Quantifying Aleatoric Uncertainty of the Treatment Effect: A Novel Orthogonal Learner
Authors: Valentyn Melnychuk
Stefan Feuerriegel
Mihaela van der Schaar
Venue: NeurIPS 2024 poster (2024)
Keywords: causal inference
treatment effect estimation
Primary area: causal_inference
TL;DR: quantification of an aleatoric uncertainty of the treatment effect at the individualized (covariate-conditional) level

Abstract: Estimating causal quantities from observational data is crucial for understanding the safety and effectiveness of medical treatments. However, to make reliable inferences, medical practitioners require not only estimating averaged causal quantities, such as the conditional average treatment effect, but also understanding the randomness of the treatment effect as a random variable. This randomness is referred to as aleatoric uncertainty and is necessary for understanding the probability of benefit from treatment or quantiles of the treatment effect. Yet, the aleatoric uncertainty of the treatment effect has received surprisingly little attention in the causal machine learning community. To fill this gap, we aim to quantify the aleatoric uncertainty of the treatment effect at the covariate-conditional level, namely, the conditional distribution of the treatment effect (CDTE). Unlike average causal quantities, the CDTE is not point identifiable without strong additional assumptions. As a remedy, we employ partial identification to obtain sharp bounds on the CDTE and thereby quantify the aleatoric uncertainty of the treatment effect. We then develop a novel, orthogonal learner for the bounds on the CDTE, which we call AU-learner. We further show that our AU-learner has several strengths in that it satisfies Neyman-orthogonality and, thus, quasi-oracle efficiency. Finally, we propose a fully-parametric deep learning instantiation of our AU-learner.

Quantifying Aleatoric Uncertainty of the Treatment Effect: A Novel Orthogonal Learner
Valentyn Melnychuk"", Stefan Feuerriegel', Mihaela van der Schaar?
'LMU Munich & Munich Center for Machine Learning (MCML), Germany University of Cambridge & Alan Turing Institute, United Kingdom "Correspondence: melny eoglnychuk@mmu.co
Abstract
Estimating causal quantities from observational data is crucial for understanding the safety and effectiveness of medical treatments. However, to make reliable inferences, medical practitioners require not only estimating averaged causal quan- tities, such as the conditional average treatment effect, but also understanding the randomness of the treatment effect as a random variable. This randomness is referred to as aleatoric uncertainty and is necessary for understanding the prob- ability of benefit from treatment or quantiles of the treatment effect. Yet, the aleatoric uncertainty of the treatment effect has received surprisingly little attention in the causal machine learning community. To fill this gap, we aim to quantify the aleatoric uncertainty of the treatment effect at the covariate-conditional level, namely, the conditional distribution of the treatment effect (CDTE). Unlike average causal quantities, the CDTE is not point identifiable without strong additional assumptions. As remedy, we employ partial identification to obtain sharp bounds on the CDTE and thereby quantify the aleatoric uncertainty of the treatment effect. We then develop a novel, orthogonal learner for the bounds on the CDTE, which we call AU-learner. We further show that our AU-learner has several strengths in that it satisfies propose hally-parametric deep learning instantiation of our AU-learner.
and, thus, quasi-oracle efficiency. Finally,
Introduction
Estimating causal quantities from observational data is crucial for decision-making in medicine 99 12
22.30 70). For example, medical practitioners are interested in estimating the effect of chemotherapy vs. immunotherapy on patient survival from electronic health records understand the best treatment strategies in cancer care. Here, common estimation targets are averaged causal quantities such as the average treatment effect (ATE) and the conditional average treatment effect (CATE), yet averaged causal quantities do not allow for understanding the variability of the treatment effect.
What is needed for the reliability of causal quantities in medicine? To obtain reliable causal quantities, one often needs to "move beyond the mean" [44] 68) and consider the inherent randomness in the treatment effect random variable. This randomness is referred to aleatoric uncertainty [17 60 110]. Quantifying the alcatoric uncertainty of the treatment effect is relevant in medical practice to understand the probability of benefit from treatment (26 600 and the quantiles and variance of the
treatment effect (3) 17 26 33 [99. As an example, averaged quantities such as the CATE would
simply suggest positive effect for some patients, while the probability of benefit from treatment can inform patients about the odds of being negatively affected by the treatment. Hence, aleatoric uncertainty of the treatment effect promises additional, fine-grained insights beyond simple averages.
Methods for quantifying the aleatoric uncertainty of the treatment effect have gained surprisingly little attention in the causal machine learning community. So far, machine learning for treatment
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Figure 1: Identification and estimation of the conditional distribution of the treatment effect (CDTE) (=our setting) compared to the (well-studied) identification and estimation of the CATE. In this paper, we focus specifically on the CDF of the CDTE, P(Y [ < 8 x)) shown in orange. Our main contribution relates to the estimation, shown in yellow. However, moving from CATE identification
and estimation our setting comes with important challenges: ① CATE (shown in green) is point identifiable but the CDTE is not (shown in blue); @ there is no closed-fon form expression of the target estimand in terms of nuisance functions and, because of that, CATE learners cannot be directly adapted for estimation; and @ CATE is an unconstrained target estimand whereas Makarov bounds (shown gray are monotonous and contained the interval [0, I].
effect estimation was primarily focused estimating averaged causal quantities 20 57
69
96
98
125 128]. Some research aims to quantify the epistemic uncertainty in treatment effect estimation [51 the total uncertainty (but without distinguishing the types of uncertainty)
114
67
76). Other works focused on the aleatoric uncertainty of the potential outcomes [13 31 65
66
94
T
or contrasts between distributions of potential outcomes (also known as distributional treatment
effects) 16 29 62
92 100 However, to the best of our knowledge, there is no comprehensive meta-learning theory for the estimation of the aleatoric uncertainty in the treatment effect.
In this paper, we aim to quantify the alcatoric uncertainty of the treatment effect at the covariate- conditional level in the form of conditional distribution of the treatment effect (CDTE). Knowing the CDTE would automatically allow one to compute the above-mentions quantities of aleatoric uncertainty, namely, the probability of benefit from treatment and the quantiles and variance of the levels.
treatment effect, at both population and
1.1 Challenges
Yet, the identification and estimation of the CDTE in contrast to CATE come with three challenges as follows (see Fig. D
Challenge ① is that the CDTE does not allow for point identifiable, neither in the potential outcomes framework nor in randomized control trials due to the fundamental problem of causal inference as counterfactual outcomes can not be observed [26] 88). We thus employ partial identification [48] to obtain bounds on the CDTE and thereby quantify the aleatoric uncertainty of the treatment effect. Specifically, we focus on Makarov bounds [87] 126 [29] that give sharp bounds for both the cumulative distribution function (CDF) and the quantiles of the CDTE.
Challenge 2 is that there is no closed-fore form expression of the target estimand in terms of nuisance functions. Because of this, existing CATE learners cannot be directly adapted to our task of estimating Makarov bounds. For example, there are no orthogonal learners in the general setting, and existing
potential outcome Y[a] refers the value an outcome variable would take for an individual under specific treatment or intervention a. Each individual has multiple potential outcomes - one for each possible treatment condition - but only one of these outcomes is observed.
In the Neyman-Rubin potential outcomes framework [109],
NNotably, (a) the distributional treatment effects and (b) distribution of the treatment effect (our setting) are both different interpretationally and inferentially. That is, (a) provide contrasts between distributions of potential outcomes and are point identifiable, and (b) work on the distribution of the difference of potential outcomes and are only partially identifiable. See Appendix A for further details.
2

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approaches only use naive plug-in estimators/leamers. Furthermore, even the derivation of the orthogonal loss is non-trivial as there no efficient influence function at hand for the Makarov bounds.
Challenge is that CATE is unconstrained target estimand whereas Makarov bounds monotonous and contained in the interval [0, 1]. Notably, any constraints of the target estimand could be violated by orthogonal learners [71 125]. Therefore, an orthogonal learner for Makarov bounds needs to be carefully adapted, especially y to perform well in low-sample settings.
1.2 Our contributions
Trestment offect, 6
In this paper, we develop a novel, orthogonal learner for estimating Makarov bounds which we call AU-learner, which allows to quantify the aleatoric uncertainty of the treatment effect. Our AU-learner addresses all of the
Patent charadtenets,
abovenmentioned challenges 1 @ Further, our AU- learner has several useful theoretical properties, such satisfying Neymma-orthoognnality and, thus, quasi-oracle efficiency (98). Finally, we propose flexible, fully- parametric deep learning instantiation of our AU-learner. For this, make of conditional normalizing flows and call our method AU-CNFs.
Treat wilect
Fotent charastenste,
x
Challinge
To summarize, contributions are as follows:
Awetoric uncertserty
Partill identiication slaaitr it uncer biirly
1. We derive a novel, orthogonal leaner called AU-learner quantify the aleatoric uncertainty of the treatment effect. For this, we estimate Makarov bounds on the CDF/quantiles of the conditional distribution of the treatment effect (CDTE).
Total
uncorta inty
Figure 2: Total uncertainty of the treat- ment effect can have different sources. Both upper and lower plots have the same total uncertainty but vastly dif- ferent aleatoric and epistemic compo- nents. Yet, aleatoric uncertainty is non- identifiable (see Challenge
2. We prove several favorable theoretical properties of our AU-learner, such Neyman-orthogonaale and, thus, quasi-oracle efficiency.
3. We propose a flexible deep learning instantiation of our AU-learner based on conditional normalizing flows, which we call AU-CNFs, and demonstrate its effective- ness over several benchmarks.
2 Related Work
In the following, we briefly summarize the existing works uncertainty quantification in the potential outcomes framework; on the identification of the CDTE; and on the estimation of Makarov bounds. For more detailed overview of literature, we refer Appendix A
Uncertainty quantification in the potential outcomes framework. The (total) uncertainty of predictive model in machine learning is generally split into (a) epistemic and (b) aleatoric uncertainty
41 50 This split is important, as it informs a decision- maker about the source of uncertainty (see Fig. 2). especially in the context of the potential outcomes framework. (a) Epistemic uncertainty was studied for predictive models targeting at identifiable averaged causal quantities, such as conditional average potential outcomes (CAPOs) and CATE [51] (2).. (b) Aleatoric uncertainty, on the other hand, is only identifiable for potential outcomes [88] Prominent methods focus on interventional
86]; (ii) density estimation [65] 66 91 94 97 102 124 and (iii) distributional distances (also known as distributional
(counterfactual) quantities such as: (i) CDF/quantiles estimation (6 13
31
43
treatment effects) estimation 16 29 62 92 100). Yet, our work differs substantially from the above,
as we aim at inferring the aleatoric uncertainty of the treatment effect, which is only partially identifiable.
CCode is available at 'Epistemic uncertainty (50) relates to the uncertainty of fitting a model on finite data and reduces to zero as data size grows. In contrast, aleatoric uncertainty originates from the inherent randomness of the outcome and is irreducible wrt. data size.
com/ comm/aaeenty1199/77 CNFs
3

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Table I: Overview of methods for estimating Makarov bounds on the CDFqquantiles of the CDTE.
Work
Covariate- conditional
Limitations
Estimaton! learner
Orthogonal
Instantiation
Plug-in
Empirical
Ruiz 4t
TT0
Plag-in
Conditional moan with hold-out residuals
Lec
77
Plug-in
Kernel density extimation
Binary ouscomae
A-IPTWINO
Random forest causal
1i
Optimization ns recumprions
A-IPT/NO
Homoskadastic Gouesian linear model
Samemova
12
Propensity known
PPTW
Our paper (AU leamer))
A A-IPTWNNO
Conditionll normalizing flows (AU-CNFF)
(A-JIPTW: (Hagmented) inverse propensity of treamment weighted; NO: Neyman-orhhggnal
Identification of the distribution of the treatment effect. Point identification of the distribution of the treatment effect (or, equivalently. joint distribution of potential outcomes) is only possible under additional assumptions on the data-generating mechanism. A common example is, e, g. invertibility of latent outcome noise (2 6]. Other works have rather focused on partial identification [48). For example, [88 proposed sharp bounds for the distribution of the treatment effect under a monotonicity assumption. Later, assumption-fin free sharp bounds were proposed for both the joint CDF of potential outcomes 26) and for the variance of treatment effect (5). both known as Fréchet- Hoeffiin bounds 39 49
Finally, [26] 28 33 proposed sharp bounds on the CDF/quantiles of the treatment effect
without any additional assumptions, so-called Makarov bounds [87] 126). Makarov bounds were
further generalized [8
129 and applied to other settings 17 25
36
59 but different from ours.
Estimation of Makarov bounds. Table I provides comparison of key methods for estimating Makarov bounds, at both covariate-conniitooaal and population levels. Existing methods build mainly upon plug-in (single-stage) estimators/eermers. Examples are methods tailored for randomized
controlled trials [26] 110 and for potential outcomes framework [17 77 110). Crucially, these methods are not orthogonal and, thus, are sensitive the misspecification of the nuisance functions. Nevertheless, we include the latter methods [17] (77) as baselines for our experiments as they use a highly flexible CDF estimator based on kernel density estimators. Some works also developed efficient estimators for Makarov bounds at the population level (analogous to the two-stage Neyman- orthogonal learners at the covariate-conditions level) but only in highly restricted settings. In particular, (60) is restricted to binary outcomes, (112) assumed known propensity score, and (54) made special optimization assumptions
F In addition, all three works (54 60 112) suggest fixing value of ó/a, which the CDF/quantiles of the treatment effect are evaluated at; when our work suggests targeting at several values of 8/a at once. Therefore, the previous methods are not applicable connitiona level Makarov bounds.
to our general setting of estimating
Research gap. To the best of our knowledge, we are the first to propose an orthogonal learner for estimating Makarov bounds on the CDF/quantiles of conditional distribution of the treatment effect.
3 Identification of Distribution of Treatment Effect
Notation. Let capital letters X, A,A,M Y, A denote random variables and small letters x, a, y, 6 their realizations from domains X, A' A. Let IP(Z) denote a distribution of some random variable Z, and let 2 = 2) be the corresponding density or probability mass function. Furthermore, T(z) = P(A = 1 X = 2²) is propensity score, (a = E(Y X = xr = a) are conditional expectations, and (y z) = PP < xz a) is conditional outcome CDF. For other conditional quantities or distributions, we use short forms whenever possible; e. e.g.. E(Y z) = B(Y X z). Further, (z)( = - f2;) is a sample average of random f(Z), where n is the sample size. We denote linear rectifier functions as [x]++ = max( 0) and [x] = min (x. 0), and sup/inf convolutions of two functions (116) (- x), fz(- x) as (f1 T (2) (6 x) suppey (1 x) fz(y 5 x)}
and (f1 (5 x) = inf (f x)) fa(y 8 x)).
Problem setup. We consider the standard setting of the Nuyman-Rubin potential outcomes frame- work [09]. That is, we have an observational dataset 'D with a binary treatment A E A (0,1}, potentially high-dimensional covariates X E X C ad and a continuous outcome Y E 3 C R. For instance, typical scenario is in cancer therapy, where the outcome is tumor growth, the treatment is whether chemotherapy is given, and the covariates include patient details like age and sex. We define a joint random variable Z (X, Y). D = {x;, dg, yi ) is sampled i.i.d. from the observational
'The work in 54) assumes the possibility of finding feasible Kantorovich dual functions to target functional (e. g., CDF of the treatment effect). By doing so, the authors are able to infer valid partial identification bounds on very general functionals; yet, the sharpness can not be practically guaranteed.

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distribution P(Z) - P(X, Y, A), where is the sample size. The potential outcomes framework
then makes three (causal) assumptions, i.e. (1) consistency: if A = a, then Y Y[a] Y: (2) overlap:
P(O S (X) s 1) 1; and (3) exchangeabillty: A (V(O),Y()) X.
Treatment effect distribution. In this paper, refer to the treatment effect A Y[1 Y random variable 6 The CATE is given by ((x) = EE 2), which is identifiable as μr (I) - no (a)
under the causal assumptions (1)(3). We interested in identifying a conditional distribution of the treatment effect (CDTE), specifically, its CDF or quantiles:
F x) ((A s 8 x) = P(Y[4] - [ 1611), ded
(1)
F (a x) = in e A a S F(8 x)).
10.
(2)
The CDF and the quantiles of the CDTE are point non non-identifiable due to the fundamental problem of causal inference, e., that the counterfactual outcome, Y[1 - -A], is never observed. This is illustrated in Fig. 3 where both conditional potential outcome distributions are identifiable as P(Y x) = PP 2;, a); but a conditional joint distribution, [0] Y x),, and the CDTE,
P(A 27, are not.
Partial identification of the CDTE. Fan al. (26) proposed pointwise sharp bounds on the CDF and the quantiles of the CDTE, so-called Makarov bounds [87] 126 [29]. Given that the outcome Y continuous, the Makarov bounds for the CDF of the CDTE are given by linearly rectified sup/inf
convolutions 116] of conditional CDFs of potential outcomes:
F(6 x)) S F(O x) S F x), (8
(3)
F x)) = [(F] (8)
=)]+ and F(8 æ) =1 [[F Fo [+)]_
where E 4, and F, (y x)) the CDF (Y z). Similarly, Makarov bounds can be formulated for the quantiles of the CDTE:
(a a) x) 12) FF 2))
(FT Fo (aaj aa),
#0
if # 1, (FT + (0.0) (a-1 [x),
ET Fo [a)
and F- ax)
(0 a) Fo (1((1)) x)
=fa=0,
FT (1 a)) Fo (01 x))
if 1,
(4)
where [0. 1], and FL x) are the quantiles of P(Y[a] x). Then, under the causal assumptions (1)-(3), conditional distributions of potential outcomes coincide with observed ones, e. ((Y]a) = IP( x. a). Notably, Makarov bounds on the CDFs are CDFs themselves. but these CDFs do not correspond to the solution of the partial identification task (which implies pointwise sharpness). We refer to Appendix for more illustrations about the inference of the Makarov bounds, the explanation of pointwise sharpness, and Makarov bounds for categorical/mixed-ry outcomes.
4 An AU-learner for estimating Makarov bounds
In the following, we develop a theory of orthogonal learning for Makarov bounds, which then gives rise to our AU-learner. For this, we first review the plug-in in learner and its shortcomings. Motivated by this, we then derive two-stage learners. Here, we first present novel CA-learner an intermediate step and finally our AU-learner. Note that both are novel but we frame our contributions around the AU-learner because of favorable theoretical properties. For notation, we use over- and underlines as to refer the upper and/or lower bound.
in, e. g., F(5 x)
4.1 Single-stage learners for Makarov bounds
Plug-in learner. A naïve way to construct estimators for Makarov bounds 17
26
77
110
is to
estimate conditional outcome CDFs, IP (v) a:), and plug them into Eq. 3
(6 x) PP.= =)]+ and Frild æ) = 1 [ x)]
(5)
The plug-in in learner for the bounds of the quantiles, Ep z)) can be obtained in similar way but where one uses Eq. Q and relies on the estimators of either the conditional outcome quantiles or inverse of the estimated conditional outcome CDFs, F- (u I).
AIss known as an individual treatment effect. The term 'individual' should not be confused with the term noovaiaate-oondtionaal which refers to the causal quantity and not the random variable itself.
5

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Chilonge
PLV
M. 2.000. 2.1
022
3
10
(b)
(b)
1.00
1:00
0.73
0.72
0 0.40
0.00
EH4 kt]
0.35
PIF (id.
0.20
119 ins F99
10
Figure 3: An example showing point non-identifiability of the distribution of the treatment effect based on the i == 7-th instance of the semi-synthetic IHDP100 dataset [46). Shown are two data- generation models, indistinguishabe in potential outcomes framework or RCTs, i. monotone,
Mm. and an antitone, MMa For both models we also plot (a) conditional densities of potential outcomes, [aY y xz)) and conditional joint laws of potential outcomes, P(Y [0), Y[1] x7);
and (b) corresponding CDFs of the CDTE (shown in blue), F(5 x7) P[Y Y [0] < 8 x7),
together with Makarov bounds (shown gray and point identifiable CATE (shown in green).
7(=>)= = [E( z7) 2.342. Non-identifiability of the CDTE is easy to see: Both data-generation models have the same conditional distributions of potential outcomes but different conditional joint laws and, thus, different CDTEs. The latter figures, (b), also demonstrate the bounds on the probability
of benefit from treatment (a special case of Makarov bounds), [[ - Y < 0 x7) E [0. 0.242). Hence, Makarov bounds are informative almost everywhers (except = (xx7).
Shortcomings. The plug-in learner suffers from two important shortcomings (96). (a) The plug-in learner does not account for the selection bias, meaning that F is estimated better for the treated population and Fo for the untreated. Hence, it could be necessary to re-weight the loss wrt. to the propensity score. A remedy is to employ inverse propensity of treatment weighted (IPTW) learner for both Fo and F2 6 43] (see Appendix C). (b) The plug-in in learner does not target the Makarov bounds directly but rather the conditional outcome distributions. Therefore, it is unclear how to incorporate an inductive bias that the Makarov bounds are less heterogeneous than either of the conditional outcome CDF's ei. the Makarov bounds depend on subset of covariates X). The second shortcoming thus motivates our derivation of two-stage learners.
4.2 Two-stage learners for Makarov bounds
In order to address the above shortcomings of the plug-in learners, a two-stage learning theory was proposed [15] 35| 72 96). Yet, the two-staged learning theory primarily built for simple target estimands (c. g., CATE) and therefore requires non-trivial adaptations by to extend to Makarov bounds, which we do in the following.
Working model & target risk. Our two-stage learners seek to find the best approximation of the ground-truth Makarov bounds, functional target estimands, by a (parametric) working model G. Formally, the working model is given g = [g(ō, x)) g 4 X' -> [0, 1]: g(., z)is non-decreasing). The best approximation g. is then obtained by minimizing a (population) target risk via g- arg mingeg C(g) In our setting, L is chosen some distributional distance between the target estimands and the working model. Specifically, we use continuous ranked probability score (CRPS) as target risk for learning Makarov bounds the CDF via
440 117
Zows(() = E ( (( X) - g x))² a6).
(6)
squared Wasserstein-2 distance target risk for learning Makarov bounds the quantiles via
Ewz (f, F (a 9' x))² 2a
(7)
"By postulating a restricted working model class, G. we might compromise on the sharpness and end up having looser bounds. Yet. this looseness bias is only relevant in the infinite data regime. In the knite-sample regime, the feasibility of the low-error estimation is much more important problem.

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The target risks in Eq. (6) and 7 cannot be directly minimized as we do not observe the ground -rouh Makarov bounds. Yet, due the identifiability results in Sec. the Makarov bounds depend on the nuisance functions, (y 2)) which can be estimated from the observational data. We perform that in the following.
Enngle algge harreca
Two alaga loarnra
From now on, we denote the ground-truth nuisance functions as 1) and their estimates as fj. Also, we make the dependence on the target risks of the nuisance functions explicit; that is, we write the target risk as [(g, 17) and the target estimands as F n;) and F (a n):
Plug-in leamer
IPTW- leamer
GA- leamer prown
AU- lcamar Ourr piwer
[a) Acdressing selection biss?
X
X
Covariate-adjusted learner. A straightforward way to es- timate and then minimize the target risk is to plug-in the esti-
(b) Targeting et Makardv bounds?
mates of conditional outcome CDFs, Fa(y æ²), into Eq. (6)
(c) Orthogonaally the loss wit. to the nussance funation?
X
and (7) respectively. This yields the so-called covariate- adjusted (CA) learner, which aims at minimizing the follow-
Figure 4: Comparison of learners for estimating Makarov bounds.
ing losses (empirical risks):
En.cus (0, n (Pu. R.)) = Po P (En(d (x:0) - -9(0, ((0, )
(8)
Em . --= (Fo (T = P.. Par J/¹ (F (ax x; - g g-' (a,x x)) da).
(9)
where we call both (p z; fi) F r (ō x)) and Ep''(o 2)
Emi (a
(a
x)) pseudo- CDFs and
pseudo-quantile respectively, and where both can be obtained from Eq.
The CA-learner addresses the shortcoming (b) of the plug-in learner from above in that loss mini- mization in the equations above targets directly at Makarov bounds. However, the shortcoming (a) of the selection bias still persists. Furthermore, a new shortcoming (c) now emerges: The losses can be highly sensitive to badly estimated nuisance functions so that Cri(g, 17) and (r f) differ significantly. Next, we develop an orthogonal learner that addresses all of the shortcomings.
One-step bias correction. In order address the we employ the concept of Nleyman--orrthoonne losses [15 35). Informally, orthogonal losses are first-order insensitive to the misspecification of the nuisance functions, which introduces many favorable properties such as quasi-oracle efficiency [98] and double robustness [127]. The CA-learner
shortcomings of the CA-learner,
losses in Eq. (8) and (9) can be made orthogonal by performing a one-step hias correction [10] 63) The one-step bias correction requires the knowledge of an efficient influence function, which has not
yet been derived for Makarov bounds (- Challenge 2). Hence, the following theorem presents one of our main theoretical results.
Theorem (Efficient inffuence function for Makarov bounds). Let P denotes P(Z) = PPXX
and let v- x) and (- x)) be argmax/argmin sets of the convolutions (F * Fo)x(- x) and (ET *Fn [[.1](-- - x). respectively. Then, under mild conditions on the conditional outcome distributions and for almost all values of E A and for all values of a E (0. (see Appendix D). average Makarov bounds are pathwise differentiable. Further, the corresponding efficient influence functions, 6, are as follows:
((((F(5) x)): (( Zin)+ + F X:n) B(E( X:n))
(10)
(<E(F>'( X)): -- (a, 2:n) + (a (X:)) ((a 1P(1|X:1) (x:))..
C(ó. 2;n) = 1(x:0)
A
(I(Y -v")= (x)) x (I(Y -6)
1-A
-8 x)
(II)
T(X)
1(Y 1(Y 1x)
1-1
(u' (w'-a+00
l(a, 7:n) x(X]
= (ur [x) 1x)1XAA=1) - 11- n(x)
H(Y =F (w" X.A= 0)
(12)
where I(X:n)= 1( X)= > 0): 3² is some value from the finite set (5 X); x² is some value from the finite set "fa. (a) X); and C(5, 7;n) and C -1 (5, Z: 11) can be then obtained
by swapping the symbols {7,> U5, 2 1]. - 0, +0} to (z. <, v u [..)) 1, + +1}.
Proof. See Appendix D
7

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In the above theorem, we use red color to show the nuisance functions of IP that are influencing the target estimand, i. e., averaged Makarov bounds. Therein, we also provide a Corollary I where we derive efficient influence functions for the target risks from Eq. (6) and 77. namely oi [g);; P).
Note infinite argmax/argmin sets. When argmax/argmin sets of the sup/inf-convolutionn infinite, average Makarov bounds are pathwise non-differentiable, and, thus, one-step bias correction not possible as statistical inference becomes non-regular (47). This result also holds for other causal quantities that contain sup/inf operators (c. g., for the policy value of the optimal treatment
strategy [84 I06).. ) Although, there exist approaches to perform inference in the non-regular setting [84), we focus solely on the regular setting where pathwise differentiability holds (see Appendix D for a discussion on the generality of such a setting).
Orthogonal leaner (AU-learner). Given the derived efficient influence function for the target risks,
we perform a 7-scaled one-step bias correction (63 of the CA-learner losses, namely, (r 1)
ylp, { (6 C()): P)}- } The latter then yields our novel orthogonal AU-learner (see Corollary 2 in
Appendix D Our AU-learner effectively resolves all the above-mentioned shortcomings (sce a
comparison Fig. 4h. Formally, it aims at minimizing one of the following losses:
E.L o (5, Fo. Fi)) P. S (6. 7) - 266, x)) dd).
(13)
LA = (R. Fu ; = Pr. S, (ax 2; - 9 (a. x)) daj.
(14)
(8, 7:4,,)) (8 A) + 7 7: and E 7: 4, F. (a x; +x0 ( 7;4),,
where C(6, ii and C (a, Z: f) are given by Eq. 1I and (12.. respectively; and y E (0, 1] is a
scaling hyperparameter. We present meta-algorithm of our AU-leamner (with the CRPS target risk) based on cross-fitting in Algorithm | (AU-learner with the W32 target risk follows analogously).
Scaling hyperparameter. The scaling hyperparameter 7 is introduced to tackle Challenge @ from above, namely, that the pseudo-CIFF term, (5, Z: f, 7), is not guaranteed to be a valid CDF for y > (both monotonicity wrt. 5 and [0, coonstraint can be violated) The same happens with the pseudo-quee pseudo-quantiles quantiles of the AU-leamer, F ATT (a, Zi 7), which could be non-monotonous wrt. a: The main intuition behind scaling is that it interpolates between the full AU-learner (r = 1). that has favorable theoretical properties; and the CA-learner (7 0), for which the pseudo-CDFss and pseudo- quantiles are valid CDFs and quantiles, respectively (we refer to Appendix + with visual examples). Hence, scaling mimics learning rate of a Newton-Raphson Raphson method (usually considered as an analogy to the one-step bias correction (34). We found fixed values for the scaling hyperparameter y to work well in all of our experiments and to improve the low-sample performance of our AU-learner.
4.3 Theoretical properties of AU-learner
Algorithm 1 AU-learner (CRPS) via cross-fitting
1: Input. Training dataset D {a., az, 8i wi scaling 7 E (0, 1]. folds K > 2, 8-grid {ō, 2
In the following, we formulate sec-
ond main theoretical result. For the re-
2: Output. Estimator Makarov bounds (6)
sults to hold, the nuisance functions 7) 3: for k E (I....,K]
K} do
> First stage
(T,, Fo, 1/)/1/ = (Tr, F ,FT need to be estimated independently from the second
4: Use (xi, as, 3ihe-1 mod K) to fit f = (#. Fo, F1)
5:
for i: k - 1 (1 mod K) do
stage model g/9 This could be done by either assuming
6:
Use â to infer the pscudo-CDF for the 5-grid:
not-toofflexbble class of models (such as Donsker class of estima- tors and fitting all the models on the same dataset D) or by using a generic approach of fittinitting [15 35).
ny (Fxr(0,, 2s; 2); 5=1 (Eq. (3)
7:
end for
8: end for
9: Fit the working model based on 8-grid: Second stage
ã = arg min geg Zau, cRe f)
Theorem 2 (Neyman--rthogonnaiiy of AU-learner (informal)). Under the assumptions of the Theorem] 7 the following holds for AU-learner from Algorithm ! with the scaling hyperparameter 7 = =1:
1. Neyman-orthogonalin Losses in Eq. (13) and Eq. (14 are first-order insensitive wrt. to the misspecification of the nuisance functions.
'The issue of geeudo-outcome violating constraints was also raised wrt. DR-learner for CATE 71
125
when the outcome space is bounded (c. g., y = [0, 1J) and for the CDFs of potential outcomes 43 Yet, the
optimal solution remains open research question.
8

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2. Quasi-oracle efficienc.. The bias from the misspecification of the nuisance functions is of second order.
We refer to the Appendix D for the detailed formulation of the theorem and the proof. Notably, the quasi-oracle efficiency (98] means that our AU-learner with (sufficienlly fast) estimated nuisance functions performs nearly identical to the AU-learner with the ground-truth truth nuisance functions.
5 Neural instantiation with AU-CNEs
We now introduce flexible fully- pallmmerraettiic instantiation of our AU-learner, which we call AU-CNFs. Therein, we employ conditional normalizing flows (CNFs) [T05] [2T] as the main backbone for our AU-learner. CNFss are a flexible neural probabilisticc method with tractable conditional densities, CDFs, and quantiles. Importantly, all three attributes the CNFs (densities, CDFs, and quantiles) can be used for both training via pack-propagation and inference [15]. which makes them a perfect model for both stages of our AU-learner.
Architecture. The architecture of our AU-CNFs is inspired by interventional normalizing flows (INFs) 94 two-stage model for efficient estimation of potential outcomes densities). Our AU- CNFs consist of several CNFs corresponding to the two stages of learning, namely a nuisance CNF, which fits the nuisance functions, (ft, Fo. fo.1i F1) or, equivalently, (ft, Fo [ )); and two target CNFs, which implement second stage working models for upper and lower bounds, G and G, respectively,
Training& & implementation. At the first stage of AU-CNFs learning, the nuisance CNF aims at maximizing the conditional log- logliikelihood and minimizing binary cross-entropy via joint loss. Then, we generate the Breudo-CDFi and pseudo- pseudo-pseudo quantiles, as described in Algorithm []] Therein, we set the öla-grid size to n48 = nu 50 and discretize the )'-space/(0, |]-interval to infer the argmax/argmin values, gF/nz Then, we proceed with the second stage of AU-CNFs learning, where we set 7 0.25 for the CRPS loss and y 0.01 for the W loss. We found the fixed values of y to work well in all of the synthetic and semi-synthetic experiments (except for the IHDP100 dataset, where the overlap assumption is violated). We use the same training data for two stages of learning, as (regularized) CNFs as neural networks belong to the Donsker class of estimators [12 We refer to Appendix F for more details on our AU-CNEs.
Experiments
We now evaluate our AU-learner. For this, we use (semi-)synthetis benchmarks with the ground-truth conditional CDFs/quantiles of potential outcomes, i. e., F. x//F= (u x). In this way, we can infer the ground-truth Makarov bounds and use them for evaluation.
Evaluation metric. We use evaluation metrics based on the target risks (as introduced in Sec 4.2) Specifically, report root continuous ranked probability score (rCRPS) and Wasserstein-2 distance (W₂) based on training data (in-sample) and test data soup-sample).
Baselines. We compare the proposed hierarchy of learners from Sec. with CNFs as backbones. These are the plug-in learner (Plug-in CNF), IPTW-learner (IPTW-CNF) )) CA-learners (CA-CNFs (CRPS// / W )) and AU-learners (AU-CNFs (CRPS/ W2)). The only relevant baseline found in the literature is a plug-in in learner based on kernel density estimation [17 77 110 9) For this, we used
distributional kernel mean embeddings (97 (Plug-in DKME), standard conditional kernel density estimation method. Details on the baselines are in Appendix G
Synthetic data. We adapt the synthetic data generator (dg = 2) from 61 93 by creating three settings with different conditional outcome distributions: normal, multi-modal, and exponential (see data generation details in Appendix HH.. In the synthetic data, the ground-truth Makarov bounds are less heterogeneous than the potential outcomes, and, hence, two-stage learners are expected to perform the best. We sample Terann E {100: 250; 500; 750; 1000} training and fess = 1000 test datapoints. The out-sample results are in Fig. 5 Here, our AU-CNFS perform the best wrt. rCRPS in the majority of settings and different sizes of training data. We also report the results wrt. W3 in Appendix
'The work of 54) focuses on averaged (population level) Makarov bounds for fixed values of 6/a and it is unclear how extend it to our setting.

---

Table 2: Results for HC-MNIST. Re- ported: median out-sample rCRPS sd / W2 + sd over 10 runs.
troethsooaal opper
erperratal oppe
Hitte
B: upper
Pla
iCRPS..
Plapia Docis
Plug-in
0,399 0.162
1
CA LA-CNEs (CRPS)
IPTW-CNF
0.385 0.501
0.382 0.045
0,986 +: 1,936
1,029 4: 0.119
cormal P:
=aaat-zzdal IB:
eqperatti beer
IPTW--C77
CA-CIT (CIPS)
CA Cllleme-
CA-CNFs (W22)
0.494 0.229
1.239
0.541
CA OXR (RT)
AU-CNEs (M/))
ALL-CN (CRPS)
0,347 0.114
0,423 0.189
0,940 0.389
All-C0os (CUPS)
40 Sanse
AIF-NNS (R1)1
B: lower
6--
(CRPS...
W2us
Plug-in CNF
0.391 0.101
1.003 +: 0.333
IPTW-CNF
0.447 0.218
1.141 :: 0.917
130
2.09
200 10.00
CA-CNEs (CRPS)
0.388 0.051
CA-UNES (M:)
0,446 0.148
1,099
+ 0.341
Figure 5: Results for synthetic experiments with varying size of training data, Rarin- in 3 settings: normal, multi-modal, and exponential. Reported: mean out-sample rCRPS over 20 runs.
AU-CNE, (CRPS)
0.357 0.097
0,948 0.344
AU-CNFs (1/2)
0.415 0.190
1.025 4: 0.520
Lower better (best in bold)
HC-MNIST dataset. HC-MNIST is nigh-dimensional semi-synthetic dataset (dN 785), built on top of the MNIST dataset [52] (see details in Appendix [H). Here, the heterogeneity of Makaroy bounds is also smaller than that of potential outcomes, reflecting inductive biases in the real world. We report the out-sample performance of different methods in T (the Plug-in DKME is omitted due to a too-long runtime). Therein, our AU-CNFs (CRPS) achieve the best performance and, thus, scale well with the dataset size and the dimensionality of covariates. Further. our AU-CNFs (W?) improve the performance of CA-CNFs (W22). In general, we observe that the loss based on the CDF distance (i. e., CRPS) has lower variance and is easier to fit. In Appendix [1 we additionally report the results for another popular semi-synthetic benchmark, IHDPI00 [46
Case study. In Appendix[] [ we provide real-world case study based on the observational dataset from 77. Therein, we demonstrate how our AU-leamnrr (AU-CNFs) can be used to estimate the effectiveness of lockdowns during the COVID-19 pandemic. We estimate the probability of benefit from intervention (a special case of Makarov bounds with ō 0). As expected, we observe drop in the incidence rate is highly probable after the implementation of a strict lockdown.
7 Discussion
Low-sample & asymptotic performance. In several experiments, especially in low-sample settings, the CA-learner or even the plug-in approach are performing nearly as well or even sometimes better than the AU-leamerr. This can be expected, as the best low-sample learner and the asymptotically best learner can, in general, be different (20), and there is no single "one-fits-all" data-driven solution to choose the former one [19]. This can be explained by too small dataset sizes or the severe overlup violations (as is the case with the IHDP 100 dataset; see Appendix D1. Yet, only our quasi-oracle efficient AU-learner offers asymptotic properties in the sense that it is asymptotically closest to the oracle (see Fig. 4. We thus argue for pragmatic choice in practice (i. e., in the absence of ground-truth counterfactuals or additional RCT data) where our AU-learner should be the preferred method for the
Makarov bounds even in low-sample data.
Future work. Our work sets a foundation for several extensions to estimate covariate-conditionn Makarov bounds. For example, the estimation of the interval probabilities (see Appendix [ of the treatment effect can provide a connection with the existing works on total uncertainty with conformal prediction []. Additionally, one might want study possible extensions of Makarov bounds tailored to high-dimensional outcomes.
Limitations & broader impact. Our work is subject to the standard assumptions of the potential outcomes framework. We further make assumptions on the outcome distribution, though these are very mild. Nevertheless, we expect our work to have a positive impact, as it will help to improve the reliability of daciision-mkknn in medicine and other safety-critical fields.
Conclusion. We are the first to offer a theory of orthogonal learning to quantify the aleatoric uncertainty of the treatment effect at the covariate-conditional level and present flexible neural instantiation.
10

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Acknowledgments
This paper is supported by the DAAD program "Konrad Zuse Schools of Excellence in Artificial Intelligence", sponsored by the Federal Ministry of Education and Research. Additionally, the authors would like thank Lars van der Laan, Dennis Frauen, and Alicia Curth for their helpful remarks and comments on the content of this paper. SF also acknowledges funding from the Swiss National Science Foundation (SNSF) via Grant 186932.
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A Extended Related Work
Partial identification and sensitivity models in the potential outcomes framework. Interventional quantities (such CAPOs and distributions of potential outcomes) and some counterfactual quantities (e. g., CATE) are point identifiable in potential outcomes framework (see Fig. 6). However, relaxation of the unconfoundedness assumption renders those quantities partially identifiable or even non- identifiable identifiable [89]. In this case, additional sensitivity models can be employed. Examples the marginal sensitivity model [1]] 23 37 38 52 53 56 [19] and outcome sensitivity model Other approaches suggest using instrumental variables 45) or noisy proxy variables Counterfactual quantities, on the other hand, are inherently non-identifiable (e. g.. joint distribution of potential outcomes [26). expected counterfactual outcomes of (un) [95]. and distribution of treatment effect (26). There is no general approach for deriving sharp bounds for partially identifiable causal quantilies, therefore the above-mentioned works are not relevant in our setting.
11
104).
42
120
Example of causal diagrams
Ladder of causation
P(Y Y x)
E Y
Mecanmneee affe!
Eg Counteriactual
P(Y x))
B(Y(a) =)
codernde del calcome
Cy Interventional
P(Y a.a)) a)
B(Y 2, a)
4 Obsemvaiooal
Figure 6: Pearl's ladder of causation 101 containing different observational, interventional, and counterfactual quantities related to the potential outcomes framework. Here, X are covariates, A is a binary treatment, and Y is continuous outcome. We also plot three exemplar causal diagrams, satisfying the assumptions of the potential outcomes framework, for each layer of causation, correspondingly. Quantities with the light gray background can be expressed with the quantities from lower layers (e. g., conditional average treatment effect) and quantities with yellow background require the information from the same layer. this paper, we are interested in the CDTE, [P - Y(0] r), shown in orange. Notably, point non-identifiability of the CDTE can be seen with parallel worlds network (the causal diagram of the counterfactual layer).
Distributional treatment effects and the distribution of the treatment effect. There two seemingly similar notions in the existing literature: (a) distributional treatment effects [16] 29 62 92]
100) and (b) the distribution of the treatment effect 26 28 33 88). However, (a) and (b) have two important differences:
(i) Interpretation. Distributional treatment effects represent the differences between different distributional aspects of the potential outcomes (e.g.. Wasserstein distances, KL-divergence, or the quantile differences) [62].. Hence, they can answer questions like "How are 10% of the worst-possible outcomes with treatment different from the worst 10% of the outcomes without treatment?". Here, the two groups (treated and untreated) of the worst 10% contain, in general, different individuals. This is problematic in many applications like clinical decision support and drug approval. Here, the aim is not to compare individuals from treated vs. untreated groups (where the groups may differ due to various, unobserved reasons). Instead, the aim is to accurately quantify the treatment response for each individual and allow for quantification of the personalized uncertainty of the treatment effect. The latter is captured in the distribution of the treatment effect, which allows us to answer the question about the CDF/quantiles of the treatment effect. For example, we would aim to answer a question like "What are the worst 10% of values of the treatment effect?". Here, we focus on the treatment effect for every single individual. The latter is more complex because we reason about the difference of two potential outcomes simultaneously. Hence, in natural situations when the potential outcomes are non-deterministic, both (a) the distributional treatment effect and (b) the distribution of the treatment effect will lend to very different interpretations, especially in medical practice. In particular, the distribution of the treatment effect (which we study our paper) is important in medicine, where it allows quantifying the amount of harm/benefit after the treatment (60). This may warn doctors about
17

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situations where the averaged treatment effects are positive but where the probability of the negative treatment effect still large.
(ii) Inference. The efficient inference of the distributional treatment effects only requires the esti- mation of the relevant distributional aspects of the conditional outcome distributions (e. g., quan- tiles) and the propensity score [62] However, in our setting of the bounds the CDF/quantiles of the treatment effect, we also need to perform sup/inf convolutions of the CDF/quantilss of the conditional outcomes distributions. Hence, while the definitions of (a) the distributional treatment effects and (b) the distribution of the treatment effect appear related, their estimation is very different.
Alternatives to Makarov bounds. Potential alternatives to Makarov bounds vary depending on the type of aleatoric uncertainty. For example, explicit or implicit sharp bounds were proposed for:
The variance of the treatment effect, Var Y [o]) (5). Here, the sharp bounds are explicitly given by Fréchet-Hoeffding bounds (39 49 and are, in general, different from Makarov bounds.
The interval probabilities, P(6 S Y Yio] s 52) [33]. The sharp bounds on the interval probabilities are only defined implicitly and are also, in general, different from Makarov bounds (see Appendix (B).
We are not aware of other bounds for measuring aleatoric uncertainty (e. g., kurtosis, skewness, or entropy). Importantly, the above-mentioned bounds on different measures of uncertainty are orthogonal to work, and we focus on the bounds on the CDF/quantiles of the treatment effect.
Efficient estimators and orthogonal learners. Efficient estimation of causal quantities (target estimands) was studied in the scope of (i) semi-parametric efficient estimation theory and, more general, (ii) orthogonal learning theory. Both theories (i) and (ii) rely on the concept of influence functions (pathwise derivatives) 66 72 122]. which allow performing a one-step bias correction for (i) a plug-in estimator of the target estimand or (ii) the population risk containing target estimand,
respectively. (i) Semi-parametric efficient estimation theory [10] 73 74 107 122] provides asymptot- ically efficient estimators for finite-dimensional target estimands (e. g., average treatment effect (ATE) and average potential outcomes (APOs)). On the other hand, (ii) orthogonal learning theory (or (functional) estimands, like, CATE and CAPO. Orthogonal learning theory (or orthogonal learners) estimates target estimands by minimizing (Neyman- Jorthogonal losses that are first-order insensitive to the misspecification of the nuisance functions. Specific examples of orthogonal learners for identifiable quantities include CAPO learners 125) and CATE learners
debiased ML) [5| 35 96 125) was developed for
18
64
71
96
98].
Estimation of partially identifiable quantities. Orthogonal learners were also proposed for bounds on partially identifiable causal quantities. Examples include different interventional quantities such, e.g., in marginal sensitivity model (99), in instrumental variables setting (79). in noisy proxy variables setting 880 118). Examples from counterfactual quantities are, e. g.g the variance of treatment effect and joint CDF of potential outcomes (Fréchet- -ootheetn- bounds) [6]. More generally, estimation and learning for partially identifiable quantities was studied in econometrics as estimation of intersection bounds [14 112 [13]. In this paper, we construct novel orthogonal learner targeting at Makarov bounds on the CD/qqaantles of the CDTE.
Total uncertainty quantification. Total uncertainty in potential outcomes framework can be generally quantified with two approaches: (i) Bayesian methods (e. g., Gaussian processes) and (ii) conformal prediction framework. (i) Bayesian methods (2 (6) allow to infer posterior predictive distributions or credible intervals and for both potential outcomes and treatment effect, but under additional identi- fiability assumptions (e. g.. an assumption of additive latent outcome noise, which renders treatment effect distribution identifiable). (ii) Conformal prediction aims at providing a valid predictive interval and treatment effect [ 55 58 67 78). The latter works either make a similar additive latent outcome "hide' non-identifiable treatment effect into the predictive interval (however, oracle predictive interval could never be reached in this case). We argue that total uncertainty needs to be split into epistemic and aleatoric explicitly provide insights the origin of uncertainty [50), especially in our setting where aleatoric uncertainty of the treatment effect itself is partially identifiable (see Fig.
and was applied to quantify total uncertainty of predicting potential outcomes [76 78 90
noise assumption
18

---

B Makarov Bounds
B.1 Bounds Construction
In the following, we provide an additional intuition on how Makarov bounds are inferred from conditional distributions of the potential outcomes, [P z). For example, Makarov bounds for the CDF the CDTE are composition of the sup/inf convolutions, applied to the conditional CDFs of the potential outcomes, and the rectifier functions (see Fig.
Makarov bounds can be inferred (i) analytically or (ii) numerically. (i) Analytic formulas were proposed for very simple distributions (c. g.. normal distribution (26)). At the same time, the (ii) numerical approach more flexible. For example, we can discretize the Y-space or [0, 1]-interval and perform ground-truth bounds with the approach (i), when the ground-truth conditional potential outcomes distributions are normal; and (ii) otherwise.
on that grid. Notably, in our experiments, we infer the
Rily A (a))
Eid ml
WilryfPiN Pola-1 22))
2.5
36
5.0
9
1++ H. mlmm.
(4)
Prly ())
Fdy 105-411
1:51 m)
3.0
3.4
2.4
2.2
11 max
Figure 7: A example of the inference of Makarov bounds on the CDF of the CDTE based on 7-th instance of the semi-synthetic IHDP100 dataset [46). The construction the upper bound is shown in (a) and (b); and the lower bound corresponds subfigures (c) and (d). The subfigures left, (a) and (c). contain the conditional CDFs of both potential outcomes, namely, Fo(- x7) and F- I7). Therein, the conditional CDF F- x7) is shifted wrt. four values of ó. The figures the right, (b) and (d), then demonstrate the corresponding Makarov bounds values for the same four values of (shown in red). We also plot the full Makarov bounds with gray color.
the
B.2 Pointwise and Uniformly Sharp Bounds
The sharpness of Makarov bounds proposed 266 has one important characteristic. Specifically, although the Makarov bounds on the CDFs, F(ò X), are CDFs themselves, they are not valid solutions to the partial identification task. This can be easily checked, i. e., the expectation wrt. to the Makarov bounds does not coincide with the point-identifiable CATE, T T(x) = E(A x):
F(8 2) /
F x) dō ( (1-F(5 x)) d8.
(1 (1-F(6 r)) ds < 7(x)
(15)
Therefore, there an important distinction between so-called pointwise sharpness and a uniform sharpness (33)
In the case of uniform sharpness, a sharp bound coincides with the solution the partial identification task. This implies that if bound is uniformly sharp, then the joint bound on the set of quantities,
19

---

evaluated in two (or more points of A or [0. 1|, is also sharp. Many known bounds (c. Fréchet-
Hoeffding bounds 39 49 and marginal sensitivity model bounds [23 37 38
119) are uniformly
sharp. Yet, Makarov bounds are only pointwise sharp.
Recent works developed uniformly sharp bounds on the CDF of the CDTE (33] However, their inference requires a special computational routine for every value of 8/a wrt. the CDF/quantiles of the CDTE. Their usage further complicated by the fact that the CDFs of the uniformly sharp bounds correspond to mixed-type discrete/continuous random variables. We show an example of the uniformly sharp bounds on the CDF for 5 0 in Fig. The uniformly sharp bounds may be useful for more complex aleatoric uncertainty quantities (e. g.. interval quantities like P( < Y [ < 52 z)) or simultaneous bounds on the variance and the CDF of the CDTE. Nevertheless, in many practical applications, pointwise sharp bounds (Makarov bounds) are enough and we focus on those our paper.
10
18
EIG
08
vinj) 2/A
[(n)
FM
30
08
Figure 8: Comparison of pointwise (Makarov) and uniformly sharp bounds the CDF of the CDTE
based on i = 7-th instance of the semi-synthetic IHDP100 dataset [46]. Pointwise (Makarov) bounds shown in gray Also, show uniformly sharp bounds inferred for In(0 xz7) = PY [1][4--Y Y [0] s
0 17). namely F (d 177) (shown in red). We display the lower bound in (a) and the upper bounds in (b). Notably, the expectation wrt. (0 27) coincides with the CATE, ( E( A 27).
B.3 Makarov bounds for mixed-type outcomes
In the following, we provide a connection of the Makarov bounds for the continuous outcome Y (the main target of our paper), with the bounds the fraction of the negatively affected (FNA) from 60
The FNA is defined for the binary outcome, Y E {0, 1].
as
FNA P(Y Y Y < 0) P(( Y s -11)=F((( -1 X)).
(16)
As suggested by (60, sharp bounds on the FNA are given by
FNA FNA FNA,
FNA -E(minn (x))(x)) and FNA E. min (mo(X), 1 - (x(x))
(17)
To connect the Makarov bounds for continuous outcome and the bounds on the FNA, we formulate the following remark.
Remark 1 (Makarov bounds for mixed-type outcome
129
Sharp bounds on the CDF of the
treatment effect for the mixed-type outcome Y F
x)
F(8 xx) F(6) x), which
F x)) [F, (Fo (o))x(( x)]+ and F(8) æ)
[[F (Fo (8) x))
(18)
where P. ( x) P(Y 1 2 A
0) probability mass function.
Notably, for (absolutely) continuous outcome Po 0, Eq.
18 matches Eq.
3 of Sec.
20

---

Remark 1
immediately yields the bounds on the FNA, proposed by [60
FNA
max
sup
(19)
max
11 0)
(20)
50
P(Y =,= -) || 0), P(Y 50 50|X -0
(21)
$0
= B max (0, -PYY= =
= - - 0) = B max (0. [(x)] -E min (0, (0,+(X))
(22)
PNA
min
inf yER
(P(Y / x P(Y = y -- 0)), 0]
(23)
= 1 + E
00 0- P(Y < X, - 0) 00 (24)
P(Y 50 X. - 1+ P(Y X, 0) P(Y 0 X, 4 1) - 1+
(25)
=B min (P(Y= 1 0). P(Y X 0) P(Y X = 1), 1 P(Y =1 =1
1)]
(26)
B min (X),
(x 11 11 (X), 1 (x)) E min (nO 1 (x))
(27)
(xo(X) and 21
21

---

C IPTW-learne
In the following, we develop improved single-stage learner, namely, an inverse propensity of treatment weighted (IPTW) learner. First, we revisit the plug-in learner by defining its estimation objective. Then, we introduce the IPTW-learner, which addresses one of the shortcomings of the plug-in learner. At the end, we mention a surprising property of the IPTW-leamer, namely, the orthogonality wrt. to target risk aiming at the potential outcome distributions.
Empirical risk of the plug-in learner. We assume that the plug-in learner (Sec. 4.1 uses the estimators of the conditional outcome CDFs, Fa, that aim at minimizing the following empirical risk:
F.) - Pm ( [[A= 0} ((Y, x))]
(28)
where .(:,)) > 0 is a probabilistic loss (c. g., negative log-likelihood, check score, or CRPS with an empirical CDF). Here, the plug-in learner has two possible variants, namely, S- and T-learne, depending on whether the conditional outcome distribution is learned by a single model or two
models [20 (69]
IPTW-learner. The IPTW-leamer addresses the selection bias of the plug-in learner. For this, additionally employs an estimated propensity score, Å. The estimated propensity score is used to re-weight the original probabilistic loss, (() in the following way:
1[A 1[4=a) a) 1(Y, 1(Y, (x))a)(11)))
LIPTW a (n) (f. F.)) = P
(29)
(X)
where ft(x)) = a +xx) + (1 -1- a) (1 +z))). The IPTW-leamer up-weights the loss of Fo and F in the treated and untreated populations, respectively.
Orthogonality of the IPTW-learner. Interestingly, the IPT'-learner from Eq. (29 can be seen as an orthogonal learner targeting at the following risk:
Cg(g) E ((Y(a),
(30)
where x)) E G is working model defined as in Sec. 4.2 This target risk aims to find the best approximation of the conditional potential outcome distribution, [a x), with the working model, g E G.
The orthogonality of the IPTW-learner wrt. the target risk aiming at the potential outcome distributions was formally proved in [125]. Therein, the authors notice that the target estimand (e. g.. the CDF of one of the potential outcomes) coincides with one of the nuisance functions (i. e., F,). Informally, the orthogonality follows from the fact that the working model, y, simultaneously fits the target estimand and the nuisance function.
The orthogonality of the IPTW-learner can also be seen by (1) performing a one-step bias-correction of Eq. (28) and (2) setting the same estimator for the working model and the nuisance func- tion, i.e., 9 = Fo. For example, if the probabilistic loss the CRPS with the empirical CDF 1Y s(, X)) = (Y S <n} - g(y, x))² dy, the (1) one-step bias-corrected loss is given by 6 43
Eon.a (.) =P,, 11A=a] a}
(f s - -g(u. dy
(X)
(Fa(y) X) glu, x)), du) dd)))dd)))))))) + / (Faly) (X) x)) alu,
(fol ((),
(31)
where (a())) a (x) + (1 (1-a) a) (1- #(x)). Then, after step (2), g=
= F., we immediately yield the IPTW-learner from Eq. (9). Similarly, for the negative likelioid loss (NLL) ((Y, x)) = - -log (Y, the (1) one-step bias- correctoott is given by [65 94
L f = (#, P)) =P., 1(A = a} ( Lo, x)+ / log llg x)P(V = I a) dy
(X)
log beg x) = :
X,A = a) dy
(32)
22

---

(1- - and is an estimator of the conditional density of where (x) a (
the outcome. Again, after (2) setting 9 =P, it is easy to see that the minimization of one-step bias corrected loss in Eq. (32) is equivalent to the minimization of the IPTWWleamer's objective in Eq. (29) (where P is used in place of F).. This is possible due to two facts: (I) both entropy terms in Eq. (32) ( - Jy g x ggy X dy, only require the minimization wrt. to the working model under the
logarithm: and (2) these entropies are minimal as the cross-entropy for any distribution minimal when evaluated with itself.
23

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D Proofs
In the following, we provide the main theoretical results of our paper. We use the following additional
notation: 8{-} is Dirac delta function, a X b means there exists C' 2 0 such that a S C b, and
X, (rrr)) means Xx I' 3 0. Also, in the following theorems, we use red color show the nuisance functions of IP that are influencing the target estimand.
D.1 Efficient influence functions
We start with deriving the efficient influence functions for the average Makarov bounds and, af- terwards, for the target risks. For that, we make two mild assumptions: (1) one the conditional outcome distributions and (2) another on the set of 5 where linear rectifiers are differentiable. These assumptions allow us to (1) handle uuppiinf-oonvlutionn as max-/min-convolutiin with a finite number of argmax/argmin values and to (2) get a derivative of the linear rectifiers.
Assumption 1 (Finite argument sets). We assume that the outcome space y is compact. Also, we assume that conditional outcome CDFs, F Faly x)), are continuously differentiable and consist of finite number of strictly concave convex regions.
Assumption D implies that sup-linf-coovolltioon are achieved by some finite set of values in V. Furthermore, Assumption] i a special case of the margin assumption (Assumption 3.2) from [112]. Yet, we find our version to be more interpretable and many regular distributions satisfy it, g.g. an exponential family, finite mixtures of normal distributions, etc.
Assumption 2 (Differentiability of linear rectifiers). Values of 8 E A are considered to satisfy differentiability of linear rectifier for some x E X if (F * (6) x) + 0.
is easy to see that, if Assumption Tholds, the new Assumption will hold for almost all 5 E A and
X, namely, P (F1 (u) (X) = 0} 0 or, equivalently, P [ (6 X)] I {
for some { >>>>>>>>0 0. Also, no additional requirements are needed for the values of a E [0, 1j. Notably,
the Assumption Di is analogous to the margin assumptions, e. g., Assumption 3.2 of (84) or Eq. (3) of [12]. Therefore, the formulation of the following theorem is not too restrictive.
Theorem (Efficient influence function for Makarov bounds), Let P denotes P(Z) = A,Y) and let v x) and ufa.nf( (-1 a:) be argmax/argmin sets of the convolutions (F1 7 F Fo)(( x) and (FT FF 10 u
x)), respectively. Then, under the mild assumption of the finite argument sets (Assumption 7y. average Makarov bounds are pathwise differentiable for values of 5 E A that satisfy the differentaability of linear rectifiers (Assumption 2 and for all values of a E (0, 1). Further, the corresponding efficient influence functions, @, are as follows:
(B X)): C(8, 2;
F X:n) B (E6 X: n)),
(33)
(B (a x)): = C- (a, Z:
F (a)
X;
(a X;:)),
(34)
C(o, Z:m) = (1
X) > 0}
A
(1(Y S v²) - -i( (161xy) 1x))
(X))
(35)
1-A
(108 -/¹ - -8 - Foly' -61 x
1- x(x)
A
1(Y S SF,' (u' [x)] u²
Ca, Zin) =(X)
P(Y =F (u- [X] X, =)
(36)
1-4 1(Y S Fo (u' a + 0 X)) (u - a -a+0)
1Xx(X) P(Y = Fo aaa-a+01 X) X, A,A=0) = 0)
where y² is some value from (6 X): u² is some value from uf (a | X): and C(6, Zin)
and C-1 (5, 7; can be then obtained by swapping the symbols (,,>, 1) 0, +0} to (t. <, " +13.
Proof. We start the proof by employing the properties of the efficient influence functions, namely, product rule and chain rule; and some existing efficient influence functions (e. g., for conditional expectation (63) ).
24

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The efficient influence function of the lower averaged Makarov bound is as follows:
(FE(F(6) = IF (F x)))
(37)
[F(F(6 x)) P(X x) + F(6 1) [ - =n)3 dx
(38)
x
(F( x)) P(X= = x) + F( x) 101x - =] - -P(X = = dr
(39)
(F( x)) P( x) F(6 x) B(F(8) x)).
(40)
x
Furthermore, the inner term is
IF
= (((FF * Fo)y x))+)
(41)
[F((FF Bo)y( x)), if FF))()) x) x)> >0. 0,
00
(42)
otherwise.
Then, the efficient influence function of the sup-convolution can be derived under the finite argument sets assumption and using the envelope theorem (we refer to Appendix C.2.6 and C.2.7 of [83) for further details):
(FF + (5) x)) = LF (F x) - Foly - 6 m))) = 1I (Fi(()) ()) Foly" - 6 x)),
yey
(43)
where yf is some value from (6 r). The latter can be expanded by using product and chain rules and the efficient influence function for the conditional expectation. Specifically, we do the following:
IF (F (=) - Fo(w" - 5 x)) = (v | =)) - (Fo(' - =))
(44)
=IF(E(L(Y S ~ 1)) (E S - 2 =0) 0))
(45)
=/, (1(1vS''))
-= (1 (P = 1)) dy
(46)
(1)
(2)
-/ (R (1 P(V P(Y = A= +1 (u S 14 5) TF x. A ==00) 01) dy.
(3)
The terms (2) and (4) yield well-known efficient influence function for CDFs, i.e.,
/ 1lM S ~ IFP(Y v =. 1)) 1 1 wn¹ - 63 IF( Ir, = 0) dy
(47)
P(X 1) (I(Y وッン - FFi 1
A81X- - x}
(1- - x)
(1 ~5¹ -6} -Foly* 8 1x))
=
(48)
Let us now consider the remaining terms (1) and (3):
TR ((((vu>')))(( 2
P(V = xx A = 00 dy
(49)
/, |- 8lw 4" 1' y 12,4 = 1) 6 (v' - o)} (F)) PPY 2 0) dy
(50)
IF (P(Y = v' xx = 1) P(Y = n¹' 6 x. = 0) = 0.
(51)
where the last equality holds due to the properties of the argmax, 1 E (5 | xx) Namely, under the necessary condition for local maximum, we have + (I (v x)) - Fo(v-8 x)) = -P(Y = =y A 1) IP(Y y - 8 x, A 0) 0. In the context of the efficient influence functions, this
25

---

means that the Makarov bounds are first-order insensitive to the misspecification of argmax/arrgmn. Interestingly, similar result was demonstrated for the efficient influence functions of the policy
values of the optimal policies 82 85
Finally, we obtain efficient influence function for the lower Makarov bound by expanding the part (+) of Eq. (40)
f IF (F(8 u)) P(X = x)) dr
(52)
fx
((Pi (1)0))((()))000)))))))))))>))) 0)
P(X= I, A= 80
Aő(X =)
-Pi(y" -B (x))
(53)
(1 (1=A)6 -
(LIY 1 8" -8) Folw" (0lw-~61220 01=1)) F(X = x) da
P(X = =,A= 0)
-C66,
(54)
where C(6, n; is defined in Eq.
The Makarov bounds on the quantiles are derived analogously. However, several last steps are different. The differences start from Eq. 4
IF [ (u" ar) ("
(u-
x)) = IIR (FF (m" (a)) - IR(F 1 x)),
(55)
where u" is some value from vfa.. (ax xx).
Now, again, the Makarov bounds for quantiles are first-order insensitive wrt. argmax/argmin values misspecification. Thus, we can employ the efficient influence function for the quantiles [3] 32
(F(FT (u" [)) IF (u" x))
(56)
A5(X - ( 1(y v B (u" x)) u
(57)
P(X x,A=1 1) (P(Y = 11 (u- xx] x,A= = 1)
(1 A) - x) [1 S (u"
2)) (u (a))
(58)
P(X =x A = P(Y F (u- x)
2,, 12,A=0)
The latter then yields the final part (*) of Eq. (40) for the efficient influence function:
LF(F (a 2)) IP(X = 2) dæ
(59)
X
= =/
AØ(X 2} ( 1(y S F [)) - u²
P(X = A= (P(Y=
(2) xx, A ==1)
1)
(60)
(1- 1)=A) - 1) ( [1 [ ' -a x)] (4'-a)
P((-=2))
P(X
P(X = 2, == ==0) 0) P(Y = F,'(u" (w--a 1 x., A= |0 0)
=C- (a,Z:n). Z:n).
(61)
where Ca, Z:n) defined in Eq. 36).
Corollary 1 (Efficient influence functions of the target risks). Let the Assumption of the Theorem
7
hold. Then, the efficient influence functions, 9. of the target risks in Eq. (6) and Eq. 7 are as follows:
/ 2 (F(6) X:n) 9(5, x)) C(6, 7:n) + (E(6 X; - 966 x)) d8
oLCnns())::
(62)
- E [(6 X;:) g(6. x)) d6)
#Lwwz0 1)) //¹ [2 (F (a
(al X;n) g (a, C- Za, Z;n) + (F-'(a) X:n) 9 " x)) da
(63)
-E (f( (F X:n) - 9 (a, X)) da
where Z; and (a, Z;n) are defined in Eq.
335 and 36 respectively.
26

---

Proof. We start by using the properties of the efficient influence function, namely, chain and product rules. Then, the efficient influence function for the CRPS risk of the lower bound as follows
o{<RRs (g): = IIF E (
(6 X) -x)- x(6x d6
(64)
TIF x
(F x) g(ō, x)) dō
P(X x) dr
[[ X) 9(6, x)) (65)
(+)
EE (
x) g x)) d6)
(66)
We then note, that the inner term of the (*) can be expanded as:
IF (L (F(6 x)) - x(),2)) dd = / IF (F(8 x) - x(6, dō
-2 (F(6 x) x() (F(( x)) dó.
(67)
Finally, the term
(x) of Eq.
(65) equals
/ IIF / (F(6 x) - d6) P(X x)) da
(68)
=2
(F(6) x) z), (F( x)) P(X x)) dō
(69)
A
2 (F x) - -6(6 C(o, m: d6.
(70)
The derivations for the upper bound and for the W3 target risk is fully analogous.
Corollary 2 (One-step cirrecorrected estimator of the target risks). Let the Assumption of the Theorem 7 holdd. Then, the r-scaled one-step bias-corrected estimator of the target risks Eq. (6 and Eg. 77 is given by the following:
LAU. CRps(g f = (tt, F..FF.F.) = =Pa (/ (E -(6, do).
(71)
LA (g 1 ' = (#, Fo [T = p f// FEA (a, Z: 7) g
(a, x)) 2 da }
(72)
(5 7; (5 (5 (: + + C(o, 7; and Fxo1 (a, Z; 7) Epj (a X; + + +
(a, Z;1),
(a,
f) and (a, Z) are given by Eq. 35 and 36 respectively; and (0, is scaling hyperparameter.
where Z;
Proof. The s-saledd one-step correceed estimator of the CRPS target risk proceeds as follows:
E CRP.CRS î) + IP,, (1))) (p}
(73)
Pn / (Fm( Z; - x)) d6}
(74)
+ yP, n0 S/A 2 (Fm(6 X; x)6,x)) T(8, + (Fp(6 X: - s(6,x))² d6
>P,, { 1/ (Ep(( x; 9(6, x))² do
(75)
-P, ( //
-(6x)x) d6} + +P { ///05/0
2 (F X; x(,,X)) C(6, Z: n) d6
(76)
27

---

The minimization of the latter wrt. is then equivalent to the minimization of
Pa / (Er(() x: Z: 9
x))² d5
(77)
The proof for the 7-scaled one-step bias-corrected estimator of the W target risk is analogous.
D.2 Neyman-orthogonallty and quasi-oracle efficiency
Now, we proceed with the second main theoretical result. Here, we use additional notation.
Definition Sheymman-orthogonliin (35 96).. A risk L is called Neyman-orthogonal if its pathwise cross-derivative equals zero, namely,
D,,Dy M)g g., 9 ni
for all g E g,
(78)
& Fff + t1h1 .... where DfF(f( F(f + th)) and Dj hR] "t
tyha)les (e2 (.... are pathwise derivatives (35, g- arg mingeg L(g, n), and nj is the ground-truth
nuisance function.
Informally, this definition means that the risk is first-order insensitive wrt. to the misspecification of the nuisance functions. Notably, the pathwise derivative in the direction of the Dirac delta distribution
coincides with the efficient influence function [34 633. i.e., D DFF( - P(Z = .]]
(F((P): PP,, where (Z =)) is the PDF of the IP
Theorem 2 (Neyman (Neyman-orthogonee orthogonality of AU-leamer). Under the assumptions of the Theorem 7 the following holds for AU-learner from Algorithm with the scaling hyperparameter ? 1:
1. Neyman-orrhogonality Population versions of the empirical risks in Eq. (11 and Eq. 71 are first-order insensitive wrt. to the misspecification of the nuisance functions, e.,
BT,, cwpssc. (9.: 17]lg- 9-. 7) for all g E G.
(79)
Dr Do (0- ,1){g= 9- v) 0 for all g
g E g,
(80)
where 9- = arg min ges EAU. EAU.CRPS(5: 1) and g, = arg ming tggeg EAU EAu.w((9 (g- .1)) 2. Quasi-oracle efficiency. The bias from the misspecification of the nuisance functions is of second order. Specifically, the following two inequalities hold:
g 1-9--12- 1-112,CPP 2 $LAU,CRPS( CAU,cRes(g-.f)
| - o 12
+ | =+122
(81)
1,RRPS
+ || #12,,
Fo -))
- (j" -)) 2,2RPs
2
i 0 + | (.) (5* -
+ -1 8 l'lli,ckes* +
1,CRP
1.CRPS
+ o" - F 0210011 2,CRPS
Foo - -Fo (0" -.) 2,CRPS
-)
1
9,
- LAu.w3 ( i) - -Lu,
2.W3
+ || =112
| o â- - -P- o a²
(82)
2,
+ || #112,
| (û" 0) -0-10 1r (Q v. +00 0)
1.W3
+ ||u - 0 31
where 7
arg nigmin mingeg EAU.CRPS(O. i):
1 = arg min, lggg EA
[s())))z = (B([(Z)) (5, Z) 1), kRCCP
(B(Ja (5 2)1* /k)) (a, 2) =
28

---

(E(f) If 2) da))(//8; I(X:n) = [[ * Fo))(( | |X) > 0}: y" is some value from v5 X): O is some value from (5 X); u' is some value from vio..yla X); and i' is some value from La.n(( X). The inequalities corresponding to the up- per Makarov bound, can be obtained by swapping the symbols 15.2,55. 4i. -0, +0}
to (,y "oo.ai -1, -1,+1). Furthermore, if the muisance functions are estimated suffi- ciently fast, i.e., 110" - 1l oz(n-1/4); [ f1oy" - -R100" 1,CRPS
= or(n=1/4);
Fo o (y" 1)) o (0" - - = or((n 1/1); || - -TLL, F1 o - [F 1 60 2,CRP
or(n 1/2): and || -if--lly T1 FFo o (5' -.) Bo (5* (0'--))(..cmm - or 1/2).. then the AU-
learner (CRPS) achieves the quasi-oracle property (analogous result holds for AU-leamner (W3)) ))- This means that the estimation error of the second stage with the estimated nuisance functions behaves in the same way as if the ground-trun -rrnn nuisance functions were used.
Proof. 1. Neyman-orthooonality The Scyman-orthogonallgg follows by the construction of the AU-learner as a one-step bias-corrected estimator. Specifically, it is easy to verify that the pathwise cross-derivative from Eq. (78) is equal to zero
Let us consider the CRPS risk. First, we find pathwise derivative wrt. working model g: d
D2 (0.. - 9.] = dt [ (F 2: - (6, x) - 2. x1)) do}
(83)
--2( [/ (FAUlő, 2: x)(6,x) f(g(6, x) ()() (0(6,x)- - ( (6,))) d6
(84)
1=
- x)) (old, x) ()) db}
(85)
Then, we find derivatives wrt. to different nuisance functions (e.g.. the propensity score):
Dr Dy DeD,aatcces((s..nll vDly 00
(86)
d dt
28
1/1 Fo (x +(=
-9- (o(8. (6, do
(87)
d dt
20 1// 7 -16, Zin = (T 1 1(# a).Fo.Fr)) (o(o.X) g (6. d6
(88)
1(X:n) A(#(X)- (1(' -Fz 1x))
27 E
(89)
(T(X))²
(1 A)(x()- x(x))
(I(Y Vv' -66- Fols" 51 ((x)) (066,)) d)
(90)
(1-x(X))2
=27 Ex
E(::m)
(T(X)) (IIY <v' -Fi 1x)) x]
(91)
E
A-A
(I(Y sn -6 -po -6 xx) |x
x(x)
x(x)) ()(6,x) (6) d6
(1 x(=)
(92)
=27 Ex
1(X:n)
P x)
(E(1(Y 20 -=)
(93)
(x(X))2
P(A 0 x
x. (B(ILY -5'-8) | -Fo
Folu - 8 1 ( (x(x)
(o(6) x)
g- (8, x))
d8
(1 =(x]}²
(94)
22 Ex 1 ((X:n)
1 0+
1 -( o (+(X) -(x)) ((x)) - 20 (6, d5 =0.
(95)
[x())
where n(X;n)=
[ * #o)s((1 / X) > 00} or 11 / X) 00: and y is some value
from /8 XX.
29

---

The derivatives wrt. the conditional CDF F1 is
Dr, D&LAU.CRPS wl gr.Fi Fi]
(96)
d
28
1/1
-..(..))) - do
(97)
dt
d
(6n ( --
1/4
(98)
dt
+ 2: (o. +0 E()))) -
1-0
=E
/
1(X:1= (F.. F.))) (1(10(0
-(0,00000))---00001000 -Fo
(99)
x(X) (IIY (0) 11 (0,0(000000)) x) (x(00(00
1-x(x) -10=
1-A
-
-9. dB
=-2B
1(X:n)
Pi (0) - Fo -0---111 x1 10
d
( -
dt
de
()==
(1(((00) FF (1)
(100)
+7
1(Y -1
-0 d -
(X)
(( x) x))
1-4 4 d BYY - (o(0 - 1x) -11)))))_))
) (o(0) x) -
1- x(x) de
2Bx / ((X:n)
Bi (37(00) 1x) - (g
T(X)
B - -8
- 60(0)) -
x di
+(X)
E - -000=-51000 (
(o) 1x))
1 m(X)
(101)
+(x)
-E[ - [ (o(o,x)- - ()( -9.6,))) d6
=- / ((x:n)(1 (F) (F) - Fi -( x)) (9(8, X) (6, x)) do
(102)
where I(X;n) 1[
X) > o} or
X) 0}: (t) is
some value from (6 X); 8- X) are the argmax/argmin sets of the convolutions (F) + + Fo) X): and (x) = follows from the the same considerations as in Eq. (51) Analogously, the pathwise derivative wrt. Fg can be shown to be equal to zero. We refer to the appendices of (96) for more details.
The Neymaan-orhoognallity of the W3 population risk can be proved in a similar fashion. First, the pathwise derivative wrt. g has a similar form Eq. 85 namely
DoLue (ol -1 -- "=0]]][[]] (FAv (a, Z:n,) g. (a, (a, g. (a, X3) da 20[
(103)
30

---

Furthermore, similarly to the CRPS target risk, the cross-derivative wrt. to the propensity score is
DrD, AlL w3 (9, 1))9 - 9, a =
(104)
[Y S S [X} | P(A=1IX)
-2 Ex
(x(x))²
(ur [x] (X)] 1)
(105)
P(A (1X)
-1
B[I &
(+(X) x(X)) (g Xa, g. 35' ()) da
=(X)²²
P(Y = Fo 10- x| X,A=
Fi (u* -
=2y Ex
(106)
((x)
P(Y= Fi (u* X) X, A)
Fo (1r a + X) X) (wr (40 - - (#(X) (#(X)- xxx)) (o (a, 9. (a) da
(()
P(Y = Fo 'u a 0 X) X, A = 0)
=2y Ex
0+ [-) (5(X)- x())
95 da =0,
(107)
((X)
where u" is some value from u ii..1(0 (a X). The cross-derivative for the upper bound follows
similarly.
Finally, to show that a cross-derivative wrt. F we make use of the following property of the derivative of the quantiles (inverse function rule):
d T (a) + (a) - (a)] da
(108)
P(Ý P F- ((P - (a))):)
where P(Y- ;t) is the density function for distribution with quantiles (a) + t(F (a) F (a)). Then, the pathwise derivative is
di P(Y = (o) -10)--0)----00-------a--a----)-- = d da [
d
{ (a)- - x ( d
1=0 da
(109)
Therefore, the cross-derivative is
D2, Du Pu wg (9.", nl(a
9. 05'
(110)
28 L. de 17 (a) x) (1"()) 10000000000000000000000000000000
x)
1(V 112_771()))3)3((((00000000711111 -
x(x) (p (1 |1 | (II
1-4 -11-)00000--4000000000--00000001 -,--,1111-0000111101010101011"13 X
(),) da
1 (xx(x) -00=00000000+000000
28 S. dp
(x) 'o' ((00)=aai)i)
+0 d 1- x)]
(1
( (0) - (x))
+-( x(X)
( p. (w + p(y - x) x,A=1)
(12)
(I1Y (00(( (0)00))))aa(())
(0 (000) - a (1
1/ S (o (2,, (01)
150
P(y P( (0 X.A=1
10
1- x(X)
di LP(Y 17, (ī- -000
(8' (0) - 03)
10 (77(1) x1)
# (() 100 ) (o ((, x) 2. (a. x3) des
F(Y= (a)
0 X) x,1=0)
31

---

1.' 81 (8 (x] -10000100-
=- 2Bx
x(x)
x(x)
| 3 x) x,A= p( x) x, X.1=1)
B - s( [
+ (aT ((7
- xt)) ने (7())
r (a) xx)xx
1- m[X) (d
(113)
1- x(X] (tt =o (a-())
R 8(Y to (0) + x)) x, - +1 (0-0)-a01x)
# [00 100 )) (0 xy - x1) x))
P(Y - (0-
22x 6.' 87 (2' (x0)
[x<(0]] - -(y==r' (a)(0) x) 11) -a [a)())
100
(0:00 1x) x1)=1)
(114)
# a1(0)
#
F(Y (a' (0 ) (0 "ra, x) x(a,x)
3 x))
=-2 1. (FT' FF |
(0 ((a,)) (a, da = 0,
(115)
where üi" (t) is some value from यي 1) (a X); and ūل ij..1 (a | X) is the argmin set of the convolution (IT + 1(F - FT Fo
(aaa) - 0 X). The cross-derivatives for the upper bound wrt. FT
and for both upper and lower bounds wrt. FF follow similarly.
Quasi-oracle efficiency. The result is direct application of Theorem I in (35 that the Assumptions 1-4 from (35) hold for the population versions of the empirical risks of our AU-learner (i.e., CRPS and W In the following, we provide the derivation of the quasi-oracle efficiency property for the population version of CRPS loss (the derivation is similar to one in (96 125)).
is easy to see
We first apply functional Taylor expansion of LAU, @ n) at the g.:
ZAU.CHes =B
E //1
(Au 2: 366 + 9. x)) as]
(116)
= EAU.CRIs( i) 28 //2 EEu(l 2:0.)) --(6 (g(5, x) - g (8.x)) a6)
(117)
+ R f 76, x) - (5. x)) a6].
Therefore, the following holds:
=L/ (3(0.) x.(6,x)) [s} = -IN Zu (o....))
(118)
2,CRPS
2E / (F. (5, (5. (66,, X) 2.(6, xi) as]
The latter term then also allows for the distributional Taylor expansion around ry:
2 J (6,
F x) + E x) - ()) x) - 2. (6) d6)
(119)
28| / (6 0: - F x)) (0(s. x) - 90. (8.xx) x)) d6)
(120)
+ 2B / (F(8) x) - (6. (g
9 (6. (6. db}
32

---

where the first term is known as a srdend-order remainder term, (2, f). and the second term equals to - -DPLAU,CRPS (0+:1) n) 9-). R2(n, fi) can be further expressed as
(n n)) = 2E S/ (F x - F(6| x)) (ō(s. x) 9. (6, x)) d8)
(121)
J a
1(x: (7l [ (LIY 10 - (00011)) I 1-4 (11 s 5) Fo(0" - 6 [1xx)]
2B
#(X)
(122)
+ + 1 xx) F0(00-81 -81x)) ((x:n)) (R x) - -FolO* -Foo( -81x))) -Fo()-0(1x)) (ă(τ, x) - 10, x3) a6
[x()
[(X) (Fi(5" xx) - (x)) - 1- x()) 1xx(X)
1- #(x) (Fo(8* - x) 100000005500000000000000000 -51
2B
-
- -(FI(0"
xx)) - Fo -8 x) - -F0 -81 xx)) ( x) - (0, x3) db
1A (I(X:)) - ((x:0)) (F' xx) - Fo l9- x))
+ 2E
(123)
+ +1(xmm) (0 1 1x) - -FO - 1 x) - FF x) - -Fo -51
x1)) (H1 -8|x))
- (0) ab
- go
J (xx:Ma(x) [ - )xx7) (F xx) - -8 (x)) (01a x) - x3) do
2E
x(x)
2F
-0 1x))) () 0-(6 x3) d6 LA (I(X:0) - -I(X:n)) (Fi(0') - -Fo(O= -81 xx) (ā () (8, (5.)) as)
/ ((x0)(1=xxx) - [[-((x]
7 1
1- x(x)) (Fo(0) x0 -
+ 2E
(124)
+ E
/1 m)x;m) 1 d²
[F x) - Fu - 81 [x]] ==
(0' - 0²)² [g(8, x) - g. (6. db
2 dyy
where nn = (F (8) X) > 0} or 1 [( * ( X)= < 0): y/ is some value from (8 | X): 0/ some value from (8 X); and 3" is a value between y* and g".
Considering there is such an e > O, for which S *(x) < (1 c), the second-order remainder term can be upper bounded with the Cauchy-Schware inequality:
(n) E hx(()) #(X11 / C1.166.) (i (X) Pr (5" x) [66.x) 2 (. db
(125)
2B B (x((()) 1) 1)- (+(X) 1/1l / (1,6(6,x) Po(O" - 01x) Ro(o -0 -010) 2. x)
2E / (6.) - 1(x, m)1 [3, 2)(6
SA Cy. (6.(6 8/ x) 8- d6
E
In #lle, ||Cr.ell.ccss |F000 in - 2,CRPS 0 g. 1=l2.cars
-
(126)
(Ihr #llea + b 11) ||Cr,al2,camss
|oo (0" -.) ) 9o 10" 160--)) 12.2c.0
| 9. 1|2
+ 2 22ICeess- 1 [| f) T(,, 1))
- 9- 112.crrs
||C /l 1 11 3 11 11, - 9. 1|2ccees'
where (6,
x) = [I(x, A) x5) = I[xx;))
1
da dya
Fi x)) - -Fu - 5 -11²)
: and
2
v=g-
(6,x) = F1 (ū" |x) Fo - 11)).
33

---

By combining the previous expressions, the following holds;
[ 9. 1!
AuL,CRP @. f)
LAn,CRoss fi) D2 Au.cRps(0..1)) -
(127)
2,CRP
1cr.ll2,ccoss ll - #11e, || o -B 08"
2,CRPS 1 - 9- 1,
|Cr,lll,,coss (I alle, by -11- | (5 --)--)-- Poo (8 3
9- 1,
2 1| ll 1l,,cess [T(,0) 1(.,))l2,.CRs
go 2.CKPS
|| E2 Iv" 8" 11 [ g- 1.ccsss
Furthermore, using the AM-GM inequality for the last three terms, for any constants 61 > 0, 82 0, 53 > 0, 64 > 0, 61 + 82 + dg + 64 < 1, we obtain:
82 dg 54 (IAn,TRPS(C (. f) -Av,cues(a... - DDT ao 9.1)
2,CRPS
1 61
|CC,nll2,cRPs 11 - -1122 | o -B, o o0" 112
(128)
cd.
2.CRPS
1 ||Cr,ll2.coPs (177) =lli, b 1 || Bo - Bo (0" -.)
co2
2.CRPS
||C2,s- 112.cRPs (1 - ()..)lll.mm.
03
1
||Cg
I1?, Ill" 0" 11,.c
204
We note that, given Assumptions as follows
and 2 the term [IT(;,)) T(: ((:;)))22ccr) can be upper bounded
= E / (x) 03 [[FF FF) X) 01) d8
[1 n) I(.w1))1..0m 2.crs
(129)
[f x) 0} - (F x) > 0} d8
(130)
/ P sign ((f x)) + sign (FF ()} do
(131)
/
(BI x) (P (o)y (81 e $ d6
P
(132)
4
9 1
((A) (f) XX (F, (o)) x)) '1 d6
4)
(133)
1
P1 o y² - r o 00 11'l + Bo (y" -) -o (0" - 4.CKPS
gd
(134)
+ 2/B8 - B -1000 2,Rrs
Fo" - - B. (3* -.) 1 2,RRPS
where& & 0 and (*) holds due to Markov's 's inequality.
34

---

On the other hand, since the term Dz C.r.A (9., n) 9.] is always positive (by virtue of the optimization), we finally yield the desired quasi-oracle efficiency:
-
| 2.CRP
(AU,CR 1) ExI.car((o.. 8
(135)
61
82
63
C 1C,,ili,cwss lht Allie [1 FFOD" - Fi 0/000 2,CRPs
cdi
1CInll2.cRrs
(| - Alle, +1 +11-11) || Fo (0" -) (0" - 2
2
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F1 uy" -fi 1 I 12.c [FO (v" (v" - Po (0 - - 2.ckus
+ 2
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264
1
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(136)
1
81
82
|Cr,91l || Alli. Pi F -f, 00 2,CRPS
e
1
|Cr,1 12,kss || - -111,, |o (i--
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||Cr,s ||Cr,--12..crs
4,CRPS
2
Fr e ey"
- fi - 1/ 2,c0ps |Fo0 (x) -F Bo (0" -.) [>/33c0mm
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20
where (6, x) = [xx, i) Cy- (6,z) ((;;))
d2 dy
F (y x) Fo 5 =1x)
and
y=
Cr.5-(6,30 = Fi |z)
Fo 8 $1z).
The quasi-oracle efficiency of the W? loss follows similarly.
35

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E
Scaling in pseudo-CDFs and pseudo-quantiles
Fig.
provides visual comparison of different pseudo-CDFs for our AU-learner, with and without scaling 7. Therein, we see that the scaling hyperparameter helps to enforce the constraints on the pseudo-CDFs (i. e., [0, 1] boundeddssss and monotonicity).
9
B: upper, Co-leaeeer
B: upper: AU-loamer (w/o sculing)
B: upper: AU-learer (w/ sealing)
2.5
Farl (x
Fould 408 0.20
1.5
0.5
1.0
B: lowerr CA-learer
B: lower; AU-learnr (w/l sculing)
B: lown; All-laanne (w/ sealing)
Erelo (x
Era suckin
Eon(6 au 0.26)
2.0
Pen CDD
1.0
-0.
-10
Figure 9: Comparison of estimated pseudo-CDFs based on i = , instances of the semi- synthetic IHDPI00 dataset [46]. Here, we compare CA-learnerss pseudo-CDFs (first column) with two variants of AU-learner: w/o scaling (7 1, second column), and w/ scaling (? 0.25, third column). The scaling hyperparameter 7 0.25 facilitates pseudo-CDD CDFs better comply with [0, bounddddsssssss and monotonicity constraints.
36

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Details on AU-CNFs
E.1 Architecture
ONF 2UA
Legend
Narunce CNT
Six cnrs/w
FC
Pulpoonetted subradook
ONF 1U0 A *
P
Kansa penamoriod ood by
Zyg (6 Ept
FC2
1X0
7 FC
NI
Eato XX]
Outsurs with diiitutione
5,000
Gerdinnt hoctiiilliinggg
veg. ryploiattion
Nuisance CNF
2. Upper and lower target CNFs
Figure 10: Overview of our AU-CNFs. AU-CNFs combine several conditional normalizing flows (CNFs), which we call a nuisance CNF and upper/lower target CNFs, The nuisance CNF is a first stage model and aims at estimating the nuisance functions, i. e., the propensity score, #a att (x) (1 a) (2)): and the conditional outcome CDFs, Fa z). Upper/lower target CNFs are the second stage working models, 9 and respectively. They aim minimizing one of the losses of AU-learner. LAU, CRPS/W?'
Our AU-CNFs allow us implement the Algorithm 1 of our AU-learner (see Fig. 10 by combining several conditional normalizing flows (CNFs) 121). It consists of a (i) nuisance CNF and (ii) two target CNFs (upper and lower). (1) The nuisance CNF aims fit the nuisance functions,
(ft Fo, F1) or, equivalently, (á, F. " (2) Upper and lower target CNFs constitute the second stage working models, namely, 9 and g, and minimize the loss of our AU-learner.
(1) Nuisance CNF. The nuisance CNF has three components, similarly to (94). These are two fully-connected subnetworks (FC, and FC2) and CNF, parametrized by 9. The subnetworks FC, and FC2 form hypernetwork, which outputs the conditional parameters, 0 = 0(X, A). This allows us flexibly model the conditional outcome distribution.
The nuisance CNF has the following joint loss for the nuisance functions: L2 = LNLL + aLz. Here, LNLL is conditional negative log-like lig-liielhood loss, L, is binary cross-entropy, and 0 is hyperparameter. We additionally employed noise regularization to regularize the conditional negative log-likelihood loss (108).
(2) Upper and lower target CNFs. The upper and lower target CNFs use the pseudo-CDFs pseudo-quantiles, generated by the nuisance CNF, and then implement second stage loss of our AU-learner. Both target CNFs have the same structure. Specifically, they have a fully-connected subnetwork, FC3, and CNF, parametrized by B. Analogously, FC, serves as a hypernetorrk so that the parameters can be conditioned on X: 3 = (XX).
To fit the target CNFs, For that, we discretize the y-space or the [0, ij-itterv of u into na values and infer argmin/argmax do the same for the A-space and
second stage loss of our AU-learner, namely, Eq.
Eq. (14)
values based on those grids. Then, approximate the integrals, the [0, -j-interval of a. The later creates ó/a-grid with ns ns/n, points. Those grids are later used for rectangle quadrature integration. Furthermore, we also regularize the target CNFs by applying the noise regularization [108)
F.2 Implementation
Implementation. We implemented our AU-CNFs using PyTorch and Pyro. For the CNFs of both stages of learning, we used neural spline flows (24) with a standard normal distribution as a base distribution. Neural spline flows build an invertible transformation based on invertible rational- quadratic splines and, thus, allow the direct inference of the (conditional) log-probability, CDF, and quantiles. Neural spline flows are characterized by two main hyperparameters, namely, a number of knots 7kknos and a span of the transformation interval, [-B, B). The number of knots, 7knotss controls the expressiveness of the flow. The span B defines the support of the transformation. In
37

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our experiments, we tune the number of knots Kknos and set the span B via a heuristic depending on sample max/min values (as y assumed be compact).
Training. To train our AU-CNFs, we make use of Algorithm T However, both first and second stage models are fit the same training data 'D without cross-fitting (regularized) CNFs as neural networks belong to the Donsker class of estimators [123). Training of our AU-CNFs proceeds as follows: (1) we fit the nuisance CNF; (2) we freeze the nuisance CNF and generate pscudo- CDFs/psendo--quatiles and (3) we train the upper and lower target CNFs. The hyperparameters are then as follows:
1. First stage. We used stochastic gradient descent (SGD) with a minibatch size by. n.NN 200 epochs and a learning rate 7/. Furthermore, we set the loss coefficient number of hidden units of FC,/FC2 and the size of the output of the FC₁ are set to 10. The nuisance CNF has the number of knots Tiknors, and where Y) are standard normalized outcomes y/- The intensities of the noise regularization for the input and the output are set to o² and oy. respectively.
1. Both the
span B (1:) - min, (ū:) +
2. Intermediate stage. We set nd 200 and nı = na = 50. Furthermore, we set the scaling hyperparameters = 0.25 for the CRPS loss and 7 = 0.01 for the W? loss. We clipped too low propensity scores (lower than 0.05).
3. Second stage. The upper and lower target CNFs also fit via SGD with the minibatch size by = 64, ne.T = 200 epochs, and the learning rate = 0.005. The intensities of the noise regularization for the input are the same as for the nuisance CNF, o. The number of hidden units of FC3 is also set to 10. The target CNFs have the number of knots twice larger than the nuisance flow, 7iknots T = 2 Minesss and the span = max; (a: ÿt) min, (a: ÿt) where Y; standard normalized outcomes y/- To further stabilize the training of the target CNFs, we employed an exponential moving average (EMA) of the target CNFs parameters [103 with smoothing hyperparameter A = 0.995.
max; a,) ÿ1) min, ((1 - a,) y,
We demonstrate the detailed training procedure of our AU-CNFs (CRPS) in Algorithm 2 (AU-CNFs (W2) follow analogously).
Hyperparameter tuning. We performed extensive hyperparameter tuning only for the nuisance CNF. The following hyperparameters are subjects to tuning: the minibatch size Rs,N. the learning rate T)s, the number of knots Nkmors, and the intensities the noise regularization, o² and oz Further details of hyperparameter tuming are provided in Appendix G The hyperparameters of the target CNFs for all the experiments are either kept fixed or are inherited from the nuisance CNF.
38

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Algorithm/ 2 Training procedure of our AU-CNFs (CRPS)
1: Input. Training datasct D = {z., as, himi scaling y E (0, 1]; hyperparameter o; number of epochs Re.N, Re.ri minibatch sizes bx, by; learning rates Yhs 7r5 intensities of the noise regularization of oy EMA smoothing x; 6-grid {6, E -)}= V-grid {y, E )
2: Init. Parameters of the nuisance CNF: FC and FC
First stage
3: for == =0te [N.,N n//b] do
4: Draw minibatch B {X, A, of size by from
5: (R, (X)) + FC (X)
6:
Noise regularization: &u (O(0, o(), (R, (R+ E,,Y Ev)
7:
((X, ( R)
8:
P(Y XX,) density of CNF with parameters 0(X,
9:
Éx(P, R) Pbx log P(Y X, A) + BCE(fA(X), A)]
10:
(PC(i+1), FC(++)) +- optimization step Lx(F, #) with the learning rate YN
end for
12:
Output. Estimator of the nuisance functions f = (+, Fo, 1
13: for == to do
Intermediate stage
14: Infer argmax/argmin of sup/inf based on the V V-grid: (15 (6, xx.)
j=1
Clip propensity scores caa-ccorretion terms [C(,,, 26 i) ns
(Eq.
i=1
16: Use r to infer the pscudo-CDF for the 6-grid: EAu (6,, 21
(Eq. 13
17:
end for
Init. Parameters of upper and lower target CNFss: FC and FC
Second stage
19:
for = to [n,,T n bt do
20: Draw minibatch B = [X, A, Y) of size br from D
21:
Noise regularization: & ~ (0, of), X + X + Eo
22:
3(X) FC (X), 3(X) FC (X)
23:
X) CDF of CNF with parameters 3(X),
[(,X) CDF of CNF with parameters ((X)
ZAU. CRPS (g. ()) Por
ng E (F (6,, Zi x)),
25:
ZAI. CRPS (]. Fo,
E (FA (6,, 1) g(d, x))}
26:
FCg (1+1)
+ optimization step wrt. ZAU. cRps (g. f)) with the learning rate ihr
27:
FC + optimization step wrt. ZAV, cups @ n) with the learning rate nr
EMA update FC (6+1) and FC((+1) with smoothing À
28:
end for
30:
Output. Estimator of Makarov bounds with CNFs: EMA smoothed ğ and ğ
39

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G Hyperparameter tuning
We performed hyperparameters tuning of the nuisance function estimators for all the baselines based on five-fold cross- rross-validation using the training dataset. For each baseline, we did a grid search wrt. different tuning criteria (see the details Table Bp. The optimal hyperparameters can be found as YAML files in our GitHub.
Table 3: Hyperparameter tuning for baselines.
Model
Sub-model
Hyperparameter
Range Value
Kereel smoothness (0) 22
0.0001. 0.001. 0.01, 0.1. 10, 20 0.0001, 0.001, 0.01,
Plug-in DKME
Regularization parameter (8) Tuning strategy
94 97
full grid search MSE of ridge regression
Tuning criterion
Numher of knots (Miam)))))
20
Intensity noise regularization (o Intensity noise regularization (o
0.0. 0.01 0.05². 0.12
0.0, 0.012, 0.05?. 0.12
nuisance CNF ( Plug-in CNF) ( IPTW-CNF)
Leamingg (nx)
0.001, 0.005
Minibarch size (hx)
32, 64
Tuning strategy
random grid search with 50 runs
Tuning critcrion
LUL
CA-CNE', AU-CNFs
Number of epochs (2,,))
200
Optimizer
(momentum 0.9)
Number of knots (Mikmm))) Intensity noise regularization
2 Tignons
d
Learning rate (?r)
target CNFs
Minibatch size (br)
64
Tuning strategy Number of epochs (Me.))
w/o tuning
200
Optimizer
SGD (momentum = 0.9)
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H
Dataset details
H.1 Synthetic data
Although the original synthetic benchmark contains hidden confounding, we include the confounder as the second observed covariate. We created three settings with different conditional outcome distributions: (1) normal, (2) multi-modal and (3) exponential. Specifically, synthetic covariates, X2, treatment, A, and an outcome, Y. are sampled from the following data generating mechanisms:
Our synthetic data generator is adapted from [61] 93 [31]9]]
X1 Unif(-2,,2) 2),
X2 (0, 1),
A ~ Bern I+expl (0. -,,,,x),,,))
(137)
(X = A2 1) X1 + A -- sin Xi + X2) (1+
Y FY (A(X),
where X1, X2 mutually independent and P,(Y [A(X), A) defined by three settings EE (1,2,3):
Pi(Y A) (x),
(138)
Mixture (0.7 (x) 0.5, (1
A =
Pa(Y A)
Mixture { (.3 (()o(X) 2.5, 0.355) + 0.4 0,5,0.7 (.oo + 2, 0.55)))
if =
(139)
P2(Y ()))) = Exp(1/ (X)))))
(140)
where (H, o?) is the normal distribution and where Exp(A) is the exponential distribution.
The synthetic benchmark allows us to infer or approximate the ground-truth Makarov bounds. For the (1) normal distribution, they are given by the analytical solution (26). namely, the CDFs of half- normal distributions. However, in the settings (2) and (3), they need to be approximated numerically. Thus, for the (2) multi-modal distribution, we only infer the Makarov bounds on the CDF, as the quantiles are not directly available for the mixture distribution. For the (3) exponential distribution, however, we can infer both the Makarov bounds on the CDF and the quantiles.
H.2 HC-MNIST dataset
HC-MNIST dataset was introduced as a high-dimensional, semi-synthetic dataset 52 based on the MNIST image dataset (75). The HC-MNIST dataset builds on Teran 60, 000 train and Phest = 10, 000 test images. HC-MNIST takes original high-ddmensional images and maps them onto one-dimensional manifold, where potential outcomes depend in a complex way on the average intensity of light and the label of an image. The treatment also uses this one mieeennnsional summary, ø. together with an additional (hidden) synthetic confounder, (we consider this hidden confounder as another observed covariate). HC-MNIST is then defined by the following data generating mechanism:
Bern(O.5))
X MNIST-image())
ceipp
PN. Mc.
1.4, 1.4
Min.
Max. Min
do
1.4((-1.4)
(-1.)
a(o;
+
(141)
sigmoid 0..5
B(0; [+)= = (U. (U.
p
A Bern a(d:T")
1- 3(dTT")
Y ((22 1) (2A 1) 122A 2(20 1)(1 0.56), 1),
where is label of the digit from the sampled image X; H.N. is the average intensity of the sampled image; Pe and a. are the mean and standard deviation of the average intensities of the images with the label and Min, = -2 + 4 c, Max, -2 + 4 (c 1). The parameter I* defines what 10
factor influences the treatment assignment to a larger extent, i.e., the additional confounder or the one-dimensional summary. We set I* exp(1). For further details, we refer to [22)
Similarly the synthetic data with the normal distribution, the ground-truth Makarov bounds for the HC-MNIST dataset are given by the analytical solution, namely, the CDFs of half nonm-l distributions 26
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H.3 IHDP100 dataset
The Infant Health and Development Program (IHDP100) [46 114) is standard semi-synthetic
benchmark for treatment effect estimation. It contains 100 train/test splits with Rarin = 672, Reest 75, and dæ 25. Yet, this dataset contains severe overlap violations, which makes the methods using propensity re-weighting unstable [20 21
The IHDP100 dataset samples synthetic outcomes from the conditional normal distribution, N(4, 1), where 14 are CAPOs provided in the dataset. Therefore, the ground-truth Makarov bounds are given by the half-normal distributions [26).
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I
Additional results
In the following, we provide additional results for our experiments with synthetic data, the results of the semi-synthetic IHDP100 benchmark, and the runtime information for all the baselines.
I.1 Synthetic data
We provide additional results of our synthetic benchmark in Fig. [ Therein, the out-of- -a--pe performance is reported wrt. W2 evaluation score for two settings, i.e., normal and exponential setting TD Our AU-CNFs achieve superior performance the normal setting and perform well the exponential setting. Notably, the IPTW-CNF also perform well in the exponential setting, mainly due to the orthogonality wrt. potential outcome distributions.
St normal
R:
8: exponential
B: upper
0.8
Play leamer
1.8
Plug-in CNP
01.0
1.0
IPTW CNF
CA-lmaaue
CA-CNEs (CRPS)
S normal
B: knoer
expouiniial
B. lower
1.0
CA-ONEs (W
AU leonerr
3.2
AU-CNE (CRPS)
AU-CNE& (W2)
0.4
Figure 11: Results for synthetic experiments with varying size of training data, Tumin- in the settings: normal and exponential setting. Reported: mean out-sample W2 over 20 runs. The results Plug-in DKME are omitted for the W22 evaluation score, as Plug-in DKME does not provide direct quantiles inference.
1.2 IHDP100 dataset
We report the and out-sample results for the IHID1100 dataset in Table 4 As expected, our CA-CNFs achieve the best performance. This happens due the severe overlap violations in the IHDP100 dataset [20 21), and learners with propensity-score re-weighting are theoretically expected to perform worse.
1.3 Runtime comparison
Table provides the runtime comparison of different methods used to estimate Makarov bounds. Therein, our AU-CNFs are well scalable.
We omitted the multi-modal modal setting, as the mixture distribution does not provide direct ground-trutt quantiles and, thus, Makarov bounds on the quantiles.
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Table Results for IHDP100 dataset. Reported: median in-sample and out-sample rCRPS + W2 + sd over 100 train/test splits. The results for Plug-in DKME are omitted for the W2 evaluation score, as Plug-in DKME does not provide direct quantiles inference.
B: upper
rCRPS
rCRPS
Wzin
W2ox
Plug-in DKME
0.718 + 0.831
0.778 +: 0.865
Plug-in CNF
0.302 1 0.269
0.317 1 0.260
0.631 1.195
0.644 1 1.237
IPTW-CNF
0.367 0.198
0.380
0.206
0.783
0.971
0.783
1.067
CA-CNE' (CRPS)
0.281 1 0.284
0.299 1 0.294
0.609
1.508
0.649
1.625
CA-CNFs (W2)
0.283 0.273
0.292 0.284
0.583
1.318
0.606
1.425
AU-CNFs (CRPS)
0.314 0.264
0.324
0.278
0.681 1.385
0.709
1.503
AU-CNFs (W2)
0.285 0.300
0.297 0.308
0.589 1.375
0.620 1.467
B: lower
rCRPS
rCRPS
W2in
W out
Plug-in DKME
0.709 = 0.827
0.757 = 0.866
Plug-in CNF
0.312 0.212
0.326 0.227
0.650 0.973
0.660
1.073
IPTW-CNF
0.384 + 0.208
0.381 0.227
0.814 1.028
0.781 + 1.193
CA-CNFs (CRPS)
0.303 0.310
0.308 0.329
0.675
1.617
0.658 1.797
CA-CNFs (W22
0.307 0.302
0.311 = 0.323
0.639 1.451
0.642 1.625
AU-CNFss (CRPS)
0.330 = 0.254
0.344 = 0.267
0.726
1.379
0.744 = 1.550
AU-CNFs (W2)
0.308 + 0.284
0.314 +: 0.304
0.642 1,362
0.647 + 1.549
Lower (best bold, best underlined)
Method
Training stages
Average duration (in mins)
Plug-in DKME
First stage
~ 10.7
Plug-in CNFs
First stage
R 1.4
IPTW-CNF
First stage
1.5
CA-CNFs
First second stages
3.3
AU-CNFS
First second stages
4.1
Table 5: Total runtime (in seconds) for different methods to estimate Makarov bounds. Reported: average runtime duration (lower is better). Experiments were carried out on 2 GPUs (NVIDIA A100-PCIE-40GB) with IntelXeon Silver 4316 CPUs @ 2.30GHz.
44

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J Case study: Lockdown effectiveness
In the following, we provide case study, where we apply our AU-learner real-world problem. Here, we want to study the effectiveness of lockdowns during the COVID-19 pandemic by using the observational data collected in the first half-year of 2020 (7.. Specifically, we aim estimate the probability that the incidence falls after the implementation of the strict lockdown, i. e., a probability of individual benefit from treatment (intervention) (PITB).
J.1 Dataset
We used multi-county data provided by (7) The outcome Y E [-7.0] 0) is defined as the relative case growth per week (in log), namely, the number of new cases divided by the number of cumulative cases. Then, the treatment A E {0, 1) is taken as an implementation of the strict lockdown one week before. We also choose three (dg 3) pre-treatment covariates X: the relative case growths from the previous week, the relative case growth from two weeks ago, and the implementation of the strict lockdown from two weeks ago. We assume that the data i.i.d. and that the causal assumptions (1)-(3) are satisfied. We filtered out observations where the number of cumulative cases is fewer than 20. As a result, we ended up with n no + n1 152 + 112 treated and untreated observations, respectively.
J.2 Results
We present the results for our case study in Fig. 12 Therein, we report two quantities: the estimated
bounds on the probability of the individual treatment benefit (PITB), IP [1] [ S 0 r), for 20 countries during week 15 of 2020. Additionally, we show bounds on population analogue of PITB, namely, a probability of the population treatment benefit, [P - [0] < 0). We estimated the bounds on the PITB with both AU-CNFs (CRPS) and AU-CNFs (W3), and both methods produced very similar results (which implies robustness of our AU-learner). For the population analogue of the PITB. we first efficiently estimated the distributions of the potential outcomes with interventional normalizing flows (INFs) (94) and then used them to infer the Makarov bounds at the population level.
Week
IND
Al AL-C00 (Cur5)
CNN (W2)
Avanag
Ammell
Anards
Usadda
Doamsur
Prance
Iniond
Irayy
Portaga
Spar
Stondo
felbbrrad foad
Unarel Klmplon
Usked Ftstas
Donatry
Figure 12: Results for the real- world case study analyzing the effectiveness of lockdowns during the COVID-19 pandemic. Reported: in-sample estimated bounds on the probability of the individual treatment benefit (PITB), P(Y Y < 0 æ)), over 20 runs for 20 countries during week 15 of 2020. Also, we show bounds on probability of the population treatment benefit, i.e., P( -(0)) < 0). These are displayed in the small figure the left. Each estimated bound is shown two boxplots; hence, we also display the epistemic uncertainty.
There are two important takeaways: (1) The bounds on the PITB are more shifted towards suggesting the drop in the incidence is highly probable after the implementation of the strict lockdown in all the studied countries. (2) The bounds on PITB are much tighter than their population analogue (e. g., average upper-lowed bound width is 0.66 for AU-CNFs (CRPS) and 0.88 for INFs). The latter implies that individualization enhances decision-making and makes the Makarov bounds on the aleatoric uncertainty tighter and, thus, more informative.
"Teee data is at pastte//asaa/aatt_pggrooossssss
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For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
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We provide the full set of Assumptions in Sec. 3and | and Appendix D The proofs for the theoretic results are provided in Appendix [
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Reviews:
[{"review_id": "gkoUOLy2Hw", "reviewer": "Reviewer_U9cv", "summary": "The authors study the distribution over the individual treatment effect for binary treatments, continuous outcomes, and observed, potentially high-dimensional confounders. The authors build up on prior work on Makarov bounds, to develop a new method to lower/upper-bound the conditional CDF and the quantile function of the treatment effect. They develop a doubly robust learner and perform a theoretical and empirical evaluation. In a series of experiments, They compare several instances of their method with a baseline based on kernel density estimation. While the bound-based methods clearly outperform the baseline, they yield rather similar results.", "questions": "I do not fully understand the role of the working model g \\in G in the CA learner. The Makarov bound is a CDF and G may include all possible CDFs; so if G is rich enough, the CA learner may return the same CDF estimate as the plugin estimator. If we restrict G, why does this cause tighter bounds?", "limitations": "NeurIPS Paper Checklist is provided; no concerns.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The authors clearly state the problem and derive a viable solution. They first argue that learning estimators of the CDFs of the two potential outcomes, and plugging them into the bound computation does not yield optimal bounds. Instead, they then propose to learn the outcome CDFs which directly target the Makarov bounds. Third, they augment this loss with a term accounting for selection bias, which yields a doubly robust learner. Finally, the authors derive an implementation based on neural normalizing flows.\n\nThe authors present a great set of supplementary material including additional discussion of related work, additional experiments, and implementation details. The material might be sufficient for a journal publication.", "weaknesses": "The plugin estimator based on a conditional normalizing flow seems to perform rather well in the experiments. The CA learner w/o bias correction seems to perform as good as the AU learner (slightly worse in Table 2, slightly better in Table 4 in the appendix). The added complexity of the AU learner, in particular, the one-step bias correction which comes with an additional tuning parameter, renders the practical value questionable. It seems that a lot of work went into the AU learner; but it might still be worth focusing the presentation more on the simpler CA learner (e.g., discussing the role of g in the main doc).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the positive review. In the following, we respond to the weaknesses and questions.\n\n### Response to weaknesses\n\nThank you. We will follow your suggestion and expand our explanation around the CA-learner. Here, we would like to give more intuition as to why we think that the AU-learner has clear benefits over the CA-learner. The reason is due to the differences in low-sample performance vs. asymptotic properties. Importantly, the best low-sample learner and the asymptotically best learner can, in general, be different [2], and there is no single “one-fits-all” data-driven solution to choose the former one [1]. Therefore (as you mentioned), in some experiments, the CA-learner or even the plug-in approach are performing nearly as well or even sometimes better than the AU-learner. This can be explained by too small data sizes or the severe overlap violations (as is the case with the IHDP dataset [6] and the results in Table 4). Yet, **only our doubly-robust AU-learner offers asymptotic properties** in the sense that it is asymptotically closest to the oracle (see Figure 4). We thus argue for a pragmatic choice in practice (i.e., in the absence of ground-truth counterfactuals or additional RCT data) where our AU-learner should be the preferred method for the individualized Makarov bounds even in low-sample data. \n\nRegarding the additional tuning parameter $\\gamma$: we found the fixed values to work well in **all of the synthetic and semi-synthetic experiments** (except for the IHDP dataset, where the overlap assumption is violated).    \n\n**Action**: We will follow your suggestion and expand our explanation of the CA-learner in comparison to the AU-learner (e.g., discussing the role of $\\gamma$). We further will clarify the distinction between low-sample performance and asymptotic properties to better motivate the relevance of the AU-learner. \n \n\n### Response to Questions\n\nIndeed, by postulating a restricted working model class, $g \\in \\mathcal{G}$, we might compromise on the sharpness and end up having looser bounds. Yet, this looseness bias is only relevant in the infinite data regime. In the finite-sample regime, the feasibility of the low-error estimation is a much more important problem. Thus, by having a projection on the restricted model class $g \\in \\mathcal{G}$, we have a significantly lower variance of estimation than the variance from the unrestricted plug-in model (as confirmed by the experiments in the paper).  \n\nThe above-mentioned bias-variance tradeoff becomes directly apparent when we assume an inductive bias that **the treatment effect is less heterogeneous than both of the potential outcomes**. Such inductive bias is widely made in the literature on CATE estimation [3-5]. In extreme cases, the treatment effect can be – on average – zero with constant Makarov bounds, while the conditional CDFs would depend on the covariates in a complicated manner. In this case, the plug-in (single-stage) learner would suffer from high variance and there is no direct way to regularize it without compromising on the fit of the conditional CDFs. We refer to Fig. 2 of [3] with an analogous example for the plug-in learner and DR-leaner for CATE estimation. Hence, this illustrates the benefits of having a restricted working model class, $g \\in \\mathcal{G}$.   \n\nImportantly, our synthetic experiments contain this inductive bias (especially, in normal and multi-modal settings). Therein, the two-stage learners systematically outperform the plug-in CNF learner.    \n\n**Action**: We will clarify the above in the revised paper.\n\n### References:\n- [1] Curth, Alicia, and Mihaela van der Schaar. \"In search of insights, not magic bullets: Towards demystification of the model selection dilemma in heterogeneous treatment effect estimation.\" International Conference on Machine Learning. PMLR, 2023.\n- [2] Curth, Alicia, and Mihaela van der Schaar. \"Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2021.\n- [3] Kennedy, Edward H. \"Towards optimal doubly robust estimation of heterogeneous causal effects.\" Electronic Journal of Statistics 17.2 (2023): 3008-3049.\n- [4] Morzywolek, Pawel, Johan Decruyenaere, and Stijn Vansteelandt. \"On a general class of orthogonal learners for the estimation of heterogeneous treatment effects.\" arXiv preprint arXiv:2303.12687 (2023).\n- [5] Vansteelandt, Stijn, and Paweł Morzywołek. \"Orthogonal prediction of counterfactual outcomes.\" arXiv preprint arXiv:2311.09423 (2023).\n- [6] Curth, Alicia, et al. \"Really doing great at estimating CATE? A critical look at ML benchmarking practices in treatment effect estimation.\" Thirty-fifth conference on neural information processing systems datasets and benchmarks track (round 2). 2021.", "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": "5l4lCd26jp", "reviewer": "Reviewer_bvQh", "summary": "The authors introduce AU-learner, a method to estimate the conditional distribution of treatment effects (CDTE) and hence capture the  variability in the treatment effect. They use Makarov bounds for partial identification and use conditional normalizing flows for estimation. Further, they show that AU-learner satisfies Neyman-orthogonality and double robustness.", "questions": "(Q1) For a binary outcome, could you comment on how your bounds relate to Kallus [58]? Why don't you consider them as a baseline (say for estimating bounds on the quantiles of the treatment effect distribution)?", "limitations": "Yes", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "(S1) The paper is well written and the authors provide a good experimental validation of the proposed method.\n\n(S2) The authors are the first to propose a doubly robust estimator for the Makarov bounds on the distributional treatment effect.", "weaknesses": "(W1) I would have liked a more comprehensive discussion of how the proposed bounds compare against existing ones in terms of sharpness. For instance, it is not clear to me if the bounds you get are sharper than [58] in the binary outcomes setting. \n\n(W2) I think it would be useful to compare the proposed methods against the non-doubly robust estimators. In practical settings both outcome and propensity models are likely misspecified, hence the main advantage of doubly robust estimators are the faster theoretical rates. In this setting (with both nuisances misspecified), it is not clear if the doubly robust version of the bounds is always better in practice and some more experiments would be nice to explore this. \n\n(Minor) In the extended related work on partial identification and sensitivity models (line 658) you could also mention more recent approaches that incorporate RCT data together with proxy and instrumental variables, e.g. the works of [1] and [2].\n\n\n[58] Nathan Kallus. “What’s the harm? Sharp bounds on the fraction negatively affected by treatment”. In: Advances in Neural Information Processing Systems. 2022.\n\n[1] Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictions. Hussain et al. AISTATS 2023.\n\n\n[2] Hidden yet quantifiable: A lower bound for confounding strength using randomized trials. De Bartolomeis et al. AISTATS 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive review and the interesting questions. It’s great that you found our paper well-written and that you appreciate the theoretical contributions.\n\n### Response to weaknesses\n**(W1)** This is an interesting question. In our paper, we adopted the Makarov bounds, which were shown to be sharp for the left-continuous definition of the CDF ($\\mathbb{P}(Y < y)$) in [1] and (very recently) for the right-continuous definition of the CDF ($\\mathbb{P}(Y \\le y)$) in [2]. This distinction does not matter in our context of (absolutely) continuous outcome, and, hence, we used the traditional, right-continuous definition. Yet, it matters for the binary, ordinal, and, more generally, mixed-type outcomes. Given the results of [2], we show how the Makarov bounds for binary variables **exactly match** the ones, proposed by Kallus for the fraction of the negatively affected (FNA) in [3]. Due to space limitations, we moved the full derivation to **the rebuttal PDF** \n\n**Action**: We will add the theoretical derivation from our above to the Appendix of our camera-ready version so that we show the correspondence of the FNA bounds and our Makarov bounds.\n\n**(W2)** Thank you for the suggestion to highlight the advantages of the doubly-robust estimators. Upon reading your comment, we realized that we should have explained our experiments better because, therein, we actually show that the doubly robust learner is almost always better in practice. In our paper, we actually report a comparison of the doubly-robust AU-learner and other non-doubly-robust (two-stage) learners: CA-learner and IPTW-learner. Notably, in **all the experiments** and for all learners, each of the nuisance functions can be considered to be **misspecified due to the low-sample uncertainty**. Should you see the need for further experiments, we would be glad to include them.  Importantly, our AU-learner is asymptotically guaranteed to achieve the best performance among the  CA- and IPTW-learners even when both the nuisance functions are misspecified, as the second-order estimation error is the multiplication of both nuisances second-order errors (=double-robustness).      \n\nAlso, we would like to mention, that doubly-robust learners are only guaranteed to be optimal asymptotically. In very low-sample regimes, another learner can be preferable [4]. Yet, to reliably check it, we would need the ground-truth counterfactuals.\n\n**Action**: We will add the above clarifications to the camera-ready version of the paper to highlight the benefits of the doubly-robust estimators.\n\n**(Minor)** Thank you – great idea! We will extend the related work in the Appendix with the above-mentioned streams of work.\n\n### Response to questions\n**(Q1)** To answer your question, we first would like to point to our answer in **(W1)** where we now show that the Makarov bounds for the binary outcome exactly match the ones proposed in [3]. **Therefore, our work can be seen as a generalization of [3] in three novel directions:** (1) from binary to the continuous outcomes; (2) from population to individualized level; (3) targeting at the bounds for multiple values of $\\delta$, not just $\\delta = 0$ as in [3]. Hence, [3] is too limited and, thus, not a relevant baseline in our setting.\n\n**Action**: We will add the above explanation to our paper.\n\n\n### References:\n- [1]  Williamson, R. C. and T. Downs (1990). “Probabilistic Arithmetic I: Numerical Methods for Calculating Convolutions and Dependency Bounds”. International Journal of Approximate Reasoning 4, 89-158.   \n- [2] Zhang, Zhehao, and Thomas S. Richardson. \"Bounds on the Distribution of a Sum of Two Random Variables: Revisiting a problem of Kolmogorov with application to Individual Treatment Effects.\" arXiv preprint arXiv:2405.08806 (2024).\n- [3] Kallus, Nathan. \"What's the harm? Sharp bounds on the fraction negatively affected by treatment.\" Advances in Neural Information Processing Systems 35 (2022): 15996-16009.\n- [4] Curth, Alicia, and Mihaela van der Schaar. \"Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2021.", "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": "9kVQzdttzP", "reviewer": "Reviewer_Cpq8", "summary": "In this paper, the authors propose a method to quantify the aleatoric uncertainty of the treatment effect.\nFor this, authors estimated Makarov bounds on the CDF and quantiles of the CDTE, and then showed, how one can build a learner, which has properties of Neyman-orthogonality and double robustness.\nThe authors proved the abovementioned theoretical properties of the resulting learner and demonstrated the usefulness of the proposed approach in a series of experiments on synthetic and real-world data.", "questions": "- What are the alternatives for Makarov's bounds?\n- Are such options like conformal predictions or plain confidence intervals somehow useful to estimate AU in the context of treatment effect? \n- Compared to the vanilla Plug-in estimator (say Plug-in-CNF), what is the additional computational overhead of AU-learner?\n- In lines 212-216, it is mentioned that CA-learned still has shortcomings (a) (and a new shortcoming (c)). In practice, how severe is the shortcoming (a) (selection bias) compared to AU-learner? Can you provide a toy example?", "limitations": "One of the limitations, not mentioned by authors, is that the approach requires (conditional) Normalizing Flows, and hence might not work well in the high dimensional scenario.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "I think the paper has the following strengths: It\n- Addresses an important problem in the field of treatment effect estimation;\n- Proposed a theoretically grounded method, that has useful theoretical properties, to quantify the aleatoric uncertainty of the treatment effect;\n- Published code;", "weaknesses": "I don't have critical concerns about the paper, but I believe the following can improve the paper:\n- I find the paper logically well-structured, but at the same time challenging to read, as it is too concentrated with technical details. I would suggest authors reconsider the narrative, concentrating more on the conceptual ideas in the main part, and moving technical details to the Supplementary;\n- The field of treatment effect estimation is quite special and narrow (from my point of view) among the machine learning community, and it is worth adding some clarifications to the terms and notation used. For example, what is $Y$? It is just said that it is a continuous outcome, without any intuitions of what it could be.\n- I feel that the paper missing the discussion of the alternatives of the proposed approach. For example, why specifically Makarov bounds were chosen, but not other possible alternatives? \n- In Table 2, it is worth explicitly writing that CNF corresponds to the Plug-in learner (as it was called throughout the paper). I would suggest Plug-in-CNF (like written for IPTW-CNF).\n- In Line 268 there is a minor typo: Should be Eq.13 and Eq. 14.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the positive review. Below, we respond to your questions.\n\n### Response to weaknesses\nThank you for the suggestions on how to improve the paper. We are more than happy to implement in the final version of the paper. \n\n**Action**: We will improve our paper as follows: \n- We realized that our paper draws upon two different streams (namely, uncertainty quantification and treatment effect estimation), which may make it harder to understand for reviewers unfamiliar with the terminology in causal inference. We will thus (1) add more background and more explanations around causal inference terminology (e.g., explaining the potential outcomes framework where $Y$ is typically the outcome of treatment), and (2) revisit the introduction so that it focuses on the conceptual ideas while delegating the technical details to a separate section. \n- We will proofread the typos and rename “CNF” to “Plug-in-CNF” (thank you for the suggestion!).\n\n_Choice the Makarov bounds:_ Potential alternatives to Makarov bounds vary depending on the type of aleatoric uncertainty. For example, explicit or implicit sharp bounds were proposed for:\n- The variance of the treatment effect , $\\operatorname{Var}(Y[1] - Y[0])$. The sharp bounds are explicitly given by Fréchet-Hoeffding bounds [1]. We briefly discuss those in Sec. 2.\n- The interval probabilities, $\\mathbb{P}(\\delta_1 \\le Y[1] - Y[0] \\le \\delta_2)$. The sharp bounds on the interval probabilities are, in general, different from Makarov bounds and are only implicitly defined [2]. We briefly discuss those in Appendix B.2. \n\nYet, we are **not** aware of other bounds for measuring aleatoric uncertainty (e.g., kurtosis/skewness or entropy). Note that the above-mentioned bounds on different measures of uncertainty are _orthogonal_ to our work, as we consider bounds on the CDF/quantiles of the treatment effect as the main target. The bounds on the CDF  provide **more information** than the bounds on the variance but are the **first step** towards developing bounds on the interval probabilities.\n\n**Action**: We will highlight different alternatives to the Makarov bounds in our paper.\n\n### Response to questions\n\n1. We are happy to discuss alternatives to the Makarov bounds (see the answer above). Yet, these are not relevant to our work, as we focus on the estimation of the individualized level bounds on the CDF/quantiles of the treatment effect.\n2. Conformal predictive intervals have a **different* objective as they are a measure of the total uncertainty (without the distinction of AU and EU). Thus, conformal predictive intervals on the treatment effect would rather relate to the **interval probabilities** with additional EU. Nevertheless, this would be a very interesting extension of our paper for future work.\n\n    **Action**: We will add this as a suggestion for future work (i.e., bridging conformal prediction and partial identification of AU).\n3. The AU-learner additionally (1) fits a propensity score model (jointly or separately with the nuisance CNF), (2) infers the pseudo-CDFs, and (3) fits a second-stage working model. Roughly speaking, the AU-learner takes twice as much time to train compared to the “Plug-in-CNF” (as the pseudo-CDFs inference time is negligible).\n\n   **Action**: We will clarify this in Appendix H.3 (Runtime comparison).\n\n4. Addressing the selection bias matters considerably in a low-sample regime for any CATE estimator [3], and, thus, also affects other casual quantities (e.g., individualized level Makarov bounds). The importance of addressing the selection bias can be understood with the **following toy thought experiment**. In our context, both the CA-learner and the AU-learner use the estimated nuisance CDFs, $\\hat{\\mathbb{F}}_a(y \\mid x)$. Here, in the presence of the selection bias, $\\hat{\\mathbb{F}}_0$ will be oversmoothed for the treated population and  $\\hat{\\mathbb{F}}_1$, for the untreated, as there are very little samples to fit the corresponding CDFs. Then, the CA-learner uses the oversmoothed nuisance CDFs as they are, but the AU-learner will aim to correct this oversmoothing. The exact error of both learners due to the misspecification of their CDFs can be then verified with the help of the pathwise derivatives. We proved that, for a doubly-robust AU-learner, it is first-order insensitive to the misspecification of the nuisance CDFs (aka Neyman-orthogonality, see Theorem 2). Additionally, we derive a pathwise derivative for the CA-learner’s risk to show that it is **first-order sensitive (non-zero)** for the setting of CATE estimation (see the **rebuttal PDF**). Hence, this demonstrates a shortcoming of the CA-learner.\n\n    **Action**: We will add the toy example to explain the shortcomings of the CA-learner. Also, we will add the derivation of the pathwise derivative for the CA-learner’s risk targeting at the original Makarov bounds to demonstrate its first-order sensitivity (see the **rebuttal PDF**). \n \n\n### Response to limitations\n\nThe validation of our method requires the datasets where the ground-truth CDF are known (and thus the Makarov bounds are known). A prominent example is HC-MNIST, which is one of the largest public datasets for treatment effect estimation. Here, our methods work well and have a reasonable runtime.\n**Action**: We will state the possible extension of Makarov bounds that are tailored to high-dimensional outcomes as an idea for future work.\n\n### References:\n- [1] Aronow, Peter M., Donald P. Green, and Donald KK Lee. \"Sharp bounds on the variance in randomized experiments.\" (2014): 850-871.\n- [2] Firpo, Sergio, and Geert Ridder. \"Partial identification of the treatment effect distribution and its functionals.\" Journal of Econometrics 213.1 (2019): 210-234.\n- [3] Alaa, Ahmed, and Mihaela van der Schaar. \"Limits of estimating heterogeneous treatment effects: Guidelines for practical algorithm design.\" International Conference on Machine Learning. PMLR, 2018.", "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": "swHvqOsOf9", "reviewer": "Reviewer_d6jE", "summary": "The authors propose a partial identification of quantiles of the individual treatment effect, which are not point-identifiable in general. The authors justifiably argue that characterizing the distribution of individual treatment effects gives a better idea of the aleatoric uncertainty in a causal-inference problem. The proposed method is doubly robust, works with heterogeneous treatment effects, and requires minimal additional assumptions about the data.", "questions": "My main question has to do with recent related work. In particular, Ji et al. [52] appear to solve a similar problem, and the quick dismissal of that approach in this paper because they \"made special optimization assumptions\" needs further discussion. It would be helpful to spell out what these assumptions are in a concrete sense. In doing so, the authors could also discuss whether these two approaches have any fundamental commonalities, or if the partial identifications are expected to be materially different. Finally, it would be nice to see [52] appear as a baseline in the empirical evaluations, although I understand this could be difficult to implement, especially since [52] is relatively recent.", "limitations": "N/A", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The problem of aleatoric uncertainty in causal effects is clearly important, the contribution is significant, and the presentation is solid and concise. \n\n * Explanations of aleatoric versus empirical uncertainty are clear.\n * The numerous diagrams are informative.\n * Algorithm 1 is also easy to understand and succinct.", "weaknesses": "* The benefit and/or novelty of the CA-learner is questionable (also reflected in the results Table 2) and I wonder if its exposition is taking up valuable space. It is an insightful point on lines 182--185 that learning Makarov bounds could benefit from inductive biases of lower heterogeneity than the conditional CDFs. However, it is unclear if the CA-learner loss really incorporates that inductive bias and if so, how much.\n\n * The conclusion is a bit grand. Arguably, previous papers like those highlighted in Table 1 have proposed robust methods for quantifying a version of aleatoric uncertainty of causal effects.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive review! It is great to hear that you found our contribution significant.\n\n### Response to weaknesses\n\n**Benefit of the CA-learner**.  We introduce the CA-learner only as an interim step to develop the full AU-learner. We follow the classical hierarchy of learners, which were developed for other individualized treatment effect estimators [1-4], namely: “plug-in leaner -> IPTW-learner/CA-learner -> DR-learner”. Additionally, the CA-learner serves as a natural ablation of the full AU-learner. \n\nRegarding the incorporation of the inductive bias: both the CA- and AU-learners are able to do so by adding a regularization to the working model at the second stage (importantly, this regularization is independent of the regularization of the nuisance models). In our instantiations of CA-CNFs and AU-CNFs, we employed an exponential smoothing of model weights [5] as a regularizer of the working model (see Appendix E.2 Implementation (Training) for further details). \n\n**Action**. We will spell out these important details in the manuscript more clearly.\n\n**Conclusion**. The methods listed in Table 1 are _either_ (a) robust but aimed at the averaged (population level) Makarov bounds, _or_ (b) aiming at the individualized Makarov bounds but are not robust. Thus, the statement of the conclusion still holds, as _neither_ the efficient influence function _nor_ doubly-robust (orthogonal) learners for general observational data were proposed for the individualized Makarov bounds.\n\n**Action**. We will be more specific in the conclusion by saying “individualized Makarov bounds” instead of “Makarov bounds”.\n \n\n### Response to questions\nYou have raised an important question. The work of  Ji et al. [7] targets at estimating the **averaged (population level)** Makarov bounds among the other, more general, non-identifiable functionals of the joint distribution of the potential outcomes. In contrast, we focus on **individualized** Makarov bounds. \n\nJi et al. also assume the **possibility of finding feasible Kantorovich dual functions** to the target functional (=special optimization assumption). By doing so, the authors are able to infer valid partial identification bounds on very general functionals. However, the **sharpness of [7] can not be practically guaranteed**. Specifically, feasible Kantorovich dual functions have to be chosen from a finite set of functions, which makes the bounds less sharp (this goes on top of the usual errors from estimating nuisance functions and the working model). Therefore, the solution of [7] is **completely impractical** in our setting, where the expression for sharp bounds already exists (Makarov bounds). \n\nAdditionally to the above-mentioned problem: there are other practical obstacles to making  [7] a relevant baseline. For example, it is unclear how to adapt it to the individualized level Makarov bounds or how to fit the working model for several grid values of $\\delta / \\alpha$  at once.\n\n**Action**. We will add more background on why the work of Ji et al. [7]  is not a relevant baseline.\n\n### References:\n- [1] Morzywolek, Pawel, Johan Decruyenaere, and Stijn Vansteelandt. \"On a general class of orthogonal learners for the estimation of heterogeneous treatment effects.\" arXiv preprint arXiv:2303.12687 (2023).\n- [2] Vansteelandt, Stijn, and Paweł Morzywołek. \"Orthogonal prediction of counterfactual outcomes.\" arXiv preprint arXiv:2311.09423 (2023).\n- [3]  Curth, Alicia, and Mihaela van der Schaar. \"Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2021.\n- [4] Valentyn Melnychuk, Dennis Frauen, and Stefan Feuerriegel. “Normalizing flows for interventional density estimation”. In: International Conference on Machine Learning. 2023.\n- [5] Polyak, Boris T., and Anatoli B. Juditsky. \"Acceleration of stochastic approximation by averaging.\" SIAM journal on control and optimization 30.4 (1992): 838-855.\n- [6] Semenova, Vira. \"Adaptive estimation of intersection bounds: a classification approach.\" arXiv preprint arXiv:2303.00982 (2023).\n- [7] Ji, Wenlong, Lihua Lei, and Asher Spector. \"Model-agnostic covariate-assisted inference on partially identified causal effects.\" arXiv preprint arXiv:2310.08115 (2023).", "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": "tu74vbzCYv", "reviewer": "Reviewer_8SMd", "summary": "The paper provides the orthogonal estimator for the distributional treatment effect (the conditional CDF of $Y[1]-Y[0]$).", "questions": "1. Are rate-doubly-robustness and Neyman orthogonality violated when the scaling hyperpameters are not $1$? \n\n2. Is the Makarov bound sharp? \n\n3. What are the practical examples of the distributional treatment effect? \n\n4. In line 178, is this phenomenon officially termed the selection bias?", "limitations": "The paper is limited to the setting where the ignitability holds.", "rating": 3, "confidence": 5, "soundness": 4, "presentation": 1, "contribution": 2, "strengths": "This paper is technically strong, demonstrating a high level of mathematic rigor and diligence. It presents an in-depth description of the proposed estimator. I think the proposed estimator is useful in practice.", "weaknesses": "Despite the strong technical details, the paper is poorly written in overall. \n\n__1. Weak motivation__\n\nA current shape of Introduction is weakly motivated. Firstly, the way that the paper motivates the problem is misleading:\n> Methods for quantifying the aleatoric uncertainty of the treatment effect have gained surprisingly little attention in the causal machine learning community.\n\nThis sentence is incorrect, since there are literatures on quantile regression and semiparametric density estimation, as reviewed in Section 2. \n\nMore importantly, the introduction doesn't provide the motivation of the problem against the following question: _why a community need the proposed estimator, given that the quantile estimator can capture the distributional treatment effect_.  \n\n\n__2. Difficult to understand due to insufficient information__\n\nAnother issue that the paper has (especially in Introduction) is its lack of back-ground information that readers may need to comprehend. Specifically, in Introduction, the paper doesn’t provide any definition or clue what CDTE is. I understand that the CDTE is defined in the caption of Figure 1 as $P(Y[1] - Y[0] \\leq \\delta \\mid x)$. However, $Y[a]$ is undefined, and this key quantity should be in the main body of the text. Also, even if Figure 1 aims to provide a whole summary of the paper, authors at Introduction have insufficient knowledge to comprehend it. In other words, Figure 1 is too detailed to be presented in Introduction section. Since Figure 1 can only be understood by those who entirely digested the paper from the beginning to the end, Introduction section is not the right position where the Figure 1 is located. I understand the goal of Figure 1, but it doesn't achieve the goal because of insufficient background information. The same issue happens to Figure 2 and Table 1. For example, in Table 1, technical terms like AIPTW, hold-out residual, optimization assumptions are undefined, so it's hard to appreciate the contribution of this paper. \n\n__3. Fuzzy description on contribution__ \n\nFirst, the terms like \"Aleatoric uncertainty\" and \"distributional treatment effect\" are used for denoting the same target quantity. Given that the \"distributional treatment effect\" is clearly describing the problem, I don't see why the authors want to use \"Aleatoric uncertainty\" as a title and employ these two words to denote the same estimand. \n\nSecond, the contribution is wrongly described. Consider this sentence: \n> AU-learner solves all of the above-mentioned challenges 1 – 3 .\n\nThe AU-learner doesn't address Challenge 1 and 2. Challenge 1 means that the distributional treatment effect is not identifiable. The Makarov's bound, _NOT_ AU-learner, is employed to address Challenge 1. Challenge 2 is actually the same as Challenge 1, since it means that there are no known nuisances-based functional for the distributional treatment effect. Again, the Makarov's bound is used to address Challenge 2. AU-learners are representing the approximated quantity of the target estimand in terms of nuisance functionals. \n\nFinally, third contribution \"flexible deep learning instantiation of our AU-learner\" is scarcely described only in Section 5. The description needs to be much improved. \n\n__4. Little focus on the real contribution__ \n\nThe real contribution of this paper, compared to the existing works in Table 1, is to provide the doubly robust conditional distributional treatment effect for the bounds of the conditional CDF of treatment effects. However, little focus and efforts have been made for this contribution. For example, if developing a doubly robust estimator is a contribution, then corresponding results such as detailed error analysis, a closed form of estimators, a detailed recipe of the proposed estimator for the specified working model, how to minimize the losses in Equations (8,9), a simple example, and assumptions should be described.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "First of all, thank you for the detailed and positive review of our paper. Below, we respond to the mentioned weaknesses and questions. Importantly, all the issues will be easily fixed for the camera-ready version of the manuscript.\n\n### Response to weaknesses\n1. We want to stress that there are several – seemingly similar – research streams connecting the aleatoric uncertainty and potential outcomes/treatment effects, yet these have **different causal quantities** with **crucial differences**:\n   - _Aleatoric uncertainty of potential outcomes._ This stream includes works on quantile regression and semiparametric density estimation. Here, the causal quantities are **point identifiable** and given by an explicit functional of the nuisance functions.\n   - _Distributional treatment effects._ This stream includes all the works, which infer and estimate the distributional distances/divergences between the potential outcomes distributions (e.g., Wasserstein distances, KL-divergence, or the quantile differences).   Here, similarly, the causal quantities are **point-identifiable**.\n   - _Aleatoric uncertainty of the treatment effect (= distribution of the treatment effect)._ Our paper is located in this stream. Therein, the casual quantities are, for example, the variance of the treatment effect, and the CDF/quantiles of the treatment effect (our paper). Importantly, they are **not point-identifiable** and given by the implicit functional of the nuisance functions.     \n\n   The **above-mentioned streams of literature contain different causal quantities with different interpretations**. For example: quantile differences != quantiles of the difference (treatment effect). Further: the distributional treatment effects != the distribution of the treatment effect. Hence, we argue that the original statement holds, given the differences in terminology. Nevertheless, we realized upon reading your comment that we need to elaborate on the differences more carefully.\n\n   **Action**. We will elaborate more carefully on the above differences in Appendix A (Extended Related Work) and thereby point out how our work is novel.\n\n2. We acknowledge that the Introduction might be hard to understand for non-causal ML practitioners. Initially, we did not want to overburden the Introduction with the exact notation and definitions (apart from the summary in Fig. 1). However, we understand that more explanations are helpful.\n\n   **Action**. We will follow your advice, and provide more explanations for the potential outcomes, $Y[a]$, and for the CDTE. Also, we will simplify Figures 1 and 2, and add the details regarding Table 2.  \n\n3. Again, we argue that the “distributional treatment effects” and “the aleatoric uncertainty of the treatment effect” are two **very different causal quantities** (see Answer to 1.). \n\n   We thank you for your feedback and agree that the sentence “AU-learner solves all of the above-mentioned challenges 1 – 3.” needs to be rephrased. Given that the Makarov bounds were already proposed in the literature, we will improve our text by shifting Challenge 1 to the fact that there were no efficient influence functions (EIFs) derived for the Makarov bounds. The latter is solved in our paper. Then, Challenge 2 would still hold, as both the Makarov bounds and its EIFs are **implicit** functionals of the (estimated) nuisance functions, i.e., we need to perform sup/inf convolutions. This not only complicates the practical implementation of the AU-learner, but also the design of the experiments, as the ground-truth has to be derived in advance or approximately inferred numerically.   \n\n   **Action**. We are happy to implement the above-mentioned changes to the final version of the manuscript. Also, we will put more emphasis on the deep-learning instantiation.\n\n4. **Action**. We are happy to reformulate our contribution so that it is centered around the derivation and implementation of doubly-robust learner (AU-learner).\n\n### Response to questions\n1. Yes, when the scaling parameter $\\gamma \\neq 1$, Neyman-orthogonality does not fully hold (the cross-derivative wrt. CDFs are not zero). Hence, rate-doubly-robustness also does not hold. \n\n   **Action**. We will expand the proof of the Neyman-orthogonality and rate-doubly-robustness in the final version of the paper, so this fact would be more visible.\n2. Yes, the Makarov bounds are sharp [1]. See the discussion in Appendix B.2 Pointwise and Uniformly Sharp Bounds.\n3. Although the distributional treatment effects are **not the target of our paper**, the examples include distributional distances between potential outcomes [2]; quantile / super-quantile treatment effects and $f$-risk treatment effects [3], etc. Our paper is focused on the CDF/quantiles of the treatment effect, which is a different causal quantity.\n4. Yes, it was referred to as a selection bias, e.g., by [4-6]. An alternative name is a covariate shift [6].\n\n### References:\n- [1] Fan, Yanqin, and Sang Soo Park. \"Sharp bounds on the distribution of treatment effects and their statistical inference.\" Econometric Theory 26.3 (2010): 931-951.\n- [2] Kennedy, Edward H., Sivaraman Balakrishnan, and L. A. Wasserman. \"Semiparametric counterfactual density estimation.\" Biometrika 110.4 (2023): 875-896.\n- [3] Kallus, Nathan, and Miruna Oprescu. \"Robust and agnostic learning of conditional distributional treatment effects.\" AISTATS. PMLR, 2023.\n- [4] Curth, Alicia, and Mihaela van der Schaar. \"Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms.\" AISTATS. PMLR, 2021.\n- [5] Alaa, Ahmed, and Mihaela van der Schaar. \"Limits of estimating heterogeneous treatment effects: Guidelines for practical algorithm design.\" ICML. PMLR, 2018.\n- [6] Johansson, Fredrik D., et al. \"Generalization bounds and representation learning for estimation of potential outcomes and causal effects.\" Journal of Machine Learning Research 23.166 (2022): 1-50.", "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 are grateful for the insightful and high-quality reviews. We appreciate seeing that the reviewers found our paper to be “well-written”, “theoretically grounded”, “demonstrating a high level of mathematic rigour and diligence”, containing numerous informative diagrams and explanations, and with “a good experimental validation”.

We provide point-by-point responses for each reviewer below. We uploaded additional proofs and clarifications in a **PDF file**. Here, we summarize our key improvements: 

1. **Theoretical explanations**. We added the connections between the Makarov bounds from our paper to the previous works, namely, bounds on the binary outcomes [Kallus, 2022] (see the **rebuttal PDF**). Also, we show that the CA-learner is first-order sensitive wrt. to the misspecification of the nuisance functions (see the **rebuttal PDF**).
 2. **Better contrast between learners**. We better clarified the distinction between low-sample performance and asymptotic properties of different learners. Also, we provided toy examples of where CA-learner can be inferior compared to our AU-leaner. Additionally, we provided more intuition on the need to use the working model, $g \in \mathcal{G}$. 
3. **Better contextualization**. We provided more context on how our work is different from several other streams in the context of the aleatoric uncertainty and causal inference. We highlighted better what are the alternatives to the Makarov bounds and why they are limited / not applicable in our case. We also listed multiple future work ideas, based on our work. Additionally, we will simplify the Introduction section and sharpen the contributions, so that the significance of the paper is more clear to the general audience.   

We will incorporate all changes (marked with **Action**) into the camera-ready version of our paper (if accepted). Given these improvements, we are confident that our paper provides valuable contributions to the causal machine learning literature and is a good fit for NeurIPS 2024.

Meta-review: The paper introduces a method called AU-learner to estimate the conditional distribution of treatment effects (CDTE) for binary treatments and continuous outcomes. The authors use Makarov bounds for partial identification and conditional normalizing flows for estimation, developing a doubly robust learner that satisfies Neyman-orthogonality. 

Pros:

+ Addresses an important problem in causal inference by quantifying aleatoric uncertainty in treatment effects

+ Develops a theoretically grounded method with useful properties (double robustness, Neyman-orthogonality)

+ Provides a comprehensive set of experiments and supplementary materials

Cons:

+ The presentation is dense with technical details, making it challenging to read and understand
Decision: Accept (poster)