = Φ). In Stage 2, we compute the lower and upper bounds on the RICB.
confounding bias could be introduced. Therefore, [()) indicates how much information is lost about sub-covariates X or about X In order to actually distinguish whether we lose information specifically about X or X=, we would need to know (x), which makes the task of representation learning for CATE obsolete. Thus, our bounds are conservative in the sense, that they grow if the information on X is lost, even though it does not lead to the RICB.
Under the assumption in Eq. (IO). bounds (Frauen et al. 2023 Oprescu et al. 2023 on the RICB are given by
(0( (x)
and
(Ø) 1 (0)
p (Φ)
(11)
with
yP(Y = y b + 5+ Ø)
y DP(Y = y a, +) dy,
sa
(c (0,0)
10,00
t00
po( s4 Φ)
UP(Y = y a) s の)
y pPYY y dy.
1(04 la.d)
where: s- (a, Φ) = (1- + -())) [() (a. = (1- (0) + +
c_ 1/(1 + - - -
C+
= P(Y
A a, (X) is
=
a ciaresentationnconddite
density of the outcome, and F (c a. o) its corresponding quantile
function. This result is an adaptation of the theoretic results, provided in (Frauen al. 2023 Oprescu et 2023). We provide more details on the derivation of the bounds in Lemma 3 of Appendix B Importantly, the bounds provided Eq. (TT) valid and sharp (see Corollary | in Appendix B).
Validity means that our bounds always contain the ground-truth CATE wrt. representations, and sharpness means that the bounds only include CATE wrt. representations induced by the observational distributions complying with the sensitivity constraint from Eq. (10).
The above bounds are not easy to compute (which motivates our neural refutation framework in the following section). The reason is that the bounds on the RICB require inference of so-called conditional values risk (CVaR) (Artzner et al. 1999 Kallus 2023). Formally, we have two CVaRs given by __ PP = y a, Φ) and IT P(Y CVaRs can be estimated directly without estimating the conditional density, like in (Oprescu et al. 2023). Yet, in our case c. and c+ depend on o, and, thus, it is more practical to estimate the conditional density, as done in (Frauen et al., (023). Given some conditional density estimator,
a, Φ) for E (8-1(c_1 a. Φ), F..F--. (c+ I a, })}
?(YY = a, Φ), we can then estimate CVaRs via importance sampling, i...,
kcj
for CVaR = f (((ea.0) P(Y = y a, +). for CVaR = f 22 (cla.,) FF = y a, o),
je.
CVaR
(12)
- [ke]+1
where (i is sorted sample from P(Y a. ().). Hence, we use a conditional density estimator in our refutation framework.
4.2 NEURAL REFUTATION FRAMEWORK FOR ESTIMATING BOUNDS
In the following, we provide a flexible neural refutation framework for estimating the bounds on the RICB (see Fig. for the overview). Overall, our refutation framework proceeds in three stages.
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Table 2: Results for synthetic experiments.
Table 3: Results for HC-MNIST experiments, Re- ported: sut-sample policy error rates (with im- provements of our bounds); mean over 10 runs. Here, 0.0005.
Reported: out-sample policy enror rates (with improvements of our bounds); mean over 10
runs. Here, meain 1, 000 and ò 0.0005.
BR.))
FRR (A BRa.
de
30
78
TARNeT
30.79%
(- 12.899)
9.22
(-37555)
TARNe
(1211(-22559))
10.15 (-3.55))
(-2628)
BNN IMMD. 0.1)
14.32%
15.41%)
16.15%
4.19%)
INN (MMI o 201
12.01% (-4 4.55
11 11.35%
(-5 5.200
30.705
(-101%)
CER IMMD; 00.1)
35.01%
14.27%)
11,929
5,586)
CFR (MMD D.3)
11.40% 1
11.05% 3.13%)
11.73%
4.77)
CIR (MMD;a-055
35.79% 11.43%)
17389
1.27%6
CIR a
15015 (8322883
12:542 (-1995))
12:975
(-5.55))
ORR (WM) 1.00
3457%
14.27%)
103555
(-7,976)
CFR (WN) 1X0)
24,556 10 (242
27.87%
10.188)
31.99% 31.5585
CER (WM) 2:00
35.189
(-1659))
19.199
(-6,2855)
$1.255 (- 10.30%)
30.75%
(-7.229)
51.83%
(-1.999))
InTTARNe:
29.51% 0.9555)
5.64% (- 0,225)
TARNNt
(2.18%--1229)
11.38% (-3.99%)
11.55% (-.344))
(WM)
33.02% 3.585)
(-4227))
RCER
21.513 9.17%)
36.14%
14,2099
CER (WM,
BACER (WM,
35.00%
10/00%)
9.43981
7.448
7.27% 1386))
4.57%)
BWCER
13.02%
3.966)
32.56%
(-6,758)
Lower bette imprroeeeet over bracla 16 srEci, konseting baidlis (n
Classcal CATE Estomatoo: N.NI 8.15,.
17.70 15.10%
Stage O. The initial stage is a naive application of existing representation learning methods for CATE estimation. As such, we fit a standard representation learning method for the CATE of our choice (e. g.. TARNet, CFR, or different variants of CFR), Stage O always contains fully-connected representation subnetwork, FCs. which takes covariates X as an input and outputs the representation, (X). Similarly. potential outcomes predicting subnetwork(s) are also fully- uully-connected, FC Together, FC and FC, aim minimize (potentially weighted) mean squared error of the factual observational data, LMSE. The representation can be further constrained via (a) balancing with an empirical probability metric, (b) invertibility, or have additional (c) loss re-weighting. These are as follows. (a) Balancing, Laa is implemented with either Wasserstein metric (WM) or maximum mean discrepancy (MMD) with loss coefficient a (Johansson et al., 2016 Shalit et al. 2017). (b) Invertibility (Zhang et al. 2020) is enforced with reconstruction subnetwork, FC3 and reconstruction loss, CRee. (c) Loss
re- we-weighting is done by employing either trainable weights (Johansson et al. 2018 2022), with a
fully-connected FC..; with representation-propenee subnetwork (Hassanpour & Greiner 2019a b).
FCa,ei or with a covariate- prropenstttp subnetwork (Assaad et al. (021). namely FC,,.-
Stage 1. Here, parameters, T(Ø), and representation-condiiion For that, train two fully-connected propensity networks, FCx,o and FC .. (or take them from the Stage O, if they were used for (c) loss re- re-weighting). Both networks optimize a binary cross- entropy (BCE) loss, C. Then, we use Eq. (TO) to compute (((x) for o: = ((z;) with z; from D. Specifically, each i'((::) is maximum over all (dx()) where ((x,) are the representations of the training sample in ô-ball around é;. Here, 8 is hyperparameter, whose misspecification only makes the bounds more conservative but does not influence their validity. In addition, we estimate outcome density with a conditional normalizing flow (CNF) (Trippe &
use the trained representation subnetwork and then estimate the sensitivity
outcome distribution, IP(Y | A = a, ((X) = Φ).
the pppresentation-codiiiion Turnrr 2018) using a fully- connected context subnetwork, FCCNF. The latter aims at minimizing the negative likelikel of the observational data, LNLI.-
Stage 2. Finally, we compute the lower and upper bounds on the RICB, as descrihed in Eq. 12
Here, the CNF is beneficial for our task, as it enables direct sampling from the estimated conditional distribution. Further details on implementation details and hyperparameter tuning are in Appendix C
EXPERIMENTS
Setup. We empirically validate our bounds the RICB For that, we use several Jsynthiiyt benchmarks with ground-truth counterfactual outcomes Y [ and [[] and ground-truth CATE (x). Inspired by Jesson et al. (202I), we designed our experiments so that we compare policies based on (a) the estimated CATE or (b) the estimated bounds on the RICB. In (a), policy based on the point estimate of the CATE applies a treatment whenever the CATE is positive, i.e. #()) (g) > 0}.
In (b), a policy x² (d) based on the bounds on the RICB has three decisions: (1) to treat, i.e., when
(6) > 0: (2) to do nothing, i.e., when (0) < 0: and (3) to defer a decision, otherwise.
"We also considered other evaluation techniques. However, the ground-trutt CATE wrt. representations is intractable; therefore, one can not employ established metrics such as coverage compare MSES directly. As a remedy, we follow Jesson Follow/essonn al. (202T) and evaluate the decision-making under our CATE, which is closely aligned with how our refutation framework would he used for making reliable decisions in practice.
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Table 4: Results for IHDPI00 experiments. Reported: out-sample policy error rates (with improve- ments of our bounds); mean over 100 train/test splits. Here, 8 0.0005.
ER.,
TARNet
3,17% (-2 (-265%
2.88% (-2.9%%
3.28% (2277
3.22% (-2.52%)
2.89% (-2.5716)
BNN IMMD: o- -0.3)
2.52% (-149-9))) 1.89%)
2 (-140%)
2.59% (-2001))
2.43% 1887))
2.2915 (-1.16%)
CPR a
1.77% (-089%)
2,09%5 (-100%)
2.25%
(-1336))
1.88% (-0.68%)
2,085
1.405)
CRR (MMD) c 00.5
2,078 [-] (.46%)
2.00% (43.56%)
2.68% (+1.39%
2.36R (+6.37%)
2.17% (4-3419)
CFR (WM: o = 1.0)
1.9% (-089%)
1.75% (-0.25%)
188% (- 1.2495)
1.8%% (-0.49%)
1,00% (-0.39%)
CFR 2.00
1.97% (-004%))
2.17% (- (-1499))
2.05%
(-121%))
2.08% (-1.29%)
2,09%5 (- (-1,39%)
IncTARNNe
2,505 (- (-195%)
3.11% (-2,47%)
2.99% (-25116)
2.75% (-2.419)
2.835 (-2229%)
RCFR
3.36%
84%)
(-5,32))
2.67%
4.60% (-3838)
1,05% (+-106))
(WM: 1.0g
9096)
(-0.69%)
1.85% (- (.5%)
(916
BWCFR (WM) 1.0)
3,57%
1-499)
3.52%
(-2.168)
3,88%
1.300)
3.90% 3.38))
4,07% (- 1,189)
Lower better. Improvement
bascline
wersening of boreline
Classial CATE
K-NN: BART: 5.07%, C.Forest 6.25%
Evaluation metric. To compare our bounds with the point estimates, we employ an error rate of the policy (ER). ER is defined as the rate of how often decisions of the estimated policy are different from the decision of the optimal policy, ((x = [x)=(xx) > 0}. For the policy based on our bounds, we report the error rate the non-deferred decisions. Hence, improvements over the baselines due to our refutation framework would imply that our bounds are precise and that we defer the right abservattions/nddiiddulll Additionally, report root precision in estimating treatment effect (rPEHE) of original methods in Appendix H
Baselines. We combine our refutation framework with the ofate-of-the-ar baselines from repre- sentation learning. These are: TARNet (Shalit et al. 2017]
2017|Johansson Johansson et al. 2022) implements a
representation network without constraints; BNN (Johansson et al.] (016) enforces balancing with MMD (ox = 0.1); CFR (Shalit et al. 2017 Johansson et al. (202)) is used in four variants of balancing with WM (a- 1.0/2.0) and with MMD (a 0.1/0.5). InTTARNet adds an invertibility constraint to the TARNet via the reconstruction loss, as in (Zhang et al. 2020).. Finally, three methods use additional loss re-weighting on top of the balancing with WM
1.0): RCFR (Johansson et al.) 2018 2022), CFR-ISW (Hassanpour & Greiner 2019a). and BWCFR (Assaad et al. (021). Also, for reference, we provide results for classical neon-n CATE estimators, i. e., k-NN regression, Bayesian additive regression trees (BART) (Chipman et al., 2010), and causal forests (C-Forest) (Wager & Athey 2018).
Synthetic data. We adapt the synthetic data generator from Kallus et al. (2019), where we add an unobserved confounder to the observed covariates so that de = 2. We sample Auain {500; 1, 000; 5, 000; 10, 000} training and Reest 1, 000 datapoints. Further details
in Appendix D We plot ground- and estimated decision boundaries in Appendix [ Additionally,
in Table we provide error rates of the original representation learning methods and improvements in error rates with our bounds. Our refutation framework achieves clear improvements in the error rate among all the baselines. This improvement is especially large, when dø so that there are both the loss of heterogeneity and the RICB.
IHDP100 dataset. The Infant Health and Development Program (IHDP) Hiill 2011 Shalit et al. 2017) is a classical benchmark for CATE estimation and consists of the 100 train/test splits with Auain 672, Thest = 75, de 25. Again, our refutation framework improves the error rates for almost all of the baselines (Table 4 We observe one deviation for CFR (MMD; = 0.5), but this can be explained by too large balancing with an empirical probability metric and the low-sample size.
HC-MNIST dataset. HC-MNIST is a semi semi-synthetie benchmark on top of the MNIST image dataset (Jesson et al., 202T). We consider all covariates as observed (see Appendix D). The challenge in CATE estimation comes from the high-dimensionality of covariates, i. e., d, 784 + 1. Again, our refutation framework improves over the baselines by a clear margin (see Table
Additional results. In Appendix E. we provide additional results, where we report deferral rates in addition to the error rates for different values of the hyperparameter 8. Our refutation framework reduces the error rates while increasing the deferral rates not too much, which demonstrates its effectiveness.
Conclusion. We studied the validity of representation learning for CATE estimation. The validity may be violated due to fow-dimensional representations as these introduce a representation-inue induued confounding bias. As remedy, introduced a novel, representation-agnosse that practitioners can use to estimate bounds on the RICB and thus improve the reliability of their CATE estimator.
refutation framework

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Published as a conference paper at ICLR 2024
Acknowledgments. S.F. acknowledges funding via Swiss National Science Foundation Grant 186932.
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A EXTENDED RELATED WORK
CATE estimation. A wide range of machine learning methods have been employed for estimating
CATE. Examples include tailored methods built on top of linear regression Johansson al. (2016).
regression trees (Chipman et al. 2010 Hill 2011), and random forests (Wager & Athey 2018)) Alaa
& van der Schaar (017)) proposed non parametric approaches based on Gaussian processes. Later,
it was shown m (Alaa & van der Schaar 201 8a b), that Gaussian processes can achieve optimal minimax rates for non-parametric estimation, but only in asymptotic regime and under smoothness conditions Another stream of literature studied meta-learners (Künzel et al. 2019; Kennedy 2003 Nie & Wager 202T Kennedy et al. 2022), which combine different nuisance function estimators; and double/debiased machine learning (Chernozhukov et al. 2017), which turns CATE estimation into two-stage regression. Both approaches can achieve optimal convergence rates CATE estimation, but, again, only asymptotically and with well well-specified models. Curth & van der Schaar (2021a) and [Curhh & van der Schaar (2021b) specifically studied neural networks as base model classes for meta-learners. According (Curth & van der Schaar 2021b), representation learning methods are one-step (plug-in) learners and they aim at best-in-class estimation of CATE, given potentially misspecified parametric model classes (Johansson et al. 2022).
Prognostic scores. Inferring prognostic scores (Hansen 2008) is relevant to CATE estimation with representation learning. The prognostic score is such a transformation covariates, which preserves all the information about a potential outcome and can be seen as a sufficient dimensionality reduction technique for CATE. For example, linear prognostic scores were rigorously studied in (Hu al.. 2014; Huang & Chan 2017). As mentioned by (Johansson et al. (022), representation learning CATE estimators aim to learn approximate non-linear prognostic scores.
Uncertainty of CATE estimation. Several works studied uncertainty around CATE estimation. For example, JJsssnn et al. (2020) studied epistemic uncertainty, and Jesson et al. (202T) additionally considered bias due unobserved confounding, which was part of aleatoric uncertainty. Our work differs from both of them, as we aim at estimating the confounding bias, present in low-dimensional (potentially constrained) representations. This bias still persists in epistemic uncertainty drops to zero.
sample regime, where
Runtime confounding. Runtime confounding (Coston al. 2020 arises when subset of covariates is not available during prediction time. This setting is highly relevant, c. g., for multi-step-aae -the--heppaaa not observed; and (questionably) some methods used balanced representations as a heuristic to circumvent this bias (Bica al., 2020 Meinychuk al.) 2022 Seedat et al. 2022 Hess al. (024). Notably, non- availability of covariates can be seen as special case of low-dimensional representations which zero-out some inputs and have identity transformations for the others. Therefore, runtime confounding is a special case of our proposed RICB.
prediction of time-varying methods where future varying covariates
"One important result of (Alaa & van der Schaar 2018b is that minimax rate of non parametric estimation of CATE does not asymptotically depend on treatment inbalance. In a low-sample regime nevertheless, addressing treatment imbalance is still important.
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B PROOFS
Lemma 1. Let X be partitioned in a cluster of sub-covariates, following independencies hold
1x=, X x², If the
IL ((),, x T X" ((),
((X), (13)
then (--) an invertible transformation.
Proof. Without a loss of generality, let us consider posterior distribution of some sub-covariate, e. g., X given a representation & (X) = o, i.e. IP (xX) = Φ). Due to the independence, the following holds
(14) where the last equality follows from the fact that ((-) is measurable function. It easy see that F(X X = x) is a Dirac-delta distribution. Therefore, for every Φ, P(X ((X) Φ) is also Dirac-delta distribution, namely puts point mass on some z". In this way, we can define an inverse of (-) wrt. ø.
P(X ((X) = Φ) P(X
+ ( = 0, X = x) = P(X X = x),
The equalities Eq. (14) also hold for other sub-covariates of X, and, thus, we can construct full inverse of (-)).
Lemma 2 (Removal of noise and instruments). Let X = {x* Xx, X", } be partitioned according the clustered causal diagram in Fig. Then, representation Φ(-) is valid if
XLXA (XX),
X ll X ((X).
(15)
Proof. First, it is easy to see that the potential outcomes are d-separated from X by conditioning on X and X i.e., X Y[a] X X" or, equivalently, PP [a] X x) P(Y X
A X r²). This is an immediate result of applying a d-separation criterion to an extended causal diagram in Fig. 3 (also known as parallel worlds network), i. e., X and X d-separate [[a] from X and X Therefore, the following holds
7" (x) [( - 0 X x)) = [ Y ===x= x X x²). =x)
(16)
Figure 3: Parallel worlds network with observed outcome and potential outcome Y[a). In the diagram, (1) XA and X d-separate Y[a] from X and X"; and (2) X and X d-separate Y|a] from A.
Furthermore, due to the invertibility of ((-) wrt. XA and X2, which follows from Eq. 15 and Lemma D we have
E(Y - Y 1== = E(Y [ ((X) (x))
(Lx)))
(17)
Second, Y[a] is also d-separated from A by X and X (see Fig. Therefore, we can employ assumptions (1)-(3 and substitute potential outcomes with observed outcomes:
E(Y(11 0
x²= X x²)
(18)
=E(Y A 1, X - مدد X x²) E(Y A 0, X xA X
2²)
(19)
- (+)))) - (
(20)
where the latter holds due the invertibility.
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Lemma 3 (MSM bounds the RICB). Let representation
x(x) satisfy the following
sensitivity constraint:
I(0)- (x(())/=((0) (fi(z)/7(zz) =()) for all x X s.t. ((x) = Φ.
(21)
where (()) 2 1 is representaioo-deeendeer sensitivity parameter.
Under the assumption in Eq. are given by
21 bounds (Frauen et al.
2023
Oprescu et al.
2023 on the RICB
(Ø) = 1 (b) (0)
and
7 (Ø) = m1(()) u6 (Ø)
(22)
with
(c-la0))
pa( (-
P y (l
y PPY = a, 0) dy
8+ J3 (c_1a.00
10,00
(o) = s+(a, Φ)
NP(Y = y a.
gP(Y = y a. dy.
sa Φ) FF (a.t)
where 8- s-(a, Φ)) = (1- (1- +()))
((1 [()) )= (0) ((6)-1)-1
c_ = 1/(11 T(1)) C+
P(Y a = P(Y = A- a x is
a representation-conniionn density of the outcome, and F= (c a, の) its corresponding quantile function.
Proof. We employ the main results from (Frauen al. 2023
First, it is easy to see, that our definition of the MSM, i e., Eq. (21 fits to the definition of the generalized marginal sensitivity model (GMSM) from (Frauen et al. 2023). Specifically, we can rearrange the terms in Eq.
(2 in the following way (see Appendix C.I in (Frauen et al. (023)):
T
for all æ E X' s.t. ((x) = 0, (23)
r) a (0) +I
x (Ø)
(1- x +
where (0, 1}. Then, apply the Bayes rule the inner ratio
(z)
P(A- = a X = x. 4()) = 4(x))
(24)
Ta (Ø)
PPA
P(X- = x ((X) - (0)
(25)
P(X = x [(X)= = ((x) (0)
P(X x(X) = ((x), A
(26)
P(X = ((X) ())===((00))))))))))))) do
where the latest equality holds, the intervention on A, do(A a), does not change the conditional distribution of P(X (xX) = (xx)). Therefore, our definition of the MSM matches the definitions
of the GMSM in (Frauen al. 2023).
Second, to derive the bound on the RICB, we refer to Theorem I in (Frauen et al. 2023), which states that the lower and upper bounds are expectations of maximally left- and right-shifted interventional distributions, respectively. The maximally left-shifted distribution, P_ a, Φ), and the maximally right-shifted distribution. F+ (Y a, 6). are defined in the following way:
P- = 0) = (a, P(Y = a, o),
if a, 6) S c_.
(27)
(1/s+(a, o)) P(Y
a)
if if a, Φ) > c.,
(1/s+(a, (a) IP(Y = a)
if a, S S.4
PY = a.
(28)
(a, P(Y = - if 0, > C+
(a. where (a, Φ) = (1- (() +T())) (., ((000)))))_((())(((((0(00 (( (1)))((0000))))))))+)))))) +()))))
e_ = ceepresentatioonccndiiiinnn density of the outcome, and Fly a, () its corresponding CDF Then, the lower and upper bounds can be obtained by taking expectations, i.c.,
is
1 ( / yP y a Φ) dy and xa(00) / y yP+ = y 1 (o.6)d dy.
(29)
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Published as a conference paper at ICLR 2024
Corollary 1 (Validity and sharpness of bounds). The bounds the RICB in Lemma 3 Jare valid and sharp. Validity implies that the bounds always contain the ground CATE wrt. representations, Lee (Ø) E (6), (6)] Sharpness implies that all the values in the ignorance interval
(1b) are induced by the distributions which comply with the sensifivity constraint from Eq. (LO).
Proof. We refer to (Frauen et 2023 for formal proof of both properties.
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Published as a conference paper at ICLR 2024
C IMPLEMENTATION AND HYPERPARAMETERS
Implementation. We implemented our refutation framework in Py and Pyro. For better compatibility, the fully-connected subnetworks have one hidden layer with tuneable number of units. For the CNP, we employed neural spline flows (Durkan et al. 2019) with a standard normal base distribution and noise regularization (Rothfuss et al. 2019). The number of knots and the intensities of the noise regularization are hyperparameters. The CNF is trained via stochastic gradient
descent (SGD) and all the other networks with AdamW (Loshchilov & Hutter 2019). Each network was trained with Piger = 5, 000 train iterations.
Hyperparameters. We performed hyperparameter tuning at all the stages of our refutation framework for all the networks based on five-fold cross- validation using the training subset. At each stage, we did random grid search with respect to different tuning criteria. Table provides all the details on hyperparameters tuning. For reproducibility, we made tuned hyperparameters available in our GitHub
Table 5: Hyperparameter tuning for baselines and our refutation framework.
Stage
Model
Hyperparameter
Range Value
Learning rate
0.001, 0.005, 0.01
Minihatch
32, 64, 128
TARNet
Weight decay
0.001, 0.01, R Rd, 1.5 Rd 2 Rd, R dg. 1.5 Rdg. 2 Rd random grid search with 50 runs factual MSE loss Adam
BNN CFR InvTARNer
Hidden units FC, (FCg-1)
Hidden units FC,
Tuning strategy
BWCFR
Tuming criterion
Optimizer
Representation network learning rate
0.001, 0.005, 0.01
Propensity network learning Minihatch size
0.001, 0.005, 0.01
32, 64. 128
Representation network weight decay Propensity network weight decay
0.0, 0.001, 0.01, 0.1
0.0, 0.001, 0.01, 0.1
Stage
CFR-ISW
Hidden units FC,
R dy. 1.5 Rd,, Rdg R Rdg. Rd dg. 1.5 Rdg. Rd random grid search with 100 runs factual MSE loss factual BCE loss
Hidden units FC.
Hidden units FC
Tuning strategy Tuning criterion Optimizer
AdamW
Learning rate Minibatch size
0.001, 0.005, 0.01 32, 64, 128
Weight decay Hidden units FC,
0.0, 0.001, 0.01, 0.1
Rd,, 1.5 Rd Rdg
RCFR
Hidden units FC,,
Rd 1.5 Rd 2 Rd
Hidden units in FC..
Rdg. 1.5 Rdg. 2 Rd
Tuning strategy
random grid search with 50 runs factual MSE loss
Tuning criterion
Optimizer
Adam
Learning rate
0.001, 0.005, 0.01
Minibatch size
32, 64. 128 0.001, 0.01,
Weight decay
Propensity networks
Hidden units FC,
R ddyjo- 1.5 Rd Rd
Tuning strategy
random grid search with 50 runs
Tuning criterion
factual BCE loss
Optimizer
Adam
Stage
Learning rate Minibatch size
0.001, 0.005, 0.01
32, 64. 128 R 1.5 Rdg. Rd,
Hidden units FCeNe
Number of knots
5. 10, 20
CNF
Intensity of outcome noise
0.05, 0.1, 0.5
Intensity of representation noise
0.05, 0.1. 0.5 random grid search with 100 runs factual negative log- likelihokodd loss (momentum 0.9)
Tuning strategy
Tuning criterion
Optimizer
R 2 (synthetic data), R (1IDP1000 HI MNIST datasets)
https://github. yenttt 997/R/C9//0 CB
18

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Published as a conference paper at ICLR 2024
D DATASET DETAILS
D.1 SYNTHETIC DATA
We adopt synthetic benchmark with hidden confounding from (Kallus et al. 2019). but add the confounder as the second observed covariate. Specifically, synthetic covariates, X1, X2, treatment, A, and an outcome, Y, are sampled from the following data generating mechanism:
11 ~ Unif( 22,
X2 ~ N(0, 1).
(30)
A ~ Bern 1+expl- (0.75 -1.76XXX22
Y N ((2 1) 11 + A -2 sin (2 -22A-11 - 1) + x -1
where X2 mutually independent.
D.2 HC-MNIST DATASET
Jesson et al. (021)) introduced a complex high-dimensional, semi-synthetic dataset based on the MNIST image dataset (LeCun 1998), namely HC-MNIST. The MNIST dataset contains Thrain 60, 000 train and frest = 10, 000 test images. HC-MNIST takes original high-dimensional images and maps them onto one- mieedimensional manifold, where potential outcomes depend in complex way on the average intensity of light and the label of an image. The treatment also uses this one- dimensional summary, Φ, together with an additional (hidden) synthetic confounder, U (we consider this hidden confounder as another observed covariate). HC-MNIST is then defined by the following data-generating mechanism:
U ~ (ern(.
X MNIST-image())
Φ = (clip
HN -ec - 1.4, 1.4
Min.
Max. Min. (-441-1.4
a(d; In*)
sug 0.. + 1
(31)
B(o: [*) == (..
1
A ~~ Bern [
+
1-1 B(o;T*)
Y ((2A 1)6 (2.A 1) 2 sin 1)6) 2(U 1)(1 + 0.5Ø), 1).
where is label of the digit from the sampled image X; J,N, is the average intensity of the sampled image; Ae and a. are the mean and standard deviation the average intensities of the images with the label с and Min, factor influences the treatment assignment to a larger extent, i.e., the additional confounder or the one-dimensional summary. We [" exp(1). For further details, we refer to (Jesson et 2021).
-2 + c, Max. -2 + (c 1). The parameter I" defines what
19

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Published as a conference paper at ICLR 2024
E ADDITIONAL RESULTS
E. PRECISION IN ESTIMATING TREATMENT EFFECT
In the following, we report results on the root precision in estimating treatment effect (rPEHE) for the baseline representation learning CATE estimators. The rPEHE is given by
V R EA= (v:[1) (0) - ( where ((0)),;(1 are sample from both potential out- comes. Tables 6 and report the results for synthetic data, IHDPI00 dataset, and HC-MNIST dataset, respectively. For comparison, we also provide the performance of oracle estimators, i. e. ground-truth CATE. Notably, there is no universally best method among all of the benchmarks.
Table 6: Results for synthetic experiments. Reported: in-sample and out-sample rPEHE; mean over 10 runs.
:EELE,
de
1000
5000
10000
300
1000
5000
10000
N-NN
0.03
0.51 0.02
0.48 0.01
0.47
0.01
0.54
0.03
0.51 0.02
0.01
0.47 0.01
BART
0.64 2 004
0.60
1: 0.03
0.51
: 0.01
0.48
2:000 0.0
0.64
: 0.04
0.60 0.03
0.51 0.00
0.48 :: 0.01
C-Porest
0.6 004
0.50 + 0:03
0.49 + 0.01
0.48 + 0.00
0.67 + 0.04
0.59 + 0.05
0
1:900000 0.00
0.48 : 0.00
TARNet
0.96 0.08
0.90 + 0:07
0.0 + 0.07
0.91 0.05
0.59 + 0.10
0.58 à 0.06
0.55 0.05
0.56 1: 0.04
BNN aM 0.1)
098 1 0.8
0.98 1: 0.06
0.97 1: 0.01
0.97 1 000
0.70 1 0.11
0.64 1: 0.11
0.63 0.08
0.59 1: 0.09
CFR (MMD: 0.1)
096 006
0.97 +: 0:05
0.91 : 0.05
0.95 + 000
061 + 0.06
0.61 4: 0.08
061 0.06
0.59 1 0.09
0.67 0.07
(WM)
0.97 0:06
0.95
0.56 0.05
0.86
0.53 0.03
CPR (WM: a 2.0)
0.9 0.08
0.96 +: 0.05
0.97 : 0.04
0.96 0.01
0.38 1 0.06
0.61 : 0.06
0.38 0.08
0.57 : 0.07
In TARRee
0.05
0.90 1: 0.07
0.87 1: 0.05
0.06
0.54 1: 0.85
0.51 1 0.03
0.51 0.04
0.49 1 0.01
1.0)
0.57 0.05
0.05
CFR-ISW (WM: a= 1.0)
0.97 1: 0.05
0.99 1: 0.10
0.96 0.01
0.54 0.03
0.52 1: 0.02
0.33
0.55 0.02
BWCFR (WM: n= 1.0g
0.97 0.04
0.96 4: 0.06
0.95 =: 0.03
0.98 0.02
0.58 - 0.10
0.54 + 0.03
0.53 0.00
0.51 : 0.02
Oracle
0,557
Lower better (est in bold. secund best undertined)
Results for E-NN BART, and C-Forest ane deplicated for differrnnt de
(PENE...
500
1000
5000
10000
500
1000
5000
10000
1.7::00
0.57
0.55
0.51
BART
000
0.62 +: 0.04
0.52 0.02
0.50
0.64
0.62 4 0.04
0.57==00
0.50 0.02
C-Forest
068 2 000
0.61 + 0.83
[151 2 0.82
0.49 0.01
0.68 + 0.63
0.61 # 0.05
0.51 0.02
0.49 2 0.01
TARNet
0.95 0.03
0.91 1: 0.07
0.39 1: 0.08
0.92 0.05
0.59 1: 0.11
0.59 1 0.07
0.55 0.08
0,57 1: 0.04
(MMD: = 0.1)
003
0.98 1 0.06
:: 0.02
0.98
0.70
0.65 4 0.11
0.6 1: 0.09
CFR (MMD: - 0.1)
093 000
0.97 + 0.05
0.93 = 0.06
0.96 + 0.02
0.62 + 0.07
0.62 # 0.08
0 000
0.59 + 0.08
CFR (MMD: 0.5))
0.6 0.03
0,99 1 0.04
0.97
1 0.02
0.98 0.02
0.66 1 0.08
0.72 1 0.07
0.67 0.09
0.67 1 0.08
CFR 1..0)
0.95 1: 0.03
0,97 1: 0.05
0.95 1: 0.00
0.95 1 0.02
0.56 1: 0.05
0.57 1: 0.06
0.1 0.08
0.54 1 0.04
(WM,
IncTARNet
: 0.4
0,90 0,06
0.53 0.04
0.52 0,03
0.02
RCFR a 1.0)
0.96 0.01
0,98 1 0.07
0.96 0.00
0.96 0.03
0.57 1 0.04
0.55 1 0.04
0.56- 0.00
0.56 1 0.04
(WM: 1.0)
1.00
0.5
BWCER (WM: -
0.96 :: 0.05
0.96 = 0.04
0.58 : 0.10
0.54 + 0.04
0.52 : 0.03
0.457
hold, secund underlined) Resulis E-NN BAKI, and duplicated different d,
20

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Table 7: Results for IHDP100 experiments. Reported: in-sample and out-sample rPEHE; mean over 100 train/test splits.
(EHEE,
de
5
10
15
20
25
K-NN
0.668 1. 0.064
0.668 1. 0.064
0.668
1: 0.064
0.668 1: 0.064
0.668 1: 0.064
BART
0.594
0.124
0.594 0.124
0,594
0.124
0.594 = 0.124
0.594
C-Forest
0.641 1 0.066
0.641 1: 0.066
0.641
0.066
0.641 :: 0.066
0.641 1: 0.066
TARNet
0.514 + 0.245
0.503 1: 0.235
0.502
0.236
0.505 0.232
0.503 0.234
BNN a=
0.495 + 0.172
0.490 + 0.175
0.500 + 0.173
0.498 = 0.172
0.469 + 0.184
CFR (MMD; 0.1)
0.463 1 0.186
0.470 1: 0.191
0.477 0.181
0.469 :: 0.188
0.476 1: 0.183
CFR (MMD; 0.5
0.462 = 0,185
0.489 + 0.190
0.503 0.185
0.492 = 0.176
0,500 0.185
CFR (WM: 1
0.463 1 0.190
0.465 1: 0.195
0.467 0,190
0.466 = 0.192
0.466 0.194
CFR (WM; a =200)
0.471 1: 0.183
0.473 1: 0.191
0,475 1 0.189
0.471 1 0.191
0.471 1 0.188
InTTARNer
0.505 + 0.258
0.537 1: 0.310
0,523 0.240
0.514 = 0.235
0,505 1: 0.226
RCFR (WM; 1.0)
0,560 1: 0,154
0.572 1: 0.147
0.499 0.233
0.653 + 0.253
0,506 0.208
CFR-ISW (WM; = 1.0)
0,466 + 0.183
0.469 + 0.189
0.464 + 0.195
0.466 = 0.195
0.471 + 0.199
BWCER (WM: == 1.0)
0.581 0.163
0.600
0.166
0.175
0.595 0.204
Oracle
0.425
Lower = better (best in bold, second best underlined)
N-NN BART,
are duplicated different de
de
5
10
15
20
25
A-NN
0.756 + 0.123
0.756 + 0.123
0.756 + 0.123
0.756 = 0.123
0.756 + 0.123
BART
0.615 + 0.133
0.615 0.133
0.615
0.133
0.615 = 0.133
0.615
0.133
C-Porest
0.667 + 0.123
0.667 = 0.123
0.667 + 0.123
0.667 = 0.123
0.667 + 0.123
TARNet
0.620 + 0.292
0.622 + 0.288
0.620
0.291
0.613 0.285
0.613 0.285
0.514 + 0.181
0,510 0.185
0.183
0.515 0.181
0.496
CFR (MMD; 0.1.)
0.492 1 0.193
0.497 1: 0.201
0,500 1: 0.186
0.493 1 0,191
0,500 1 0.190
CER (MMD;
0.493 0.191
0.502 1 191
0.191
0.509 1: 0.185
CFR a.
0,499 + 0.202
0,500 + 0.211
0.495 + 0.203
0.497 = 0.204
0,498 + 0.204
CER cx
0.500 + 0.198
0.49 + 0.201
0.503 0.202
0.499 = 0.203
0,500 0.196
TarTARNet
0.563 + 0.269
0.610 + 0.350
0.618 + 0.287
0.605 = 0.277
0.595 + 0.267
RCFR (WM: c 1.0)
0.601 + 0.202
0.612 + 0.198
0,535 + 0.238
0.676 0.281
0,580 + 0.256
CFR-ISW (WM; a= 1.0)
0,501 1: 0.195
0.501 1: 0.203
0.496 1: 0.207
0.497 1: 0.208
0.495 0.210
BWCER (WM: = 1.0)
0.599 1 0.182
0.618 0.190
0.604 0.184
0.601 1: 0. 190
0,731 0.988
Onacle
Lower hetter (hest in bold, second hest underlined)
Rosults R-NN, BART, and C-Forest duplicated for different de
Table 8: Results for HC-MNIST experiments. Reported: in-sample and out-sample rPEHE; mean over 10 runs
iPEHE,
:PEHE...
d A-NN
1.06
0.01
1.06 +
0.01
1.10
0.01
BART
0,89 1 0.24
0.89 0.24
0.89 1 0.24
0.89 1 0.25
0.89 1 0.25
0,89 1 0.25
C-Forsst
0.82 0.00
0.00
0.82 - 0.00
0.00
0.82 +
0.82 0.00
TARNet
0.71 = 0.02
0.66 1 0.01
0.67 + 0.03
0.72 - 0.02
0.70 + 0.01
0.71 0.02
BNN (MMD; 0.1)
0,75 4 0.03
0.73 0.03
0.93 + 0.14
0.77 0.03
0.74 + 0.03
0,93 :
CFR (MMD 0.1)
0.73 = 0.03
0.67 - 0.01
0.69 - 0.03
0.74 0.02
0.71 + 0.01
0.73 0.02
CFR (MMD, 0.5))
0.86 :: 0.23
0.74 : 0.02
0,80 + 0.15
0.86 0.13
0.76 +: 0.02
0,78 0.05
CFR o=M.a=100
1.10 = 0.17
1.21 = 0.14
1.37 - 0.23
1.10 0.17
1.19 + 0.14
1.28 0.14
CFR (WM; 2.0))
1,40 0.31
1.33 0.09
1,30 + 0.05
1.28 0,10
1,30 +: 0.06
1.29 129:000 0.06
Inv TARNet
0.01
RCFR (WM, = 1.0)
1.83 - 1.48
2.78 1: 4.05
3.68 +: 3.43
1.24 1: 0.29
2.66 4: 4.33
3.05 372
CFR-ISW (WM; a
1.29 0.12
1.25 0.04
1,28 1 0.06
1.25 0.05
1,26 1. 0.08
1.27 0.07
BWCER (WM: - 1.0)
2.92
2.39
1.31 + 0.05
1.35 + 0.05
1,46 : 0.45
1,32 + 0.06
1.35 + 0.05
Oracle
0,522
0.513
Lower better bold. second best undertined)
Results for A-NN BART, and C-Forest are duplicated for differeut d,
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E.2 POLICY ERROR RATES AND DEFERRAL RATES
Here, we examine the trade-off between the improvement in the error rates and the increase in the deferral rates, which appear after the application of our refutation framework. One way do so is
through policy error rale vs. deferral rale plots, as in (Jesson et al., 202T). In Figures 4 [| and 6 we report the error rates and the deferral rates for synthetic data, IHDPI00 dataset, and HC-MNIST
dataset, respectively. Therein, the bascline representation learning methods for CATE are shown as points located the vertical line with a deferral rate of zero. Each of those points is then connected with a point corresponding to our bounds performance. As result, we see that our refutation framework achieves an improvement in the error rates with only a marginal increase in the deferral rates.
2
1-NN
01-
SART
03
0.4
2
26
4
CTT 1W52
CFR
C3T (W-)
032
(403-- 1100
JWOTT (W33
00 0.2
6.4
0.6 0.0
3.2
81 8.5
3.0
100
112
0.5
3.5
0.5
0.4
100
100
1.0
Figure 4: Policy error rate vs. deferral rate plot for synthetic data. Reported: out-sample performance of baseline methods connected with the performance of our refutation framework; mean over 10 runs. Here, 5 varies in the range {0.0005, 0.001, 0.005, 0.01, 0.05), which corresponds to several scatter points per line.
10
35
0.00
0.00
E-N
0.00
BAIET
(.Rursl
TARlNet
0.00
RNN (MMT)) o 0.1
Oifirrl
CSR (MMI2
21
25
OER (W3E
0.00
LOR (WM) R.]
Tw<TARN
BCFR (W3G
3.06
UWAESE (W3): 1.0.
0.07
8.00
Figure 5: Policy error rate vs. deferral rate plot for IHDPI00 dataset. Reported: out-sample performance of baseline methods connected with the performance of our refutation framework; mean over 100 train/test splits. Here, 8 varies in the range (0.0005, 0.001, 0.005, 0.01, 0.05]. which corresponds to several scatter points per line.
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JABI
(000
1932 28)
1931
3030
Figure 6: Policy error rate deferral rate plot for HC-MNIST dataset. Reported: out-sample performance of baseline methods connected with the performance of our refutation framework; mean 10 runs. Here, å varies in the range {0.0005, 0.001, 0.005, 0.01, 0.05), which corresponds to several scatter points per line.
E.3 DECISION BOUNDARIES
Fig. and we provide plots of the decision boundaries the baselines (using the synthetic benchmark). Therein, we plot both the ground-truth boundaries and boundaries, estimated with our refutation framework in combination with different representation learning methods for CATE. On the left side (dg 1), the low-dimensional representation induces the same time loss of heterogeneity and confounding bias. We see that our bounds remain approximately valid for CATE wrt. representations, for different variants of CFR and Inv InvTARNet. On the right side (dg = 2), there is no loss of heterogeneity but still some hidden confounding induced. Yet, our bounds are valid for both CATE wrt. representations and covariates for all the baselines. Additionally, we observe that the tightness of the estimated bounds highly depends on the representation learning baseline, e. g.. the bounds on the RICB are the tightest for TiRRNNe, as its representations introduce almost no confounding bias.
when the dimensionality of the representation is misspecified, especially
Enbardd
CATT
Riice Raack
Figure 7: Decision boundaries for the synthetic dataset. Ground-truth and estimated boundaries shown with hatches and colors, respectively. Training samples (Murain = 10, 000) are shown too to highlight treatment imbalance. Here, 0.01.
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OPA (MKG)
Tourd
CTA CTR-WVa--4 20 4-1
CH -20) mm
Triting work
x-00 Righ en
Tondal
Troked
A-4
Drarnc Ca
Queic
Denndd
Eeng fig
Catooled
RCFR WWl
DATS
Koll lig-a-nn x-ila
Tuatne
4.50
4.5
0.0
CT NOW (FF1 18)
C
Toand
Toals
Teandl
-01
20c7z (Mlc 2-1
(W3
Salorte
caogle Toard
Tearne
-03
Figure 8: Decision boundaries for the synthetic dataset. Ground-truth and estimated boundaries are shown with hatches and colors, respectively. Training samples (Murain = 10, 000) are shown too to highlight treatment imbalance. Here, 0.01.
24


Reviews:
[{"review_id": "58NqmYT85N", "reviewer": "Reviewer_a9aZ", "summary": "This paper addresses the problem of induced confounding that occurs in neural network based conditional average treatment effect estimation as a result of representation learning that operates over a lossy reduced dimension embedding. The authors propose to account for the confounding by leveraging sensitivity analysis. In particular the authors use the marginal sensitivity model and provide bounds on the CATE. A framework is then introduced to estimate the proposed bound within a neural network training flow. A set of experiments are provided which validate the efficacy of the proposed approach.", "questions": "Given the relative difficulty of CATE estimation in small sample regimes, as the authors point to, it would seem that there are a number of settings where representation based CATE estimation is inappropriate. Given this it would be useful for the authors to compare the bounds provided here and contrast to non-NN based approaches (e.g., BART / causal forests) to give a sense of the relative loss in precision due to the representation induced confounding.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "This paper addresses a very important, and often overlooked, aspect of representation learning for causal effect estimation. The authors do a commendable job of describing the circumstances under which we should expect to incur bias due to representation induced confounding, and clearly delineate them from existing approaches which don't suffer from the same issues. The proposed sensitivity analysis is intuitive and the authors do a nice job of describing it's integration into the neural network training process.", "weaknesses": "The largest weakness I see is the same as what is commonly shared throughout the sensitivity analysis literature, namely that practitioners must place assumptions on the extent of confounding.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "6o4NvmhuzD", "reviewer": "Reviewer_Pw4L", "summary": "Estimating conditional average treatment effect (CATE) estimation widely uses low-dimensional representation learning, which can lose information about the observed confounders and thus lead to bias.\nIn this paper, the authors propose a new framework for estimating bounds on the representation-induced confounding bias (RICB). To summarize, the contributions are three-fold:\n1.\tCATE from representation learning methods can be non-identifiable due to RICB.\n2.\tThe authors propose a representation-agnostic framework to perform partial identification of CATE.\n3.\tThe authors demonstrate the effectiveness of our bounds together with a wide range of state-of-the-art CATE methods.", "questions": "1. According to the definition of $X$, $X=\\{X^\\emptyset,X^a,X^y,X^\\bigtriangleup\\}$. At the same time, $X$ is independent of $X^\\emptyset$, $X^a$, $X^y$ ,$X^\\bigtriangleup$ conditioning to $\\Phi(X)$. It is strange to claim Eqn. (4).\n2. In the example “Representations with removed noise and instruments”, the authors claim that under Eqn. (5), the validity follows from the d-separation in clustered casual diagram and Appendix B. In appendix B, only investigations related to the example “Invertible representations” are presented.\n3. I suspect the equality of $\\mathbb{E}(Y[1]-Y[0]|X=x)=\\mathbb{E}(Y[1]-Y[0]|X^\\bigtriangleup=x^\\bigtriangleup, X^y=x^y)$ and $\\mathbb{E}(Y[1]-Y[0]|X^\\bigtriangleup=x^\\bigtriangleup, X^y=x^y)= \\mathbb{E}(Y[1]-Y[0]|\\Phi(X)=\\Phi(x) $ in Eqn. (6) under Eqn. (5). Could the authors provide more details?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper is technically sound and well-organized.", "weaknesses": "It seems that the notations/symbols are not defined correctly. For example, in the section of notations, the authors claim that $\\mu_a^x(x)=\\mathbb{E}(Y|A=1,X=x)$, but $\\mu_a^x(x)$ should be $\\mathbb{E}(Y|A=a,X=x)$. In the same paragraph, the authors claim that $\\mu_a^\\phi(\\phi)=\\mathbb{E}(Y|A=1,\\Phi(X)=\\phi)$, but $\\mu_a^\\phi(\\phi)$ should be $\\mathbb{E}(Y|A=a,\\Phi(X)=\\phi)$. In addition, the authors define $\\pi_a^x(x)= \\mathbb{P}(A=a|X=x)$. I wonder why the authors do not simply $\\pi_a^x$ or $\\pi_a(x)$. Problem arises when the authors introduce overlap assumption. The authors claim that $\\mathbb{P}(0<\\pi_a^x(X)<1)=1$, but I cannot obtain $\\pi_a^x(X)$ from the definition. Indeed, in the definition of $\\pi_a^x(x)= \\mathbb{P}(A=a|X=x)$, The two “x”s in $\\pi_a^x(x)$ should be mapped to “x” in $\\mathbb{P}(A=a|X=x)$. Nevertheless, when $\\pi_a^x(x)$ is changed to $\\pi_a^x(X)$ or $\\pi_a^X(x)$, the mapping procedure is not clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "64yud8sH9Y", "reviewer": "Reviewer_8op8", "summary": "This paper tackles the problem of confounding bias induced by learning representation of confounder for CATE estimation. The authors proposed a framework for estimating bounds on the induced confounding bias. A neural framework is used to compute the bounds.", "questions": "- Could you provide some intuition about learning the representation of all the covariates together instead of the confounder?\n- If learning representation of covariates inducing bias is unavoidable, how does the bias compare with bias due to finite-sample? e.g., How does it compare with the non-representation learning approach?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper presents a problem that is novel and related to the representation of learning for CATE, which is a prominent research direction.\n- A detailed analysis of representation-induced bias is provided.\n- Both real-world and synthetic experiments are performed with the proposed framework.", "weaknesses": "- The motivation for employing CDAG is not quite clear. \n- No theoretical proof of the proposed bounds.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "yOMnCTLUS8", "reviewer": "Reviewer_FevT", "summary": "This paper discusses a setting where a representation function $\\phi(X)$, (a generalization of propensity score $\\pi(X)$), is available while part of X is unobservable. That is, instead of following the typical approach of choosing X from only observables (expecting $\\phi(X)$ to be a balancing score) and discussing the potential effects of unobservable covariates, they follow the approach of considering X as all covariates including even unobservable covariates. At the same time, they assume that $\\phi(X)$ value is available (while part of X is not observable)\n\nIn the typical approach, when we cannot assume that $\\phi(X)$ is a balancing score, we may suffer confounding bias. In the exact same way in their approach, when we cannot assume that $\\phi(X)$ (in their case it is called the representation function) includes enough information about unobservable covariates, we may suffer a bias. (in their case it is called the representation-induced confounding bias or RICB)\n\nThe authors identify RICB, and propose a technique to estimate the bound.\nComprehensive simulation studies were followed.\n\n\n---------------------------------------------------------------------------------------------------------------------------------------------\n**Replying to the public discussion***\n\nTitle: \"2021 NeurIPS paper ... assumes that all confounders are X\" is what I am saying.\n\nHi Alicia,\n\nThank you for participating in the discussion.\nYes, your paper assumes that all confounders are included in X, which is observed. This is your paper's assumption 1, which states that there are no unobserved confounders.\n\nOn the other hand, this paper's setting is different. \nI quote myself above:\n\"This paper discusses a setting where a representation function $\\phi(X)$, (a generalization of propensity score $\\pi(X)$), is available while part of X is unobservable. That is, instead of following the typical approach of choosing X from only observables (expecting $\\phi(X)$ to be a balancing score) and discussing the potential effects of unobservable covariates, they follow the approach of considering X as all covariates including even unobservable covariates. At the same time, they assume that $\\phi(X)$ value is available (while part of X is not observable)\"\n\nMy concern is that, the authors are saying that their setting is making a typical assumption (which is not true) and then citing your work as the one of the works that make typical assumption (which is true). \n\nThanks,\nReviewer FevT.\n\n---------------------------------------------------------------------------------------------------------------------------------------------\n**Replying to the public discussion 2***\n\nHi Alicia,\n\nI understand what you are saying - I think what AC pointed out is correct. \nI modified my score, as I have no further concerns. \n\nThank you very much again for participating in the discussion.\n\nMany thanks,\nReviewer FevT", "questions": "In terms of Weakness 1: Could you please give a clear motivation of choosing X to include unobservable but considering $\\phi(X)$ as observable, and then reframing the confounder bias we know as RICB in the newly proposed setting? Does the fact that we are dealing with representation learning make some difference? I just want to try to understand.\n\nIn terms of Weakness 2: How tight is the bound for some popular special cases, especially for the settings you did your experiment? Are they practically good?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Simulation studies are quite comprehensive.\nTheoretical bounds has been proposed. \nThe paper is very well written. It was pleasant to read.", "weaknesses": "\\textbf{1. Motivation of their approach}\n\nAs discussed in the Summary part of this review, for me it was hard to understand why we need a new approach of choosing X. The concept of RICB is, in essence, equivalent to confounder bias but formulated in a different choice of X. For example, in the traditional way of choosing X as only unobservables and talking about $\\phi(X)$ not being a balancing score, potential effect of unobservable covariates not being included as X can be discussed. So I am not sure about the potential benefit of choosing X to include unobservable covariates.\n\n\\textbf{2. Bounds}\nTheoretical bounds provided should be appreciated, but I cannot be sure how strong this theoretical bound is only from current version of the manuscript.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (spotlight)