Identifying Treatment Effects under Unobserved Confounding by Causal Representation Learning
Authors: Pengzhou Abel Wu
Kenji Fukumizu
Venue:  (2021)
Keywords: VAE
variational autoencoder
Representation Learning
treatment effects
causal inference
Unobserved Confounding
identifiability
CATE
ATE

Abstract: As an important problem of causal inference, we discuss the estimation of treatment effects under the existence of unobserved confounding. By representing the confounder as a latent variable, we propose Counterfactual VAE, a new variant of variational autoencoder, based on recent advances in identifiability of representation learning. Combining the identifiability and classical identification results of causal inference, under mild assumptions on the generative model and with small noise on the outcome, we theoretically show that the confounder is identifiable up to an affine transformation and then the treatment effects can be identified.  Experiments on synthetic and semi-synthetic datasets demonstrate that our method matches the state-of-the-art, even under settings violating our formal assumptions.

Under review as a conference paper at ICLR 2021
IDENTIFYING TREATMENT EFFECTS UNDER UNOB- SERVED CONFOUNDING BY CAUSAL REPRESENTA- TION LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
As an important problem of causal inference, we discuss the estimation of treat- ment effects under the existence of unobserved confounding. By representing the confounder as latent variable, we propose Counterfactual VAE, a new variant of variational autoencoder, based on recent advances in identifiability of representa- tion learning. Combining the identifiability and classical identification results of causal inference, under mild assumptions on the generative model and with small noise on the outcome, we theoretically show that the confounder identifiable up to an affine transformation and then the treatment effects can be identified. Ex- periments on synthetic and semi-synthetic datasets demonstrate that our method matches the statc-of-the-art even under settings violating our formal assumptions.
INTRODUCTION
Causal inference (Imbens & Rubin 2015 Pearl 2009). i.e, estimating causal effects of interven- tions, is fundamental problem across many domains. In this work, we focus on the estimation of treatment effects, e.g., effects of public policies consisting of binary labels for treatment/ / control (non-treated outcome, and other covariates. The fundamental difficulty of causal inference is that we never have observations of counterfactual out- comes, which would have been if we had made another decision (treatment or control). While the ideal protocol for causal inference randomized controlled trials (RCTs), they often have ethical and practical issues, or are prohibitively expensive. Thus, causal inference from observational data indispensable, though they introduce other challenges. Perhaps the most crucial one confound- ing: there might be variables (called confounders) that causally affect both the treatment and the outcome, and spurious correlation follows.
new drug, based a of observations
Most of works in causal inference rely the unconfoundedness assumption that appropriate co- variates are collected so that the confounding can be controlled by conditioning on or adjusting for those variables. This is still challenging, due systematic difference of the distributions of the variates between the treatment and control groups One classical way of dealing with this difference re- weighting (Horvitz & Thompson 1952). There are semi- carimparamettic methods, which have bet- ter finite sample performance, e.g. TMLE (Van der Laan & Rose 2011), and also paramarrreticc tree-based, methods, e.g. Causal Forests (CF) (Wager & Athey 2018). Notably, there is recent rise of interest in representation learning for causal inference starting from Johansson et al. (016)).
There are a few lines works that challenge the difficult but important problem of causal inference under unobserved confounding. Without covariates we can adjust for. many of them assume special structures among the variables, such as instrumental variables (IVs) (Angrist et al., 1996). proxy variables (Miao et al. (0088), network structure (Ogburn) 2018), and multiple causes (Wang & Blei, 2019). Among them, instrumental variables and proxy (or surrogate) variables are most commonly exploited. Instrumental variables are not affected by unobserved confounders, influencing the out- come only through the treatment. On the other hand, proxy variables are causally connected to unobserved confounders, but not confounding the treatment and outcome by themselves. Other methods use restrictive parametric models (Allman et al.] 2009), or only give interval estimation (Manski 2009 Kallus al. 2019).

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In this work, we address the problem of estimating treatment effects under unobserved confounding. We further discuss the individual-leve treatment effect, which measures the treatment effect condi- tioned on the covariate, for example, on patient's personal data. To model the problem. regard the covariate as a proxy variable and the confounder as a latent variable in representation learning.
Our method partieularly exploits the recent advance of identifiability of representation learning for VAE (Khemakhem et al.] (020). The hallmark of deep neural networks (NNs) might be that they can learn representations of data. is desirable that the learned representations are interpretable, that is, in approximately the same relationship to the latent sources for each down-stream task. A principled approach to this identifiability. that is, when optimizing our learning objective w.r.t. the representation function, only a unique optimum will be returned. Our method builds on this and further provides the stronger identifiability of representations that needed causal inference.
The proposed method is also based firmly on the well-established results in causal inference. In many works exploiting proxies, it assumed that the proxies are independent of the outcome given the confounder (Greenland, 1980; Roothman et al. 20088 Kuroki & Pearl 2014). This also motivates our method. Further, our method naturally combines a new VAE architecture with the classical result of [Rosenbaum & Rubin (1983) regarding the sufficient information for identification of treatment effects, showing identifiailitt proof both latent representations and treatment effects.
The main contributions of this paper by a new VAE architecture for estimating treatment effects under unobserved confounding; 2) theo- retical analysis of the identifiability of representation and treatment effect; 3) experimental study on diverse settings showing performance of state-of-the-arr
follows: 1) interpretable, causal representation learning
2 RELATED WORK
Identifiability of representation learning. With recent advances in nonlinear ICA, identifiabil- ity of representations is proved under number of settings, e.g., auxiliary task for representation learning (Hyvärinen & Morioka 2016 Hyvärinen et al. 2019) and VAE [Khemakhem et al.) 2020). Recently, Roeder et al. (2020) extends the the result icluude a wide class of state-of-the-a deep discriminative models. The results are exploited in bivariate causal discovery (Wu & Fukumizu 2020) and structure learning (Yang al., 2020). To the best of our knowledge, this work the first to explore this new possibility in causal inference.
Representation learning for causal inference. Recently, researchers start to design representation learning methods for causal inference, but mostly limited to unconfounded settings. Some methods focus on learning a balanced covariate representation, e.g., BLR/BNN (Johansson al.. 2016), and TARnet/CFR (Shalit et al. 2017). Adding to this, Yao al. (2018) also exploits the local similarity of between data points. Shi et al. (2019) uses similar architecture to TARnet, considering the importance of treatment probability. There are also methods using GAN (Yoon et al. 2018
GANITE) and Gaussian process (Alaa & van der Schaar 2017). Our method adds to these by also tackling the harder problem of unobserved confounding.
Causal inference with auxiliary structures. Both our method and CEVAE (Louizos et al. 2017) motivated by exploiting proxies and use VAE as learning method. However, CEVAE assumes a specific causal graph where the covariates should be independent of the treatment given the con- founder. Further, CEVAE relies the assumption that VAE can recover the true latent distribution. Kallus et al. (2018) uses matrix factorization to infer the confounders from proxy variables, and gives consistent ATE estimator and its error bound, Miao et al. (2018) established conditions for identification using more general proxies, but without practical estimation method. Note that, two active lines of works machine learning exist in their own right, exploiting IV (Hartford et al.) 2017) and network structure (Veitch et al. 2019).
3 SETUP AND PRELIMINARIES
3.1 TREATMENT EFFECTS AND CONFOUNDERS
Following Imbens& & Rubin (003)). we begin by introducing potential outcomes (or counterfactual outcomes) y(), 0, 1. (l) is the outcome would observe, if we applied treatment value t.
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Note that, for unit under research, which factual treatment we have applied. This is the fundamental problem of causal inference.
observe only of y(0) or y(), corresponding to
We write expected potential outcomes, conditioned covariate(s) x
(x [t)(())
x). The estimands in this work are the causal effects, which are Conditional Average Treatment
Effect (CATE) and Average Treatment Effect (ATE) defined by
r(x) μn (x) fo( ATE = B(T(=))
(1)
CATE can be understood as an indiidual-leved treatment effect, conditioned on high dimensional
and highly diverse covariates.
In general, we need three assumptions for identification (Rubin 2005). There should exist variable 2 E IR" satisfies ignorability (y(0), (()) and positivity (v2, f: p(t z/z =) > 0), and also given the consistency of counterfactuals (y = y(t) if = t) (See Appendix for explanations). Then, treatment effects can be identified by:
J[ppppll p(ulz, f)ydy)p(z x)dz (2) The second equality uses the three conditions. We say that strong ignorability holds when we have both ignorability and positivity. In this work, we consider unobserved confounding, that is, we assume the existence of confounder(s) z, satisfying the three conditions, but it is (partially) unob-
[x(x) = (y(((((t) x x z)) = E(E = =))= J(
served.
The following theorem adapted from Rosenbaum & Rubin (1983) is central to causal inference and we will use it for motivating and justifying our method. Such function b(z) is called a balancing score (of z). Obviously, the propensity score e(z) := p(t 1|2), the propensity of assigning the treatment given 2, is a balancing score (with f' be the identity function).
Theorem 1 (Balancing score). Let b(z) be a function of random variable z. Then LlZz)b(z) if and only if f(b(z) = p(t = 1 1z) == e(z) for some function f (or more formally, e(z) is b(z)- measurable). Assume further that satisfies strong ignorability, then so does b(z).
3.2 VARIATIONAL AUTOENCODERS
Variational autoencoders (VAEs) (Kingma et al. 2019) are a class of latent variable models with latent variable z, and observed variable y is generated by the decoder pe py(z)z) The variational lower bound of the log-likelitne likelihooo is written as:
log p(y) 2 log p(y) DKL. (g(z(y))lp(zzz)) Ezng log (y(l) - DKI. (oo
(3)
(y
where the encoder qo (zly) ) is introduced to approximate the true posterior p(zly) and DKL denotes KL divergence. The decoder pe and encoder are usually parametrized by NNs. We will omit the parameters 0, Φ in notations when appropriate. Using the reparameterization trick (Kingma & Welling 2014) and optimizing the evidence lower bound (ELBO) Ey~r(C(v)) with data D, we train the VAE efficiently. Conditional VAE (CVAE) adds conditioning variable c to 3 (See Appendix for details).
As mentioned, identifiable VAE (iVAE) (Khemakhem et al. 2020) provides the first identifiability result for VAF, using auxiliary variable u. It assumes y y_u Lulz, that is, p(y|2. p(y|z). The variational lower bound is log p(ylu) 2 Egzg log ()g()))(2) Dal u) (2/1)))))
(y.
where f(z) +, e is additive noise and has exponential family distribution with sufficient statistics T and parameter A(u). Note that, unlike CVAE, the decoder does not depend on due to the independence assumption.
Here identifiability means that the functional parameters (f, (,T. can be identified (learned) up to a simple transformation.
4 VAE ARCHITECTURE FOR CAUSAL REPRESENTATION LEARNING
'This allows the existence of observed confounders in x. As we will see, since z is the latent variable(s) for VAE and is leamned from covariates x by the VAF, it can contain all confounders in principle. Our method will extract the confounding part of x into z.
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ply'zcc)
The probabilistic model of our VAE follows naturally from the right most side of (2). where the involved distribution p(y, z z/x, t) = p(y|z, x, ()p(z//,t)) t). If treat covariate proxy and assume further the conditional independence xxx 2, t as in most work ex- ploiting proxies, have
y
p(yle)
p(miu)
p(yis)
ply, p(yle 2,, ()p(z)x()) t)
(5)
A natural next step is to design VAE for this joint distribution, which learns to recover a causal representation of By "causal", mean the representation can be used to identify estimate treat- ment effects. Recovering the true confounder z would be great, but this is not required, as shown in Theorem 1 which says, for iden- tification of treatment effects, we only need to have b(z), a causal representation of z, which contains the information of propensity score e(z), the part of z that is relevant to treatment assignment.
p(a)x,
Figure 1: Graphical models of the decoders. From top: CVAE, iVAF, and CFVAF. The encoders for the VAFs are similar: they take all observed variables and build approximate posteriors.
We are a step away from the VAE architecture now. Note that (5) has similar factorization with iVAE: p(y, z//) = p(y|z)p(z from y ulz, meaning we can use our covariate
auxiliary vari- able u in iVAE from the independence assumption of proxy variable. Further from the conditioning on t in (5), we design a VAE architecture as a combination CVAE and iVAE, with treatment and covariate x as conditioning and auxiliary variable, respectively. The ELBO can be derived as
peyp//xx x, t) 2 Dogg(y/xx x, t) (g(lxx y. ()lx((),, y,t)
= Ezzg log plyl =. DKL. (g(2) y, y, p)) x, t)) == (x
As in iVAE, the decoder drops the dependence on x. We name this architecture the Counterfactual VAE (CFVAE). Figure depicts the relationship of CVAE, iVAE, and CEVAE.
We detail parameterization of CFVAE. The decoder Pf. jz, t), conditional prior Ph. t), and encoder 4r a . xz) t) are factorized Gaussians, i.c., a product of I-dimensional Gaussian distribu- tions. This not restrictive if the mean and variance are given by arbitrary nonlinear functions. y zzz IT
- (y,s fz)9;). z/x,t I (22 h4 k2), z|x,y,t~ NV st).
= (f.g h k) and Φ = (r, s) are functional parameters given by NNs which take the respective conditional variables inputs (e.g. h = (h,(x,t))²)
IDENTIFYING REPRESENTATION AND TREATMENT EFFECTS
In the following, we will show that CFVAE can identify the latent variable up to affine transfor- mation (Sec. 5.T]. it can learn a balancing score as a causal representation, and its decoder is a valid estimator for potential outcomes (Sec. 5.2).
5.1 IDENTIRIABILITY REPRESENTATION
In this subsection, we show that CFVAE can identify latent variable z up to an element-wise affine transformation when the noise on the outcome is small. Based on this result, we can gain insight on how to make CFVAE learn balancing score.
Our starting point is the following theorem showing the identifiability of our learning model, adapted from Theorem in Khemakhem et al. (2020), by adding conditioning on
Theorem 2. Given the family po(y, z z|x, 1) specified by 5 and for t = 0, 1, assume
1) f. (z) = (fi(z,t)) is injective;
2) [z) = y .. is constant (i.e. 9;(2,t)=0y.. = oy,,e
3) Ax(x) = (h(x, t), k(x, T) which is seen as random variable, is not degenerate.
We specified factorized Gaussians in and they show good performance in our experiments. But Corol- lary and Theorem 3 can be extended more general exponential families, see Khemakhem al. 2020
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Then, given t. the family is identifiable up to an equivalence class. That is, for t 0,1, if Po, (y|x,t t= = Po; (y xx, { we have the relation between parameters
(y.) = Av.: + b, b, (y.:
(8)
where ply:) = plylt), A, is an invertible n-square matrix and b, is n-vector.
Similarly Sorrenson et al. (2019). we can further show that A, = diag(a:) is a diagonal matrix. By a slight abuse of symbol, we will overload to make shorthand for equations like (8), e.g., can be written as f (y) = A(f (y))]r. Note that, by definition of inverse, we also have f Alt.
The importance of model identifiability can be seen more clearly in the limit small noise on y. Corollary [ can be easily understood by noting that after learning with small noise y, the encoder and decoder both degenerate to deterministic functions: in 77) g' = s' = 0, and (z. y) = ft (y). Note that, we only assume the VAE learns observational distribu- tions Po; (y|x, t = t) the same as the truth, but this leaves room for latent distributions different to the truth.
Vx, 21 = r/
Corollary 1 (Identifiability of representation). For = O, 1, assume 1) yy - O and 2) CFVAE can learn distribution po; = Pa., then the latent variable z and the mean parameter f. y y can be identified up to an element-wise affine transformation: z A(z') t and f f Alt.
This is strong result for learning interpretable representation, but it is not enough for causal infer- ence. To see this in a principled way, we recall the concept of balancing score. The recovered latent z' in Corollary 1 jis not a balancing score, due to the different A, for t 0, 1. If z' were balancing score, we would have tlLz/zz. However, given z' = x², z = diag( a. )z² + b, is a deterministic function of t, contradicting with t zllz (A more concrete analysis can be found in Appendix.) This example also suggests that we will have a balancing score if we can get rid of the dependence on L. The next subsection discusses some assumptions on remove the "It" in Corollary L
5.2 IDENTIFICATION TREATMENT EFFECTS
The following definition will be used in Theorem 3 The importance of this definition is immediate from Theorem | that is, if a balancing covariate is also function of z, then is a balancing score. Definition 1 (Balancing covariate). Random variable x is a balancing covariate of random variable z z/x We also simply say x is balancing (or non- balancing if it does not satisfy this definition).
Given that a balancing score of the true confounder is sufficient for strong ignorability, natural and interesting question is that, does a balancing covariate of the true confounder also satisfies strong ignorability? The answer is no. To see why, and also better understand the significance of Theorem 3 we give Proposition indicating that a balancing covariate of the true confounder might not even satisfy ignorability. We also refer readers Appendix 8.5 where we examine two important special cases of balancing covariate, one of those is noiseless proxy, which might not satisfy positivity.
Proposition 1. Let be a balancing covariate of z. satisfies ignorability and y(0), (() then x satisfies ignorability.
Given this proposition, we know assumptions are weaker than ignorability, and method can work under unobserved confounding (x might not satisfy ignorability). Note the independence y(O), (x)_Lxx t in this proposition implies y ylLx|z, assumed by CFVAE. From the decomposi- tion rule of conditional independence, we have y(). [1)__xx|z, = vt(y(t)_llx/zz) t) - Vt, f(y(t)_xx/zz = f), 4 independence in total. Using only two of them, and with the consistency of counterfactual, we have vt(y(t) [Lx/z, = t) + Vt(zi t) = y_ll//z, t. In general, the other two independence y(0) = 1 and y(1) y_Lx|z,1 + y1)),x())_Lx((,,.
0 do not hold, and thus
As shown in the following theorem, for any balancing covariates, CFVAE can identify treatment effects, by learning a balancing score of the true confounder as latent representation. Note that
00 (f:: h4, k!) is another set of parameters giving the same distribution, which is learned by VAF. In this paper, symhol (prime) always indicates parameters (variables, etc.) learned/recovered by VAE.
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assumption 3) is nontrivial, it requires that the true data generating distribution is in the learning model. And thus identification of treatment effects follows from the identifiability of our model.
Theorem (Identification with balancing covariate). Assume
1) the same as Theorem 2 and Corollary
2) h l in depend on x but not t (i.e. he(xr, t) ha(2:) and same for k);
3) The data generating process can be parametrized by family po po z|x, t) specified above;
4) z satisfies strong ignorability and x is a balancing covariate of z.
Then, for both t = 0, 1, we have z = diag( a yz' + = A(z'), and z' satisfies strong ignorability. We identify the potential outcomes by ))5()))
f) (r. x))
The result may be
easily understood following. Now with the same A for both treatment groups f, given observation the counterfactual prediction given by CFVAE is the same as truth: y -i-((z() fi-t r A(AA- = f y(1 - t) (Also compare this Appendix 8.4). We can identify potential outcomes, using (x, V1) = 2 in the last equality, by
(x(x) E(y(O)x x) = E(y/|x x)
(2) =x)
(9)
Note that counterfactual assignment 7 may may not be the same factual L. The algorithm
for estimation CATE and ATE is as following. After training CFVAE, we feed data D = {xx ye) = (x, y. }}} into the encoder, and draw sample from q(z') x a, = y. t) = 8(z' r (x. y1) (ō denotes delta function). Then, setting & E {0. 1} in the decoder, feed posterior sample {z} (a, }. we get counterfactual prediction p(y' |z' z = f) = 8(y² fz())) Finally, we
estimate ATE by taking average Ep(y' -1),, and CATE by E (D)x=n) (vi v6), adding conditioning on æ.
A caveat is that (9) requires post-treulment observation y/- Often, it is desirable that we can also have pre-treatment prediction for new subject, with only the observation of covariate x = æc. conditional prior p(z ((z (x) as a pre-treatment predictor for z': input and draw
To this end, sample from pzz') x = x) instead of q, and all the others remain the same. Since the ELBO has a KL term between p(z') xx) and q. the two distributions should not be very different, and will also have sensible pre- treatmettett estimation of treatment effects.
Although method works under unobserved confounding, it still formally requires small outcome noise and balancing covariate. However, experiments show our method can work very well with large outcome noise, and the covariates can be non-balancing and also directly affect the outcome, including general proxies, IVs, and even networked data.
6 EXPERIMENTS
As in previous works (Shalit et al. 2017 Louizos et al. (017), we report the absolute error of ATE CATE == (1) - y(0) - Ep(1/ 1 No 1.). and the square root empirical PEHE (Hill 2011)
ÉPEHE == Ep (v(1) v(0) - (v1 2)))² for individual-level treatment effects.
Unless otherwise indicated, for each function f.g. h k. r, 8 in we use a multilayer perceptron (MLP) that has 3*200 hidden units with ReLU activation, and h, k depend only optimizer with initial learning rate 10-4 and batch size 100 is employed. More details on hyper- parameters and experimental settings are given in each experiment, and are explained in Appendix.
The Adam
All experiments use etrly-stopping of training by evaluating the ELBO on a validation set. We evaluate the post-treatment performance training and validation set jointly (This is non-trivial. Recall the fundamental problem of causal inference). The treatment and (factual) outcome should not be observed for pre-treatment predictions, we report them on testing set.
6.1 SYNTHETIC DATASET
We generate data following randomly generated in range
10 with y 1-dimensional and 3-dimensional. H4 and d; 0.2 0.2) and (0, 0.2), respectively. The functions h, k. I are lin- The outcome model is built for the treatments separately, i.e.
with random coefficients.

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fz, f 0, We generate two kinds of outcome models, depending the type of fi: linear and nonlinear outcome models use random linear functions and NNs with random weights, respectively.
I (mi; os); 2/x ((((()),
(10)
x/x, Z Bern Bern(Logistic(((x:)) z)));
y/z, y N(CTT
We adjust the outcome and proxy noise level by a, re- spectively. The output of fi is normalized by C2 Var 10)) (f((z)). This means we need to use 0 < a < 1 to have reasonable level of noise on y (the scales of mean and variance are comparable). Similar reasoning applies to 7/x; outputs of h have approximately the same range of values since the functions'' coefficients are generated by the same weight initializer.
Ours Go Sontoonding
GEWAE Contooneing
Dure red
Oue re?
Cur
Agnonia re
We experiment on three different causal settings (indi- cated in Italic). To introduce x as IV, we generate another 1-dimensional random source w in the same way as x, and use w instead of to generate z/w (h((w), (w))) Besides taking inputs x, in 1, we con- sider two special cases: := 1(x) (x fully satisfies ignor- ability) and / == ((z) (unobserved confounder Z and non-balancing proxy x of z). Except indicated above, other aspects of the models are specified by 10 See Appendix for graphical models of these three cases.
Figure 2: Plots of recovered (x) true (y) la- tent the nonlinear outcome. Blue: f= O, Orange: t = 1. a, B = 0.4. "no." indicates index among the 100 random models.
In each causal setting, and with the same kind of outcome models and noise levels (a, 3), we eval- uate CFVAE and CEVAE 100 random data generating models, with different sets of functions k., in (TO). For each model, we sample 1500 data points, and split them into 3 equal sets for training, validation, and testing. Both the methods 1-dimensional latent variable in VAE. For fair comparison, all the nyper-parametert including type and size of NNs, learning rate, and batch size, are the same for both the methods.
Figure B shows our method significantly outperforms CE- VAE on all cases. Each method works the best under ig- norability, as expected. The performances of our method on IV and proxy settings match that of CEVAE under ig- norability, showing the effective deconfounding. Figure 2 shows our method learns highly interpretable representa- tion as an approximate affine transtormation of the true latent value. To our surprise, CEVAE also possible to achieve this when both noises small, though the quality of recovery is lower than CFVAE. The relation- ship to the true latent is significantly obscured under IVs, because the true latent is correlated to IVs only given t, while model it by p(z' (x) as required by Theorem
Nonlinear
Nonlinesr
0.5
0.4
0.3
0.2
0.2
noise
y
Ours conf.
CEVAE cont. CEVAE ig.
. Ours inst.
Ours ig.
We can see our method and also CEVAE are very robust w.r.t. both outcome and proxy noise. This may due to the good probabilistic modeling of the noise by VAE. Still, we can see in Appendix that the noise level affects how well we recover the latent variable.
Figure 3: Pre-treatment VEPEHE on non- linear synthetic dataset. Error har 100 random models. We adjust one of the noise levels a, B each panel. with another fixed to 0.2. See Appendix for results on lin- ear outcome. Results for ATE and post- treatment similar.
6.2 IHDP BENCHMARK DATASET
The IHDP dataset Hill 2011 is widely used to evaluate
machine learning based causal inference methods, e.g. Shalit et al. (2017 : Shi et al. 2019). Here, ignorability holds given the covariates. See Appendix for detailed descriptions. Note, however, that this dataset violates assumption y y x/z, since the covariates x directly affect the outcome. To overcome this, we add two components introduced by (hhalit et al. (2017) into our method. First, build two outcome functions fz (z), / 0, in our learning model ( using two separate NNs.
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Second, add to ELBO (6) regularization term, which is the Wasserstein distance (Cuturi 2013) between the learned p(z |x, = 0) and p(z'x, x t = 1). We find higher than 1-dimensional latent variable in CFVAE gives better results, very possibly due to the mismatched latent distribution: the confounder race is discrete but we use Gaussian latent variable. We report results with 10-
dimensional latent variable.
As shown in Table I the proposed CFVAE matches the state-of-the-art methods under model mis- specification. This robustness of VAE was also observed by Louizos et al. (2017). where they used 5-dimensional Gaussian latent variable to model a binary ground truth. And notably, without the two additional modifications, our method has the best ATE estimation and overall the best among generative models (better than CEVAE and GANITE by large margin).
Table 1: Errors on IHDP. "A/B" means mrc-trrattmeem/poot-rraaaa
prediction. The mean and std are calcu- lated over 1000 random draws of the data generating model. "Results with the two modifications. The results 1.0=.0s/ 0 2.04. Bold indicates
without the modifications method(s) that are significantly better than all the others. The results of the other methods are taken from Shalit ct al. (2017), except GANITE Yoon et al. 2018) and CEVAE (Louizos et al. 2017).
CATE .21 01/ /.172.01 and VEPEни
Method
TMLE
BNN
CFR
CF
CEVAR
GANITE
Ours
CATB
NA/30
42 03/37 os 27 o1 40 08/184 46 02/344 01 .49 43/43 oo .3g 03/.30
VEPRHR 2
12.22.
o677 on 3.8 2/882
262 12.7..
24g.//19.
.77..021.69
6.3 POKEC SOCIAL NETWORK DATASET
Pokec (Leskovec & Krevl 2014 is a real world social network dataset. We experiment on a semi- synthetic dataset based Pokec, which was introduced in Veitch al. (2019), and use exactly the same pre-processing and generating procedure. The pre- pre-processed network has about 79,000 vertexes (users) connected by 1.3 x 10 undirected edges. The subset of users used here are restricted to three living districts that are within the same region. The network structure is expressed by binary adjacency matrix G. Following Veitch al. (2019), we split the users into 10 folds, test on each fold and report the mean and std of pre-treatment ATE predictions. We further separate the rest of users (in the other 9 folds) by 6 : 3, for training and validation. Table 2 shows the results. Our method is the best compared with the methods specialized for networked data. We report pre-treatment PEHE of our method in the Appendix, Veith et al. (2099) does not give individual-level prediction.
Table 2: Pre-treatien treatment ATE on Pokee. Ground truth is "Unadjusted" estimates ATE by Ev(y1) Ey(yo), ). "Parametric" a stochastic block model for networked data (Gopalan & Blei 2013). "Embed-" denotes the best alternatives given hy Veitch et al. (201)). Bold indicates method(S) that are significantly hetter than all the others. 20-dimensional latent variable in CFVAE works better, and its result is reported. The results of the other methods taken from Veitch al. 2019)
Unadjusted
Parametric
Embed-Reg.
Embed-IPW
Ours
Age
4.34 1 0.05
4.06 1 0.01
2.77 1 0.35
3.12 0.06
2.08 0.32
District
4.51 1 0.05
3.22 1 0.01
1.75 1 0.20
1.66 1 0.07
1.68 1 0.10
Join Date
4.03 = 0.06
3,73 + 0.01
2.41 + 0.45
3.10 = 0.07
1.70 + 0.13
Each user has 12 attributes, among which distric,, age, or join date is used confounder z to build 3 different datasets, with remaining 11 attributes used as covariate x. Treatment t and outcome y are synthesised as following:
Bern(g(z)); y=t+ 10 0.5) + +,, N(0, 1)
(11)
Note that dist sissrrionn ct of 3 categories; age and join date are also discretized into three bins. g(z) maps these three categories and values to {0.15, 0.5, 0.85).
Some assumptions to justify our method may not hold in this dataset. The important challenges are 1) x obviously does not satisfy ignorability, and 2) large outcome noise exists. On the other hand, given the huge network structure, most users can practically be identified by their attributes and neighborhood structure, which means can be roughly seen
deterministic function of G. x. Then, G, x can be, as defined by us, noiseless proxies of z (see Appendix 8.5). CFVAE is then ex- pected to control for the confounding to large extent and able to learn balancing score based on
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Theorem if we exploit the network structure effectively. This idea is comparable to Assump- tions and in Veitch et al. (2019), which postulate directly that a balancing score can be learned in the limit of infimite large network.
To extract information from the network structure, we use Graph Convolutional Network (GCN) (Kipf Welling 2017) in conditional prior and encoder of CFVAE. difficulty is that, the network G and covariates X of all users are always needed by GCN, regardless of whether is training, validation, testing phase. However, the separation can still make sense if we take care that the treatment and outcome are used only in the respective phase, e.g., (ym, (m) of a testing user m is only used in testing. See Appendix for details.
DISCUSSION
In this work, we proposed new VAE architecture for estimating causal effects under unobserved confounding, with theoretical analysis and state-of-the-art performance. To the best of our knowl- edge, this is the first generative learning method that provably identifies treatment effects, without directly assuming that the true latent variable can be recovered. It is achieved by, on the one hand, noticing we only need the part of latent information that is correlated treatment assignment, and, on the other hand, exploiting the recent advances that the latent variable can be recovered up to trivial transformations in a broad class of generative models.
Despite the formal requirement, the experiments show our method is robust to large outcome noise. Theoretical analysis of this phenomenon an interesting direction for future work. related the- oretical issue is that, while Khemakhem et al. (2020) assumes fixed distribution of noise on y. we observed that, in most cases, allowing the noise distribution to depend on z, improves performance. Extending identifiability to conditional noise models is also an interesting direction.
When the latent model is misspecified (Sec. 6.2 and 6.3), our method still matches the state-of-the- art, though we cannot see apparent relationship between recovered latent variable and the true one. would be nice to see the learned representation indeed preserves causal properties under model misspecification, for example, by some causally-speciaalzed metrics, e.g. Sutter et al. (2019). Given the fact that all nonlinear ICA based identifiabiltyy requires an injective mapping between the latent and observed variables, theoretical extensions discrete latent variable would be challenging.
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Jinsung Yoon, James Jordon, and Mihaela van der Schaar. GANITE: Estimation of individualized treatment effects using generative adversarial nets. In International Conference Learning Rep- resentations, 2018 URL https ///o:s eww net/ forum?i Fowummiinnyyyyooooo d=ByyWWeWX

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8 APPENDIX
8.1 PROOFS
Proof of Corollary We need the consistency of our VAE to learn a observational distribution equaling to the true one in the limit of infinite data, so that the learned parameters O2 is in the equivalence class of O, defined by ( This can proved (Khemakhem et al. 2020, Theorem 4) by assuming: 1) our VAE is flexible enough to ensure the ELBO is tight (equals to the log likelihood of our model) for some parameters; 2) the optimization algorithm can achieve the global maximum of ELBO (again equals to the log likelihood).
In this proof, all equations and variables should condition on t. and we omit the conditioning in delta function:
notation for convenience. In the limit of σy + 0. the decoder degenerates p(y z) = 5(y - f fz)), we have y = f() and y' = ff'z'). From the consistency of VAE, y should have the same support as y'. For all y in the support, there exist a unique z and a unique z' satisfy
f()) fz'z (use injectivity). Substitute y = f(z) into the l.h.s of (8), and = f'(z') into the r.h.s, we have x diag(a)z' b. The relation is one-to-one for all z, so we get z A(z'). Similar result for f follows.
Proposition 2 (Properties of conditional independence). For random variables w, x, y, We have Pearl 2009 1.1.55):
x x w/y, = JL z
l (Contraction).
x x.W,, = X_yy w, z ( (Weak union). y|z = x (Decomposition).
Proof of Proposition 1. This proof will use the above three properties of conditional independence. We first write our assumptions in conditional independence, as A/. tllz|x (balancing covariate), A2. w lllz (ignorability given z), and A3. w.lxx where w == (1)))
Now, from A2 and A3, using contraction, we have w.lxx, t/z, then using weak union, we have w w z. From this last independence and Al, using contraction, we have t w/x. Then t.ll t.l follows by decomposition.
Proof of Theorem 2. From the proof of Theorem I in Khemakhem et al. 22020), we know A., b depend on only through h k. But we assume h. do not depend on t. So we have z = A(zz) for both 0, 1. From Theorem 1, z' is a balancing score of z, and satisfies strong ignorability. Here, we proceed a bit different from (2). we have
(x(x)) E(E(y(f) x)) / ='y
==,x= x)=ppppp//>> =z))d
f Elylz' 2 t, D)((='x x= xx)dz' = / plul x (
(12)
Compare the rightmost side to 2). note that, there no conditioning on in p(='(ar) because we the strong ignorability given z' (and consistency of counterfactuals) in the third equality, after expanding the outer expectation.
From the consistency of VAE, p(yl=', "0)l"',,, xx,) 5(y ff(2))) and g(z v,t) 8(z' - -r((2, (()())))) p(zz [x, 4,4) 8(z' (y.)) where (ar,y)) (ar, y, 1) is data point. And p(2'(a))
E. I p(z' xr y. t)p(y, f/x)dy- =.I, 5(z' (v:))() fly)dy. We have
[z(x)= = 1 (fi(2') Σ 8(z -
(13)
= =E SA F: =
"This is the statistical consistency of an estimator. Do confuse with the consistency of counterfactuals.
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We should note that p(z'|az |xx v. por (2/ æ, y,t = pe: (y y, 2 2,
t) / Pe: (y y, 2'|x æ,t t)dz' might not be equal the truth p(z|ar y, t) po, (z|x a, y.t= (in partieular it is possible that f. + fi), but they are in the same equivalence class
in the sense that 02 0, should satisfy (8)
Also note that the learning of inverse mapping f: in the encoder q is enforced by consistency (g(z'(az [x, y, = p(z'|x, y, t)), we can just use MLP for r' in the encoder, and we will have r ft- if the MLP is flexible enough to contain f/ Similar situations prevail when identifiability is achieved by nonlinear ICA (Hyvärinen & Morioka, 2016 Hyvärinen et al. 2019 Khemakhem et al. 2020
8.2 ON THE THREE IDENTIFICATION CONDITIONS
Ignorability given z means there is no correlation between factual assignment of treatment and counterfactual outcomes given 2, just as the case RCT. Thus, can be understood as an- confoundedness given 2, and z can be seen as the confounder(s) we want to control for. Positivity says the supports of p(t = t/x = æ).t t = 0, should be overlapped, and this ensures there are no impossible events in the conditions after adding
and the expectations from observational data. Finally, consistent counterfactuals are well defined: given assignment of treatment t = f, the observational outcome should take the same value as the potential outcome ytt).
thus be estimated
8.3
CONDITIONAL VAE
By adding conditioning variable (usually a class label), Conditional VAE (CVAE) (Sohn al.] 2015 Kingma et al.) 2014) can give better reconstruction of observation of each class. The varia- tional lower bound
loyp(y/c) Exng logply/ (appp(y/cccc c) (g c))p(()c)))lllll))) = Lov
(14)
The conditioning in the prior is usually omitted, since the dependence between and the latent representation is also involved in the encoder
8.4 IDENTIFIABILITY OF REPRESENTATION IN SEC. 5.1 IS NOT ENOUGII
Consider how the recovered z' would be used. For a control group (t 0) data point (xx, y. 0), the real challenge is to predict the counterfactual outcome y(1). Taking the observation, the encoder will output posterior sample point z1 fo Ao ) (with zero outcome noise, the encoder degenerates to delta function: q(z 0) 8(2 - f--(())). Then, we should do counterfactual inference. using decoder with counterfactual assignment ( = 1: v/ ((() = fi Ai (Ar (An'(zz)), (za)). This prediction can be arbitrary far from the truth y(1) = fi (zo), due to the difference between Ai and Ao. More concretely, this is because when learning the decoder, only the posterior sample of the treatment group (t 1) is fed to fí, and the posterior sample difffrentt to the true value by the affine transformation A1, while it is Ag for 2/1
(v))).
8.5 TWO SPECIAL CASES BALANCING COVARIATE.
Definition 2 (Noiseless proxy). Random variable x is noiseless proxy of random variable if z is a function of x (z (x)).
Noiseless proxy is special case of balaneing covariate because if x x. is given, we know w(x) and w a deterministic function, then p(z|x x) = p(z|x = x, = 6(2 w(x)). Also note that, a noiseless proxy always has higher dimensionality than z, or at least the same.
Intuitively, if the value of x is given, there is no further uncertainty about z, so the observation of æ may work equally well to adjust for confounding. But, we will soon, noiseless proxy of the true confounder does not satisfy positivity.
Definition (Injective proxy). Random variable x is an injective proxy of random variable z if is an injective function of (x x(z). X is injective).
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Injective proxy is again special case of noiseless proxy, since, by injectivity, 2 x 'xx), i.e. is also a function x.
Under this very special case, that is, if x is an injective proxy of the true confounder z, finally have x is a balancing score and satisfies strong ignorability, since x is a balancing covariate and function of z. To this in another way, let f
x and b X in Theorem [ then f(x) = f()(2) = e(z). By strong ignorability of x, (2) has a simpler counterpart f4g (4, = B(y(U)xx x) = E(y|x = t). Thus, regression of y on (x; t) will give a valid estimator of
CATE and ATE.
However, noiseless but non-injective proxy is not balancing score, in particular, positivity might not hold. Here, simple regression will not do. This is exactly because w is non-injective, hence multiple values of that cause non-overlapped supports of p(t f|x =), 0, might be mapped to the same value of z. An extreme example would be t I[x > 0). = |x]. We can see p(t t(x) are totally non-overlapped, but Vt, = + 0 p(t = t|z 2) = 1/2.
8.6
DETAILS AND ADDITIONAL RESULTS FOR EXPERIMENTS
8.6.1
SYNTHETIC DATA
p(x/w)
p t) y
p(x/x) p(y/a,t)
ply t) y
z
X
p(t Xx
x(t(x)
p(t
Figure 4: Graphical models for generating synthetic datasets. From left: IV, ignorability given x, and non- balancing proxy Note that in the latter two cases, reversing the arrow between does not change any independence relationships, and causal interpretations of the graphs remain the same.
Linear
Linear
Interestingly, linear outcome models seem harder for both methods, maybe because the two true linear outcome models for 1 = 0, distinguish and learning outcome models. Note that after generating the outcomes and before the data used, we normalize the distribution of ATE of the 100 generating models, so the errors on linear and nonlinear settings are basically comparable.
1.4
more similar, and it is harder to
1.3
1.2
1.1
1.0
0.0 0.2 0.4
088
0.8 D.8.C.
022
04
08
0.8
You can find more plots for latent recovery at the end of the paper.
prany moise
Ours ig.
CEVAE conf.
+ Ours insl.
Ours conf.
CEVAE
linear synthetic dataset. Error bar on 100 random models. We adjust one of a, 8 at a time. Results for ATE and post- trst-treatment are similar.
Figure 5:
VEPEHE
8.6.2 IHDP
IHDP is based on an RCT where each data point repre- sents child with features about their birth and moth-
ers. Race is introduced as confounder by artificially removing all treated children with nonwhite mothers. There are 747 subjects left in the dataset. The outcome is synthesized by taking the co- variates (features excluding race) input, hence ignorability holds given the covariates. Following previous work, we split the dataset by 63:27:10 for training, validation, and testing.
8.6.3 POKEC
GCN takes the network matrix G and the whole covariates matrix X := (xf"" 2 D where user number, and outputs representation matrix R, again for all users. During training, select the rows in R that correspond to users in training set. Then, treat this training representation matrix as if it is the covariates matrix for non-networked dataset, that is, the downstream networks in conditional prior and encoder are the same as in the above two experiments, but take (Rm.." where Rn expected input. And we have respective selection operations for validation and testing.
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We can still train CFVAE including GCN by Adam, simply setting the gradients of non-seleted rows of to O.
Note that GCN cannot be trained using mini-batch, instead, perform batch gradient decent using full dataset for each iteration, with initial learning rate 10-2 We use dropout (Srivastava et al, 2014) with rate 0.1 to prevent overfitting.
The pre-treatment VEPEHE for Age, District, and Join date confounders are 1.085, 0.686, and 0.699 respectively, practically the same the ATE errors.
8.7 ADDITIONAL PLOTS SYNTHETIC DATASETS
See next pages.
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Figure 6: Plots of recovered-tru true latent under unobserved confounding. Rows: first 10 nonlinear random models, columns: praxy noise level.
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Figure 7: Plots of recovered-true true latent under unobserved confounding. Rows: first 10 nonlinear random models, columns: outcome noise level.
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Figure 8: Plots of recovered-tru latent when ignorability holds. Rows: first 10 nonlincar random models, columns: praxy noise level.
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Figure 9: Plots of recovered-true latent when ignorability holds. Rows: first 10 nonlincar random models, columns: outcome noise level.
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Figure 10: Plots of recovered-true latent when ignorability holds. Conditional prior depends Rows: first
10 nonlinear random models, columns: outcome noise level. Compare the previous figure, we can see transformation for t 0, 1 are not the same, confirming our Theorem
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Figure 11: Plots of recovered true latent on /Vs. Rows: first 10 nonlinear random models, columns: outcome noise level.
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Reviews:
[{"review_id": "YXLZnKRH9Hm", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "3342 Identifying Treatment Effects Under Unobserved Confounding\n# Summary\n\nThe authors propose a representation learning method for estimating causal effects in the presence of unobserved confounding when covariates that act as proxies for a latent confounder are available. The authors connect the problem to some recent results on the partial identifiability of VAE and non-linear ICA models. The authors lay out some conditions under which the causal effect may be identifiable and propose a VAE-based model, CFVAE, that enforces some of these conditions on the estimated model. They compare CFVAE to a previously proposed VAE algorithm, CEVAE, and other methods that are designed to work under unconfoundedness.\n\n# Feedback\n\nIdentification of causal effects in the presence of proxy variables for unobserved confounders is an important but subtle problem. However, this means that the bar for making contributions in this area needs to be high. Because the conditions for identification can’t be falsified in a particular application, making unclear statements about when the effects of interest are and are not identifiable is very important. Readers who misunderstand will only experience silent failures and make poor decisions as a result.\n\nUnfortunately, I don’t believe this paper meets this bar of clarity. I think that the authors explore some interesting connections to recent work on identifiability in latent variable models, and understanding what these results imply for causal inference is important. In particular, the observation that one might estimate a decoder up to a _different_ affine transformation in the treated and control arms seems like a useful insight. But the results in the paper are incomplete, disorganized, and in some cases wrong. I’ll list out a few general points here, then discuss some substantive issues with the paper’s specific argument.\n\n## General points about identifiability arguments\n\n### Identifiability is not a property of the method\n\nIdentifiability is a property of the _data generating process_ and not a property of the model being used to do estimation. If the causal effect of interest is not identifiable in the process that generated the data, then using an “identifiable model” to estimate the causal effect will not solve the problem.\n\nThe exposition in the paper seems to argue that using the right model will make the causal effect identifiable. This may just be a matter of unclear writing, but this is a broader point of confusion in the ML community, so it’s important that the authors be clear on this point.\n\nHere, I think it would make sense for the authors to state what their assumptions are about the data generating process, separately from the parameterization of their model. Reading the paper, the distinction between these two layers of assumptions was unclear.\n\n### Relaxing identifying assumptions is not an option\n\nIn the same vein, if there are assumptions that the authors need to make to eliminate identification failure modes, then showing that the model “works” when those assumptions are relaxed does not inspire confidence. Unlike standard model misspecification which can be detected and debugged based on observables, making the wrong identifying assumptions in a model can result in silent failure, where the model can fit the observed data perfectly, but return a causal effect estimate that converges to the wrong place, or doesn’t converge at all.\n\nIf one is able to relax the identifying assumptions and still see success in experiments, this means either (a) the assumptions were unnecessary, or (b) the experiments did not probe the method well enough.\n\n### Assumptions needs to be stated clearly, with implications clearly highlighted\n\nWhen making identification arguments, assumptions play the role of eliminating equally plausible causal explanations of the observed data, until only the true one can remain (if the assumptions are true). These are not the kinds of assumptions that eliminate exotic corner cases; instead, they eliminate cases like the most obvious explanation for the observed data like the absence of unobserved confounding. Bounding arguments like the Manski and Kallus et al papers that are cited in the introduction construct the full set of causal explanations that identifying assumptions must narrow down to a point.\n\nAll of this is to say that clearly stating assumptions, and the cases they eliminate, is essential for any identification argument. In the paper as it is written now, many assumptions are made implicitly or in passing, and it is unclear which assumptions are made for illustration (e.g., the noise on the outcome going to zero) and which assumptions are essential for the argument in general. The assumptions are always framed as “mild” and do not highlight situations that the assumptions eliminate (i.e., in which cases they would fail to hold).\n\n## Specific Concerns\n\n### Balancing covariate implies that the naive estimator “just works”\n\nThe primary identifiability result of the paper involves the assumption that the observed covariates are “balancing covariates”, satisfying t \\indep z | x. The authors argue that this is a weaker condition than requiring that x satisfy unconfoundedness. This may be true, but the gap between the two assumptions is, at most, a set of knife-edge violations of faithfulness. In terms of estimating causal effects, simply doing the standard covariate with x and ignoring z would give the right answer.\n\nThis can be shown in two ways. First, graphically, t \\indep z | x implies that there is no backdoor path through z from t to y when you condition on z. So z doesn’t induce any non-causal association between y and t. Secondly, using the standard adjustment formula:\n\n\\mu_t(x) = E[ E[Y | X = x, T = t, Z = z] ]\n= \\int_z E[Y | X = x, T = t, Z = z] p(z | x) dz\n= \\int_z E[Y | X = x, T = t, Z = z] p(z | x, t) dz  (using the balancing covariate property)\n= E[Y | X = x, T = t]\n\nIn particular, the naive regression function E[Y |  X = x, T = t] only fails in cases where p(z | x, t) \\neq p(z | x); i.e., when the distribution of the latent variable is different in the two observed treatment arms even after conditioning on x. The balancing covariate property eliminates this possibility.\n\nThis also means that the VAE model specified can only generate data where the latent variable z does not introduce confounding.\n\n## Other Concerns\n\n * The adjustment formula in equation (2) is wrong. The last integral should be with respect to p(z | x), not p(z | x, t) (see the argument above).\n\n * There is a substantive difference between estimating individual level causal effects given the observed outcome (a counterfactual query) versus estimating the CATE. The authors do divide this into “pre-treatment” and “post-treatment prediction”, but the counterfactual query presents additional identification questions. In particular, whether the reported expectation is correct depends on Cov(y(1), y(0) | z, t), which is never observable. None of the identifying assumptions in the paper make any arguments about this quantity, so the models in the paper are making implicit strong assumptions here.\n\n * The f^{-1}(y) notation in the paper is very unclear in the case that there is actually outcome noise for y. The function f relates the latent z to the expectation of y, not y itself. When y includes independent noise, the distribution of f^{-1}(y) does not yield the marginal distribution of z; you need to deconvolve the independent noise in y, which is non-trivial.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "V0icFyEY3f", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\n\nThis paper provides a method for using a VAE with proxy variables to estimate CATE in a model with latent confounding by recovering a conditional distribution over the latent confounders. Building upon results from Khemakhem et.al. 2020, the confounding can be identified if the latent variable is parameterized by an exponential family distribution dependent on the proxy variable, and the outcome variable is an injective function of the latents with (small) additive noise. Conceptually, the paper is similar in goal to Louizos et al. 2017, with the main difference seeming to be a stronger theoretical base.\n\nReasons for score:\n\nOverall, I would rate this paper as a borderline accept. The work seems to provide a strong theoretical background for VAE-based models to be used for identification purposes with latent confounding when proxy variables are present, allowing to explicitly specify conditions under which approaches modelling confounders for adjustment can work. I find this an interesting result that validates the intuitions behind previous works. My main concern with the paper is an unclear comparison to Louizos et. al, as well as an unclear specification of the necessary assumptions, which only becomes clear after reading Khemakhem et. al. Finally, given that this work seems to be a direct combination of these two previous works, I have some lingering doubts about the result's novelty, which would be greatly helped if the paper included a more detailed comparison.\n\nPros:\n\n- As mentioned, provides a strong theoretical backing for a VAE-based approach to CATE, giving a clear set of conditions under which such approaches can be used\n- Provides convincing experimental evidence that the proposed method, CFVAE, compares favorably with existing approaches.\n- Very clearly lays out the relevant background literature, allowing to cleanly place this work in context.\n\nCons:\n\n- The difference between CEVAE and CFVAE should be clarified. For example, the statement \"CEVAE assumes a specific causal graph where the covariates should be independent of the treatment given the confounder.\" seems to contradict the end of page 3 in Louizos et. al., where it looks like they state that there can be a direct edge from X to t. Could the authors comment on this?\n- While the fact that the paper was using results from Khemakhem et. al. was made very clear, I needed to go to the original paper for a clear specification of the precise conditions required for the method to work (exponential family, outcome is function of latent with small additive noise, etc). It would be useful if these conditions were listed clearly and centrally in the paper, without assuming familiarity with this previous work.\n\nComments:\n\nI think it might be useful to include CEVAE in Fig 1, and add it to the discussion in section 3.2/4. In general, showing the precise difference between the two VAE formulations would go a long way towards making the contribution's novelty easier to understand.", "extra_scores": {}, "extra_text": {}}, {"review_id": "GBfpIq1F2s", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a new variant of VAE for estimating conditional treatment effects under unobserved confounding. It provides theoretical results on the identifiability of the confounder and the treatment effects under some assumptions. Experimental results are provided to demonstrate the performance of the proposed method. \n\nPros:\n\n-The paper addresses an important problem\n\n-The paper provides a theoretical analysis of the identifiability of representation and treatment effect.\n\nCons:\n\n-I’m confused by the role of $x$ and its relation with the unobserved confounder $z$. In the works that do not consider unobserved confounders, I believe the covariates $x$ in the treatment effect $\\mu_t(x)$ are the observed confounders and assumed to satisfy the ignorability condition. In this paper, are you assuming there exist no observed confounders? That would be a very unreasonable assumption.\n\n-I think the ignorabiltiy assumption made before Eq. (2) is not correct. For (2) to hold, you’d need (y(0), y(1) independent t |z,x) to hold. I believe the propensity score should be p(t=1|z,x).\n\n-The paper made several assumptions that look to me quite arbitrary, including (y indep x|z, t) and (t indep z|x). I’m not sure they are consistent with each other or with the ignorabiltiy assumption. What graphical models satisfy all these assumptions? The CFVAE structure in Figure 1 does not look like satisfying all these assumptions.\n\nOverall, I vote for reject. I think the paper made some unreasonable assumptions.\n\nOther comments:\n\n-What is the graphical structure of the data-generating model in (10)?\n\n-Why only compare with CEVAE on the synthetic datasets but not other methods?\n\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "6qLBw9ATBYh", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThe present paper introduces Counterfactual VAE (CFVAE), a generative learning method to estimate treatment effects under a latent unconfoundedness assumption. It builds on variational autoencoders (VAE) to learn causal representations. The authors provide identification results using recent results on nonlinear ICA (Khemakhem et al., 2020). They show that the confounder is identifiable up to an affine transformation.\nThe main contributions of this work are presented in Sections 5.1 and 5.2, where they derive identifiability of the treatment effect via identifiability of representation.\nThe theoretical claims are complemented with various synthetic and semi-synthetic experiments which also show that the proposed models can compete with and in certain cases improve upon state of the art.\n\nRecommendation:\nReject. In summary, I am rather convinced that the contribution of this paper is important, but lacks clarity in its arguments and clarifications w.r.t. its positioning in the standard causal inference framework which it claims to make a new contribution to and its causal model/underlying assumptions about confounding are not stated clearly enough. I will read the rebuttal carefully and am willing to increase the score if the authors address the raised concerns.\n\nStrong points: \n - This work provides a simplified and yet still practically useful framework that leverages recent nonlinear ICA result to provide an identification result in the causal inference framework.\n - The simulations, especially in Section 6.1 are well presented and commented.\n\nIssues/Points that require clarification:\n - The positioning w.r.t. CEVAE and more generally w.r.t. to classical unconfoundedness assumptions is not very explicit in that the authors first state that their method requires weaker assumptions than CEVAE but from the identifiability discussion (Sec. 5.2) and the experiments (Sec. 6.1) it seems that the models are more different and not a weaker, resp. stronger version of each other. A simplified causal graph for CFVAE (maybe together with one for CEVAE) could make the comparison more explicit.\n - To my understanding, the balancing covariate assumption implies that $z$ is not directly related to $t$ (meaning in a causal graph there would be no arrow from $z$ to $t$) but only through $x$. So if one were to adjust for the treatment bias using only $x$, this should be sufficient to estimate the treatment effect, for instance by inverse propensity weighting. Thus again, could the authors provide a causal graph representing their assumptions about the role of each variable, most importantly $z$ and $x$, similar to Figure 1 of Louizos et al. (2017)?\n - The theory seems solid for the univariate case. How challenging would an extension to a multivariate case (i.e., multiple latent confounders) be? As a first step, have the authors looked at the empirical behaviour in case of multivariate confounders? \n - Related to the previous point, are there some concrete examples where the assumption of a latent univariate confounder is plausible/sufficient to capture all confounding?\n - The main claim of this work relies on results from Khemakhem et al. (2020), it would be helpful to have at least 1-2 sentences summarizing the main idea/result for identifiability of the iVAE in this cited article, so that the present work is more self-contained.\n - Concerning the pre-treatment prediction (p.6), have the authors verified empirically how their proposed alternative in the absence of post-treatment observation $y_t$ performs?\n - Would it be possible to add the parametric probabilistic PCA based approach by Mao et al. (2018) to the experiments? To my knowledge, this is the only work that gives a proved consistent estimator in the case of latent confounding. Also, this reference should be added in the related work section.\n\nMinor comments (that did not impact the score):\n - p. 1: \"effects of public policies or clinical trials\" $>>$ replace clinical trials (clinical trial is a mean to estimate the effect of a new drug/treatment, not something that has an effect in itself)\n - p. 2: In many work $>>$ In many works\n - p. 2: if we apply $>>$ if we applied\n - p. 3: casual effects $>>$ causal effects\n - p. 3: CATE can be understood is an $>>$ CATE can be understood as an\n - p. 3: introduce/define $D_{KL}$ notation\n - p. 5: introduce/define $\\delta$ notation\n - p. 6: check caption of Figure 2.\n - p. 7: descriptoins $>>$ descriptions\n\nReferences:\n - Nathan Kallus, Xiaojie Mao, and Madeleine Udell. Causal inference with noisy and missing covariates via matrix factorization. In Advances in Neural Information Processing Systems, pp 6921–6932, 2020.\n - Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational autoencoders and nonlinear ica: A unifying framework. In International Conference on Artificial Intelligence and Statistics, pp. 2207–2217, 2020.\n - Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. Causal effect inference with deep latent-variable models. In Advances in Neural Information Processing Systems, pp. 6446–6456, 2017.\n\n===================\n\nPost Rebuttal Update:\n\nWhile I think the proposal is interesting, I still think the proposed methodology lacks a clear presentation of the studied (causal inference) problem and statement of its underlying assumptions, as it has also been pointed out by other reviewers. Despite some clarifications from the authors I still vote for rejection as I believe the paper requires a major revision.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers noted that this is an important, interesting but difficult topic. They appreciated that the authors clarified their assumptions in the theorem statements. Nevertheless, they recommend the authors to detail in depth when the method work better than the method where only the covariates are adjusted. They still think that the paper would require major modifications to be considered for publication hence the decision is rejection the paper. 

Decision: Reject