Debiased Causal Tree: Heterogeneous Treatment Effects Estimation with Unmeasured Confounding
Authors: Caizhi Tang
Huiyuan Wang
Xinyu Li
Qing Cui
Ya-Lin Zhang
Feng Zhu
Longfei Li
JUN ZHOU
Linbo Jiang
Venue: NeurIPS 2022 Accept (2022)
Keywords: Deconfounding
Heterogeneous treatment effects estimation
Confounding entropy
Boosting
Causal tree

Abstract: Unmeasured confounding poses a significant threat to the validity of causal inference. Despite that various ad hoc methods are developed to remove confounding effects, they are subject to certain fairly strong assumptions. In this work, we consider the estimation of conditional causal effects in the presence of unmeasured confounding using observational data and historical controls. Under an interpretable transportability condition, we prove the partial identifiability of conditional average treatment effect on the treated group (CATT). For tree-based models, a new notion, \emph{confounding entropy}, is proposed to measure the discrepancy introduced by unobserved confounders between the conditional outcome distribution of the treated and control groups. The confounding entropy generalizes conventional confounding bias, and can be estimated effectively using historical controls. We develop a new method, debiased causal tree, whose splitting rule is to minimize the empirical risk regularized by the confounding entropy. Notably, our method integrates current observational data (for empirical risk) and their historical controls (for confounding entropy) harmoniously.  We highlight that, debiased causal tree can not only estimate CATT well in the presence of unmeasured confounding, but also is a robust estimator of conditional average treatment effect (CATE) against the imbalance of the treated and control populations when all confounders are observed. An extension of combining multiple debiased causal trees to further reduce biases by gradient boosting is considered. The computational feasibility and statistical power of our method are evidenced by simulations and a study of a credit card balance dataset.

Debiased Causal Tree: Heterogeneous Treatment Effects Estimation with Unmeasured Confounding
Caizhi Tang!*
Huiyuan Wang
Xinyu Li
Qing Cui!
Ya-Lin Zhang'
Feng Zhu!
Longfei Li
Jun Zhou
Linbo Jiang!
1 'Ant Group, School of Mathematical Sciences, Peking University {caizhi tcz, cuiqing eq, cq. lyn. 2on.,.yl,,. zyl,zhufeng zhu longyao .llo,11ff,i. JIf,jun.zhoujun zhoujun, xi xiaobo..... Bantgrouu com (huiyuan. wang, xinyu. 11] @pku. edu.
Abstract
Unmeasured confounding poses significant threat to the validity of causal infer- ence. Despite that various ad hoc methods are developed to remove confounding effects, they subject to certain fairly strong assumptions. In this work, consider the estimation of conditional causal effects in the presence of unmea- sured confounding using observational data and historical controls. Under interpretable transportability condition, we prove the partial identifiability of condi- tional average treatment effect the treated group (CATT). For tree-based models, a new notion, confounding entropy, is proposed to measure the discrepancy intro- duced by unobserved confounders between the conditional outcome distribution of the treated and control groups. The confounding entropy generalizes conventional confounding bias, and can be estimated effectively using historical controls. We develop a new method, debiased causal tree, whose splitting rule is to minimize the empirical risk regularized by the confounding entropy. Notably, our method integrates current observational data (for empirical risk) and their historical controls (for confounding entropy) harmoniously. We highlight that, debiased causal tree can not only estimate CATT well in the presence of unmeasured confounding, but also is a robust estimator of conditional average treatment effect (CATE) against the imbalance of the treated and control populations when all confounders are observed. An extension of combining multiple debiased causal trees to further reduce biases by gradient boosting is considered. The computational feasibility and statistical power our method are evidenced by simulations and study of a credit card balance dataset.
Introduction
Conditional average treatment effect (CATE), also known as heterogeneous treatment effect, can characterize the causal effect of certain treatments on the outcome over the whole population as function of measured covariates. Another closely related concept, conditional average treatment effect on the treated group (CATT), is more relevant when we focus on the effect of treatments for the sub- sub-population which actually exposed certain specific treatments; for instance, we are more interested in the effect on children of taking anti-depressants during pregnancy for the people who
"Equal contribution. t "Corrssonndigg author.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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actually took the drug (Boukhris et al. 2016). The interpretable and informative nature has made it great importance to estimate CATE and CATT from observational or experimental data in wide spectrum of applications, including medical decision-making and personalized medicine (Glass et al. 2013; Ohermeyer and Emanuel 2016), evaluating advertising and socioeconomic policies (Breslow
and Johnson 1993] Athey 2017 Gordon et al. 2019) and large-scale randomized experiments for
recommendations (Li et al. 2010 Gilotte et al. 2018 etc.
In the absence of unmeasured confounding, there exist lots of methods developed to estimate CATE
(CATT), including tree-based methods (Athey and Imbens 2016 Wager and Athey 2018 Athey
et al. 2019), , meta-learners (Künzel et 2019), R-learners (Ne and Wager] (021), doubly robust learners (Kennedy 2020), and so on. Basically, such an unconfoundedness condition hinges on an investigaaor's s ability to accurately measure all related covariates, which is questionable in practice.
especially when there exist covariate-dependent measurement errors (Wacholder 1995; Tekwe et al., 2019). biased treatment a allocations (Kalish and Begg 1985 Tran and Zheleva 2019) and batch effects (Gagnon-Bartsch and Speed 2012 Goh et al. 201 7). In the presence of unmeasured confounding,
however, the identifiability of CATE or CATT cannot be ensured in general, leading to failures of the aforementioned estimation procedures.
To control the effect introduced by confounders, various methods have been developed, e.g., sensitivity analyses (Rosenbaum and Rubin 1983), instrumental variable (IV) approaches (Angrist et al. 1996
Lin et al. 20155, and proximal causal learning (Kuroki and Pearl 2014] Miao et al. 2018 Shi
et al.] 2020j. Nonetheless, several limitations of these methods hinder their direct application in CATE/CATT estimation. For example, we cannot derive point estimation from sensitivity analyses,
and even valid IV's can only partially identify CATE in principle (Swanson et al., 2018). Although
ATE and ATT can be identified in proximal causal learning (Cui et al.] 2020), the assumption that the
measured covariates are correctly classified into three types of proxies requires prior knowledge.
Recently, a line of work proposed to combine randomized trials (Concato el al. (2000) and observa- tional studies adjust confounders, which typically requires transportability (Pearl and Bareinboim 2011) of randomized controlled trial to external observational studies (Rothwell) 2005). For ex- ample, Yang et al. (2020) proposed to use the confounding function which summarizes the impact of confounders to estimate heterogeneous treatment effects, where the treatment effect function is assumed to be transportable. For a comprehensive review, see Colnet et al. (2020). Problematically, many observational studies do not have related randomized trials due to practical and methodological criteria.
issues such as cost and restrictive inclusion/exclusic
In this work, we move further in the direction of data fusion, without requiring experimental data. Instead, we consider the case where covariates and the outcome are collected at multiple timestamps, before and after the treatment. We take two timestamps 1 E {41 (22 < (2), as an example, where the treatment is imposed at the time t2. Following the potential outcome framework (Rubin) 1978). denote by Y, = DY, + (1 - D) the observed outcomes at the time t, where Y:! is the potential outcome for the treatment D = d E (0. We remark that Y,, >r(0)) since the treatment does not take place at 4. Let X and U be time-invariant covariates and unmeasured confounders, respectively. Our work is motivated by the following identity
E(V,")) D DD=1XX = x) E(Y,"))
D = 0, = =x) Fly! - y(()) D = D=1,X=0
CATT
+ (0) (0)
1- = x) E (0) D= 0, X =
Vt
Confounding Blas,
which holds regardless of U. In general, the confounding bias (Bareinboim and Pearl 2012 Yang ee al. (020) b, does not vanish due to U. However, the smaller b, is, the closer CATT could be to the difference in conditional expectations. We then propose a tree-based method, which, at high level, aims to (i) find an optimal partition of the feature space such that the confounding bias is smallest within each small region, and (ii) estimate the left hand of [.. Despite that the confounding bias is unavailable in general, we can represent it with tree-based models using data collected at t since
Yr, = y,(*) and it is known which individual would receive the treatment. the optimal partition that leads to the smallest historical confounding bias changes "mildly" over time, i.e., it is transportable, the partition derived at time 4 also nearly optimal at 12. The two steps can proceed simultaneously,
and we calibrate and formalize the basic idea in Section 2

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Our contribution is summarized as follows: (i) we introduce a new notion, confounding entropy, to characterize the discrepancy between the conditional distributions of the treated and control groups that generalizes the confounding bias, and propose a tractable and scalable method, debiased causal tree, to estimate CATT from observational data before and after the treatment by developing a new criterion for node splitting, which minimizes both empirical risk and confounding entropy at the same time; (ii) to further reduce the bias, we integrate multiple causal trees by gradient boosting, which we refer to as Gradient Boosting Causal Tree (GBCT); (iii) under the unconfoundedness condition, show that GBCT is robust to imbalance of the treated and control populations and outperforms other compared methods, which is obvious in severely imbalanced cases.
2 Methodology
We first introduce the problem setup, related notation, and basic ideas behind our proposed method.
2.1 Problem Sctup
To begin with, we consider the case of binary treatment where = E 1) denotes the treatment status. Let y:()) E I denote the potential outcome at the time i E {41,.... [mm] for the treatment D= d, X E 1]" denote a time-invariant p-dimensional covariate and denote a possibly time- varying unmeasured confounder vector. We maintain the classic SUTVA (Rubin| (980) assumption that no interference between units and no hidden variations of treatments occur. We impose the
treatment at the time t = mm and the observed outcome Yt,. = DY + (1 Dyy(", before the
treatment occurs, the historical outcomes Yt, following condition.
(-(U) 1/ 1 < k S m - 1. Here, we further impose the
Assumption 1. For d E [0, P(D d X) 0 and y(d) TT LL (X,,U).
Given triplets of i.i.d. samples {(X;, {Y D.) our goal is to estimate the CATT ( DD= 1, = x). In general, 17" is not identifiable due to unmeasured
77 (z) = _( km confounding; however
prove that it is partially identifiable under certain transportabilitt condition and Assumption [ In the following, we denote the indicator function of event A by I(A).
Equation implies that, 17" (xx 8i., (x) = E(Y ()) 1- = x)
E (0)
D = 0, x)
km the confounding bias b (x) = E(Y!!!) D = 1,
tm
= =x) -
0 = - 0. In
general, [b. (X)] 2 cu > 0 due to unmeasured confounding. Proper stratification (Kernan 1999,, notably, can help mitigate confounding effects by introducing multiple subgroups within which confounders remain almost constant. By exploiting historical controls that contain information of unmeasured confounding, we propose to find the optimal partition of the feature space, e.g. (Q,))=1
with _ = [o, 1}2, such that bi... (Q,) = F((Y:()) (0) fm
= 1, X E Q2) F.(Y (0)
D= = 0, X
Q₃) Av 0, for each j = 1,.... T or alternatively,
(Q...... 22) argmin
bi (Q₂).
(2)
Q j=1
Once (Qi,... ,ū) being derived, by D.. we can choose
<xx) = Σ S.., (Q)){T E Q3)
(3)
1=1
to approximate 17" (-), where &.,. (Q)) E D E - E(y (0)
E(Y
D D 0, Qj)
constant function that approximates 6,,, (z) for x E Q3. In 3 choosing a constant function to approximate d (x) within Qj is only for simplicity, and local polynomials of any degree are applicable. As expected. ry(z) close to (x) provided that 2 implics maxi< by (Q;) x 0. major difficulty of solving 2 the unavailability by (Q,), S S g. since --(0) counterfactual for D = 1. Fortunatcly, since D, is known for each i and all units stay untreated
3

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before tm. b(() for all t E ft.....,,t.... are identifiable. Instead of 2
consider
m-11
(Qi",.... ,...,,9'1 = argmin Qussisn
by (Q₂).
(4)
k=1
The partition =0;" of the feature space yields minimal historical confounding bias. If confounding effects are independent of time, Q7 = Q7 holds for each j = I.....g. q. We impose the following condition to ensure transportability from the historical optimal partition its current counterpart.
Assumption 2. For the partition (Q1") defined in
4.
B(t) = log { Σラ= b((())) tt
B(t) R is Lipschitz continuous function with parameter L; that is,
- ( B(T2) < L/m - T2l, for any TI, 72 2 0.
Assumption 2 is testable empirically for / E [0, (m,) provided that historical controls are collected at sufficiently many timestamps. Under Assumption 2. we prove in the Appendix that Q7 ~ Q in effect; that is, the solution to 4 is transferrable or Transportable the current data source. The partial identifiability of 17" (x) is justified by the following theorem.
Theorem 1. Under Assumption for the partition {Q)")) of [0, 1]2 defined in 4 it holds that
|n(x) (x) -x))) sup #e 10.
Cr
ΣΣ
by (100))
1/2
max sup (n" 17" 15I592EQ""
-1) (2,")/ (5)
m k=
where 17 (Q)") FY y D= 1, E Q7²)) and Ct. depends only T. and Imm l-
The first term in denotes the confounding bias, and hinges on the transportability assumed in Assumption 2 while the second term measures the error of using a piecewise constant function to approximate 1f" (-). Theorem [ shows that 17" (.) is identifiable up band; this property that is referred to as partial identification (Swanson et al. 2018). Interestingly, the width of band in our case can be close enough to zero under stronger assumptions. For example, the confounding bias can be small if the stratification {Q can effectively adjust for confounding, and under certain smoothness condition, the approximation error decays to zero rapidly as the maximal volume of Q," tends to zero. Our goal is to develop a computational feasible method to solve 4 and estimate vy" (z) as in substituting Q2 with Q for each j.
1
2.2 Debiased Causal Tree
When it comes to stratifying or segmenting the feature space, the perhaps mostly used algorithms are tree- bese methods (James et al. 2021). including classification and regression tree (CART)
(Breiman al. 1984). random forest (Breiman 1996), and gradient boosting trees (Friedman 2001).
A regression tree T(z; Q. μ4) consists of two components; a set of leafs that partition the feature space, and the corresponding parameter μ nQ)) (m(2,)). where Q, is the region partitioned by the j- th leaf, = S g. and q is the total number of leafs. Typically, given a partition Q and pairs of i.i.d samples {(X,, (1))) ) the optimal data driven estimator is T(x; Q, (2) = ΣE= Ifa E (Q,)) where
p(q,) = (I 2,D0
Σ 1
v1
(6)
We define the auxiliary trees T(x; 0 to represent confounding biases, where 0.d) (0(..) such
that 0.4) o(d)) (Q₂) fits the conditional mean E(( D d, X E Q2) for d = 0, 1, < I. and
define the main tree T(x; Q. ((()) to learn CATT, where je (d) (1(50) such that μ (d) (Q,)
&m fits the conditional mean ( = d, X E Q2) for 0, Auxiliary and main trees share the same spplitting rules.

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Confounding entropy. Treating the partition {Q)}}=ı as parameter to be optimized, by 6 we estimate b,())) for each t E 1m-11 by
1(x, Q; Ki X, D; (7)
(Q,) - - (2)) (2) 0 (Qz). where
=
For simplicity, we write (Q₂)) as 023 Plugging 6, (Q,)
into
we obtain the estimator
of the optimal partitions. Nonetheless, by (Q₂) well as its population counterpart b,( only summarizes information of the first-order moment, and thus can give rise to partitions that yield estimators with large variances. This motivates to model the discrepancy between Pt,,,j
((y D = X E {3) and Pro.j = P(y, | = 0, X E Q2) using symmetric cross entropy (Wang et al. 2019) defined as H(Pr.1.j. ((Pr...PP.o.) Ep.. [log[ (r,0,s]] E...,, [log(P...))) which we reter to as the confounding entropy for time and region Q3. For each & < Em. parametrize the working densities of Pt,,, and Pr,o, by p(y:; (2(1) and p(ye: 0-(1)) respectively. In our case, = d, X E Q;) for both d = 0, 1. The estimator of symmetric cross entropy between Pt.1 and Pe.oj can be obtained by cross-fitting the negative log-li lglliielihood; that is,
the parameter the conditional mean 0- = E(Y,
{ EXEEQ,
Dj Do [To: X, E Q3. Dj =11
(1 - D,) log((((.) 0 I(i: X, E Qj. D, = 0}|
(,)) = E
[Tx:
+
(8)
Note that for two distributions and Q with density functions p and respectively,
(()) log (())
+ Eq ((() log p(x)
H)) EEp + EEgflog al (x)) + Ep
p(X) p(X) f (x()) ((x)
= Entropy(PP) (P) + Entropy(Q) + KL( + KL(,QQ)
Thus, confounding entropy can not only measure discrepancy between distributions, but also control the uncertainty/complextty.. which avoids overfitting. As concrete example, for isotropic Gaussian
distributions, log(p(ve 0.15) - -(y -0 2 d ===01. =, 1,
D. 0.)
(1- Hi X, X, Q3 = 0}
(Q,)) = 2 03)
Σ :X:EQ
(9)
Ui: X1 Dz 13
where, the first term is exactly (Q,) defined in 7 and the second term is the of sample variances of 9(0) and 0: Therefore, both confounding bias and variance can be emphasized. In the end, assuming historical data collected at distinct timestamps are independent, the confounding T.ctt,
(,5
entropy of all historical controls H(TT mT Pt.., Thct.,. 1T - Pt.o.)) ) estimated by
Q
H. (Q,), He. (Q,) is defined in 9
(10)
We remark that TTp can be interpreted as marginally independent composite likelihood estimator Varrn et al. 2011). which can be readily generalized to model potential temporal correlation among
(Ve [mm k However, for simplicity, we presume no temporal dependencies.
Estimating CATT.
On the basis of 6 we then can estimate r)" (z) with
Th (x) T(x; Qu, (2()) ( (Q))) T(x; Q (1))
(Qn)), ((()((())
(Q)= (T( (11)
where by (Q,) = Exxeo, i. I(D. = d))/((;: X, E Qq. = d)l, = 00 for
any given Q. and Q 14 denotes the minimizer of Ii(Q) in (T)). Here, the superscript
"h" denotes the fact that only historical data are used to learn Q" This two-stage estimator n (x) may suffer from potential heterogeneity between historical and current data sources, since splitting mechanism of 7h (I) is independent of current data set {(X, Yut.. D.) It relies crucially
on the transportability of historical optimal partitions (Q²")
19 Jz=1- To leverage information from the current data, we also use trees T(x; Q, (1() (Q)) to learn F((:(d) |
d, X x)) Let
fm
mi( D)Q) -DT(x; Q. () (Q)) + (1 (x; Q, ()) We aim to find a Q such that mxx D; Q) can predict Yú, well meanwhile H{()) being small.
5

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Regularized Regression Tree Estimator. For any mroblem-specifer loss function E(; )), e.g.. squared loss for regression and cross-entropy for classification, we propose the following regu- larized empirical risk minimization:
Di; Yes..) + )
Q arggin
(12)
where denotes the tuning parameter, and the CATT estimator Tx:; Q (Q) (xx pz (Q)).
given by 7)x)
=
We remark that there exists trade-off between data fidelity of the current source and transportabilitty of historical sources for the choice of A. When À + oc, 77-) is reduced to the naive estimator, ignoring the information of current data. In contrast, À means that we only use current data to learn CATT, which, under the unconfoundedness condition, is similar in spirit to causal tree estimators proposed in Wager and Athey (018)) and Athey and Imbens (2016). For the optimal choice of X, our method minimizes the confounding bias introduced by unmeasured confounders that deteriorate the performance of conventional causal trees, and thus the name "debiased causal tree" follows.
2.3 Ensemble of Debiased Causal Trees by Gradient Boosting
Ensemble is commonly used method improve the performance of tree-based models. Boosting and bagging are two representative methods, where the former is designed to reduce bias, and the latter focuses on variance reduction. Existing tree-based methods developed for treatment effect estimation combine multiple trees to make a random forest, since unregularized decision trees are prone to overfitting, especially when the tree in employment deep. In our case, the objective function 12 includes a regularization term. As a consequence, our estimator tends to be underfitting instead of overfitting, particularly when the working densities that define { are misspecified and/or the tuning parameter À is relatively large. This motivates us to choose the boosting framework to improve the performance of single debiased causal tree. We define the boosting of trees as
F(x; 2x. M&) = Ei= T(x; (2144),, where Qs = { (QI...... 2s) = = denotes the set of partitions of each tree, and Mr- = ( p4 } = M [1x] denotes the set of corresponding
parameters. To highlight the difference, we use O = (0I..... 0x) } to denote parameters of auxiliary
tree models that measure confounding entropy.
At the k-th iteration, we have already obtained k 1 trees, ((4) (z; 2x-1, M( learn E(Y, (d)
D d, = 22, 0, I, whose difference yields CATT, and k 1 trees, F.ld)) (x: 2k-1, to learn EYY, D d, X = x), t < tm. whose difference yields confounding bias. Notice that the trees for confounding bias share the same splitting regions with trees for CATT. In the following, we illustrate how to obtain the k-th tree.
Parallel to a single debiased causal tree, we first show a refined representation of confounding entropy by integrating multiple trees. Following the gradient boosting algorithm (Friedman 2001p. the k-th tree [x;; Qu, o()) manages to fit the residual, (X.) - p_(d) (X,; _ p with D. =
for each d 0, 1. Denote by (((l) (Qx) (o(c) (Qk.)) the local mean for any partition Qs. where
0.ti) (Qk.s) is defined similarly to 7 with Y.. being replaced by residuals. Similar to we use the cross mean squared error to represent the confounding entropy
g
(Y.
(Yr,
(()) (X; 2x. @())²² + D. P(O (X,; 2 %
i.)Q.) = Σ 1=1
[it: D. = -0
[i:: D. = =1
(13)
where 2x = - @ = (4))) u ( and y! denotes Y,, with D, = d.
suffices consider the integration of trees learn drn, (x). Again, the k-th tree T T(x; Q p
(d)
manages to fit the residual, R!!)) (d) p(0) (X,; Qt-1. M
k-1) with D, = d. Let p (d) (Q²)
(o (Qk.)) )) be the local mean for any partition Qk. where (d) (Qk.s) is defined similarly to

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with y(d) being replaced by 1 The objective function for the k-th tree is displayed as follows: ftm.
Q = argmin
(R T(X;; Qk ( (Qx)))
A
12-1
Σ
H... (00)
(14)
Qx
7
m
Update 2: - 2 (Qx). and M (d) = M
f (Qx) = 0, 1. The CATT estimator at the k-th iteration is f)) = F (a; 25, M (()) (2; _r M
(4)
k-1
We refer to our proposed method as Gradient Boosting Causal Tree (GBCT). We remark that the objective function 41 greedily minimized by gradually adding the tree. Also, speed up the computation and stabilize the algorithm, we follow XGBoost to approximate the objective function with second-order Taylor expansion and exploit squared Eg regularization. See [Chen and Guestrin (011)) for details.
3 Applicability of Debiased Causal Trees Without Confounders
In this section, we discuss the applicability of debiased causal trees without confounders. Despite that the confounding bias (x) = 0 for any and x without unmeasured confounding, numerically, the term (Q₂) in 7 may not vanish, especially when the minimal sample size is small. This issue is substantive when treated and untreated units are imbalanced. Notably, the imbalance can significantly degrade the performance of T-learners and S-leareers (e.g.. see Künzel al. 2019). Given the historical training set {(X,, Y.,, D:)) === the smaller empirical confounding entropy is for the partition Q, the more balanced samples can within each small region. Here, for simplicity, we consider m 2 such that only one historical data source is used. Let n.a (Q,) | D, d, X; E H
be the number of samples in Q, with D; = d. We formalize the above intuition with the empirical confounding bias ((Q) defined in (T)) with working distribution being Gaussian.
Proposition 1. Suppose there are no unobserved confounders, and if , ~ Y2 with mean μ4 and variance of independently for a given t < tm. then, under the Gaussian specification of working densities, it holds that E [[))) 2 + + (mo((,) (mo( + nI (n():)
(Qz))-
By Proposition 1 and the harmonic mean inequality, to reduce A())), suffices to find a Qj such that mo(Q,) ni (Q₂), especially for Q, such that no())) nı } is small. Numerical analysis can be found in Figure I in the Appendix, which indicates that, our proposed debiased causal tree is robust imbalance of treatments.
4 Experiments
We conduct experiments with both simulated and real data for a better understanding of our proposed strategy. We investigate two scenarios in simulation studies: one with and one without unobserved confounding. When implementing GBCT, we subtract the empirical pre-treatment confounding bias from effect estimates as trick to improve performance, i.e., 7))) (r) (2;; ②x, M())
(() 2t, ( m-1 Em= Ege 3 b.., (Q). We evaluate GBCT against state-of-the- causal inference algorithms: (i) meta learners (Künzed et al. 2019), including TLearner, SLearner and Xlearner; (ii) causal forests, which are forest-based methods to model the treatment effects, includ- ing generalized causal forest (CRF, Athey et al. 2019), doubly robust forest (DR-RF RF, Bang and Robins 2005) and double machine learning forest (DML-RC RF, Chernozhukov et al. 2018). These benchmarks
implemented by the third-party library ECOML (Keith et al. 2019).
For benchmark methods, both covariates X and pre-treatment outcomes are used input features. In contrast, our proposed GBCT employs alternative way of exploiting historical controls, i.e., building models on X while treating pre-treatment outcomes as labels de-bias via the confounding entropy. To make a fair experimental comparison, we set all hyperparameters using the same strategy. The number of trees in ensemble models (including boosting and bagging) is 200, the sub-sample ratios of instance and feature are 0.8 and 0.6 respectively, and the learning rate is 0.3. The maximum depth of each tree in forest-based (GRF, DML-RF and DR-RF) and boosting based (meta learners and GBCT) methods is 10 and 3, respectively, where it should be noted that due to the respective characteristics
7

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of the bagging and boosting frameworks, the trees in random forests are generally deeper. Meta learners employ LightGBM as base models and causal forests utilize sub-sampled honest random forest (Athey and Imbens
2016).
4.1 Simulation Studies
Data Generating Process. Here we shall give conceptual introduction our data generating processes, whose implementation details can be found in Supplementary Materials. We generate a p-dimensional confounder vector X from Σe T(G = s)W( E,), where G E {1.2. ... S} is hidden variable following a predetermined discrete distribution, and indicates which latent cluster each subject belongs to. The treatment- treatment-independeer independent feature vector W of p-dimension is developed similarly. We then generate the potential outcomes by the following factor augmented auto regressive process of order one, for each < k S m,
y:10) =pYi (1 (a W) f(X, We)) +e), +e), y! = y(()) g(X) + E1: 5x
is the individual- sppecfci auto-correlation coefficient following truncated normal distribution (Mpp o²: [o, 1J). X is the time-varying factor, the initial value y = 3Xx, W). are specified functions described in the supplementary material, and e, & mutually independent noise items subject to (0, 1). To ensure the samples are drawn from the stationary distribution of the data generating process, we use outcomes with time larger than a threshold ty observations.
where P E [0,
(()),
We assign treatment for each unit according to its latent cluster, such that P P 0 G s) = (-1)²m for = 1, S. where & E [0, 1] is specified value. The parameter measures the degree of balance between the treated and control populations; as e rises from zero to one, the imbalance gradually increases. In particular, / = O means the treated group has the same population as the control group (i.e., randomized controlled trial), whereas b = means all treated group's units are from different latent clusters than the control group's. ss In following experiments, we set = 2, Up = 0.7, a. = 0.2, and the dimension of feature space 2p = 20. A total of 20000 samples are generated and randomly split into training and validation sets 10 times.
No Unmeasured Confounding Scenario. The confounding vector X blocks the backdoor path from treatment to potential outcomes. Thus, given X. potential outcomes y:(1) is independent of D for d E 1]. Suppose that we observe all confounders X, and apply GBCT as well as benchmarks estimate the conditional treatment effects. We compare our proposed method with benchmark algorithms on the mean absolute error (MAE) of conditional average treatment effects, i.e., MIAECATEE n -11 Ei= (m( - where 7/2 is the true value of the ith individual's 's treatment effect. Table D shows the results. As we can see, GBCT outperforms the benchmark algorithms in most cases, especially when ó larger, suggesting that it beneficial to reduce the empirical confounding bias in case of no unmeasured confounding. As é increases towards one, except for GBCT and GRF, other methods generally lead severe bias and significant performance degradation due low overlap. Moreover, we perform ablation experiments to compare GBCT with its two variants: the variant GBCT-ND without using the confounding entropy (())), and the other variant GBCT-B using solely the first term b3 (2) in 2 as H,)),) construct the de-bias loss. The results presented in Table [ demonstrate the important role of our proposed confounding entropy in making the estimates less sensitive to the degree of imbalance between the treated and control populations.
Unmeasured Confounding Scenario. We consider the scenario where the confounding vector X are not fully observed. For estimation and evaluation, we retain only the last five dimension of the confounding vector and discard the rest. When & O, the treatment is assigned randomly, resulting in no confounding bias. However, as á approaches one, the impact of confounding on treatment assignment grows stronger. Therefore, a large & suggests substantial confounding bias due to unmeasured confounding. In this case, most traditional methods may fail to adjust for bias without further assumptions, while in contrast, GBCT can approximate CATT well. We use Σm= Dilii (X.)) Tjl as the criterion to evaluate the performance of algorithms, and report the results in Table I As expected, GBCT yields smaller mean absolute errors than other methods, and the gap becomes more pronounced as & increases, indicating that GBCT can offer more robust and compelling solution for estimating conditional treatment effects in the
MAECATT = (Σi1)
8

---

Table I: MAE (dea of each algorithm the simulated data. Scenario represents the absence of unmeasured confounding, whereas II represents the presence of unmeasured confounding. In addition, o /o indicates whether GBCT is statistically superior/inferior to the benchmark algorithms (pairwise t-test at 0.05 significance level).
cansal Forests DMI-RF
Slearne
Tleamer
Xlearner
DR-RF
GRCT
GBCEND
0.2
1226
0.00
1.16
0.07
1.06 1: 0.05
1.23 1. 0.01
1.28 4: 0.01
1.26 1: 0.01
1,05 0.01
1.10 0.00e
1.06 0.01
0.5
1446
0.13m
1.50
0.17
1.19 0.10
1.29 1: 0.02
1.34 1: 0.05
1.31 1: 0.00
1.25 0.05
1.30:100 0.002
1.31 = 0.01
0.46
1.51 0.255
1.48
1.39
1.41+
0.03e
0:22e
0.21 1.43
1.80 0.44e
1.44 0.08%
1.41
1.40++0.00
1.43 0.100
1
4.56
= 0.13+
3.28
1,59
2.46
0.90
4.73 + 0.06+
16.02 = 9.70
1.06=
1.71 + 0.06
3.80 0.57
2,85 0.34
0.2
1488
0.07
1.97
0.53+
1,54
0.25
1.60 1: 0.02m
1.66
0.02
1:26 0.00
1.07 1 0:01
1.11 1.111
1.08 0.02
1.30
1.90
1.70 1.75
0.05m
1.31
0.04
0.00le
1.72
2.07
3.94
3.31
2.14 0.44
1.40
1.34 0.04e
1.32 0.00
0.9
2.30
0.07
2.63
0.29
2:35
0.23+
2.13 + 0.03+
3.10 0.12e
1.72 0.00
1.38 0.00
1.92 0.04e
1.45 0.08
3.84 0.24
1.32
4: 3.49
4.84
4: 2.94
5.20 4: 0.06
2.84 2 0.48
2.38 d: 0.17
1.81 0.10
3.19 3.19 0:775
2.65 0.76
presence of unmeasured confounding. We also compare the performance of GBCT with GRF the single-tree scale on removing confounding bias, showing that our method can be more effective, see the Appendix. In the ablation experiment, the results confirm the effectiveness of our proposed confounding entropy for reducing bias, especially in the presence of strong unmeasured confounding.
Table 2: Description of the credit card balance dataset. The biased observational dataset only consists of samples among the underlined instances. The features consist of the time-invariant covariates and the balance of credit card account over the last eight months.
Number of Instances
Risk
Number of Features
2k
3k
6k
medium-risk
389477
390928
391211 459443
391215
87+8
low-risk
456773
459762
460352
Table 3: The mean absolute eтror MAEATE (mean s.d.) of each algorithm on the credit card balance dataset. The left and right sides are trained on the biased observational and the trial data, respectively. We treat the users with no increase in credit line as the control group.
Biased 3k
RCT 3k
risk
2k
6k
2k
6k
medium-risk
49.1 39.8
106.5 + 119.7
178.6 + 116.4
92.6 = 75.3
75.2 = 55.6
76,9 + 39.1 53.9 : 34.7
TLeamer
low-risk
57.0 :: 39.0
104.8 + 122.2
197.7 1 87.4
87.8 76.9
51.8 4 55.4
medium-risk
88.7 + 129.0
85.0 + 139.3
99.6 132.9
98.8 106.4
93.0 = 81.2
93.0 = 48.7
XI eaame
low-risk
62.4 129.0
99.4 139.3
127.1 132.9
97.0 106.4
81.5 = 81.2
98.1 = 48.7
medium-risk
207.3 26.3
253.4 37.8
261.9 59.4
12.9
23.4 9.7
24.7 + 10.9
SLeamer
low-risk
95.1 :: 28.4
100.4 33.77
45.1 +: 39.6
41.3 8.4
59.0 +: 29.2
100.9 :: 34.9
medium-risk
249.6 + 9.3
317.7 12.2
408.6 9.4
8.6 + 7.5
14.2 + 7.73
48.4 + 8.8
DR-RF
low-risk
149.0 + 6.5
162.8 +: 8.9
151.2 + 11.1
35.0 5.0
31.7 +: 6.9
55.5 +: 9.4
DML-RF
medium-risk low-risk
> -99.9
999.9
> 999.9
8.5 1 6.7
15.7 - 8.6
56.4 1 8.6
> 999.9
> 999.9
> 999.999
34.5 = 4.9
32.8 = 6.4
62.4 - 9.3
GBCT
medium-risk low-risk
10.6 + 6.1
27.5 + 13.6
61.5 +: 29.5
13.2 = 7.6
8.0
14.2 9.2
8.4 5.8
8.6
19.3 = 22.5
21.7 5.2
1.
2.9
4.2
Real Data: Credit Card Balance Dataset
The dataset comes from a randomized controlled trial (RCT) by a commercial finance company aimed at assessing users' heterogeneous responses increasing credit line of credit card? The trial employs a stratified random assignment design with strata based on risk, dividing users into low-risk and medium-rist risk. Within each stratum, users are randomly assigned one of four treatment groups:
$1. The dataset does not contain any Personal Identifiable Information (PII). 2. The dataset desensitized and encrypted. 3. Adequate data protection was carried out during the experiment to prevent the risk of data copy leakage, and the dataset was destroyed after the experiment. 4. The dataset is only used for academic research. it does not represent any real business situation.

---

increasing credit line by 0, 2000, 3000 or 6000 converted to some currency. In this study, the outcome of interest the balance of the credit card account in the following month after treatment, i.e., the total amount of debt owed to the credit card. In addition, we observe 87-dimensional covariates at the individual level, as well as outcome values over the last eight months.
In real-world financial scenarios, however, increasing credit line always favors low-risk users; such preference can lead to confounding bias in estimation. In order to assess the performance different methods in the real-world scenario, we artificially construct biased observational data by retaining only medium-rist, users with no credit line increase and low-risk users with credit line increase. See Table 2 for details of the trial dataset and the biased observational dataset.
We interested in the causal effects of increasing credit line balance. Unlike simulated data, we do not know all potential outcomes for each user, so we cannot use MAFсATE for evaluation. Fortunately, within each risk stratum, due to the randomness of treatment allocation, the ground-truth of the average treatment effect is approximately the mean difference between any two treatment groups in the trial dataset, denoted as TJATE. Note that in RCT, the average treatment effect is equal to the average treatment effect on the treated group. Therefore, we compare our proposed method GBCT with the benchmarks on the mean absolute error of the average treatment effect, MAEATE 1 Em= f(X,) - MATE - To assess variability, we randomly split the entire samples into two folds and repeat this step ten times, each time using one fold unbiased test data and the other to construct the training data. Given the estimates obtained from the training data, mean absolute errors are calculated separately in each risk stratum of the test data.
The results are reported in Table 3 It is impressive to observe that on both trial and biased data, GBCT significantly outperforms the benchmarks in terms of mean error and standard deviation. Although the benchmarks DR-RF and DML-RF do not perform poorly on the trial data, they suffer severely from bias regarding the observational data; a possible reason is the low overlap between the low-risk and medium-risk strata. As a comparison, GBCT suffers substantially less from bias than others. These findings in line with those of the prior simulation studies. Therefore, recommend using GBCT in practice to gain a more stable and robust estimate of causal effects.
5 Discussion
In this work, we propose simple yet powerful method, debiased causal tree, to remove confounding bias caused by unmeasured confounding in the estimation of causal effeets (e.g., CATT). Our method only hinges the availability and transportability of historical controls, i.e., outcomes of units before the treatment, to ensure CATT to be partially identifiable. This allows our method to be readily applicable in a wide variety of scenarios, especially in clinical trials (Viele et al. 2013). Moreover, method can be generalized to integrate observational data and external control (Li et al. 2021 2022), with much weaker transportability condition than existing work (Shi al.
2022). Nonetheless, our
method suffers from several limitations, e.g., time-invariant confounders assumption and failure to provide principled uncertainty quantification as Bayesian causal forests do (Hahn et al. 2020), which needs future investigation.
Acknowledgments
This work is supported by Ant Group. We thank the three anonymous referees for their constructive suggestions and thoughtful comments, which helped to improve the quality of the paper.
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13


Reviews:
[{"review_id": "taXM0wHqrMk", "reviewer": "Reviewer_hbGq", "summary": "In the observational data, this paper consider the case where covariates and the outcome are collected at multiple timestamps, $t \\in \\{ t_{1}, \\ldots, t_{m} \\}$, and only implement treatment $D=d, d \\in$ {$0,1$} at the last timestamp $t_m$. In this case, the historical controls means that all treatments are $D=0$ at the timestamp $t \\in ${$ t_{1}, \\ldots, t_{m-1} $}. The treatment sequence is an known sequence, i.e.,{$0,0,\\ldots, 0, d$}.\n\nUsing observational data and historical controls, this paper proposes a new notion confounding entropy measure the discrepancy between the conditional outcome distribution of the treated and control groups. Then, they use confounding entropy as splitting rule to divide the data into multiple subgroups and estimate  conditional average treatment effect on the treated group. \n\n----\nI thank the authors for addressing my comments. I increase my score to 5.", "questions": "- Using observational data and historical controls, can we directly and accurately estimate potential control outcomes with a large amount of historical control data? Is the estimation of conditional average treatment effect on the treated group still a challenge? \n- Lines 126-127, what is the relationship between Theorem 1 and the identification of instrumental variables? \n- Why not use observed outcomes on the treated group when estimating CATT?\n- The data generation mechanism, and estimators $\\{\\alpha(\\cdot), \\beta(\\cdot), f(\\cdot), g(\\cdot)\\}$ are not detailed in this paper.", "limitations": "I don't believe that this method proposed in this paper can reduce the bias from unmeasured confounders that are independent with observed confounders. Besides, in practical, time-invariant covariates and unmeasured confounders assumption is a strict condition that is difficult to satisfy. ", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\nThis paper considers a intersting setting about historical controls, and proposes a new notion confounding entropy measure the discrepancy between the conditional outcome distribution of the treated and control groups. The computational feasibility and statistical power are evidenced by simulations and credit card balance dataset. \n\n**Weaknesses**\n\n- Using observational data and historical controls, we can directly and accurately estimate potential control outcomes $\\hat{Y}^0$ with a large amount of historical control data. The conditional average treatment effect on the treated group can be easily obtained: $CATT = \\mathbb{E}[Y|D=1,X] - \\mathbb{E}[\\hat{Y}^0|D=1,X]$. \n- In Lines 98-102, proper stratification [26] can help mitigate confounding effects from observed confounders but can't reduce the bias from unmeasured confounders that are independent with observed confounders.\n- In practical, time-invariant covariates and unmeasured confounders assumption is hardly satisfied in a causal scenario. \n- In Lines 104-107, without any prior knowledge, neither constant function nor polynomial function  guarantees robust results. \n- Checklist in Lines 492-493: The Assumption 1 is not the full set of assumptions. SUTVA, positive/overlap assumption are missing in this paper. The authors should reframe the assumptions required. \n- Figure 1 and Figure 2 in Appendix, what data was used to draw this picture? the authors did not give a clear explanation. \n- Not Reproducible. The data generation mechanism, and estimators $\\{\\alpha(\\cdot), \\beta(\\cdot), f(\\cdot), g(\\cdot)\\}$ are not detailed in this paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "cS1zdgtuxHE", "reviewer": "Reviewer_ZPaK", "summary": "The paper considers heterogeneous treatment effect estimation when you have longitudinal observations of outcomes pre-treatment. The main contributions are (i) a partial identification result of the CATE in this setting, (ii) a new objective function for finding optimal tree splits, and (iii) a boosting procedure that appears to work well on simulated and real-world data. \nThe tree splitting criteria seeks a recursive partition of the space of measured confounders that minimizes the sum of two terms. The first term measures the error in fitting the observed post-treatment outcomes using piece-wise constant step functions (i.e. a regression tree). The second term aims to regularize towards partitions for which the distributions of outcomes for treated and control subjects in the pre-treatment period are thought to be small. Under a technical assumptions about how similar confounding bias in the pre-treatment period is to confounding bias in the post-treatment regime, the authors show that they can partially identify the CATE in the potential presence of unmeasured confounding.\n\n=== \n\nAfter the response period, I have raised my score to 6.", "questions": "* Can you prove hat regularizing with \\hat{H} instead of the sum of squared biases actually results in estimators with smaller variance?If not, you should provide, at a minimum, more justification for not using the sum of squared biases (i.e. the righthand side of Equation 4) as the regularizer in Equation 12. Better would be an empirical demonstration that \\hat{H} yielded better point estimates or better uncertainty quantification than using the sum of squared biases\n\n* Relatedly, why did you try to minimize the symmetric cross-entropy between $P_{t,1,j}$ and $P_{t,0,j}$? There are several other choices of distributional differences/distances/divergences. Does closeness in this distance imply closeness in moments? \n\n* To what extent does your framework allow treatment assignment to depend on the potential outcomes in the pre-treatment period? One can readily imagine scenarios where the decision to treat is based on past history. It would be useful to clarify explicitly whether you need to assume treatment is assigned independently of past outcomes.\n\n* Relatedly, what specific causal assumptions do you need to make in order to establish Theorem 1? Presumably you are assuming some sort of positivity assumption and SUTVA. You likely do not assume strong ignorability (i.e.$ (Y(1), Y(0))$ and $D$ are conditionally independent given $X$) and it may be helpful to point this out.\n\n* How does one set $\\lambda$ in practice (generally) and how did you set it in your experiments? Is it just through cross-validation?\n\n* Does your method provide any uncertainty quantification? As presented, I don't think it can easily. It may be useful to compare the proposed method to Bayesian Causal Forests, which does provide a type of uncertainty quantification while still flexibly targeting the CATE. \n\n", "limitations": "The authors do not address any negative societal impacts nor can I easily imagine any for this particular work.\nThe authors do not explicitly mention any methodological or theoretical limitations. I would encourage them to elaborate on some ways to extend their results in the discussion, e.g., by discussing potential relaxations of specific assumptions or implementation choices.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: I commend the authors for an original approach to the difficult problem of estimating the CATE in the potential presence of unmeasured confounding. On the whole, I thought the paper was well-written and reasonably clear. I think that the proposed method has the potential to be quite useful to practitioners.\n\nThat said, my enthusiasm for the paper is tempered by several concerns about specific modeling and implementation choices and the corresponding assumptions. I will summarize these concerns here and expand on them in the Questions section. \n\n* I am somewhat concerned about the transition from Equation 8 to Equation 9. As written, the assertion in line 159 about the log density of $P_{t,1,j}$ being $-(y_t - \\theta^{*(d)}_{j})^{2}$ is true only when $y_t$ is normally distributed with unit variance. You seem to implicitly assume that no matter the partition $Q_j$ or value of $D,$ the outcomes are normally distributed with known and fixed variance. I find the fixed & known variance assumption somewhat unrealistic. \n\n* Additionally, since $\\theta_{t,j}^{*(d)}$ is typically unknown, you plug in the sample mean in Equation 8. This may be reasonable when $Q_{j}$ contains a large number of treated and control observations. However, it seems much less reasonable when there are not many of either.  As such the expression in Equation 9 may not be a particularly accurate estimate of the true symmetric cross-entropy between $P_{t,1,j}$ and $P_{t,0,j}.$ I'd go further and say that the expression in Equation 9 represents the symmetric cross-entropy between $P_{t,1,j}$ and $P_{t,0,j}$ under extremely restrictive conditions, namely that the $y_{t,j}$'s are centered around the corresponding sample mean with variance 1 for all choices of $Q_{j}.$\n\n* I found the motivation for regularizing based on historical cross-entropy loss somewhat lacking. The identification result suggests that one should strive to make $\\sum_{k = 1}^{m-1}{\\sum_{j = 1}^{q}{b_{t_{k}}^{2}(Q_{j}^{**})}}$ as small as possible. You write in lines 150-151 that minimizing this expression can \"give rise to partitions that yield estimators with large variance.\" It is not at all clear that your suggested regularizer avoids this pathology.  \n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "QoLooHaheHq", "reviewer": "Reviewer_J9bc", "summary": "This paper proposes a new notion, confounding entropy, to measure the discrepancy between the treated and controlled groups. Based on this, the authors propose a new method, the debiased causal tree (more specifically, GBCT), to estimate treatment effects.", "questions": "1. What are the differences between your work and \"Recursive partitioning for heterogeneous causal effects\" \"Estimation and inference of heterogeneous treatment effects using random forests\" \"Generalized random forests\"?\n\n2. Why is confounding entropy valid? Is it like the individual version of the integral probability metric? Is there any connection?\n\n3. Why do you need to compare GBCT with meta learners? They are just ways to estimate treatment effects, but the base models can be chosen as linear/tree/neural nets. Similarly, why DML-RF? DML is a doubly robust estimator, and the base models for nuisance parameters can be chosen as linear/tree/neural nets, and you choose RF. In my view, you only need to compare all the regression trees and random forests models that are related to causal inference and analyze the source of gain produced by confounding entropy.\n\n4. In Section 4.2, I don't find any source of data. Where is the real-world dataset and code to reproduce your experiments? I hope code/datasets can be released in the rebuttal stage.\n\n5. I am confused about the Biased dataset in Table 3. Is the control group increasing the credit line by 0? Why increasing the credit line for the low-risk group can bring bias into both low-risk and medium-risk groups? How does the RCT experiments conduct? Is it randomly assigning 0/2k/3k/6k to some people, where 0 is control and 2k/3k/6k is treat?\n\n6. In my view, the most important part is to analyze the gain of confounding entropy, which is missing in other tree/forest-based causalML models. You are not developing a new triply robust estimator or a novel meta learner, so I think there is no need to compare with doubly robust estimators or other meta-learners. So, I think 4-5 well-known tree models for causalML are enough as baselines.", "limitations": "The authors do not discuss their limitations. But I think more extensions can be done based on this work.", "rating": 9, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Originality: 5. This paper gives a new insight into causal inference with tree-based methods. The confounding entropy impresses me a lot.\n\nQuality: 5. This paper is well written, especially for the mathematical parts.\n\nClarity: 4. It would be great if the context can be connected with former works closely. Readers may be very familiar with Athey's works on regression trees and causal forests, but the content in Section 2 is too dense.\n\nSignificance: 3.5. I don't think the improvement is as fair and significant as the paper states. I think the improvement should lie in comparing with the tree-based or forest-based causal ML methods.\n\n___________Discussion period_________________\n\nSince the authors have added adequate experiments to support their theoretical contributions. I strongly recommend an acceptance. I encourage other reviewers to have more discussions in case we miss a good paper. I thus raise my score by 1.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept