Title: Learning Counterfactual Outcomes Under Rank Preservation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal Inference

Nonparametric Identifiability

Individualized Treatment Effects
Abstract: Counterfactual inference aims to estimate the counterfactual outcome at the individual level given knowledge of an observed treatment and the factual outcome, with broad applications in fields such as epidemiology, econometrics, and management science. Previous methods rely on a known structural causal model (SCM) or assume the homogeneity of the exogenous variable and strict monotonicity between the outcome and exogenous variable. In this paper, we propose a principled approach for identifying and estimating the counterfactual outcome. We first introduce a simple and intuitive rank preservation assumption to identify the counterfactual outcome without relying on a known structural causal model. Building on this, we propose a novel ideal loss for theoretically unbiased learning of the counterfactual outcome and further develop a kernel-based estimator for its empirical estimation. Our theoretical analysis shows that the rank preservation assumption is not stronger than the homogeneity and strict monotonicity assumptions, and shows that the proposed ideal loss is convex, and the proposed estimator is unbiased. Extensive semi-synthetic and real-world experiments are conducted to demonstrate the effectiveness of the proposed method.

Learning Counterfactual Outcomes Under Rank Preservation

Abstract

Counterfactual inference aims to estimate the counterfactual outcome at the individual level given knowledge of an observed treatment and the factual outcome, with broad applications in fields such as epidemiology, econometrics, and management science. Previous methods rely on a known structural causal model (SCM) or assume the homogeneity of the exogenous variable and strict monotonicity between the outcome and exogenous variable. In this paper, we propose a principled approach for identifying and estimating the counterfactual outcome. We first introduce a simple and intuitive rank preservation assumption to identify the counterfactual outcome without relying on a known structural causal model. Building on this, we propose a novel ideal loss for theoretically unbiased learning of the counterfactual outcome and further develop a kernel-based estimator for its empirical estimation. Our theoretical analysis shows that the rank preservation assumption is not stronger than the homogeneity and strict monotonicity assumptions, and shows that the proposed ideal loss is convex, and the proposed estimator is unbiased. Extensive semi-synthetic and real-world experiments are conducted to demonstrate the effectiveness of the proposed method.

1 Introduction

Understanding causal relationships is a fundamental goal across various domains, such as epidemiology [1], econometrics [2], and management science [3]. Pearl and Mackenzie [4] define the three-layer causal hierarchy-association, intervention, and counterfactuals-to distinguish three types of queries with increasing complexity and difficulty [5]. Counterfactual inference, the most challenging level, aims to explore the impact of a treatment on an outcome given knowledge about a different observed treatment and the factual outcome. For example, given a patient who has not taken medication before and now suffers from a headache, we want to know whether the headache would have occurred if the patient had taken the medication initially. Answering such counterfactual queries can provide valuable instructions in scenarios such as credit assignment [6], root-causal analysis [7], attribution [8, 9, 10, 11], as well as fair and safe decision-making [12, 13, 14, 15, 16].

Different from interventional queries, which are prospective and estimate the counterfactual outcome in a hypothetical world via only the observations obtained before treatment (as pre-treatment variables), counterfactual inference is retrospective and further incorporates the factual outcome (as a post-treatment variable) in the observed world. This inherent conflict between the hypothetical

and the observed world poses a unique challenge and makes the counterfactual outcome generally unidentifiable, even in randomized controlled experiments (RCTs) [5, 8, 13, 17].

For counterfactual inference, Pearl et al. [8] proposed a three-step procedure (abduction, action, and prediction) to estimate counterfactual outcomes. However, it relies on the availability of structural causal models (SCMs) that fully describe the data-generating process [18, 19]. In real-world applications, the ground-truth SCM is likely to be unknown, and estimating it requires additional assumptions to ensure identifiability, such as linearity [20] and additive noise [21, 22, 23]. Unfortunately, these assumptions are hard to satisfy in practice and restrict the applicability.

To tackle the above problems, several counterfactual learning approaches have been proposed with respect to different identifiability assumptions. For example, Lu et al. [24], Nasr-Esfahany et al. [25], and Xie et al. [19] established the identifiability of counterfactual outcomes based on homogeneity and strict monotonicity assumptions [23, 26]. The homogeneity assumption posits that the exogenous variable for each individual remains constant across different interventional environments, and the strict monotonicity assumption asserts that the outcome is a strictly monotone function of the exogenous variable given the features. In terms of counterfactual learning, [24] and [25] adopted Pearl's three-step procedure that needs to estimate the SCM initially. In addition, [19] proposed using quantile regression to estimate counterfactual outcomes that effectively avoid the estimation of SCM. Nevertheless, it relies on a stringent assumption that the conditional quantile functions for different counterfactual outcomes come from the same model and it requires estimating a different quantile value for each individual, leading to a challenging bi-level optimization problem [27].

In this work, we propose a principled counterfactual learning approach with intuitive identifiability assumptions and theoretically guaranteed estimation methods . On one hand , we introduce the simple and intuitive rank preservation assumption, positing that an individual's factual and counterfactual outcomes have the same rank in the corresponding distributions of factual and counterfactual outcomes for all individuals. We establish the identifiability of counterfactual outcomes under the rank preservation assumption and show that it is slightly less restrictive than the homogeneity and monotonicity assumptions used in previous studies.

On the other hand , we further propose a theoretically guaranteed method for unbiased estimation of counterfactual outcomes. The proposed estimation method has several desirable merits. First, unlike Pearl's three-step procedure, it does not necessitate a prior estimation of SCMs and thus relies on fewer assumptions than that in [24] and [25]. Second, in contrast to the quantile regression method proposed by [19], our approach neither restricts conditional quantile functions for different counterfactual outcomes to originate from the same model, nor does it require estimating a different quantile value for each unit. Third, we improve the previous learning approaches by adopting a convex loss for estimating counterfactual outcomes, which leads to a unique solution.

In summary, the main contributions are as follows: (1) We introduce the intuitive rank preservation assumption to identify the counterfactual outcomes with unknown SCM; (2) We propose a novel ideal loss for unbiased learning of the counterfactual outcome and further develop a kernel-based estimator for the ideal loss. In addition, we provide a comprehensive theoretical analysis for the proposed learning approach; (3) We conduct extensive experiments on both semi-synthetic and real-world datasets to demonstrate the effectiveness of the proposed method.

2 Problem Formulation

Throughout, capital letters represent random variables and lowercase letters denote their realizations.

Structural Causal Model (SCM, [28]). An SCM M consists of a causal graph G and a set of structure equation models F = { f 1 , ..., f p } . The nodes in G are divided into two categories: (a) exogenous variables U = ( U 1 , ..., U p ) , which represent the environment during data generation, assumed to be mutually independent; (b) endogenous variables V = { V 1 , ..., V p } , which denote the relevant features that we need to model in a question of interest. For variable V j , its value is determined by a structure equation V j = f j ( PA j , U j ) , j = 1 , ..., p , where PA j stands for the set of parents of V j . SCM provides a formal language for describing how the variables interact and how the resulting distribution would change in response to certain interventions. Based on SCM, we introduce the counterfactual inference problem in the following.

Counterfactual Inference . Suppose that we have three sets of variables denoted by X,Y, E ⊆ V , counterfactual inference revolves around the question, 'given evidence E = e , what would have happened if we had set X to a different value x ′ ?". Pearl et al. [8] propose using the three-step procedure to answer the problem: (a) Abduction : determine the value of U according to the evidence E = e ; (b) Action : modify the model M by removing the structural equations for X and replacing them with X = x ′ , yielding the modified model M x ′ ; (c) Prediction : Use M x ′ and the value of U to calculate the counterfactual outcome of Y . In this paper, we focus on estimating the counterfactual outcome for each individual. To illustrate the main ideas, we formulate the common counterfactual inference problem within the context of the backdoor criterion.

Problem Formulation. Let V = ( Z, X, Y ) , where X causes Y , Z affects both X and Y , and the structure equation of Y is given as

Let Y x ′ denotes the potential outcome if we had set X = x ′ . The counterfactual question, 'given evidence ( X = x, Z = z, Y = y ) of an individual, what would have happened had we set X = x ′ for this individual", is formally expressed as estimating y x ′ , the realization of Y x ′ for the individual. Here, we adhere to the deterministic viewpoint of [28] and [8], treating the value of Y x ′ for each individual as a fixed constant. According to Pearl's three-step procedure, given the evidence ( X = x, Z = z, Y = y ) for an individual, the identifiability of its counterfactual value y x ′ can be achieved by determining the structural equation f Y and the value of U X for this individual. This is the key idea underlying most of the existing methods.

For clarity, we use y x ′ to denote the realization of the counterfactual outcome Y x ′ for a specific individual with observed evidence ( X = x, Z = z, Y = y ) .

3 Analysis of Existing Methods

In this section, we elucidate the challenges of counterfactual inference. Subsequently, we summarize the existing methods and shed light on their limitations.

3.1 Challenges in Counterfactual Inference

The main challenge lies in that the counterfactual value y x ′ is generally not identifiable, even in randomized controlled experiments (RCTs). By definition, y x ′ is a quantity involving two 'different worlds" at the same time: the observed world with ( X = x, Z = z, Y = y ) and the hypothetical world where X = x ′ . We only observe the factual outcome Y x = y but never observe the counterfactual outcome Y x ′ , which is the fundamental problem in causal inference [29, 30]. This inherent conflict prevents us from simplifying the expression of y x ′ to a do-calculus expression, making it generally unidentifiable, even in RCTs [8]. Therefore, in addition to the widely used assumptions such as conditional exchangeability, overlapping, and consistency [1], counterfactual inference requires extra assumptions to ensure identifiability. Essentially, estimating y x ′ is equivalent to estimating the individual treatment effect y x ′ -y x , while the conditional average treatment effect (CATE) E [ Y x ′ -Y x | Z = z ] represents the ATE for a subpopulation with Z = z , overlooking the inherent heterogeneity in this subpopulation caused by the noise terms such as U X [13, 31, 32, 33, 34, 35, 36].

3.2 Summary of Existing Methods

We summarize the existing methods in terms of identifiability assumptions and estimation strategies.

We first present an equivalent expression of Eq. (1) using ( Y x , Y x ′ ) . Eq. (1) be reformulated as the following system

where U x and U x ′ denote the values of U X given X = x and X = x ′ , respectively. The exogenous variable U X denotes the background and environment information induced by many unmeasured factors [8], and thus U x and U x ′ account for the heterogeneity of Y x and Y x ′ in the observed and hypothetical worlds, respectively. These two worlds may exhibit different levels of noise due to unmeasured factors [32, 34, 37]. For identification, previous work [19, 24, 25] relies on the key homogeneity and strict monotonicity assumptions.

Assumption 3.1 (Homogeneity) . U x = U x ′ .

Assumption 3.2 (Strict Monotonicity) . For any given ( x, z ) , Y x = f Y ( x, z, U x ) is a smooth and strictly monotonic function of U x ; or it is a bijective mapping from U x to Y x .

Assumption 3.1 implies that the value of U X for each individual remains unchanged across x . Assumption 3.2 implies that Y x is a strict monotonic function of U x in the subpopulation of ( X = x, Z = z ) . In Assumption 3.2, the smoothness and strict monotonicity of f Y ( x, z, U x ) are akin to a bijective mapping of Y x and U x and serve the same purpose, so we don't distinguish them in detail.

Lemma 3.3. Under Assumptions 3.1-3.2, y x ′ is identifiable.

For estimation of y x ′ , following Pearl's three-step procedure, [24] and [25] initially estimate f Y and U X for each individual. However, estimating f Y and U X needs to impose extra assumptions, such as linearity [20] and additive noise [22]. In addition, [19] demonstrate that y x ′ corresponds to the τ ∗ -th quantile of the distribution P ( Y | X = x ′ , Z = z ) , where τ ∗ is the quantile of y in P ( Y | X = x, Z = z ) (See the proof of Lemma 3.3 or Section 4.1 for more details). Based on it, the authors uses quantile regression to estimate y x ′ , which avoids the problem of estimating f Y and U X . Nevertheless, this method fits a single model to obtain the conditional quantile functions for both the counterfactual and factual outcomes. Thus, its validity relies on the underlying assumption that the conditional quantile functions of outcomes for different treatment groups stem from the same model. In addition, it involves estimating a distinct quantile value for each individual before deriving the counterfactual outcomes, posing a challenging bi-level optimization problem.

4 Identification through Rank Preservation

We introduce the rank preservation assumption for identifying y x ′ . From a high-level perspective, identifying y x ′ essentially involves establishing the relationship between Y x and Y x ′ for each individual. Pearl's three-step procedure achieves this by estimating f Y and U X .

4.1 Rank Preservation Assumption

Our identifiability assumption is based on Kendall's rank correlation coefficient defined below.

Definition 4.1 (Kendall [38]) . Let ( x 1 , y 1 ) , ..., ( x n , y n ) be a set of observations of two random variables ( X,Y ) , such that all the values of x i and y i are unique (ties are neglected for simplicity). Any pair of ( x i , y i ) and ( x j , y j ) , if ( x j -x i )( y j -y i ) > 0 , they are said to be concordant; otherwise they are discordant. The sample Kendall rank correlation coefficient is defined as

where sign ( t ) = -1 , 0 , 1 for t < 0 , t = 0 , t > 0 , respectively. For any two random variables ( X,Y ) , we define ρ ( X,Y ) = 1 , if ρ n ( X,Y ) = 1 for all integers n ≥ 2 .

The ρ n ( X,Y ) also can be written as 2( N c -N d ) /n ( n -1) , where N c is the number of concordant pairs, N d is the number of discordant pairs. It is easy to see that -1 ≤ ρ n ( X,Y ) ≤ 1 and if the agreement between the two rankings is perfect (i.e., perfect concordance), ρ n ( X,Y ) = 1 .

Assumption 4.2 (Rank Preservation) . ρ ( Y x , Y x ′ | Z ) = 1 .

Assumption 4.2 is a high-level condition that establishes a connection between Y x and Y x ′ . This assumption is satisfied in many common scenarios, as illustrated below.

Causal models with additive noise: Y = g ( X,Z ) + U for an arbitrary function g .

Heteroscedastic noise models: Y = g ( X,Z ) + h ( X,Z ) U for arbitrary functions g and h , with h ( X,Z ) > 0 denoting the conditional standard deviation of Y given ( X,Z ) .

For the individual with observation ( X = x, Z = z, Y = y ) , we denote ( y x = y, y x ′ ) as its true values of ( Y x , Y x ′ ). Assumption 4.2 implies that for this individual, its rankings of y x and y x ′ are the same in the distributions of P ( Y x | Z = z ) and P ( Y x ′ | Z = z ) , respectively. Therefore, we have

Since y x = y is observed and the distributions P ( Y x | Z = z ) and P ( Y x ′ | Z = z ) can be identified as P ( Y | X = x, Z = z ) and P ( Y | X = x ′ , Z = z ) , respectively, by the backdoor criterion (i.e., ( Y x , Y x ′ ) ⊥ ⊥ X | Z ). Therefore, we have the following Proposition 4.3 (see Appendix A for proofs).

Proposition 4.3. Under Assumption 4.2, y x ′ is identified as the τ ∗ -th quantile of P ( Y | X = x ′ , Z = z ) , where τ ∗ is the quantile of y in the distribution of P ( Y | X = x, Z = z ) .

Proposition 4.3 shows that Assumption 4.2 can serve as a substitute for Assumptions 3.1-3.2 in identifying y x ′ . Unlike Assumptions 3.1-3.2, Assumption 4.2 is simple and intuitive, as it directly links Y x and Y x ′ for each individual. To clarify the relationship between Assumption 4.2 introduced by this work and Assumptions 3.1-3.2 from previous work, we present Proposition 4.4 below.

Proposition 4.4. The proposed Assumption 4.2 is strictly weaker than Assumptions 3.1-3.2.

̸

Proposition 4.4 is intuitive, as correlation (Assumption 4.2) does not necessarily imply identity (Assumption 3.1). To illustrate, consider a SCM with X ∈ { 0 , 1 } , Y 1 = Z + U 1 , Y 0 = Z/ 2 + U 0 , U 1 = U 3 0 . In this case, ρ ( Y 0 , Y 1 | Z ) = 1 , but U 1 = U 0 . Nevertheless, Assumption 4.2 is only slightly weaker than Assumptions 3.1-3.2 by allowing U x ′ = U x . Specifically, we can show that if U x is a strictly monotone increasing function of U x ′ , Assumption 4.2 is equivalent to Assumption 3.2, see Appendix A for proofs.

4.2 Further Relaxation of Strict Monotonicity

In Definition 4.1, we ignore ties for simplicity. However, when the outcome Y is discrete or continuous variables with tied observations, ρ ( Y x , Y x ′ ) will always be less than 1. To accommodate such cases, we introduce a modified version of the Kendall rank correlation coefficient given below. Definition 4.5 (Kendall [39]) . Let ( x 1 , y 1 ) , ..., ( x n , y n ) be the observations of two random variables ( X,Y ) , the modified Kendall rank correlation coefficient is define as

where T x is the number of tied pairs in { x 1 , ..., x n } and T y is the number of tied pairs in { y 1 , ..., y n } . We define ˜ ρ ( X,Y ) = 1 , if ˜ ρ n ( X,Y ) = 1 for all integers n ≥ 2 .

Compared with Definition 4.1, one can see that ˜ ρ ( X,Y ) adjusts ρ ( X,Y ) by eliminating the ties in the denominator, and ˜ ρ ( X,Y ) reduces to ρ ( X,Y ) if there are no ties.

Assumption 4.6 (Rank Preservation) . ˜ ρ ( Y x , Y x ′ | Z ) = 1 .

Assumption 4.6 is less restrictive than Assumption 4.2 as it accommodates broader data types of Y . To illustrate, consider a dataset with four individuals where the true values of ( Y x , Y x ′ ) are (1 , 1) , (2 , 1 . 5) , (2 , 1 . 5) , (3 , 2 . 5) . In this scenario, ∑ 1 ≤ i<j ≤ n sign (( y i,x -y j,x )( y i,x ′ -y j,x ′ ) = 5 , T Y x = 1 , T Y x ′ = 1 , resulting in ρ ( Y x , Y x ′ ) = 5 / 6 and ˜ ρ ( Y x , Y x ′ ) = 5 / ( √ 6 -1 · √ 6 -1) = 1 .

Assumption 4.6 also guarantees the identifiability of y x ′ .

Proposition 4.7. Under Assumption 4.6, the conclusion in Proposition 4.3 also holds.

5 Counterfactual Learning

We propose a novel estimation method for counterfactual inference. Suppose that { ( x k , z k , y k ) : k = 1 , ..., N } is a sample consisting of N realizations of random variables ( X,Z,Y ) . For an individual, given its evidence ( X = x, Z = z, Y = y ) , we aim to estimate its counterfactual outcome y x ′ , i.e., the realization of Y x ′ for this individual.

5.1 Rationale and Limitations of Quantile Regression

For estimating y x ′ , Xie et al. [19] formulate it as the following bi-level optimization problem

̸

where l τ ( ξ ) = τξ · I ( ξ ≥ 0) + ( τ -1) ξ · I ( ξ < 0) is the check function [40], the upper level optimization is to estimate τ ∗ , the quantile of y in the distribution P ( Y | X = x, Z = z ) , and the lower level optimization is to estimate the conditional quantile function q ( x, z ; τ ) ≜ inf y { y : P ( Y ≤ y | X = x, Z = z ) ≥ τ } for a given τ . Then y x ′ can be estimated using q ( x ′ , z ; τ ∗ ) .

We define two conditional quantile regression functions,

By Eq. (2), y x ′ can be expressed as q x ′ ( z ; τ ∗ ) with τ ∗ being the quantile of y in the distribution of P ( Y x | Z = z ) , i.e., P ( Y x ≤ y | Z = z ) = τ ∗ . Lemma 5.1 (see Appendix B for proofs) shows the rationale behind employing the check function as the loss to estimate conditional quantiles.

Lemma 5.1. We have that

There are two major concerns with the estimation method of [19]. First, it only fits a single quantile regression model for q ( X,Z ; τ ) to obtain estimates of q x ( Z ; τ ) and q x ′ ( Z ; τ ) . When the two conditional quantile functions q x ( Z ; τ ) and q x ′ ( Z ; τ ) originate from different models, this method may yield inaccurate estimates. Second, it explicitly requires estimating the quantile τ ∗ for each individual before estimating the counterfactual outcome y x ′ .

Inspired by [41], a simple improvement is to estimate q x ( z ; τ ) and q x ′ ( z ; τ ) separately. For example, for estimating q x ( z ; τ ) , the associated loss function is given as

where p x ( z ) = P ( X = x | Z = z ) is the propensity score, ˆ p x ( z ) is its estimate. Likewise, we could define R x ′ ( f, τ ) by replacing x with x ′ . Then the estimation procedure for y x ′ involves four steps: (1) estimating p x ( z ) ; (2) estimating q x ( z ; τ ) by minimizing R x ( f, τ ) for a range of candidate values of τ ; (3) identifying the τ ∗ in the candidate set of τ , that corresponds to the quantile of y in the distribution P ( Y | X = x, Z = z ) ; (4) estimating y x ′ using q x ′ ( z ; τ ∗ ) , where q x ′ ( z ; τ ∗ ) is obtained by minimizing R x ′ ( f, τ ∗ ) . Despite this four-step estimation method that allows q x ( Z ; τ ) and q x ′ ( Z ; τ ) to come from different models, it still needs to estimate a different τ ∗ for each individual.

5.2 Enhanced Counterfactual Learning Method

To address the limitations mentioned above in directly applying quantile regression and improve estimation accuracy, we propose a novel loss that produces an unbiased estimator of y x ′ for the individual with evidence ( X = x, Z = z, Y = y ) . The proposed ideal loss is constructed as

which is a function of t and the expectation operator is taken on the random variable of ( Y x , Y x ′ ) given Z = z . The proposed estimation method is based on Theorem 5.2.

Theorem 5.2 (Validity of the Proposed Ideal Loss) . The loss R x ′ ( t | x, z, y ) is convex with respect to t and is minimized uniquely at t ∗ , where t ∗ is the solution satisfying

Theorem 5.2 (see Appendix B for proofs) implies that given the evidence ( X = x, Z = z, Y = y ) for an individual, the counterfactual outcome y x ′ satisfies y x ′ = arg min t R x ′ ( t | x, z, y ) under Assumption 4.6. Importantly , the loss R x ′ ( t | x, z, y ) neither estimates the SCM a priori, nor restricts q x ( z ; τ ) and q x ′ ( z ; τ ) stem from the same model, and it does not need to estimate a different quantile value for each individual explicitly.

To optimize the ideal loss R x ′ ( t ; x, z, y ) , we first need to estimate it, which presents two significant challenges: (1) R x ′ ( t | x, z, y ) involves both Y x and Y x ′ , but for each unit, we only observe one of them; (2) The terms E [ | Y x ′ -t | ∣ ∣ Z = z ] and E [ sign ( Y x -y ) ∣ ∣ Z = z ] in R x ′ ( t | x, z, y ) is conditioned on Z = z , and when Z is a continuous variable with infinite possible values, it cannot be estimated

by simply splitting the data based on Z . We employ inverse propensity score and kernel smoothing techniques to overcome these two challenges. Specifically, we propose a kernel-smoothing-based estimator for the ideal loss, which is given as

where h is a bandwidth/smoothing parameter, K h ( u ) = K ( u/h ) /h , and K ( · ) is a symmetric kernel function [42, 43, 44] that satisfies ∫ K ( u ) du = 1 and ∫ uK ( u ) du = 1 , such as Epanechnikov kernel K ( u ) = 3(1 -u 2 ) · I ( | u | ≤ 1) / 4 and Gaussian kernel K ( u ) = exp( -u 2 / 2) / √ 2 π for u ∈ R . Then we can estimate y x ′ by minimizing ˆ R x ′ ( t ; x, z, y ) directly.

Proposition 5.3 (Consistency) . If h → 0 as N → ∞ , ˆ p x ( z ) and ˆ p x ′ ( z ) are consistent estimates of p x ( z ) and p x ′ ( z ) , and the density function of Z is differentiable, then ˆ R x ′ ( t | x, z, y ) converges to R x ′ ( t | x, z, y ) in probability.

Proposition 5.3 (see Appendix B for proofs) indicates that ˆ R x ′ ( t | x, z, y ) is a consistent estimator of R x ′ ( t | x, z, y ) , demonstrating the validity of the estimated ideal loss. The loss ˆ R x ′ ( t | x, z, y ) is applicable only for discrete treatments due to the terms I ( x k = x ′ ) and I ( x k = x ) . However, it can be easily extended to continuous treatments, as detailed in Appendix C.

It is well known that kernel-smoothing-based estimators suffer from scalability issues in highdimensional settings (i.e., the high-dimensional covariates) [42]. Therefore, for implementation, we avoid applying kernel functions directly to the original covariates. Instead, we first learn a lowdimensional representation of the covariates, and then apply the kernel-smoothing-based estimator to this representation to learn the counterfactual outcomes.

6 Experiments

6.1 Synthetic Experiment

Simulation Process. We generate the synthetic dataset by the following process. First, we sample the covariate Z ∼ N (0 , I m ) and the treatment X ∼ Bern( π ( Z )) , where Bern( · ) is the Bernoulli

(a) Sim-10 (In-sample)

(b) Sim-10 (Out-sample)

(c) Sim-40 (In-sample)

(d) Sim-40 (Out-sample)

10

10

Ours

6

Ours

Ours

Quantile-Reg

Quantile-Reg

8

Quantile-Reg

8

Quantile-Reg

EPEHE

4

4

6

6

4

4

2

2

2

0

1.5

2.0

2.5

3.0

1.5

2.0

2.5

3.0

1.5

2.0

2.5

3.0

Q

1.5

2.0

2.5

3.0

a:HeterogeneityCoefficient

α:HeterogeneityCoefficient

α:HeterogeneityCoefficient

α:HeterogeneityCoefficient

Epanechnikov

ESCFR

Quantile-Reg

3

3H3d3

2

1

1.0

3.0

5.0

7.0

9.0

h:KernelBandwidth

(a) Sim-10 (In-sample)

Epanechnikov

ESCFR

Quantile-Reg

2

1

1.0

3.0

5.0

7.0

9.0

h:KernelBandwidth

(b) Sim-10 (Out-sample)

Epanechnikov

ESCFR

Quantile-Reg

2

1.0

2.0

3.0

4.0

5.0

h:KernelBandwidth

(c) Sim-40 (In-sample)

Epanechnikov

ESCFR

Quantile-Reg

2

1.0

2.0

3.0

4.0

5.0

(d) Sim-40 (Out-sample)

distribution with probability π ( Z ) = P ( X = 1 | Z ) = σ ( W x · Z ) , σ ( · ) is the sigmoid function, and W x ∼ Unif( -1 , 1) m , Unif( · ) is the uniform distribution. Then, we sample the noise U 0 ∼ N (0 , 1) and U 1 = α · U 0 to consider the heterogeneity of the exogenous variables, where α is the hyper-parameter to control the heterogeneity degree. Finally, we simulate Y 1 = W y · Z + U 1 and Y 0 = W y · Z/α + U 0 with W y ∼ N (0 , I m ) . We generate 10,000 samples with 63/27/10 train/validation/test split and vary m ∈ { 5 , 10 , 20 , 40 } in our synthetic experiment.

Baselines and Evaluation Metrics. The competing baselines includes: T-learner [45], X-learner [45], BNN [46], TARNet [47], CFRNet [47], CEVAE [48], DragonNet [49], DeRCFR [50], DESCN [51], ESCFR [52], CFQP [18], and Quantile-Reg [19]. We evaluate the individual treatment effect estimation using the individual level Precision in Estimation of Heterogeneous Effects (PEHE):

where ˆ Y i (1) and ˆ Y i (0) are the predicted values for the corresponding true potential outcomes of unit i . It is noteworthy that ϵ PEHE is tailored for individual-level evaluation and counterfactual estimation, which is different from the common metric [47] given by 1 N ∑ N i =1 [(ˆ µ 1 ( X i ) -ˆ µ 0 ( X i )) -( µ 1 ( X i ) -µ 0 ( X i ))] 2 , where µ 1 ( X i ) -µ 0 ( X i ) := E [ Y (1) | X ] -E [ Y (0) | X ] are the true CATE, and ˆ µ 1 ( X i ) -ˆ µ 0 ( X i ) is its estimate. Both in-sample and out-of-sample performances are reported in our experiments. In addition, we run all experiments on the Google Colab platform. For the representation model, we use the MLP for the base model and tune the layers in { 1 , 2 , 3 } . In addition, we adopt the logistic regression model as the propensity model. We tune the learning rate in { 0 . 001 , 0 . 005 , 0 . 01 , 0 . 05 , 0 . 1 } . For the kernel choice, we select the kernel function between the Gaussian kernel function and the Epanechnikov kernel function, and tune the bandwidth in { 1 , 3 , 5 , 7 , 9 } .

Performance Analysis. The results of estimation performance are shown in Table 1. Our method stably outperforms all baselines with varying covariate dimensions m , demonstrating the effectiveness of the proposed method. In addition, we investigate our method performance with violated assumptions on rank and uncorrelated covariates. Specifically, we modified the data generation process to explore the performance of our method under correlated covariates by sampling the covariate Z ∼ N (0 , Σ m ) , where the ρ ij in Σ m is max(0 . 01 , ρ | i -j | ) . The results are shown in Table 2. The results show that our method still outperforms the baseline methods. Moreover, we further explore the effect of heterogeneity degrees on the performance of the proposed method, as shown in Figure 1, where one can see that as the heterogeneity degree increases, our method stably outperforms the Quantile-Reg in terms of PEHE. Finally, we examine the effect of different kernels and bandwidths,

as shown in Figure 2, our method stably outperforms the Quantile-Reg and ESCFR methods with different kernels and bandwidths.

6.2 Real-World Experiment

Dataset and Preprocessing. Following previous studies [47, 48, 53, 54], we conduct experiments on semi-synthetic dataset IHDP and real-world dataset JOBS. The IHDP dataset [55] is constructed from the Infant Health and Development Program (IHDP) with 747 individuals and 25 covariates. The JOBS dataset [56] is based on the National Supported Work program with 3,212 individuals and 17 covariates. We follow [47] to split the data into training/validation/testing set with ratios 63/27/10 and 56/24/20 with 100 and 10 repeated times on the IHDP and the JOBS datasets, respectively.

Evaluation Metrics. Following previous studies [47, 48, 54], besides ϵ PEHE, we also use the absolute error in Average Treatment Effect (ATE) for evaluation, which is defined as ϵ ATE = 1 N | ∑ N i =1 (( ˆ Y i (1) -ˆ Y i (0)) -( Y i (1) -Y i (0))) | . Weuse √ ϵ PEHE and ϵ ATE to evaluate performance on the IHDP dataset. For the JOBS dataset, since one of the potential outcomes is not available, we evaluate the performance using the absolute error in Average Treatment effect on the Treated (ATT) as ϵ ATT = | ATT -1 | T | ∑ i ∈ T ( ˆ Y i (1) -ˆ Y i (0) | with ATT = | 1 | T | ∑ i ∈ T Y i -1 | C ∩ E | ∑ i ∈ C ∩ E Y i | . We also use the policy risk R Pol = 1 -( E [ Y (1) | ˆ Y (1) -ˆ Y (0) > 0 , X = 1] · P ( ˆ Y (1) -ˆ Y (0) > 0)+ E [ Y (0) | ˆ Y (1) -ˆ Y (0) ≤ 0 , X = 0] · P ( ˆ Y (1) -ˆ Y (0) ≤ 0)) , where T, C, E are the indexes of treatment sample set, control sample set, and randomized sample set, respectively.

Performance Comparison. The experiment results are shown in Table 3. Similar to the synthetic experiment, the Quantile-Reg method still achieves the most competitive performance compared to the other baselines. Our method stably outperforms all the baselines on both the semi-synthetic dataset IHDP and the real-world dataset JOBS, especially in the out-sample scenario. This provides the empirical evidence of the effectiveness of our method.

7 Related Work

Conditional Average Treatment Effect (CATE). CATE also referred to as heterogeneous treatment effect, represents the average treatment effects on subgroups categorized by covariate values, and plays a central role in areas such as precision medicine [57, 58, 59, 60], policy learning [61, 62], and recommender systems [63, 64, 65]. Benefiting from recent advances in machine learning, many methods have been proposed for estimating CATE, including matching methods [66, 67, 54, 68], tree-based methods [69, 70], representation learning methods [46, 47, 49, 50, 52], and generative methods [48, 53]. Unlike the existing work devoted to estimating CATE at the intervention level for subgroups, our work focuses on counterfactual inference at the more challenging and fine-grained individual level.

Counterfactual Inference. Counterfactual inference involves the identification and estimation of counterfactual outcomes. For identification, [71] provided an algorithm leveraging counterfactual graphs to identify counterfactual queries. In addition, [72] discussed the identifiability of nested

counterfactuals within a given causal graph. More relevant to our work, [19] and [24] studied the identifiability assumptions in the setting of backdoor criterion under homogeneity and strict monotonicity assumptions. Several methods focus on determining its bounds with less stringent assumptions, such as [10, 13, 73, 74]. In addition, [11] proposed a method for identifying the joint distribution of potential outcomes using multiple experimental datasets.

For estimation, [8] introduced a three-step procedure for counterfactual inference. Many machine learning methods estimate counterfactual outcomes in this framework, such as [6, 18, 24, 25, 75, 76, 77]. Recently, [19] employed quantile regression to estimate the counterfactual outcomes, effectively circumventing the need for SCM estimation. In our work, we extend the above methods in both identification and estimation. Recently, counterfactual inference methods have been extensively applied across various application scenarios, such as counterfactual fairness [78, 79, 80, 81, 82, 83], policy evaluation and improvement [14, 84, 85, 86], reinforcement learning [24, 87, 88, 89, 90, 91, 92], imitation learning [93, 94], counterfactual generation [76, 95, 96, 97], counterfactual explanation [98, 99, 100, 101, 102], counterfactual harm [13, 14, 103, 15], physical audiovisual commonsense reasoning [104], interpretable time series prediction [105], classification and detection in medical imaging [106], data valuation [107], etc. Therefore, developing novel counterfactual inference methods holds significant practical implications.

8 Conclusion

This work addresses the fundamental challenge of counterfactual inference in the absence of a known SCM and under heterogeneous endogenous variables. We first introduce the rank preservation assumption to identify counterfactual outcomes, showing that it is slightly weaker than the homogeneity and monotonicity assumptions. Then, we propose a novel ideal loss for unbiased learning of counterfactual outcomes and develop a kernel-based estimator for practical implementation. The convexity of the ideal loss and the unbiased nature of the proposed estimator contribute to the robustness and reliability of our method. A potential limitation arises when the propensity score is extremely small in certain data sparsity scenarios, which may cause instability in the estimation method. Further investigation is warranted to address and overcome this challenge.

Acknowledgments and Disclosure of Funding

This research was supported by the National Natural Science Foundation of China (No. 12301370, 623B2002), the BTBU Digital Business Platform Project by BMEC, the Beijing Key Laboratory of Applied Statistics and Digital Regulation, Academy for Interdisciplinary Studies at BTBU, the BTBU Research Foundation for Youth Scholars (No. BRFYS2025), NSF Award No. 2229881, AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program, and the Al Deira Causal Education project.

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A Proofs in Sections 3 and 4

One can show Lemma 3.3 by a similar argument of the proof of Theorem 1 in [19]. For the sake of self-containedness, we provide a novel proof of it.

Lemma 3.3 Under Assumptions 3.1-3.2, y x ′ is identifiable.

Proof of Lemma 3.3. First, the distributions P ( Y x | Z = z ) and P ( Y x ′ | Z = z ) can be identified as P ( Y | X = x, Z = z ) and P ( Y | X = x ′ , Z = z ) , respectively, by the backdoor criterion (i.e., ( Y x , Y x ′ ) ⊥ ⊥ X | Z ) of the setting.

Then, according to the model (1), we can equivalently write

and Y and U X in model (1) can be expressed as Y = ∑ x ∈X I ( X = x ) · Y x and U X = ∑ x ∈X I ( X = x ) · U x , where X is the support set of X and I ( · ) is an indicator function. Assumption 3.1 implies that U X = U x = U x ′ conditional on Z , i.e., Y x = f Y ( x, z, U X ) , Y x ′ = f Y ( x ′ , z, U X ) .

Finally, for the individual with observation ( X = x, Z = z, Y = y ) , we denote ( y x , y x ′ ) as the true values of ( Y x , Y x ′ ) for this individual. For this individual, we can identify the quantile of y x in the distribution of P ( Y x | Z = z ) = P ( Y | X = x, Z = z ) , denoted by τ ∗ . Let u τ ∗ be the true value of U X for this individual, it is the τ ∗ -quantile in the distribution P ( U X | Z = z ) , then we have

which implies that for this individual, its rankings of y x and y x ′ are the same in the distributions of P ( Y x | Z = z ) and P ( Y x ′ | Z = z ) , resepcctively. Thus, y x ′ is identified as the τ ∗ -quantile of the distribution P ( Y x ′ | Z = z ) = P ( Y | X = x ′ , Z = z ) .

Proposition 4.3 Under Assumption 4.2, y x ′ is identified as the τ ∗ -th quantile of P ( Y | X = x ′ , Z = z ) , where τ ∗ is the quantile of y in the distribution of P ( Y | X = x, Z = z ) .

Proof of Proposition 4.3. For the individual with observation ( X = x, Z = z, Y = y ) , we denote ( y x , y x ′ ) as the true values of ( Y x , Y x ′ ). Assumption 4.2 implies that for this individual, its rankings of y x and y x ′ are the same in the distributions of P ( Y x | Z = z ) and P ( Y x ′ | Z = z ) , respectively. Therefore,

Since y x = y is observed and the distributions P ( Y x | Z = z ) and P ( Y x ′ | Z = z ) can be identified as P ( Y | X = x, Z = z ) and P ( Y | X = x ′ , Z = z ) , respectively, by the backdoor criterion (i.e., ( Y x , Y x ′ ) ⊥ ⊥ X | Z ), we can identify the quantile of y x in the distribution of P ( Y | X = x, Z = z ) , denoted by τ ∗ . Then

which yields that θ is identified as the τ ∗ -quantile of P ( Y | X = x ′ , Z = z ) .

The following Proposition 4.4 ∗ serves as a complement to Proposition 4.4.

Proposition 4.4 ∗ Under Assumption 3.1, or more generally, if U x is a strictly monotone increasing function of U x ′ , Assumption 4.2 is equivalent to Assumption 3.2.

Proof of Proposition 4.4. According to the model (1), we can equivalently write

Suppose that U x is a strictly monotone increasing function of U x ′ (Assumption 3.1, i.e., U x = U x ′ , is a special case of it). Under this condition, we next prove sufficiency and necessity, respectively.

First, we show that Assumption 3.2 implies Assumption 4.2. If Assumption 3.2 holds, then Y x is a strictly monotonic function of U x , and Y x ′ is a strictly monotonic function of U x ′ . Since U x is a strictly monotone increasing function of U x ′ , then Y x is a strictly increasing monotonic function of Y x ′ , which leads to Assumption 4.2.

Second, we show that Assumption 4.2 implies Assumption 3.2. If Assumption 4.2 holds, then given Z = z , Y x is a strictly increasing function of Y x ′ . When U x is a strictly monotone increasing function of U x ′ and note that

which implies that f Y is a strictly monotonic function of U X , i.e., Assumption 3.2 holds. This finishes the proof.

Proposition 4.7 Under Assumption 4.6, the conclusion in Proposition 4.3 also holds.

Proof of Proposition 4.7. This can be shown through a proof analogous to that of Proposition 4.3.

B Proofs in Section 5

Recall that l ( ξ ) = τξ · I ( ξ ≥ 0) + ( τ -1) ξ · I ( ξ < 0)

Lemma 5.1 We have that

(i) q x ( Z ; τ ) = arg min f E [ l τ ( Y x -f ( Z ))] for any given x ;

Proof of Lemma 5.1. We prove q x ( Z ; τ ) = arg min f E [ l τ ( Y x -f ( Z ))] , and q ( X,Z ; τ ) = arg min f E [ l τ ( Y -f ( X,Z ))] can be derived by an exactly similar manner. We write

To obtain the conclusion, note that l τ ( Y x -f ( Z )) is always positive, it suffices to show that

for any given Z = z . Next, we focus on analyzing the term E [ l τ ( Y x -f ( Z )) | Z = z ] . Given Z = z , f ( Z ) is a constant and we denote it by c , then

where g ( y x | z ) denotes the probability density function of Y x given Z = z .

Since the check function is a convex function, differentiating E [ l τ ( Y x -c ) | Z = z ] with respect to c and setting the derivative to zero will yield the solution for the minimum

Then let ∂ ∂c E [ l τ ( Y x -c ) | Z = z ] = 0 leads to that

that is, c = q x ( z ; τ ) . This completes the proof of Proposition 5.1.

Theorem 5.2 (Validity of the Proposed Ideal Loss). The loss R x ′ ( t ; x, z, y ) is minimized uniquely at t ∗ , where t ∗ is the solution satisfying

Proof of Theorem 5.2. Recall that

Let g ( y x | z ) be the probability density function of Y x given Z = z . By calculation,

and

we have

Since

R x ′ ( t | x, z, y ) is a convex function with respect to t . Letting ∂ ∂t R x ′ ( t | x, z, y ) = 0 yields that

That is, R x ′ ( t | x, z, y ) attains its minimum at t = q x ′ ( z ; τ ∗ ) , where τ ∗ is the quantile of y in the distribution P ( Y x | Z = z ) .

Proposition 5.3. If h → 0 as N → ∞ , ˆ p x ( z ) and ˆ p x ′ ( z ) are consistent estimates of p x ( z ) and p x ′ ( z ) , and the density function of Z is differentiable, then

where P - → means convergence in probability.

Proof of Proposition 5.3. For analyzing the theoretical properties of ˆ R x ′ ( t ; x, z, y ) , we rewritten ˆ R x ′ ( t ; x, z, y ) as

where the capital letters denote random variables and lowercase letters denote their realizations. This is slightly different from that used in the main text.

When ˆ p x ( z ) and ˆ p x ′ ( z ) are consistent estimates of p x ( z ) and p x ′ ( z ) , to show the conclusion, it is sufficient to prove that

We prove equation (5) only, as equation (6) can be addressed similarly.

Note that

we analyze the denominator and numerator on the right side of the equation separately. For the denominator, it is an average of N independent random variables and converges to its expectation E [ K h ( Z k -z )] almost surely. Let g ( z k ) be the probability density function of Z k , and g (1) ( z k ) is its first derivative. Since

when h → 0 as N →∞ , the denominator converges to g ( z ) in probability.

Next, we focus on dealing with the numerator, which also converges to its expectation.

Define m ( Z ) = E [ | Y x ′ -t | ∣ ∣ Z ] and m (1) ( Z ) is its first derivative, then the right side of equation (5) is m ( z ) , and

Thus, when h → 0 as N →∞ , the numerator converges to g ( z ) in probability.

Combining equations (7), (8), and (9) yields the equality (5). This completes the proof.

C Extension to Continuous Outcome

When the treatment is continuous, we can estimate the ideal loss with the following estimator

which is a smoothed version of the estimator

defined in Section 5. In addition, by a proof similar to that of Proposition 5.3, we also can show that ˜ R x ′ ( t ; x, z, y ) P - → R x ′ ( t ; x, z, y ) .

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new approach for individual-level counterfactual inference that does not require a known structural causal model (SCM). The authors introduce a \"rank preservation\" assumption to identify the counterfactual outcome. Building on this, they propose a convex ideal loss function and an unbiased kernel-based estimator. Theoretical analysis shows their assumption is no stronger than those used in prior work, and experiments on semi-synthetic and real-world data demonstrate the method's effectiveness.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n- The paper is well-written, and its arguments are clear.\n- A significant advantage of the proposed method is its ability to perform counterfactual inference without relying on a known SCM.\n- The approach is supported by theoretical results that formally establish its properties.\n\n**Weaknesses:**\n- The paper's core contribution relies on the rank preservation assumption. While the introduction briefly explains it, it lacks a discussion or illustrative examples to motivate why this assumption is appropriate and practical in real-world scenarios. Adding this would significantly strengthen the paper's introduction.\n- There are some minor typographical and organizational issues that should be addressed:\n    - On line 184, \"a SCM\" should be corrected to \"an SCM\".\n    - Colons appear to be missing in several places where definitions are introduced (e.g., lines 94, 159, 194).\n    - The placement of Section 6 after Table 1 disrupts the flow; it would be more logical to place it before the table.\n\nQUESTIONS:\n- The proposed loss function is a key component of the method. Could the authors provide more intuition and discussion behind its design? \n- The paper claims that the method can be extended to continuous treatments and that Proposition 5.3 remains valid in this setting. However, a formal proof for this claim is not provided. Could the authors please discuss on this point?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

comments: SUMMARY:
The authors introduce a novel condition (assumption 4.6) for counterfactual identifiability in RCTs that is a strict relaxation of the common "homogeneity + monotonicity" requirement. Roughly, this condition demands that "for any two individuals sharing the same observable pre-treatment features $Z = z$, if one individual fares better under any treatment $x$ they fare better under all treatments". The authors develop a consistent estimator for counterfactual learning under this condition.

STRENGTHS AND WEAKNESSES:
The authors' condition cuts explicit dependence on $U_X$, and, in my view, directly captures the intuition used when justifying the monotonicity condition. This is a significant improvement over a classical result and the authors clearly communicate the advantage of their approach through examples and experiments.

As for weaknesses, I have a few comments: 
- I do not agree with line 169 stating arbitrary heteroskedastic models satisfy the condition. If $h(X, Z)$ changes sign between $X = x$ and $X = x'$ for some $Z = z$ then the assumption is violated since we have $h(x, z) h(x', z) < 0$. This is an important distinction since the authors' assumption does not allow a treatment to be beneficial to some individuals with the same $Z = z$ and harmful to others. 
- The authors should make the non-technical statement of their assumption explicit, rather than describing it in solely in terms of the rank coefficient.

QUESTIONS:
The authors did not benchmark their estimator in an example where "homogeneity + monotonicity" is satisfied. How would their estimation strategy would compare to the classical quantile-based estimator in this setting?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-06-24

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an approach for individual-level counterfactual inference without requiring structural causal models (SCMs). The approach is based on the introduction of the Rank Preservation Assumption which posits that confounders account for all the correlation between potential outcomes, with correlation defined in terms of Kendall’s rank coefficient. Under this assumption, the authors derive a convex loss function and an unbiased estimator for the counterfactual outcome. They  propose a kernel-based estimator and demonstrate its performance through theoretical analysis and empirical evaluation on both semi-synthetic and real-world datasets.\nCompared to existing methods, which typically require estimating an SCM along with a monotonicity assumption (i.e., the potential outcome is a monotonic function of unobserved variables) or assume equality of conditional quantile functions, this paper’s approach relies on  weaker identifiability conditions  and simpler estimation procedure.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper addresses the complex problem of counterfactual inference problem by introducing the rank preservation assumption. The authors prove that this assumption is strictly weaker than homogeneity and strict monotonicity, which are commonly used in prior work.  Based on this assumption, the authors derive a theoretically loss function that avoids SCM estimation and bi-level optimization, making their method  practically implementable. The proposed estimator also demonstrates strong empirical performance across multiple benchmarks and is backed by consistency, unbiasedness, and convergence guarantees.\n\nDespite these contributions, the paper could benefit from more details on the assumption, in particular a more intuitive explanation of the rank preservation assumption (Assumption 4.2). While the definition is mathematically rigorous, its practical implications may not be immediately obvious. A brief paragraph and example are provided, but this may not be sufficient to help readers build strong intuition. Including additional pedagogical material to discuss in depth the different assumptions would significantly improve accessibility.\n\nIn addition,  the exposition of the paper and in particular in Section 5,  the derivation and approximation of the ideal loss, could be streamlined. The writing is mathematically dense and is difficult to follow. Distinguishing between the intuition, formal derivation, and implementation steps could enhance clarity.\n\nQUESTIONS:\nThe authors claim that Assumption 4.2 holds for common models such as those with additive noise  $Y = g(X, Z) + U$ or heteroscedastic noise $Y = g(X, Z) + h(X, Z)U$. Could you elaborate on why this is the case?\n\n- How can practitioners assess or empirically validate the rank preservation assumption in real-world applications?\n\n- In the experiments, the proposed method appears to outperform all competitors consistently. This is somewhat surprising, as methods usually have specific regimes where they excel and others where they do not. Could you provide some intuition for these results? Have you explored other scenarios or settings? Did you use simple methods for estimating nuisance functions for the baselines? This is particularly surprising in settings where the assumption is violated, such as in Table 4 of the appendix, where some competitive methods are missing. Also, in non-simulated settings, how do you choose or tune the bandwidth parameter?\n\n- The reliance on kernel-based estimation could limit scalability in high-dimensional or large-sample settings. How do you handle scenarios with many continuous confounders? In particular, in your experiments, the estimator appears quite stable even when the covariate dimension reaches 40; could you clarify why?\n\nminor remarks:\n- Finally, the abstract states that rank preservation is “not stronger” than homogeneity and strict monotonicity, whereas the main text (e.g., Proposition 4.1) shows that it is in fact strictly weaker. Clarifying this language in the abstract would help avoid potential confusion for readers.\n- In line 120, the references cited do not seem directly relevant to the point made. Highlighting heterogeneity does not necessarily imply that one must estimate individual-level effects rather than conditional effects. One could, for instance, use an extreme form of stratification with very small group sizes instead.\n\nLIMITATIONS:\nThe paper provides limited guidance on how to assess the plausibility of the rank preservation assumption in practice. As it is untestable from observational data, incorporating  a discussion or any diagnostic tools  would strengthen the method’s practical value.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses individual-level counterfactual outcome estimation by introducing a new identifiability assumption called rank preservation, claiming it is weaker than the traditional homogeneity and strict monotonicity assumptions. Building on this, the authors propose an ideal loss function for unbiased counterfactual estimation and develop a kernel-based estimator for empirical implementation. Theoretical results, including identifiability proofs, unbiasedness properties, and consistency analysis, are provided. The paper also critiques quantile regression-based approaches and demonstrates empirical results on semi-synthetic and real-world datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The paper addresses a challenging problem in counterfactual inference where full knowledge of the SCM is unavailable, a realistic scenario in many applications.\n- Introducing the rank preservation assumption provides an alternative path to identifiability that appears, theoretically, to relax previous requirements like strict monotonicity.\n- The proposed ideal loss circumvents limitations of previous quantile regression methods, and the bi-level optimization.\n- The authors provide theoretical analysis for the proposed estimator.\n- Experimental results demonstrate consistent improvements over baseline methods.\n\n**Weaknesses**\n\n-  All identifiability assumptions (including this) are restrictive\n\nWhile the authors rightly critique homogeneity and strict monotonicity assumptions for being strong and hard to verify, the proposed rank preservation assumption is similarly unverifiable in practice. Whether ranks of factual and counterfactual outcomes align across individuals is fundamentally untestable, especially when counterfactuals are unobserved. The assumption may be mathematically weaker, but its practical plausibility remains speculative.\n\n*Suggestion*: The paper would benefit from a more candid discussion acknowledging that all identifiability assumptions in counterfactual inference, including rank preservation, impose unverifiable structure on unobserved quantities. Comparing empirical consequences of assumption violations (as partially done in Appendix E) is a good direction that could be expanded.\n\n- Reliance on propensity score estimation and kernel choices\n\nThe method depends on accurate estimation of propensity scores and bandwidth parameters for kernel smoothing. Theoretical sections highlight this, but practical guidelines for tuning and robustness checks are minimal.\n\n*Suggestion*: More systematic sensitivity analyses, especially in high-dimensional settings, would clarify the method's reliability. Exploring alternatives to kernel smoothing for scalability would also improve practical value.\n\nQUESTIONS:\nsee suggestions above\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}