Causal Inference with Conditional Front-Door Adjustment and Identifiable Variational Autoencoder
Authors: Ziqi Xu
Debo Cheng
Jiuyong Li
Jixue Liu
Lin Liu
Kui Yu
Venue: ICLR 2024 poster (2024)
Keywords: Causal Inference
Variational AutoEncoder
Front Door Criterion
Primary area: causal reasoning
TL;DR: We propose CFDiVAE to learn the representation of the Conditional Front-Door adjustment variable with the identifiable Variational AutoEncoder.

Abstract: An essential and challenging problem in causal inference is causal effect estimation from observational data. The problem becomes more difficult with the presence of unobserved confounding variables. The front-door adjustment is an approach for dealing with unobserved confounding variables. However, the restriction for the standard front-door adjustment is difficult to satisfy in practice. In this paper, we relax some of the restrictions by proposing the concept of conditional front-door (CFD) adjustment and develop the theorem that guarantees the causal effect identifiability of CFD adjustment. By leveraging the ability of deep generative models, we propose CFDiVAE to learn the representation of the CFD adjustment variable directly from data with the identifiable Variational AutoEncoder and formally prove the model identifiability. Extensive experiments on synthetic datasets validate the effectiveness of CFDiVAE and its superiority over existing methods. The experiments also show that the performance of CFDiVAE is less sensitive to the causal strength of unobserved confounding variables. We further apply CFDiVAE to a real-world dataset to demonstrate its potential application.

Published as a conference paper at ICLR 2024
CAUSAL INFERENCE WITH CONDITIONAL FRONT-DOOR ADJUSTMENT AND IDENTIFIABLE VARIATIONAL AUTOENCODER
Ziqi Xu 12, Debo Cheng 1,8, Jiuyong Li 1.8 Jixue Liu , Lin Liu & Kui Yu
University of South Australia Data61, CSIRO {firstname lastn.am waw.ameeeaiaaaduuuuuuuu 2 ziqi xuêdata61. csiro.au" yukui@hfut.. edu. cn
Hefei University of Technology
ABSTRACT
An essential and challenging problem in causal inference is causal effect estimation from observational data. The problem becomes more difficult with the presence unobserved confounding variables. The front-door adjustment is an approach for dealing with unobserved confounding variables. However, the restriction for the standard front-door adjustment is difficult to satisfy in practice. In this paper, we relax some of the restrictions by proposing the concept of conditional front-door (CFD) adjustment and develop the theorem that guarantees the causal effect identi- fiability of CFD adjustment. By leveraging the ability of deep generative models, we propose CFDiVAE to learn the representation of the CFD adjustment variable directly from data with the identifiable Variational AutoEncoder and formally prove the model identifiability. Extensive experiments on synthetic datasets validate the effectiveness of CFDiVAE and its superiority over existing methods. The experi- ments also show that the performance CFDiVAE is less sensitive to the causal strength of unobserved confounding variables. We further apply CFDiVAE to a real-world dataset to demonstrate its potential application.
INTRODUCTION
Estimating causal effects a fundamental problem many application areas. For example, pol- icymakers need to know whether the implementation of policy has a positive impact on the community (Athey] 2017 Tran et al. 2022). and medical researchers study the effects of treatments on patients (Petersen & van der Laan 2014).. Randomised Controlled Trials (RCTs) (Fisher 1936)
are considered the golden standard for estimating causal effects. However, RCTs are diffiuult to implement in many weal-world cases due to ethical issues high costs (Deaton & Cartwright 2018). For example, it would be unethical to subject an individual to a condition (e.g., smoking) if the condi- tion may have potentially negative consequences. Therefore, many methods have been developed to infer causal effects from observational data. Most of the methods assume no unobserved variables affecting both the treatment and outcome, i.c., the unconfoundedness assumption (Imbens & Rubin 2015), and follow the back-door criterion (Pearl 2009) to determine valid adjustment variable for unbiased estimation.
A graphical view of the typical cases in causal effect estimation is shown in Fig. I A simple case that satisfies the unconfoundedness assumption is illustrated in Fig. Ta In this case, the causal effect can be unbiasedly estimated by back-door adjustment (Pearl, 2009). Fig. Tb Fig. Ic and Fig. [ show three cases where the unconfoundedness assumption is not satisfied. The IV (instrumental variable) approach has been extensively studied and commonly used to deal with the case shown in Fig. [b] However, in practice, IV is not always available. In this case, if there exists a standard front-door adjustment variable, e.g., ZSED indicated in Fig. Tc the standard front-door adjustment provides an effective approach to dealing with unobserved confounding variables.
However, the requirement for valid standard front door adjustment variable is too strict, which hinders their practical application. In this paper, we aim to relax the requirement by considering more practical setting shown in Fig. ld. Different from the standard front-door adjustment

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IV
W
W
W
W
ZSDD
ZeFD
(a)
(b)
(c)
(d)
Figure 1: Typpccl cases in causal effect estimation. T is the treatment; Y is the outcome; W the observed confounding variable; U is the unobserved confounding variable; IV is the instrumental variable; ZSFD is the standard front-door adjustment variable; and ZCED is the conditional front-door adjustment variable
setting in Fig. Tc. we allow the interaction between observed confounding variable (W) and the mediator (ZCFD), and we call ZCFD a conditional front-door (CFD) adjustment variable. This is a more practical setting. For instance, referring to Fig. Id smoking (I)) does not directly affect lung cancer development (Y) but mediated through tar in lungs (ZerD). For each patient, their other attributes such age (W) directly affect smoking, tar in lungs and lung cancer development. In this case, the standard front-door adjustment cannot be used since ZCED is no longer a standard front-door adjustment variable because it does meet the standard front-door criterion (Definition 3), since, there is an unblocked back-door path from 7 to ZCFD (T <- W - ZCrD), and a back-door path from ZCFD to (ZCFD t W Y) which is not blocked by T.
Additionally, it is unrealistic to assume that users always know a CFD adjustment variable in advance and thus it is desirable find a CFD adjustment variable from observational data. In this paper, we propose a novel method, CFDiVAE, which is based on the identifiable VAE technique (Khemakhem al. 2020) learn the representation of a latent CFD variable from its proxy. We consider is practical to assume the existence of proxies of CFD adjustment variable. For instance, in the above example, the investigator may not observe tar patients' lungs but they may observe the proxy variables, such as the results of patients' follow-up sputum tests and urine tests.
This paper advances the theory and practical use of causal inference in the presence of unobserved confounding variables through the following contributions:
We address the challenge of estimating causal effects in the presence of unobserved confounding variables using the front-door adjustment. We relax some of the restrictions of the standard front- door adjustment by introducing conditional front-door (CFD) adjustment to enable more practical use of front-door adjustment. We also develop the theorem that guarantees the causal effect identifiability of CFD adjustment.
We propose a novel method, CFDiVAE, that provides an effective data-driven method for deal- ing with unobserved confounding variables in causal effect estimation. We further provide the theoretical guarantee of the identifiability of the CFDiVAE.
We evaluate the effectiveness of CFDiVAE on both synthetic and real-world datasets. Experiments with synthetic datasets show that CFDiVAE outperforms existing methods. Furthermore, we apply CFDiVAE to a real-world dataset show the application scenarios and its potential.
2 PRELIMINARIES
We capital letter to represent variable and lowercase letter represent its value. Due page limitation, we provide the definitions of directed acyclic graph (DAG), causal path, non-causal path, markov condition, faithfulness, d-separation and d-connect Appx.
This paper is focused on estimating the average treatment effect as defined below.
Definition (Average Treatment Effect (ATE)). The average treatment effect of treatment, denoted as T, on the outcome of interest, denoted as Y, is defined as ATE = E[Y do(T= E(Y) do(T = 0)), where do() is the do-operator and do(T = represents the manipulation of the trealment by setting its value to L (Pearl 2009).
When the context is clear, we abbreviate do(T t) as do(t). In order to allow the above do() expressions to be recovered from data, Pearl formally defined causal effect identifiability (Pearl 2009) (7.77) and proposed two rell-known identification conditions, the dook criterion and front-door criterion.
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2009)), A set of variables ZBD satisfies the back-door Definition 2 (Back-Doo) Criterion (Pearl
criterion relative to an ordered pair of variables (T, Y) in a DAG G if: (1) no node in ZBD is a descendant T;; and (2) ZBD blocks every path between T and Y that contains arrow into T.
back-door path is a non-causal path from T Y. They have been recognised as "back-door" paths because they flow backwards out of T, i.e., back-door path points into T.
Theorem (Back-Door Adjustment (Pearl (009)). If ZuD satisfies the back-door criterion relative to (T, Y), then the causal effect of on Y is identifialee and is given by the following back-door
adjustment formula (Pearl 2009
P(yldo(t)) EEPEy P(y P(ZBD)
(1)
Ziat
Definition 3 (Front-Door Criterion (Pearl 2009 ). A set of variables ZSED is said to satisfy the
(standard) front-door criterion relative to an ordered puir of variables (T, Y) in a DAG if: (1) ZSED intercepts all directed paths from T to Y: (2) there is no unblocked back-door path from T to ZSFD: and (3) all back-door paths from ZSED to are blocked by T.
Theorem 2 (Front-Door Adjustment (Pearl 2009)). If ZsED satisfies the (standard) front-door criterion relative to (T,Y), then the causal effect of ' on Y identifiable and is given by the following standard front-door adjusiment formula (Pearl 2009):
P(y]do(())
Σ P(y 1 ZSFD) ((T)
t),
(2)
zgm.t
where t' is distinct realisation of treatment.
3 CONDITIONAL FRONT-DOOR ADJUSTMENT
In this section, we present the definition of conditional front-door criterion and the theorem showing that the average causal effect of treatment T' outcome Y is identifiable via conditional front-door adjustment. The causal effect of T on Y is identifiable if the quantity p(y do(t)) can be computed uniquely from any positive probability of the observed variables (Pearl) 2009). We formally define the conditional front-door criterion follows:
Definition 4 (Conditional Front-Door (CFD) Criterion). A of variables ZerD is said to satisfy the conditional front-doo criterion relative to an ordered pair of variables (T,Y) in a DAG G if: (1) ZeFD intercepts all directed paths from T to Y; (2) there exists a set of variables W, called the conditioning variables of ZEED, such that all back-door paths from T to ZCED are blocked by W; and (3) all back door paths from ZEFD to are blocked by {T) U.W
Fig. Id provides an illustration of CFD criterion, where ZCFD satisfies the criterion, and W the conditioning variable of ZCFD. The following theorem provides the theoretical guarantee of the identifiability of the causal effect of T on Y via CFD adjustment and gives the adjustment formula. Theorem 3 (Conditional Tront-Door (CFD) Adjustment). If LCED satisfies the CFD criterion relative to (T,Y). the causal effect of T on Y is identifiable and is given by the following CFD adjustment formula:
P(yldo(t))
Σ
Ply ZCFD. P( P(ZCFD
[t, ((w),
where l' is distinct realisation of treatment.
Proof of the above theorem is provided in Appx. B.1
In practice, we often do not have given CFD adjustment variable, hence in the next section, we develop CFDiVAE to learn the representation of CFD adjustment variable using its proxies.
4 THE PROPOSEd CFDIVAE METHOD
4.1 PROBLEM SETUP
We assume data is generated based on the DAG & in Fig. where T'is is the treatment variable, Y is the outcome variable, U is the unobserved confounding variable, X1, X1,.., X,, are the proxy of ZCTD
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(n 2 3), the latent CFD adjustment variable whose representation is be learned and used for CFD adjustment, and WI.... Wm are the observed confounding variables and are the conditioning variables of ZCFD (m 2 1). When the context is clear omit the subscript.
In our problem setting, we assume that the observed confounding variable W and the proxy variable X are naturally separable. We believe this assumption is easy to satisfy in practice since W is pre-treatment variable (measured before treatment assignment) while X is the proxy of the post-treatment variable, which is always collected after treatment assignment. For instance, with the example in the Introduction, as previously mentioned, W can be a patient's 's age, and X can be the results of some follow-up tests after the treatment has been applied, such sputum and urine tests.
Xn
ZcFD
The DAG represent- ing the data generation mecha- nism assumed in this paper.
Figure 2:
To clarify, the latent variable (i.e., ZCFD) refers to the variable that is not measured, but its proxies (i.e., XI,.., Xn) are measured. On the other hand, the unobserved confounding variable (i.e., U) is not measured and has proxy. Latent variables and the existence of their proxies are commonly assumed by data-driven causal inference methods (Louizos et al.] 2017 Zhang et al. 2021 Cheng ctt al. 2022) and it is a practical assumption. In addition to the previous example where follow-up medical test results can be a proxy for tar in lungs, another example would be the case when we not able to measure a person's economic status, common solution is rely the proxy
variable such as postcode (Angrist & Pischke 2009 Montgomery et al. 2000).
The efficacy of our method is contingent upon the quality of the proxy for the CFD adjustment variable, which is similar to most proxy based methods for causal inference. As discussed in (Goodman) 1974 Kruskal 1976 Allman al. 2009), the model can be identified if the latent variable has least three independent proxies. Miao et al. (2018) build upon these assumptions, shifting the focus to causal effect estimation, posit that the causal effect could still be identified using only two independent proxies, even if the latent variable was not fully identified.
We summarise the problem setting as follows, including the assumptions made and the objective of the paper.
There are always at least three independent proxy variables available for the latent CFD adjustment variable; the noisy level of the proxy variables should be small; and the distribution of the latent CFD) adjustment variable should restrict to the exponential family distribution.
Assumptions.
Objective. Given a joint probability distribution P(X, W,T,Y Y) that is generated the undertying
DAG in Fig. where U and ZCrD are not measured. Suppose that X = (X1,..., X,) (n 2 3) is the proxy of the latent variable ZCFD and W (W1,... Wm) } (m 2 1) is the observed confounding variable. The goal of CFDiVAE is to learn the representation of ZCFD-
For the simplicity of notation and without causing confusion, in the rest of the paper, ZCHD to represent the learned representation of the latent variable ZCHD in Fig. 2 unless otherwise stated.
4.2 REPRESENTATION LEARNING
In this section, we introduce the details of CFDiVAE for learning ZCFD. CFDiVAE learns a full generative model p(X, ZCFD T, W) = p(X ZCFD)p(ZcFD T, W) and an inference model q(Zrr T, X)
To guarantee the identifiability of CFDiVAE, we take T and W as additionally observed variables to approximate the prior p( ZCFD based works in (Louizos et al. 2017; Zhang et al. 2021, Cheng et al. 2022), we assume the prior p(ZcrD T, W) follows the Gaussian distribution.
T. T. (Khemakhem al.) 2020). Following existing VAE-
In the inference model, we design the encoder Z T, W. X) that serves as the variational approximation of the posterior over the target representation, and the variational approximation of
the posterior is defined as, g(ZFF T, W, X) Pacm N(μ Aizmy 0² 02 o22m,) ). where fuzen and 03 are the means and variances of the Gaussian distributions parameterised by the neural networks for ZCrD-
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The generative model for X is defined as, p(X ZcrD) I1 where px, = (CFFD) 0x, g(ZcFD,) Dx is the dimension of X, and g(.) is a neural net- work parameterised by its own parameters.
M(x, | p== fxx,, xxx,)
Then the evidence lower bound (ELBO) for the above inference and generative models is follows: [X| ZCFD)] plg DKL|o(ZCFD) T, W. P(ZCFD T, W)),
.MCFDIVA
where :I KL divergence term.
4.3 MODEL IDENTIFIABILITY ANALYSIS
In this section, we provide the identifiability analysis of CEDiVAE. CFDiVAE is identifiable if the following implication holds.
v(0,0', 6') pe ZCFD T, (.W per (X, ZCFD T, W) 0 o'
(5)
Let (f,S, X) be the parameters of the following conditional generative model:
po (X, ZCD
T, W) = pr(X Zcm))s,. T, W),
(6)
and we define:
(x ZCFD) Pa (X f(ZCFD)).
(7)
This means that the value of X can be decomposed as = f(ZcFD) E, where is an independent noise variable with probability density function (=))). However, method also applies to non-noisy proxy variable and this case X f(ZcrD).
The function f in Eq. 7 injective and the should be small. For the prior Ps. (ZCrD T, W), have the following assumption, i.e., conditionally factorial, where each element of ZCFD restricts to an exponential family distribution given T and W.
The probability density function is given by:
DzcD (c (T, W) czp
h Σ
(ZCD T,W)
II
(8)
where Q; is the base measure, (T, W) is the normalising constant and S; (S.,,, S.,)) sufficient statistics and A(T, W) = (T, "i,,...A,,, W)) are the corresponding parameters depending on T' and W. and & the dimension of each sufficient statistic.
Following the work in [Khemakhem et al. 2020), ler X E Rd and ZCED E R" (n S d), we have the following theorem about the identifiability of CEDiVAE.
Theorem 4. Assume that the observational data are generated according to Eq. E.
8 with
(f, X) and the following hold: (1) The function in Eq. 7 injective. (2) The set (X E X (p((X) = 0} has measure zero, where Pe is the characteristic function of the density Pe defined Eq. 7 (3) The sufficient statistics Sr,j in Eq. are differentiable almost everywhere, and (Stj)ısigR are linearly independent
parameters
any subset of X of measure greater than zero. (4) There exists nkk + 1 distinct points (T, W W)o,,.. .... (T, nkk such that the matrix L = (A(Ti, W1) X(To, Wo), A(Tnk, Wak) - X(To, Wo)) of size nk x nk is invertible. Then (f) are ~ A-identifiabl..
the parameters
This theorem guarantees the identifiability of the generative model in Eq. Proof of the theorem is provided in Appx. B.2 and more related definitions are available in Appx. A.2 We also provide an analysis in Appx. B.3 to show the validity of the learned (transformed) CFD adjustment variable.
5 ATE ESTIMATION
After learning ZCFD. we can obtain unbiased estimation of the ATE by using the CFD adjustment. Due to the page limit, we show how this is done with discrete data in Algorithm I in Appx. C For continuous data, the methods would vary depending the data generation models. We provide the basic solution for data generation under a linear model in Appx. C For data generated under nonlinear model, we will explore this future work.
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6 EXPERIMENTS
In this section, we first demonstrate the correctness of representation learning, Then, we compare the performance of CFDiVAE with the benchmark methods for estimating causal effects and validate that CFDiVAE can unbiasedly estimate the causal effects and its performance is not sensitive to the change of the causal strength of the unobserved confounding variable. We also show its feasibility when the dimension of the learned representation is mismatched with the dimension of the ground truth CFD adjustment variable. Finally, we apply CFDiVAE to a real-world dataset and demonstrate its potential application. We also provide additional experiment the analysis of model identifiability in Appx. D.3
The source code available in the Supplementary Material.
6.1 EXPERIMENT SETUP
Table 1: Methods for comparison.
We compare CFDiVAE with a number of benchmark meth- ods, including traditional and VAE based causal effect esti- mation methods. Note that all those methods are based on the back-door adjustment. FindFDSet and ListFDSets (Jcong) et al. 2022 Wienöbst et al., 2022 are used in the experi- ments since they are front-door adjustment based methods.
Name
Open-Source
LincarDRL
Chemozhukoy et
2018)
EL
CausalFores!
Wagez Allay
2018
EeonML
ForesiDRL
(AIney al.
2019
LcanML
Xl.earner
Kinnzll
2119
KemelDML
Nie & Wager
202
MoonM
CEVAE
GilHbb
FindFDSet aims to find the standard front-door adjustment variable in given DAG, while ListFDett is an extension and it can enumerate the standard front-door adjustment sets.
TEDVAE
Zhang mi
202T
GilHab
FindFDSet Ocong c1
2022
GilHab
List
Jeong
GilFlbb
The implementations of CEVAE and TEDVAE are retrieved from the authors' GitHuu, the imple- mentations of other methods are from EconML (Keith Battocchi] 2019)), and the implementations of FindFDSet and FindFDSet are retrieved from the authors' GitHub as shown in Table A detailed description of the comparison methods is shown in Appx.
For evaluating the performance of CFDiVAE and the benchmark methods, we use the Estimation Bias |(B - 3)/3| x 100% as the metric, where 3 the estimated ATE and the ground truth.
The evaluation of estimated causal effects with un- observed confounding variables relies on synthetic datasets since no ground truth causal effects available for real-world datasets (Louizos et al. 2017; Zhang et al. 2021; Cheng al.) 20022). Synthetic datasets used in the evaluation are generated based on the causal graph (mechanism) shown in Fig 2 More details on data generation are provided in the Sup- plementary Material. To avoid the bias brought by the data generation process, we repeatedly generate 30 datasets with a range of sample sizes (denoted as N), including 0.5k, 1k, 2k, 4k, 6k, 8k, 10k and 20k. For each method, we report the average (mean) estimation bias over the 30 datasets, together with the standard deviation.
N 10k
0.2
Leamed Reposeeration Truli
0.0
Figure 3: Probability Density Functions of the ground truth and the learned representation, where the horizontal axis represents the value and the vertical axis represents the density.
6.2 CORRECTNESS OF THE LEARNED REPRESENTATION
In this section, we conduct experiments to validate the correctness of the learned representation. Since we use synthetic datasets, we know the ground truth of the CFD adjustment variable. To evaluate the correctness of the representations learned by CFDiVAE, we compare the probability distribution of the learned representation against the distribution of the corresponding ground truth
CFD adjustment variable. Due to the page limit, we only show the result of N=10k. As shown in Fig. 3 the distribution of the learned representation is close the distribution of the ground truth, which indicates that CFDiVAE can learn accurate representation of the CFD adjustment variable. More results reported in Appx. D.2

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Table 2: The estimation bias (%) of CFDiVAE and comparison methods under different N values.
0.5k
lk
2k
4k
6k
8k
10k
20k
Lin
21,90 a 5.13
21.56 3.82
21.47 3.28
2.08
21.59
# 1.78
21.88 + 1.41
21.89 = 1.31
21.38 0.90
CausalForest
2.233
21.39
21.63
1.80
21.88
1.33
21.94
ForestDRL
21.90 4.99
21.58 3.55
3.336
21.79
21.54
1
1.80
21.88
1.38
21.89
1.28
2
XLearn
5.14
21.83 3.33
21.59
21.86
1.39
21.88
= 1.30
2
KemeIDML
19.57 5:38
19.63
19.79
20.38 2
20,24
1.35
20.64
1.25
20.27
CEVAE
102.63
104.31 7.82
20.50
3105 = 4.95
2.09
23.97 6.05
21.29
6.81
2.77
TEDVAL
98,91 17.37
70.73 a 16.94
26.67 3.588
24.63 a 2.28
22.84 1.85
22.67 a: 1.61
22.63 a 1.23
2
FindFDSet
15.79 8.96
16.15 7.77
6.
16.58 + 3.11
15.64 + 2.40
16.44 2.36
16.49 + 2.65
16.53 + 1.63 16.39 + 1.60
LiatDDett
15.79 + 8.82
15.97 7.72
16.20
+ 6.10
16.43 + 3.10
15,52 + 2.42
16.31 * 2.32
16.36 + 2.60
CFDIVAE
86.29 a 6.21
39.72 + 31.47
8.87 10.68
4.57 3.03
2.58 1,96
2,32 + 1.47
2.97 2.09
2.14 # 3,38
6.3 PERFORMANCE OF ATE ESTIMATION
In this section, we evaluate the performance of CFDiVAE in estimating the ATE compared with benchmark methods. As demonstrated in Table 2 CFDiVAE outperforms all other comparison methods when the sample size is 2k or larger. This outcome is anticipated. All traditional and VAE-based comparison methods employ back-door adjustment to estimate ATE, utilizing W as the back-door adjustment variable. The estimation bias for comparison methods arises from the unobserved confounding variable U. To achieve unbiased estimation based on back-door adjustment, all back-door paths between I' and Y must be blocked, which is unattainable due to the unobservable nature of U. Our proposed method, CFDiVAE, avoids the limitations of back-door adjustment.
For this set of experiments, the PC algorithm implemented in the causal discovery toolbox TETRAD (Ramsey et al. (0118) is used to generate DAGs as inputs for FindFDSet and ListFD- Sets. These two methods achieve better performance compared back-door adjustment methods. However, there is still some bias resulting from causal structure learning and errors introduced by proxy variables. We note that our method performs well only when the sample size is sufficiently large, which is also observed in other VAE methods.
6.4 IMPACT OF THE CAUSAL STRENGTH OF UNOBSERVED CONFOUNDING VARIABLE
We also conduct experiments verify the effectiveness of CFDiVAE with respect to different causal strengths of the unobserved confounding variable. For this set of experiments, the causal strength is varied by adjusting the coefficient of the path U + Y. The sample size for this experiment is fixed at 10k. We multiply a scaling factor to the coefficient (i.e., 3u,y) to realise the different causal strength levels of the unobserved confounding variable. For example, 0.0 means that there no unobserved confounding variable, and 2.0 means that the coefficient doubles the original value. The range of the scaling factor is set as [0.0, 2.0] and the step increment is set as 0.2.
The results are shown in Fig. 4 When the causal strength is zero, i.e., no unobserved confounding variable, the back-door adjustment based comparison methods each achieve their own best perfor- mance since in this case, all confounding variables observed and their performance is solely determined by their capabilities in correctly identifying or learning the correct back-door adjustment variable. With the increase of causal strength, there is clear downward trend in the performance of the back-door adjustment based comparison methods, indicating that the back-door adjustment cannot handle unobserved confounding variables. In contrast, the front-door based methods retain an unchanged trend in estimation performance. CFDiVAE achieves and maintains an estimation bias of around 3%. The result is expected as CFDiVAE is based on the CFD adjustment, which can cope with unobserved confounding variables.
6.5 SENSITIVITY TO REPRESENTATION DIMENSION
In real-world applications, a common situation that the dimension set for the representation does not match the dimension of the ground truth CFD adjustment variable. In this section, we analyse the sensitivity of CFDiVAE to the representation dimension. In the following, Dr represents the dimension of the learned representation, DR represents the dimension of the ground truth CFD adjustment variable, and Dx represents the dimension of the proxy variable. We apply CFDiVAE with various dimension settings, i.e., DR E {2, 42 8). The results are shown in Table 3 We see that CFDiVAE achieves its best performance in most sample sizes when D, = Dx, which is similar to the observation shown in (Wu& & Fukumizu, 2022). When Dr. # DR and Dr. * Dx, the performance
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50%
45%
LincarDRL
CausaLFocce oeest
ForestDKL
XLeam
KemelDML
40%
CEVAE
TEDVAE
DindFDSet
ListFDSets
CFDiVAE
35%
30%
25%
20%
15%
10%
5%
11
0%
0.2
Figure 4: Results with different scaling factor, where the horizontal axis represents the scaling factor and the vertical axis represents the estimation bias (%).
Table 3: The estimation bias (%) of CFDiVAE dimension mismatch different N values. The best results are shown in boldface.
Du-De-Dx
1k
8k
20k
82.31 + 8.83
11.99
10.7 17.07
9.52 3.08
9.86
2.54
10.35 + 4:25
9.88 1.36
2-2-
78.16 4.99
12.85
10.96
5.88
8.83 6.02
5.94 4.22
5.46 3.62
5.37 2 6.82
4.16 8.90
4-2-4
76.69 + 4.44
23.21 + 15.38
19.33 12.31
7,98 = 6.98
5.73 5.45
7.45 = 3.36
3.90 a 4.87
1.77 1.8
1-4-8
79.94 8.98
22.12 18.63
12.09 + 4.66
13.73 3.58
14.24
15.07 2.86
14.33 + 2.64
14.83 1.74
74.31
9.49
11.54 3.75
9.84
8.19
6.10 1.83
4-4-8
7.716
19.04 + 11.12
12.89 16.47
9.90 = 5.15
8.74 5.69
6.78 3.99
4.50
4,45 1.75
8-4-8
69.66 8.20
31.97 a 12.11
38.96 12.17
11.73 4.09
8.05 6.67
6.07 3.75
4.42 2.46
4.44 2.00
1-8-16
75.38 12.60
27.92 22.61
15.86 6.77
14.46 17.05
15.18 4.85
16.62 3.73
16.64 3.34
17.65 2.29
4-8-16
72.33
8.3.16
63.95 11.41
27.47 + 13.96
29.00 26.66
11.01 11.10
10:00 =: 7.61
6.27
6.69 + 4,29
7.42 3.14
of CFDiVAE also maintain acceptable level. In all cases, the performance of CFDiVAE is superior to the comparison methods (Appx. [D.4] more results). Hence, when the dimension of the ground truth CFD adjustment variable is not accessible, can safely set Dr Dx.
6.6 CASE STUDY ON A REAL- DATASET
In this section, apply CFDiVAE to detect discrimination the real-world dataset, Adult. The dataset is retrieved from the UCI repository (Dua & Graff 2017) and it contains 11 attributes about personal, educational and economic information for 48842 individuals. We the sensitive attribute sex as T, income as Y, and age, race and native_country as W.
With causality-based discrimination detection, consider that there is direct diserimination if the sensitive attribute has large enough direct causal effect the outcome (above given threshold r), and there is indirect discrimination if the sensitive attribute has large enough indirect causal effect on the outcome and the mediator is also a sensitive attribute (Zhang et al. 2018).
Appx. D.5 the green path represents the direct path from sex to income, and the blue paths represent the indirect paths passing through marital-statue status. The discrimination threshold 7 is set as 0.05. By computing the path-specific effects, [Zhang al.) (2017a) estimated that direct effect = 0.025 and indirect effect = 0.17,, which indicated no direct discrimination but significant indirect discrimination.
Following the causal network in Fig.
We aim to estimate the causal effect of sex income using representation learning and conditional front-door adjustment. We simplify the causal network (Fig. 7 Appx. D.5) to fit our method with a latent stereotype shown in Fig. g in Appx. D.5 The discrimination is not a direct result of sex, but a direct result the stereotype. We assume the proxy of the stereotype accessible, and they marital-status relationships. edu level, hours_per_weed occupation, workclass, in this example. Stereotype is not a standard front-door adjustment variable because the causal path from stereotype income is not blocked by However, the stereotype is CFD adjustment variable, since there
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no back-door paths from sex to stereotype and all back-door paths from stereotype income are blocked by age, race and native_country (adding sex into this adjustment set will not invalidate the result). By using CFDiVAE, obtain that ATE 0.176, which is consistent with the previous estimate (0.175). The direct effect is ignored since it is very small.
RELATED WORK
Over the past few decades, researchers have proposed many methods for estimating causal effects from observational data. These methods generally fall into three categories: methods based on back-door adjustment, instrumental variables (IVs) and front-door adjustment, respectively.
Methods based on back-door adjustment are the most widely used, and most of these methods need to assume that all confounding variables are observed. For example, several tree-based methods (Athey
& Imbens 2016 Su et al. 2009 Zhang al. (2017b) have been designed to estimate causal effects by designing specific splitting criteria; meta-learning (Künzel et al. 2019) has also been proposed to utilise existing machine learning algorithms for causal effect estimation. Recently, methods using deep learning techniques to predict causal effects have received widespread attention. For example, CEVAE (Louizos et al. 2017) combines representation learning and VAE to estimate causal effects; TEDVAE (Zhang et al. 222T)) improves on CEVAE and decouples the learned representations to achieve more accurate estimation; 3-Intact- VAE (Wu & Fukumizu 2022) based on the identifiable VAE technique which models a prognostic score with a latent variable to identify and estimate individualised treatment effects in situations of limited overlap.
Methods based on IVs have also received lot of attention. Most IV based methods require users to nominate valid IV, such as the generalised method of moments (GMM) (Bennett al. 2019). kernel-IV regression (Singh et al. 2019) and deep learning based method (Hartford et al. 2017
When there are no nominated IVs in the data, some data-driven methods are developed to find (Yuan
al. 2022) synthesise (Burgess & Thommson 2013 Kuang et al. 2020) an IV or eliminate the influence of invalid IVs by using statistical strategies (Guo et al. 2018 Hartford et al. 2021).
Front-door adjustment approach is rarely studied in the literature. There are only few methods for finding appropriate adjustment sets by following the standard front-door criterion (Jcong et al. 2022 Wienöbst al. 2022). These methods require given DAG and aim find and enumerate possible standard door adjustment variables in the DAG.
The methods based on back-door adjustment cannot handle unobserved confounding variables. IV based methods can cope with unobserved confounding variables, but the availability of known IVs itself strong assumption. Existing front-door adjustment methods restricted the standard front-door adjustment variable. We relax the restriction of standard front-door adjustment and develop CFDiVAE to learn a CFD adjustment variable for unbiased ATE estimation in the presence of unobserved confounding variables.
CONCLUSION
Summary of Contributions. In this work, we proposed the conditional front-door adjustment, which less restrictive than the standard front-door adjustment and proved that the average causal effect is identifiable via the proposed conditional front-door adjustment. Our proposed method CFDiVAE leverages the identifiable VAE technique learn the representation of the conditional front-door adjustment variable from data directly, and we have shown that the identifiability of the learned representation theoretically guaranteed. Extensive experiments have demonstrated that CFDiVAE outperforms the benchmark methods. We have also shown that CFDiVAE is insensitive to the causal strength of the unobserved confounding variable. Furthermore, the case study has suggested the potential of CFDiVAE for real-world applications.
Limitations Future Works. The efficacy of the CFD adjustment and the CFDiVAE method rely on set of assumptions. We acknowledge that such assumptions may not hold in some scenarios. Notably, the robustness of CFDiVAE is largely dependent on the availability of a substantial sample size. Presently,
solution is only for continuous data within linear models and discrete data. In the future, our endeavors will be directed toward relaxing these assumptions. Additionally, we aim to evolve the CFD adjustment technique for application continuous data with nonlinear models.

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ACKNOWLEDGMENTS
We wish acknowledge the support from the Australian Research Council Discovery Project 200101210. Ziqi Xu is supported by the University Presidents Scholarship (UPS) of the University of South Australia. Kui Yu is supported by the National Natural Science Foundation of China under Grant 62376087. We are grateful for the valuable feedback provided by all anonymous reviewers.
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A BACKGROUND
A.1 CAUSALITY
Let - (V, E) be directed acyclic graph (DAG), where V is the set of nodes and E is the set of edges between the nodes. A path π between nodes V1 and V, comprises sequence of distinct nodes < VV...., Vn > with every pair of successive nodes being adjacent. A node V lies on the path if V belongs to the sequence < Vi,..., Vn >
A path is causal if all edges along it all in the same direction such as V1 - Vn. A path that is not causal is referred to as a non-causal path.
Assumption (Markov Condition (Pearl 2009)). Given a DAG g = (V, E) and P(V), the joins probability distribution of V, 9 satisfies the Markov Condition vV; V, V, is probabilistically independent of all of its non-descendants, given Pa(Vi), the set of all parent nodes of V..
Assumption 2 (Faithfulness (Spirtes et al., 2000)). A DAG g (V. E) is faithful to P(V) iff every conditional independence present in P(V) is entailed by G and satisfies the Markov Condition. P(V) is faithful to G isf there exists G which is faithful to P(V).
When the Markov condition and faithfulness assumption are satisfied, we can use d-separation to read the conditional independence between variables entailed in the DAG g.
Definition 5 (d-separation (Pearl 2009)). A path in DAG is said to be d-separated (or blocked) by a set of nodes Z iff (1) the path π contains a chain V; + Vk + Vj or a fork V, t Vv - Vú such that inverted fork (or collider) V, Vi Vj
the middle node VE is in 7, or (2) the path ıt contains
such that Vg is not in and no descendant of Vg is in Z.
Let =
DAG, and P(V) is the probability distribution over V. In the DAG G, a set of nodes Z is said d-separate V; and Vj if and only if Z blocks every path between V; to Vj: otherwise, a set of nodes is said to d-connect Vú and Vj. When the Markov Condition and Faithfulness assumption satisfied by G and P(V). (V; JL VG Z) if Z d-separates V; and Vj. and (V; JK V2 Z) d-connects V, and V,,
-V, (V, E)
Theorem 5 (Rules of do-Calculus (Pearl 2009)). Let G be the DAG associated with causal model, and let P(-) stand for the probability distribution induced by that model. For any disjoint subsets of variables T, Y, 7, and W, we have the following rules.
Rule I. (Insertion/deletion of observations):
Ply do(t), 2, w) Ply do(t), if T Z (T in 97.
Rule (Action/observation exchange):
P(y do((), do(z), w) P(y do(l), w)) if (Y 11 2 T in GTZ- Rule (Insertion/dellttion of actions):
Pyy do(t), do(=), w) Pyy do(t), w), (Y 1L Z T, W) in S7,Z(W)
where Z(W) is the nodes in Z that are not ancestors of any node in W in GT-
A.2 MODEL IDENTIFIABILITY
We define two equivalence relations on the set of parameters O.
Definition 6. Let ~ be the equivalence relation on θ defined as follows:
A) ~ (f.S, X) +
=A,c (((F(z))) = (x)) c, Vx E X,
where (f, S, 7) are the parameters obtained from some learning algorithm that perfectly approximates the marginal distribution of the observations, A is an invertible nk x nk matrix, e is a vector, and X is the domain of X.
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B PROOFS
B. PROOF OF THEEREM
Proof. We compute P(y do(()) by using Theorem under the DAG 9 in Fig. Fig. 5 shows the subgraphs that are needed for the derivations in the following.
P(y do(t)) can be expanded as:
P(y do(t)) Σ
P(ZCro do( ECED, do(())
(10)
2C1
We first compute Pyy ECED. do(t)), which can be expanded as follow:
Ply ECED, do(()) Σ P(y do(l), ECED, w) ZCrD)
(11)
The first part: P(y do((), ZCFD, w) Ply (((z)))d(()cF); w); since (Y IL ZCFD T, W) in G Zez (Rule 2 in Theorem
P(y do(l), do(=cro), = P(y do (CFD), since (Y ZCFD, W) g Iemi (Rule 3 Theorem 5
P(y do(zcrD), w) Σ P(y (, do(zcrd); w)
Ply do(zcED); 1', w) Ply ECFD, ,,),
since (Y ZCFD ()) Gz (Rule 2 in Theorem
P(t' do((cro), w) = P(t') w),
since (T IL ZCFD W) 9 Zem(W] (Rule 3 in Theorem 5
P(y do((), 2CFD, w) = Σ Ply [', () w)
(12)
The second part: P(w do(t), ECFD) = P(w, ZCFD do(t)) P((CF) | do(t))
P(w, #CED do(()) = P(ZCED I, P(w)
P(zFFD (,w)P(w) P(EcED do(t))
P(w] (o((), zrd)
(13)
P( t, w) P(w)
Thus, Ply ECED, do(t))
Σ
P(y I 2CED, P(t( w)
(14)
P do(t))
w..t
We take Eq. 14 into Eq. and get,
P(zCD t, P(w) P(ECED do(())
PΦy do(t))
Σ
P(ZFED do(t)) Plylt 2 p) w)
ZCeD
P(zcrD 1, w) P(w) P( do(())
Σ
P(ZCED do(()) 2CFD P(t w)
(15)
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W
X
W
X
W
X
CFD
CFFD
CFD
SZCFD
W
x
X
ZCFD
ZCFD
U
ST,ZCED
GTZCED
Figure 5: Subgraphs of G used in the derivation of causal effects.
Finally, get,
Ply do(t)) = Σ
Plylt', <CFD, P(t P(CCED t, P(w)
(16)
where e' is distinct realisation of treatment.
B.2 PROOF OF THEOREM
Our proof is based on the proof of Theorem in (Khemakhem et al. 2020
Proof. Suppose have two sets of parameters (f, A) and S, such that po (X, ZcrD T, W) = (X ZerD T, W). Then:
Ps,A (ZFD T, W)P{(XX ZCFD)
Bam
PS [T,)), ZCFD)dZer
ZeFD
P8.A Ps,1(Zem (ZCD T, W)pe(XX - ((Zcr)))dzo
Zero
P T, -
/ (X) :())p(XX - X)dx
= -/ ps ((X) |T (X04 X)dx
(17)
We denote the volume of matrix vol A, and when A is full column rank, vol VdetAT A. denotes the Jacobian, and we make the change of the variable X = f(ZerD) on the left hand side, and X f(7CDD) the right hand side.
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From Eq. 17 we have:
Pf '(X) TT
= (X) *(X) | ()vol--(X)
(18)
By taking the logarithm on the both sides of Eq. 18 and replacing Ps,A by its expression from Eq. 8 we get:
log vol./,- +
Σ
(log Qi(ft *(X) log (), +
Σ j=1
S (Z,
i=1
log vol.4-(X) + Σ
(log Qih '(x)) - log ()( +
Bis (T,)) (19)
Let (T, W).,.................. (T, nk be the points provided by Theorem (4), and define X(T, W) = A(T, W) XToo Wo). We plug each of those (T, W", in Eq. 19to to obtain nk 1 such equations.
We subtract the first equation for (I, W) from the remaining nk equations to get for L,... nk:
(T Wo) Zi(Ti Wi)
(X)), A(Ti, Wi)) + Σ log
Σ log (To, Wo)
(x)), ((T) Wi))
(20)
ZT Wi)
Let L the matrix defined in Theorem (4), and L similarly defined for (L is not necessarily (To. Wo (T. and the vector of all for 1, 1.... nk.
invertible). Define bj Σ, log 2T Wo (T;
Then, Eq. 20 can be rewritten as:
L
E' +
(21)
We multiply both sides of Eq. 21 by the transpose of the inverse of L from the left to get:
= +c,
(22)
where = L- and = L -.
By definition of S and according Theorem 1 (3), its Jacobian exists and is an nkk x n matrix of rank
n. This implies that the Jacobian o f² exists and of rank a and so is A. have two cases: (1) If & = A is invertible since A is matrix of rank n; (2) If k > 2, A is also invertible. We have the following proof for (2):
Define X f ((X) and S,X))) (X., (X.)..)). For each [I,..., n] there exist points ,.,..., X such that (x(() (X are linearly independent.
Firstly, we proof the above statement. Suppose that for any choice of such k points, the family (S,()),,.,,,(
never linearly independent. That means that S. of R of dimension at most
(IR) ) is included in subspace Let h non zero vector that orthogonal to S(RR). Then for all 0. By integrating find that (S, (R), h) const. Since this is true
X R. we have (R), h) for all e R and for a * 0, we conclude that the distribution is not strongly exponential, which contradicts hypothesis.
Secondly, we prove is invertible. Collect those points into vectors (X!,.., X*), and con- catenate the à Jacobians Jx(X') evaluated at each of those vectors horizontally into the matrix Q (Js(X)), ..., Js Js(X*) (X*)) (and similarly define Q as the concatenation of the Jacobians of S(F {(X)) evaluated at those points). Then the matrix Q is invertible. By differentiating Eq. 22 for each X we have:
= ==Q
(23)
The invertibility of Q implies the invertibility of A and Q, which completes the proof.
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B.3 THE VALIDITY OF THE LEARNED CFD ADJUSTMENT VARIABLE
Let us consider the linear causal relationships where affects Y through the (conditional) front-door adjustment variable ZCED, as represented follows:
ZCFD = aT + <
(24)
Y = 3ZCFD +
(25)
where EZOED and ey are the error terms, assumed to be mean-zero, independent random errors. The causal effect of T on Y is commonly computed as the product of the path coefficients, i.e., ATE o;3.
Following the discussion in Section 4.3 our proposed method CFDiVAE guarantees the model identifiability, which means that the learned latent CFD adjustment variable ZCFD = (X) ) is equal to the ground truth CFD adjustment variable ZCFD = f (X), up to a linear invertible transformation (through an invertible matrix A) and point-wise nonlinearities (in the form of S and S), that is,
ZCFD A7
(26)
Substituting the above expression for ZCFD into Eq. 24 we get,
AZCFD = aT + EZerD
(27)
Multiplying both sides of the Eq, 27 by A we get,
A AZCED ZCFD TA + A A
(28)
Substituting the expression for ZGrD shown in Eq. 26 into Eq. 25 we get,
Y BAZCED EY
(29)
Eq. 28 shows that the causal effect of T on ZCFD is aA and Eq. 29 shows that the causal effct of ZCFD on is 3A, the causal effect of T Y via ZOFD is the product of the coefficients, ie., A 3 A3- a,A A = ax3.. This means that using the learned (transformed) CFD adjustment variable will give us the same ATE as using the groundtruth CFD adjustment variable.
C DESCRIPTION OF ATE ESTIMATION
For discrete data, we show the solution in Algorithm | For the continues data with linear model, we assume the data generation as followed,
We see that the solution is aligned with Eq.
Zer = CZem + Br, T + 3w. W + é
(30)
= ey + Sy.zem ZCDD + Sw,y W + uu,y + ey:
(31)
where denotes intercept, denotes error, and ZCED is the learned representation by method.
The ATE of on Y is the product of coefficients 34 Zem and By Zem The coefficients are obtained
with the following process (Tchetgen & Shpitser 2012 Bar 2018):
Step 1: ZerD is regressed on and W. This gives us the coefficient 3T. and E| T, W).
Step 2: Using E|ZerD T. W), we estimate the noise CZcw ZEED E|ZerD T, W]. Regress eZem on Y. This gives us the coefficient 3y.2em- Noise c2mm is only introduced at ZeED. and is independent of the unobserved confounding variable U.
Although the solution for continuous data with the linear model does not mirror Eq. 3 it is still consistent with the (conditional) front-door adjustment shown in Theorem 3 As know, the
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Algorithm 1 CFD Adjustment for ATE Estimation with Discrete Data
1: Input: T, Y. W, and ZCED
2: Output: ATE of on
3: Initialise E[ [do('' 0)] and B[Y]do(T 1)]
4: for each value do
5: Estimate the marginal probability P(t)
6: end for
7: for each combination of T, W, and ZerD do
8: Estimate the conditional probability P(zceolt, w)
9: end for
10: for each combination of T, W, Y, and ZCFD do
Estimate the conditional probability P(ylt, w, 2CFD)
12: end for
for each value Y. W. and 7CTD do
14: for each value of T do
15:
doo(' = 0)] += yPyyt &CFD. w) P(<CFD T = 0, Pttw)
16:
[To = 1)] += yP 2CED. w) ((CFF T 1, ((t
17:
end for
18:
end for
19:
Compute the ATE as E|Y dT 1)]
do = =0]
20:
return ATE
front-door adjustment consists of two back-door adjustments. In our case with continuous variables, these two back-door adjustments are implemented by regression adjustments. as shown Eq. 30 and Eq. 31 Specifically, for the first backdoor adjustment, we use regression adjustment to get the causal effect of 7 on ZCrD (i.e., (r, adjusting on W, as shown in Eq. 30 and Step of the process; for the second backdoor adjustment, we also use regression adjustment as shown in Eq. [3 to obtain the causal effect ZCFD on adjusting on W and However, for this regression adjustment, since U is unobserved, we follow the approach in (Tchetgen & Shpitser] 2012 Barr 2018) to firstly obtain the error e EZTD. which is independent of U, and regress EZem on instead. In this way, we can eliminate the confounding bias caused by U. This is shown in Step of the process.
D EXPERIMENT
D.1 DESCRIPTION OF THE COMPARISON METHODS
LinearDRL Chernozhukov et al. 2018): A double machine learning estimator with low- dimensional linear regression the final stage.
CausalForest Wager Athey (2018): causal forest estimator combined with the double machine learning technique for conditional average treatment effect estimation.
ForestDRI. Athey et al. (2019): generalised random forest and orthogonal random forest based estimator that uses doubly-robust correction techniques to account for covariates shift (or selection bias) between the treatment.
XLearner Künzel et al. (2019): A meta-learning algorithm that utilises supervised learning methods (c.g., Random Forests and Bayesian Regression) for the analysis of conditional average treatment effects.
KerneIDML Nie & Wager (02II: A specialised version of the double machine learning estimator that uses random fourier features and kernel ridge regression for the analysis of conditional average treatment effects.
CEVAE Louizos et al. 2017): A deep learning based method that leverages latent variable modelling, specifically Variational Autoene,,er, to estimate causal effect from observational data, even in the presence of latent confounders.
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N 0.5k
N- 1k
N 2k
0.3
0.2
0.1
0.0
N 6k
N Sk
0.3
0.3
02-
0.2
0.2
0.0
N 10k
N 20k
50k
0.4
0.3
0.2
0.1
0.1
0.1
Learned Representation
Ground Truth
Figure 6: Probability Density Functions of the ground truth CFD adjustment variable and the learned representation, where the horizontal axis represents the value and the vertical axis represents the density.
TEDVAE Zhang et al. 22021: A deep learning based method that learns the disentangled repre- sentations of confounding, instrumental, and risk factors using VAE for accurate treatment effect estimation.
D.2 MORE RESULTS OF THE EXPERIMENTS SECTION 6.2
In this section, we compare the probability distribution of the learned representation of the CFD adjustment variable with the distribution of the ground truth CFD adjustment variable under different sample sizes. As shown in Fig. 66] the distribution of the learned representation close to the ground truth distribution, which indicates that the proposed method CFDiVAE can learn the accurate representation of the CFD adjustment variable from its proxy.
D.3 ANALYSIS OF MODEL IDENTIFIABILITY
Our proposed method CFDiVAE takes T and W as additional observed variables to approximate the prior p( ZCFD T. W). In this section, we apply two partially identifiable VAE models, i.e., T- CFDiVAE and W W-CFDiVAE, and the original VAE comparison methods. 7-CFDiVAE is partially identifiable VAE model that takes T as the additional observed variable to approximate p( ZerD T); W-CFDiVAE is partially identifiable VAE model that takes W the additional observed variable to approximate p(Zerp W): the original VAE does not take any additional observed variable to approximate yl (CCD) The ELBOS for these methods defined as:
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Table 4: Results of model identifiability analysis.
0.5k
1k
2k
4k
6k
8k
10k
20k
MyA (Bq. 54
88,93
14.67
58.41
22.46
77.71
9.117
79.89 + 7.00
32,92 a 31.20
7.66
5.72
4.55 9.19
MI-CFIVAE (5g
70.09 + 20.50
77,48
12.65
81.48 2 8.19
15.94
22.84
10.25
22.99
4.60
3.89 4.56
1.77 4 1.41
Mw CRDWAE
68.89
18.89
85.83 4.64
29.13 21.00
5.49 3.83
6.49
17.82
3.14
3.14 2.35
1.69
1.55
Merrva (Eg
MI-CEDIVAE E., P(p (( X
7CFD)]
DKL ( CFD
T, X) (CFFD 7)
(32)
MW-CFDIVAF E, 1l MVAE El,
ZCFD)
DKL. ( ZCFD
W, X) || ((ZD W)]
(33)
[CFD]] DKL ( ZCFD
X) p(ZFFD)
(34)
The results are shown in Table We see that CFDIVAE achieves the best performance since it
uses all additional observed variables. The performance of 7-CFDiVAE is slightly lower than the performance of CCFDiVAE since W has more additional information than T (the dimension of W is generally higher than the dimension of 7)). The original VAE which does not use any additional observed variable achieves the worst performance.
D.4 MORE RESULTS OF THE EXPERIMENTS IN SECTION 6.5
In this section, evaluate the performance of CFDiVAE and the comparison methods when the dimension for the representation does not match the dimension the ground truth CFD adjustment variable. Table Table shows the results for DR 2, Table Table[5b] shows the results for DR 4, and Table 5c shows the results for DR 8. We note that CFDiVAE achieves its best performance when Dr. Dx in most sample sizes. Hence, in general case, can safely set Dr. Dx get acceptable causal effect estimation.
D.5 CAUSAL GRAPHS INVOLVED IN SECTION 6.6
race
age
SeX
coive_couutry
marital status
edu level
occupation
hours_pee per week
workelass
relationship
income
Figure 7: The causal network for the Adult dataset: the green path represents the direct path, and the blue paths represent the indirect paths passing through marital status (Zhang et al. 2017a).
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Table 5: The estimation bias (%) of CFDiVAE and comparison methods under different values. CFDiVAE-D, DD----D--D-DD denotes apply CFDiVAE to specified setting, where D2 represents the dimension of the learned representation, DR represents the dimension of the ground truth CFD adjustment variable, and Dx represents the dimension of the proxy variable.
(a) Estimation bias (%) when DR
0.5k
1k
2k
4k
8k
10k
20k
LRL
27.05 +:03::7 7,56
24.58 6.39
26. + 4.19
24.53 3.01
25.01 2.90
24.92 + 1.61
25:40 1.56
25.42
1.12
CausalForest
ForesiDLL
2
2.35
26.20
2.154
24.95
24.96 1.60
25.38
25.43
1.13
XLeam
27.15
24.62 6.29
2.994
2.07
25.02
24.94 1.59
25.37
25:40 1.11
KemelDDLL
2.83
22.01
6.29
24.23
23.57 3.05
23.57
24.06
1.58
24.22 1.08
CEVAE
102.05 2:222
104.47
9.799
104.04
22.15
7.16
39.88
10.46 32.34
23.62
11.21
34,20
6.46
TEDVAE
93.05
21.04 69.77
24.61
11.13
29.14 2.81
26.55 2.99
1.50 25.98
26.22
26.03
1,30
11.99
+ 17.07
9.52
9.88 1.36
CFDiVAE-2-2-4
78.16 4,99
12.85
10.96
6,90 + 5.88
8.83 + 6.02
5.94 4.222
5,46 + 3.62
5.37 + 6.82
4.16 8.90
CFDIVAE-4-2-4
76.69 4.44
23.21 15.38
12.31 # 19.33
7,98 + 6.98
5.73 5:44
7.45 + 3.36
3.90 4.87
1.77 1.08
(b) Estimation bias (%) when Dg 4.
0.5k
Ik
4k
6k
10k
20k
LinearDRI.
30.76
33.04
3.30
31.97
1.98
CausalForest
32.40
31.34 11.30
33.41
7.79
32.88
4,225
+2.49
33.18
32.11 3.36
31.94
1,93
ForesDRL
32.42
31.21
32.78
8.07
32.28 4.31
3.74
32.92
3.44
32.01
1.97
XLeam
32.88 12.93
31.14 10.54
32.81 + 7.65
4.37
30.32 3.69
32.98 3.33
32,00
KemelDML
28.12
27,83
29.48
24.24
31.19
30.48
3.17
30.60
1.98
CEVAE
10.94
TEDVAE
99.99 = 16.35
75.63 + 40.63
44.30 21.22
35.52 3.88
35.33 3.24
33.82
3.44
33.12 12.22
LYDIVAE-I-4-3
79.94 8,98
22.12 18.63
12.09
4.62
15.07 4.44 3,55 2.86
14.33 2.64
14.83 1.74
4
74.31 + 6,90
16.38 8:40
9.49 + 5.02
11.54 3.75
9.84 a
8.19 4.86
6.85
6.10 1.83
73.16 C-D
5:70
19.04 11.12
12.89 16.47
5.10
8.74 5.69
3.92
4,50 2,70
4.45 1.75
CFDIVAE-8-4-8
69.66 + 8.20
31.97 12.11
38.96 12.17
11.73 4.00
8.05 = 6.67
6.07 3.75
4
4.44 2.00
Estimation bias (%) when DR
1k
2k
10k
LincarDRL
54.02 30.87
48.22
47.84
13.83
9.29
46.80
7.52
47.17 4.97
47.56
CausslForest
$1.95
49.64 1 17.88
47.48
13.26
47.06
1 9.55
47.65
= 7.71
2.13
48.85
47.47 3.54
FuresDDLL
31.22
47.92
17.51
47.76 13.55
48.46 9.33
46.74
7.53
7.53
47.20 4.83
48.60
47.59
XLeam
30.76
48.08
47.74
4..26
46.88
4..15
4622
4.224
KemelDML
+:.33
43.16
17.91
42.93 4: 13.83
4 7.86
44.59 a: 4.881
46.30
106.92
TEDVAE
22.10 98.58
74.62 + 55.07
60.77 + 37.72
80.72 11.28
7.74 59.95
51.93
52.78 6.30
49.85
CEDIVAE I 16166
75.38
12.60
27.92
22.61
15.86 6.77
14.46 17.05
15.18 4.85
16.62
16.64
CFD
72.33
11:25
19.45 + 13.21
12.89 + 10.46
11:71 12.01
12.28
5.88
10.36
9.26 5.88
7.47 +
CFDIVAE-8-8-16
63.95
11.1
27.47
29.00
11.01 11.10
10.00
7.61
6:299
3.14
CHDIVAE-16-8-16
56.84 2 11.64
22.67 + 12.68
17.15 19.49
10.60 9.62
7.86 2: 6.56
8.28 2:7.5
6.23 a 4.15
2.99
age
native_country
sex
stereotype
income
(marital_status, ... relationship)
Figure 8: Simplified DAG for Adult dataset.
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E REPRODUCIBILITY
Table 6: Details of the parameter settings in CFDiVAE.
Parameter
Value
Parameter
Value
Parameter
Value
Reps
30
Num Layers
3
wd
le-4
Epoch
30
Ir Ird
le-3
Batch_Size
256
0.01
In this section, we provide more details of the experimental setting and configuration for repro- ducibility purposes. CFDiVAE is implemented in Python (Van Rossum & Drake Jr 1995) libraries PyTorch (Paszke et al., 2019) and Pyro (Bingham et al., 2019). The code for data generation is written in R (R Core Team 2021). We provide the parameter settings of CFDiVAE in Table 6 The descriptions of the major parameters are provided below:
Reps: the number of replications each set of experiments runs.
Epoch: one Epoch is when an entire dataset passed forward and backward through the neural network once.
Batch_Size: the number of training examples present in single batch.
Num Layers: the number of hidden layers.
Ir: the learning rate.
Ird: the learning decay.
wd: the weight decay.
22


Reviews:
[{"review_id": "gFWIqtG4Co", "reviewer": "Reviewer_ysmt", "summary": "This paper proposes a new method of Average Treatment Effect estimation and representation learning through the Conditional Front-Door Criterion (CFD) combined with a variational autoencoder model (CFDiVAE).\n\nThe authors start with the standard front-door criterion scenario and relax it by allowing backdoor paths between the treatment and the mediator as well as between the mediator and the outcome. The second relaxation is allowing unmeasured confounding variables between treatment and outcome. A background on the front- and back-door criteria is given after which the authors proceed to state the first contribution of the paper, the Conditional Front-Door adjustment (CFD). The CFD is the same formula as the standard front-door adjustment but with additional conditioning on W (the backdoor paths between the treatment and the mediator). Subsequently, the second contribution of the paper, the CFDiVAE model, is described. It follows existing generative VAE work (such as (Louizos et al.)) and adjusts it to the assumed generative model of CFD. CFDiVAE learns a representation of the mediator, given observed back-door paths (confounding variables) between the treatment and the mediator, and allowing for potential unobserved confounding variables. A thorough analysis of model indentifiability is provided. The paper concludes with a series of experiments. First, estimation bias of ATE is compared to competing causal-VAE methods. CFDiVAE beats the other methods for large enough dataset sizes. Secondly, a short assessment of CFDiVAE on a real dataset is performed.", "questions": "Would considering relaxations of the front door criterion as mentioned in Pearl, Causality, 2009, page 83, Figure 3.1, result in any major changes to the model (I assume the inference model might have to be adjusted)?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper tackles the important problem of causal inference with observational data. It is well-written and easy to follow. The methods the paper builds on (front-door criterion) are useful in many areas, from model interpretability to algorithmic fairness. The first contribution (CFD) is a correct relaxation of the front-door criterion, while the second one (CFDiVAE), while building on existing VAE methods, is well analyzed. . The experiments show that, assuming a linear generative model with a mediator, CFDiVAE estimates ATE better than methods that do not model a mediator directly.", "weaknesses": "The main weakness of the paper is its limited novelty.\n\nThe main contribution, the Conditional Front-Door Adjustment is a straightforward extension of the front-door criterion (where everything is conditioned on the backdoor path W between the treatment and the mediator; W is assumed to be observed). It, in fact, boils down to plugging in the front door criterion into the backdoor criterion.\n\nOne could argue that any concrete DAG which allows for the relaxation of the front door criterion with the do calculus (as mentioned in Pearl, Causality, 2009, page 83, Figure 3.1) can form a basis for a similar model.\n\nThe second contribution, the CFDiVAE generative model, builds heavily on similar models present in literature. The generative and inference part follow standard VAE procedures adjusted to the assumed underlying SEM.\n\nWhile the specific combination of CFD with a generative VAE-based model is novel, it also does seem incremental.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ciS7EVVzNT", "reviewer": "Reviewer_A5AY", "summary": "In this paper, the authors propose a method to relax some of the restrictions by introducing the concept of conditional front-door (CFD) adjustment and developing the theorem that guarantees the causal effect identifiability of CFD adjustment. The authors further propose CFDiVAE to learn the representation of the CFD adjustment variable directly from data with the identifiable Variational AutoEncoder. They apply CFDiVAE to a real-world dataset to demonstrate its potential application.", "questions": "The authors could apply the method to additional real-world datasets, conduct a comparison with existing methods, and provide the analysis to strengthen the paper.  The author also needs to address the errors in the section reference.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The writing of the paper is good.  The method potentially has broad applications.\n2. The authors provide theoretical analyses for the proposed approach.  The reviewer did not check the details due to the review time limit, but it looks sound.", "weaknesses": "1. Additional results on real-world datasets and analysis could strengthen the paper.\n\n2. The section references in the supplemental file are confusing.  E.g., the title of section C.4 refers to ‘Section 4.5’ in the main text. However, section 4.5 in the main text does not exist. This causes trouble for the readers.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uPk91C8ByA", "reviewer": "Reviewer_s6a3", "summary": "**Post rebuttal update** The final version is much better than the original submission. I **raise the score to 6, conditional on** that the following point is fixed in the (possible) camera-ready version. *I believe one main concern of reviewer BvSP is also related to this.*\n\n>The exponential family prior eq11 is too general to be useful. Examine the identifiability in eq25 again, the sufficient statistics S are involved. Only in some cases like Gaussians, we can get around this problem. See [1] and reference therein.\n\nFor further improvement, my concern about the outcome regression on the Adult dataset was that here the proxies also affect the outcome. That was why I am asking “is the process in Sec 5 still applicable” for the Adult dataset?\n\n**End update** \n\nThe paper considers an extension of the front-door adjustment (FDA), called conditional front-door (CFD) adjustment, where there is a (conditional) variable that affects the adjustment variable in the FDA. An identifiable VAE is employed to recover the adjustment variable from the proxy. Directly based on previous results, the identifiability and adjustment equation of CFD is proved, and the identifiability of the proposed VAE is also proved. Experiments show favorable results.", "questions": "The exponential family prior eq11 is too general to be useful. Examine the identifiability in eq25 again, the sufficient statistics S are involved. Only in some cases like Gaussians, we can get around this problem. See [1] and reference therein.\n\nAs is observed in [1], setting the representation dim(Z) to dim(X) is better than dim(Z)=1, and sometimes even better than using the true dim of the adjustment variable. It is better to add experiments in this regard.\n\n*An important reference is missing*. Intact-VAE [1] is also based on iVAE and it is more related to the proposed VAE than any other compared methods, although it considers a different causal setting. In fact, many points mentioned in this review are discussed in [1], e.g., the noise level, the form of the identifiability, and the representation dim. And it is good to add experimental comparison.\n\n[1] Wu, Pengzhou Abel, and Kenji Fukumizu. \"beta-Intact-VAE: Identifying and Estimating Causal Effects under Limited Overlap.\" International Conference on Learning Representations (2022).\n\nI believe only the link W → Z is necessary but not W→T and W→Y. In fact, the Adult dataset does not have W→T.\n\nIt should be made clearer in Contributions and Conclusion that it is the FDA, which is a well-established result, that deals with unobserved confounding in this approach, and the VAE learns the adjustment variable but does not do the adjustment using eq3. The CFD is a slight extension of the original FDA.\n\n*Some unsupported/misleading statements*.\n\n“it is often impossible for a CFD variable to be given in practice” This seems strong and needs a reference/discussion.\n\n“there exist unobserved confounding variables and the standard front-door adjustment is no longer applicable” This reads as if FDA does not deal with unobserved confounding but the proposed VAE does.\n\n”the latent variable (i.e., ZCFD) refers to the variable that is not measured, but its information is captured by its proxy” As mentioned, a proxy does not capture all the information of the latent variable, though it might capture *enough* information for the causal effect; this requires challenging analysis and specific assumptions as in Miao 2018.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The front-door adjustment (FDA) is an important and understudied approach in the ML community.\n\nConsidering a conditional variable in front-door is a meaningful extension.\n\nUsing iVAE to recover the FDA variable is novel and interesting.", "weaknesses": "*It is very hard to recover the adjustment variable from the proxy (if the proxy noise is not very small)*\n\nThe proxy setting is challenging in itself, and the only rigorous identification result under this setting I know is (Miao et al 2018), which depends on two types of proxies and has several assumptions about the proxies.  ref Miao, Wang, Zhi Geng, and Eric J. Tchetgen Tchetgen. \"Identifying causal effects with proxy variables of an unmeasured confounder.\" Biometrika 105.4 (2018): 987-993.\n\nIn fact, the current theory is only meaningful under small proxy noise that is \\epsilon in eq10.  If we examine the identifiability of iVAE closely, we see, as in eq25 in the current paper, only $f^{-1}(X)$ is identified (up to deterministic transformation) but not the latent variable. And, $f^{-1}(X)=f^{-1}(f(Z)+\\epsilon) \\neq Z$ if $\\epsilon \\neq 0$. See the Questions for more related points.\n\nHowever, if $\\epsilon = 0$, then X and Z are related by an injective function, and it is not very meaningful to say X is a proxy of Z, because the whole point of proxy setting is that there is *unmeasured* information of Z not captured in X.\n\nI do not require to fix this weakness in the rebuttal, but it should be discussed and made explicit in the revised version.\n\n*Separated, limited, and unclear outcome estimation*\n\nThe connection between the front-door setting and the proposed VAE is weakened, because the outcome is not modeled by the VAE, and the outcome estimation is done in a standalone step, as in Sec 5. Importantly, how does the 2-step process here implement the adjustment eq3? How exactly the ATE is computed? Moreover, the adjustment and outcome variables follow linear models; this might not be an inherent limitation in the approach, but this limitation in the current work should be mentioned in the Conclusion. Finally, for such an important component of the approach, citing a web article makes me nervous, please find and cite an original reference (a research paper).\n\n*Experiments are not sufficient or clear.*\n\nThe major problem is that there are no comparisons to methods designed for the front-door setting. And this renders the good result in Fig 4 not very meaningful. I cannot see why we need to exclude FINDFDSET and LISTFDSETS because they require a known DAG; when you build the synthetic dataset, the DAG is known anyway? And I doubt there are no other methods for the FDA. If you really cannot fix this weakness, it is necessary to mention this in the Conclusion.\n\nAnother important issue is that, as in Table 1, the proposed method requires a huge amount of data to perform well. In previous work, the methods usually train on sample size < 1k, but the proposed method is worse than other methods when sample size < 1k. This should be examined seriously and would be a major weakness of the approach if this is a general observation. \n\nThe experiment on Adult is very confusing to me. First, the “proxy” X also affects Y as shown in the Appendix, can we still say this is a “proxy”? In particular, is the process in Sec 5 still applicable? Second, a reader cannot understand the discussion of direct and indirect causal effects without looking into the Appendix. You need to mention X (eg., marital status) also affects Y and is the “mediator” here. Third, it seems to me that “significant indirect discrimination … through the indirect paths via marital status” and “significant discrimination against sex through the stereotype” are contradictory but not consistent?\n\n*I will read the rebuttal and revised paper and raise my score to 6 if the issues/questions in Weaknesses are addressed. Some further points for improvement are as below.*", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YgNcVy9aqT", "reviewer": "Reviewer_BvSP", "summary": "POST REBUTTAL UPDATE\nOverall the paper has clearly improved during the rebuttal phase. The assumptions and limitations of the method are much better discussed and the changes in the experiments have improved them. That said, I fully concur with reviewer s6a3 that discussion on sufficient statistics with which the theory holds should still be included. I will update my score to 6.\nPOST REBUTTAL UPDATE END\n\nThe paper considers front-door adjustment for causal effect estimation. In particular, the setup involves the following variables: T (treatment), W (observed confounders), U (unobserved confounders), Y the outcome, Z (front-door adjustment variable), and X proxy variable for Z. The article makes two main contributions: 1) The article proposes a conditional front-door adjustment, which differs from the standard front-door adjustment by having an additional edge from the observed confounders W to the front-door adjustment variable Z. The paper shows how in this situation the standard front-door adjustment must be modified to estimate the causal effect (Theorem 3). The proof is based on the assumption that the front-door adjustment variable Z is observed. 2) The second contribution is a conditional VAE method for the situation where the front-door adjustment variable Z is not observed, to estimate it from the proxies X (conditionally on variables W and T).", "questions": "1) Could you comment on the weakness number 1, please?\n\n2) Section 6.2. Can you clarify what you mean by ground-truth density function for the representation? With simulated data, don’t you know the representation variable exactly for each data point, which would allow you to compare the estimated and the ground-truth values directly (e.g. using a scatter plot)?\n\n3) What is the dimension of the proxy variable in the simulated experiments?\n\n4) Can you clarify if you used the linear method in Section 5 to estimate the causal effect? How is this compatible with the conditional VAE model that parametrizes the model for the front-door adjustment variable with a neural network?\n\n5) What would happen if you assumed a larger than correct dimension of the latent variable?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1) The first contribution, the conditional front-door adjustment, seems intuitive, interesting, and useful. To the best of my knowledge this is novel (though I haven’t read all the papers about front-door adjustment). The conditional front-door adjustment is presented clearly and the proof seems mathematically correct.\n\n2) The VAE approach seems a reasonable attempt to estimate the front-door adjustment variable from the proxies.\n\n3) The empirical results demonstrate that the proposed approach clearly outperforms methods that do not do any front-door adjustment at all. Sensitivity analysis for assuming an incorrect number of latent variables (smaller than correct) was a nice bonus.", "weaknesses": "1) There seems to be a gap in the theoretical part. The model identifiability analysis in Section 4.3 shows that the front-door adjustment variable Z obtained with VAE is identifiable up to a transformation (more or less the result from Khemakhem, 2020). But Theorem 3 is based on the assumption that the front-door adjustment variable Z is observed. It is not clear if Theorem 3 is still applicable if it is used for a transformed Z.\n\n2) The empirical comparison includes as baselines only methods that do not use the proxies of the front-door variable at all, and consequently each of those has poor performance. It would be better to think how the proxies would be used if the new method did not exist, for example by using the proxies directly for front-door adjustment? The existing front-door adjustment methods (by Jeong, Wienobst) were not included in the comparison because they assumed that the DAG is known. However, also the present method assumes that the causal graph is the one presented in Figure 2 and all simulations assume this correct structure (except for a possible mismatch in the latent dimension).\n\n3) Some small inconsistencies in the notation, at least: Equation (4): LHS has T and W but RHS does not. Equation (6): formulas for the mean and variance have j on the LHS but not on the RHS.\n\n4) In the real-world example, the ATE estimate is 0.176, which is very similar to the previous estimate 0.175 (from Appendix C.5), which seems to imply that the new method does not provide too much novel insight in this case (but of course it is a good demonstration that it is consistent with previous estimates). In general, it would be nice to see some example where the heavy VAE machinery is really needed, which might require more complex, e.g., higher-dimensional, proxies, to make a convincing case.\n\nOverall, I liked the first contribution, the conditional front-door adjustment, but I found the conditional VAE a bit confusing, possibly breaking the theory, and whose usefulness was not demonstrated very convincingly. I’m not sure but that paper might have been stronger without the VAE aspect altogether. At the very least the possible gap in the theory should be addressed. If it is not possible to fix the gap, then the theory (regarding the identifiability of the VAE) could be moved to the supplement and replaced by more convincing empirical analysis, and the text could be updated accordingly.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)