Coupling Generative Modeling and an Autoencoder with the Causal Bridge
Authors: Ruolin Meng
Ming-Yu Chung
Dhanajit Brahma
Ricardo Henao
Lawrence Carin
Venue: NeurIPS 2025 poster ()
Keywords: Causal Effect Estimation
Proxy Causal Learning
Causal Bridge
Primary area: probabilistic_methods
TL;DR: We present a new theoretical perspective for the causal bridge that allows a deeper understanding of the impact of its assumptions on causal estimates while yielding an autoencoder formulation that  improves causal effect estimation.

Abstract: We consider inferring the causal effect of a treatment (intervention) on an outcome of interest in situations where there is potentially an unobserved confounder influencing both the treatment and the outcome. This is achievable by assuming access to two separate sets of control (proxy) measurements associated with treatment and outcomes, which are used to estimate treatment effects through a function termed the *causal bridge* (CB). We present a new theoretical perspective, associated assumptions for when estimating treatment effects with the CB is feasible, and a bound on the average error of the treatment effect when the CB assumptions are violated. From this new perspective, we then demonstrate how coupling the CB with an autoencoder architecture allows for the sharing of statistical strength between observed quantities (proxies, treatment, and outcomes), thus improving the quality of the CB estimates. Experiments on synthetic and real-world data demonstrate the effectiveness of the proposed approach relative to state-of-the-art methodology for causal inference with proxy measurements.

Coupling Generative Modeling and an Autoencoder with the Causal Bridge
Ruolin Meng
Ming-Yu Chung
Dhanajit Brahma Ricardo Henao
Lawrence Carin
Duke University
(ruolin. meng, ming-yu. Baung,dhanajai brahma,r henao, Lcarin)@duke edu
Abstract
We consider inferring the causal effect of treatment (intervention) outcome of interest in situations where there is potentially an unobserved confounder influ- encing both the treatment and the outcome. This is achievable by assuming access to two separate sets of control (proxy) measurements associated with treatment and outcomes, which are used to estimate treatment effects through a function termed the causal bridge (CB). We present a new theoretical perspective, associated assumptions for when estimating treatment effects with the CB is feasible, and a bound on the average error of the treatment effect when the CB assumptions are violated. From this perspective, then demonstrate how coupling the CB with an autoencoder architecture allows for the sharing of statistical strength between observed quantities (proxies, treatment, and outcomes), thus improving the quality of the CB estimates. Experiments on synthetic and real-world data demonstrate the effectiveness of the proposed approach relative state-of-the-art methodology for causal inference with proxy measurements.
1 Introduction
Estimating the causal effect of a treatment on an outcome is crucial in various domains, but the presence of unobserved confounders can hinder accurate inference [36] 44]. Traditional methods often
rely on strong assumptions, such as the absence unobserved confounders 43
40
Other approaches
have assumed available instrumental variables with specific properties 2 51 19
However, these
assumptions may not always hold in practice, leading to biased estimates of causal effects.
A promising approach to address this challenge is the use of proxy variables 17 30 28 also known
negative control variables. These are variables that are affected by the unobserved confounder, but do not necessarily directly influence the treatment or outcome. Consequently, leveraging information from these proxies, we can gain insight into the underlying causal mechanisms and potentially mitigate the bias caused by unobserved confounders.
Recent work has introduced the concept of a causal bridge function, which uses two sets of proxy variables, one related to the treatment and the other to the outcome, to estimate causal effects 30 31 52 This approach has shown promising results, but there is still room for improvement in terms of both theoretical understanding and practical implementation.
This paper builds the causal bridge framework and makes several key advances. We provide refined theoretical analysis, clarifying the assumptions and conditions under which the causal bridge function yields accurate causal effect estimates. Furthermore, we introduce novel learning approach that leverages the power generative models to enhance the estimation of the causal bridge. Our approach enables the sharing of statistical strength between observed variables. leading to more robust and accurate causal inference. Finally, we extend the causal bridge framework to handle survival outcomes, common type of data in biomedical applications.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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In summary:
1. We develop a novel framework for causal inference with proxy variables, building upon the causal bridge function. Specifically:
(a) We re-examine the assumptions underlying the causal bridge function, providing a new bound on the average error of the treatment effect when the assumption that the causal bridge is independent of the treatment proxy is violated (Section 4
(b) We introduce a new formulation for learning the causal bridge that utilizes generative models to sample from the conditional distribution of the outcome proxy given the treatment proxy and treatment (Section 6). This approach allows for more efficient and flexible estimation compared to previous methods that rely estimating conditional expectations.
(c) We propose an autoencoder architecture that enables the sharing of statistical strength between observed variables (Section 6 leading to improved estimation of the causal bridge and more accurate causal effect estimates.
(d) We extend the causal bridge framework to handle survival outcomes (Section 6 broadening its applicability important real-world problems.
2. We validate our framework through experiments on synthetic and real-world datasets (Section 7 including comparison with a randomized control trial (RCT). Our results illustrate the implications of our assumptions and demonstrate the effectiveness of our approach compared to state-of-the-art methods for causal inference with proxy variables.
Related Work
Traditional approaches for dealing with unobserved confounders often involve sensitivity analysis [16] 15), which assesses the robustness of causal effect estimates to different assumptions about the unob- served confounder [4] 39 22 [6 23). Another common approach is the use of instrumental vari- ables (IVs), variables that influence the treatment but are independent of the unobserved confounder [ 2 19 51]. However, finding valid IVs can be challenging in practice.
Y
Figure I: Graphical model for the causal-inference problem. U is the unobserved confounder, X is the treatment, Y is the outcome of interest, and Z and W are treatment and outcome controls, respec- tively. The dashed lines represent dependencies that may may not be present.
More recently, there has been growing interest in using proxy variables to address unobserved confounding. Early work focused on categorical data and unobserved confounders (25), showing that causal effects can be estimated under certain conditions. The concept of causal bridge function 30
31 52 24 8] extends this idea to more general settings, using two sets of proxies to estimate causal effects. These studies have provided valuable theoretical insights and practical tools for causal inference with proxy variables.
Deep learning has also been increasingly applied causal inference, enabling the development of more flexible models. For example, deep-generatin models have been used to improve IV analysis
55 4 Our work builds 19 and to address unobserved confounding in various settings 28
these advances, leveraging generative models to enhance the estimation of the causal bridge function. We note that advantage of proxy variable methods in the unobserved confounding setting is that it requires weaker assumptions than an IV approach. Specifically, IV methods assume that the instrumental variable and the unobserved confounder are independent and the latter influences the treatment but not the outcome.
Our work draws inspiration from and contributes to several active research areas within causal inference, particularly those focused on handling unobserved confounders and leveraging proxy variables. Although research builds upon these foundations, it makes several novel contributions, as elucidated in the previous section.
3 Background on the Causal Bridge
Let X and represent treatment (intervention) and outcome, respectively. In the examples we consider, Y will be continuous and X could be real or categorical (with binary being an important special case). It is assumed that and are both dependent on an unobserved confounder U, as
illustrated in Figure [ As discussed in 25 35 30
31 52), to perform inference in the presence

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of the unobserved U, control (proxy) measurements are assumed to be available. which are also dependent on U. Specifically, Z is treatment control variable, and W an outcome control variable. As shown in Figure [ the outcome may depend on W and the treatment may depend on Z. We assume for simplicity that there are no additional covariates, but such could be included if available, as discussed in [31 [2].
We denote the domains of (Y, X, Z, W) as (V, X, 2, W), respectively.
For the graphical model in Figure we make the following assumptions, which are consistent with those in 30 31 :
Assumption (A1) (Latent Ignorability):
(a) 1 XUU, Vo E X
Assumption 2 (A2) (Negative Control Outcome):
W T XIU and W J U
Assumption (A3) (Negative Control Treatment):
T X) and Z WIU
These assumptions underscore that (Al) the dependence of X on Y is manifest only through U: (A2) the control outcome W does not directly influence treatment X; and (A3) the treatment control Z does not directly influence the outcome Y (or the control outcome W).
We now introduce a bridge function b(W, x) BI 52 14).
Theorem
14 If there is a solution to the Fredholm integral equation V E X' and 2 E 2,
E Ir, 2) ((b((,,)))))) 2),
(I)
then
E(Y zz U) B(((W,,) and do = r)] = B(b(W,))]
The form of Theorem was presented and proven in 14, and requires multiple additional assumptions to ensure that there is solution to
33 52) and importantly, completeness, described below.
Assumption (A4) (Completeness) For any gquare-integrable function g(-) and for any I, E[g(U)) x] = 0 almost surely if and only if g(U) 0 almost surely. And for any square- cittrr-ittegrable function h and for any x, [[h(Z))W,, x] almost surely if and only if h(Z) = 0 almost surely.
Theorem 14 Under AI-A4, and other technical conditions, there exists a solution to ①
The proof and further details concerning Theorem are presented in [4]. The above assumptions (and other technical requirements) are relatively abstract; therefore, in Section beloo we explore the properties that must hold for the existence of solution to II. The insights from this analysis will motivate our modeling approach in Section 6
Implied Properties of the Bridge Function
Conditions needed for the bridge function One may express 1 as
x. x.) = / |au EYR, x,) (2) E((|,,)) = dw (W,,,, <)W, z), bo
Note that the generalized bridge function by,W, x, 2) is a function of z, violating the assumed form of bbW, .)) in (Ι). which is independent of
Proposition 1 For the equality in ① to hold, either bo,,,2,2) 2) is independent of 2, i.e. boW,, 2) = b(W, x),, or b(W, x)) = b,, J(W, and E[f(W, 2)|x,, =| 0 V (s, z). Although by (W, x, 2) is not independent of z, the sum bo + f(W, x, 2) is.
One possible way that bwW,, xx,2 2) could be independent of 2, and therefore equal to b(W, x), is if P(U x, 2) p(U)W,,) .). However, we posit that p(U 2) p(UlW,,) does not hold in general. Another possible way for bo (W, x. 2) to be equal to b(W, xx) is if
dU E(||,, W. W.U)UU
aU B(Y W, (U),
3

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which implies that the z-dependence in bu (W, I. 2) is removed after performing the expectation wrt p(Ui,,, 22, 2). While this is possible, we also do not make this assumption. We therefore posit that in general, one cannot assume that bu (W, 2) is independent of 2, and therefore we conjecture that the second condition in Proposition I must hold, i.e., b(W, x)) # bo x 2)
Proposition 2 If holds, then for all (z, z). the bridge function b(W,z) z)) satisfies
[w-mW//aa) (WW, x)] (w)o( (W, x))]
where (W, x2 2) f dU E[Y, W, UPP x2, 2). Thus, b(W, yields the correct conditional expectation of | [2, 2] when integrated over p(W|z,2)
The hierarchy implied from the above analysis may be summarized concisely as follows.
P(U x, + a)) bo (W, # b(W,))
(4)
Blbo(W, x, 2)|2, 2] Eb,(W, 2|
(6)
(5)
B(Y 2) [E (W, x 2)|2r, s].
(7)
Note that 7 is by definition and is not an assumption. Moreover, condition 66 is required for | to hold. This interpretation of the bridge function a(, as matching 6o (w, z) in expectation
rather than a pointwise (distributional) match is consistent with relaxed identification frameworks
discussed in the literature of nonparametric instrumental variables [3 or expectations replace the exact operator equalities.
where moment conditions
Bound on the Estimation Error for the Causal Bridge We present the following new information- theoretic result, which leverages the analysis in the previous section by relating the error the fit to ① with the relative information between (U, W, Z). The proof is provided in Appendix B.
Theorem 3 (Average Error r for the Causal Bridge) Assume that the function Y Iz. W,U] is
C-Lipschitz in U, and that U is almost surely supported on bounded set with | exists a bridge function b(W, x) for which
S R. There
Bz=p( E Y pr 2]
Ew= b(W, SCR- /2T(UZ2W,
(8)
One such bridge function is
b(W.z) 2)) EV w.x) E||,,, W,U],
but others may exist that achieve tighter bounds.
Examining the statement of Theorem 3, one may be tempted to suggest proxies Z that are independent of U, but the completeness assumption (A4) concerning B[g(U)) x] disallows it. Instead, Thcorem 3 states that W should be low-noise representation of U, in the sense discussed in 22 concerning proxies (W, Z) as "noisy" measurements of U.
Corollary 1 (W as a noisy nonlinear mapping of U) Assume that W VUUU) + where V Rdd Rdw is invertible, continuously differentiable (C') ) function, and is independent of (U, Z, X), with zero mean and covariance matrix T. I. With additional (typical) regularity conditions provided in Appendix B.2 there exists constant Co > 0 such that, for sufficiently small d.
I(U: ZWW x) S Coo
The complete statement of Corollary 1 and its proof are provided in Appendix B.2 While the assumption of an invertible (U) may seem strong, this same assumption was made previously for a similar setup in 255 where discrete observations and proxies were considered (see Eq. (4) in (25)..
Theorem 3 establishes that the quality of the bridge approximation critically depends on the condi- tional mutual information [U; Z[W, z) and Corollary I provides conditions under which this mutual information becomes small, namely when W is low-noise nonlinear observation of U through an invertible and smooth transformation. This insight motivates practical modeling choices, namely, to construct effective bridge functions, one should design or select proxies W that capture the latent confounder U with minimal distortion and noise. In practice, this suggests that proxy variables with low measurement error and stable relationships to unobserved confounders are particularly valuable. Moreover, by ensuring that W is an informative (albeit noisy) transformation of U. Corollary 1 justifies the feasibility of approximately solving the Fredholm integral equation in ( enabling effective learning of causal bridge functions even in the presence of complex, nonlinear, confounding.

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In order to illustrate the results of Theorem 3 and Corollary 1, we consider a structural equation model (SEM) for the generative process in [ Appendix for details). The SEM shown in Appendix Figure is consistent with the model for Corollary 1, but we now assume (U) is linear function. We make this simplification along with letting U and the noise terms for W, Z, X and Y be Gaussian with variances dU, dw. Zz o and Øy, respectively, to be able to obtain closed-form expressions for the quantities of interest (see Appendix C
Commiatian between Maan mof and J1W waraaaa
Figure 2 shows the results for the relative approx- [ [z. Z]] imation error r(n) E EZ>n((Z)>) [n/ [
5,00
vs. T(U / ax) both averaged over X, for
ay = 10, 02 ow = {0.1, 0.25, 0.5, 0.75, 1},
and x 0.1, from which we that 7j increases with (U: x) and ow, consistent with Theo- rem 3. We provide more details and results varying in Appendix D Interestingly, when σx oz and the SEM coefficients are I, both 7 = and I(U; Z[, x) = 0, which is special case formal- ized in Lemma in Appendix
0:00
0.03
Notal motmaton1222
Figure 2: Relative error (r(1)) vs. mutual infor- mation (I(U; 2|W, x)) both averaged over X. Each line represents value of az for increasing values of ow {0.1, 0.25, 0.5, 0.75, 1). = 0.1, which are consistent with [UU a))
5 Generative Model for the Bridge Function
Concerning the assumed form b(W, x) in Theorem 3, while p(U|W,,,,)) was replaced with
P(U(W,)) z), [x U)) was retained from (w,,, xz 2). The latter conducive for analysis. However, it is possible that a better bridge b(W, x) may be learned without retaining <, W, U), and E(((W,,))a, We therefore can
yielding smaller expected difference between E(Y replace 2 with the following general form which we use in our model:
B(W x)) = f dU g(0, W. WUU))(,),,.)
(10)
which allows b(W, 2)) to not be restricted to g(r, W,U) = E2, W.U]. while still being able to
produce the desired E[Y|ær, z). Both g(, W, U)) and p(U 2:) are jointly learned when solving (1).
Note that this setup effectively suggests generative latent-variable model with encoder p(U [ W,4
and decoder for (X, Z), which we discuss in detail in Section 6 In practice do not claim the modeled (U],,,) x) represents truth, as final predictions are based on b(W
We seek model the form of the bridge in (0). From that perspective, a generative model can be constructed to draw the samples of U needed for implementing (I). In this context, and using (IO),, consider uj ((U],,,ar 2) with wj p(Wla, 2). One may consider learning function kWW,,, xr <). such that Mj = h(w,, xr, (j,, where wy ~ p(W| and (for example) Ej ~ >N(0, 7), where 7 is the identity matrix and hence is drawn from isotropic Gaussian noise of the chosen dimension. Such a generative model dWW, x. c) has been widely considered, e.g., in generative adversarial networks (GANs) and its generalizations 18
32. In practice, using Proposition
2, we may model
E(Y Ir, 2) = [r, Ep (z, W, W,h(W,,,e)))
(11)
b(War)
which is motivated by modeling b(W,z) x) in such a way that the inrectatioo-matchiinn property of (1) holds, but explicitly defining b(W, a)) functional expectation over p( (W, a:) as in (I.. modeled here by samples drawn through (WW, x, e), with a general integrand gy (z, W, U). The benefit of this model is most prominent when it is coupled with an autoencoder for (X, Z), in which h(W, x,, e) is shared, thus enhancing statistical strength. This strategy is discussed next.
6 Learning Setup
Assume that we have access to set of data Dj = {(1,, zi, wi)]i=i, from which a generative model can be learned to draw samples from p(W Jr. 2). The details this model depend on the data characteristics, and more details are presented when discussing the experiments in Section 7
5

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Note that we need not model p[ 2), rather in general, we seek to model the capacity to generate samples from this conditional distribution, e.g., using conditional GAN 32
We also assume access to data D2 = ((x, 2,, v))]=,.NN which may or may not be explicitly connected to D1. We use D2 within the Fredholm integral in ① with which we will solve for b b(W, with conditional expectations wrt p (W I 2) performed by drawing samples from the generative model developed with Dj. We note that D) is not necessary; however, is specified weaken the requirement of having a common dataset for which all observed variables (W, Z, X, Y) are available. This distinction is especially useful in situations where the dataset for which the outcome is available D2 is not the same or is (much) smaller than D1.
Bridge for the Outcome Let g0y (I., W, W,W, z,, e)) represent model for E(Y|,, W,U), where we model samples of the unobserved confounder as U = heg (W, x; e), fy and By are the model parameters with subscripts Y and U highlighting that the model is connected to expected outcomes and the unobserved confounder, respectively. To solve (Τ. we seek to minimize the following loss.
Lov = Σ> 1= (y:- Ep(W m,A;) Eptey (I,, W, hag (W, IW e))])²
(12)
If only the outcome Y is modeled, then in practice we replace Eptc) (a, W, heg (W, x e))] with b(W, x)). e.g., a neural network with inputs xW) which is understood to have parameters 8y.
The objective implied by Lay involves two steps: (i) develop a generative model to draw samples from p(W (x,z) =) using D1, and (1i) use this model within the minimization of Loy for 0y to approximate the expectation wrt W. However, note that these two steps are followed in sequence, which should be distinguished from the iterative and alternating two-step approach developed in [52].
Within the context of our discussion after (10), through Wy ~ p[W|,,(, 2) and e, ~ p(e), we seek to simulate samples from p(U [ar, 2) as u, = hō. (wy L. G)). The unobserved U is assumed to affect both the treatment X and Z as illustrated in Figure T When the number of samples N is relatively small, there may be an opportunity to improve statistical strength by also modeling { ((f;;22))=11 )=1,,, both of which are also functions of U (not only Y).
Autoencoder for the Treatment and its Control Analogous to the aforementioned model for Y, we can model [. (X U, =) = gox (U, 2) and [ZZ (U) = gez (U). In the context of an autoencoder, we as- sume that uj hew TI. e,) with wy p(W 1x,x, 22) constitutes means of encoding observations (z,, 24) into a latent feature space represented by conditional samples uy (2,, 2)). Connected to the approximation considered in Proposition 2, we assume that the expectations of gBx (U, and goz (U) wrt pU,,,, xz, 2) may be replaced by expectations wrt O(U,, x) with W ~ (W [x 2). This implies the same conditions on P(U W, xr, 2), e.g., that the expectation of P(U xx 2) depends only on (W,x), and that go, (U, 2) and gez (U) have the same class of dependence on U as göv (I, W, U), e.g., they could be linear in U. However, although these functions may be linear in U, U itself may be nonlinearly related to W (see Corollary 1). From this angle, we consider the additional losses:
Lox EN (I: - (p(W) Ep(e) 190x (hau (W, 24, <), 2))1) Loz = E (2+ 1aes-Epppiia 2. Expce) Ig0z (hog w., (,))))
(13)
where we emphasize that the function hev (W. xx. c) parameterized by fy is shared between the models of (Y, Z). ideally improving the quality of the learned hag (W, x,, c). Note that when producing causal estimates, we only need the learned gor (z., W, h(W, x. e)), i.e.
doo = =)) = Ep(w) Bpcc) lavy [x. W, hog
(14)
However, by jointly seeking to minimize Lay + Lox + Loz, it is hoped that the quality of the model for haw (W, x, e) will be improved, therefore also improving the causal-effect estimates relative to the outcomes. In practice, weighting can be used in the sum of losses, reflect their relative scale which depends on (X, Z, Y). as discussed in Section 7 Moreover, in terms of implementation, we still first build generative model for p(W [z, 2) using D2 and then optimize the parameters of the shared encoder fy, bridge By and autoencoding components (0x, 8z) } using D2.
Connection to the Causal Effects VAE (CEVAE) The composite model, employing the cumulative loss function Lay + Lax + Lag, shares characteristics and motivation with the CEVAE developed in (28 We note that recent work by [38 52 has highlighted limitations of the CEVAR. Some of the challenges with existing CEVAE research are related to the difficulty of modeling posteriors like p(U 2).

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The proposed framework differs from the CEVAE in several key ways. Within the context of (TO). W is assumed to be strong proxy for U. and therefore the model to draw samples from p(W 2 2). based on observed [1r 24, w;) provides strong information about p(U] |z, 2) that was not available to the CEVAE Moreover, within our autoencoder, we only model the latent confounder with (2i, 2,) and not the outcomes y4 as in the CEVAE. The expected conditional outcome [Y|22, =) is modeled via the causal bridge function, for which we have theoretical support (Theorem 3).
Finally, note that within the CEVAE one must set a prior p(U) with which the Kullback-Leibler (KL) divergence is computed relative to p(U] 2,). In practice, it can be difficult to set such a prior; thus we avoid this complication by simply designing an autoencoder, instead of variational autoencoder. Rather than employing a KL term for regularization of the posterior, we use Loy and our model to draw samples from ([ (W2 [2), both of which provide regularization on the inferred posterior.
The extension of our approach a CEVAF-type setup is relatively straightforward, using our definition of p(U] z) within the CEVAE. Importantly, in such a setting. the CEVAE models (X. Z), and Y is handled via the Fredholm equation for the bridge. The main difference between CEVAE version of our approach is the inclusion of a prior p(U) and the inclusion of a KL term between p(U) and [U(]2x,)) Nevertheless, we did implement such modified CEVAE formulation, in addition to the simpler autoencoder setup discussed above. We found that the KL term added significant difficulty and undermined the reliability of CEVAE-based predictions. We do not show results for an CEVAE-like version of our model (with KL term) in Section 7 because they were numerically unstable and sensitive to the way pUU) was set (like shown in (52)). In contrast, we found that our approach trained well and yielded reliable results. However, in Section
7 we do discuss and compare
to results from the original CEVAE model (28.
Bridge for Survival (Time to Event) Outcomes So far we have considered continuous outcomes Y with standard squared error loss Cay. We now consider survival outcomes, which are of special interest in a wide range of scenarios where causal inference is used in practice, constituting a novel application of the causal-bridge framework. Specifically, we consider outcomes of the form (Y, E), where Y is the observed time Y = min C)2, T is the time at which the event of interest occurs, C the follow-up time, and the observed-event indicator. If for a given sample, y = < c is said that the event of interest is observed and 1, otherwise, right-censored. Here we assume that censoring is not informative, ie.., I CIX. Extensions to other forms of censoring are possible within our framework, but left as future work. Unlike for continuous outcomes, we are not interested in modeling the (expected) value of Y through [ = x)). Instead, we are interested in [[x]do(X = x)), i.e., the risk function defined as the contribution of observed covariates on baseline hazard function, iee, X(Y| x) = At exp(b(W, x)), where (t) is the baseline hazard and exp(b(W,,)) is the risk function, conveniently written in terms of bridge function. The casual estimate of interest for binary treatments, X = {0, 1} is the hazards ratio (HR) defined
< t, e = O and the event is
X(|)WXX= exp(
HR-- XYIW, X 0) (0p 0))
(15)
Optimizing (I wrt the hazards function is achieved by maximizing the partial likelihood of the model similar to the Cox proportional hazard model [13] using
Lay Σ Pi log Σ =1,=1
explri)): Pi Ey(W(a Beee (si, W, Wao (W, a.,, (W,)),)))]
J:8,
(16)
where we do not need to account for the baseline hazards (t because it does not depend on the treatment X or the outcome control W. Note that the autoencoder version of this model is consistent with the above definition, except that the loss for X is changed to cross entropy.
Recently, [54] proposed an approach that also modeled the hazard function using the bridge function. However, they do not make the proportional hazard assumption like we do. Rather, they impose a rigid form for the bridge function (Eq. (31) in (54). That work considered experiments on real data
(SUPPORT), as we do. However, [54] the ground truth for the estimate of the causal effect not
available, while in our experiments in Section 7 we compare to an RCT.
7

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7
Experiments
All models were developed using Py and each experiment can be executed in few minutes on a Tesla V100 PCLe 16 GB GPU. The source code used here can be found at https://githh:n com/ ruol aaa_mmeeeg/aausaDArrrgggggttttt
7.1 Synthetic data
We first demonstrate the performance of the proposed method on two synthetic datasets introduced in [52]. and we perform the same experiments as considered there.
Data The first experiment considers the Demand data introduced by [1]]. Details of the data generation process are found in Appendix E..1 The second experiment considers the dSprite data introduced by 29 Details of the data generation process can be found in Appendix F.2 For both datasets, we consider two sample size settings, 1000 and 5000, for a single dataset [ 2, W, v)}. consistent with the setup in [52].
Metries We estimate the average causal effect using the bridge function with (14) and compare it to the ground-truth causal effect in terms of the mean squared error (MSE). Note that this is only possible with synthetic data since the datasets used to train the model only have access to specific (factual) combinations of { z, y) i.e. we do not also have values of {(z, y)} for which the treatment takes values different (counterfactual) than those observed. The test points are evenly distributed over the treatment variable X when calculating the out-of-sample MSE. Additional details are provided in Appendix H This setup replicates the experiments introduced by [52]
Models considered We compare our method to the deep feature proxy variable (DFPV) method of [52]. which is considered the prior state of the art. In the results below, we consider the following
model configurations: i) The original DFPV method [52]. ii) The same underlying DFPV model (and
hyperparameters) but replace iterative learning (52) by first estimating p( W [x, =). from which we
sample when learning the bridge function. This setup allows examination of the benefits of learning to sample from p( [[] [z, 2). with no change to the form of b(W, x)) from DFPV. iit) The causal bridge (CB) model learned with (12) and using the p( [ xx 2) sampling model. it) The combined bridge and autoencoder (CB + AE) model with (12) and (13) i.e., Lay + Lex + Caz. Moreover, we also consider CEVAE 28 neural maximum moment restriction (NMMR) with U- or V-statistic optimization so NMMR-U and NMMR-Y V, respectively [24]. and Kernel Alternative Proxy (KAP) [8. Concerning the new methods discussed in Section 6 (the last two methods above). our model architectures are relatively simple and not considerably more complex than those used by DFPV [52). This is done to demonstrate that the performance
our model variants is not attributed to overly complicated architectures. The details of the neural networks and hyperparameter tuning for all models are provided in Appendix F. Concerning the last two models discussed above: CB learned based on modeling Y alone and CB + AE is based on the combined loss for modeling (Y, X, 2), the exact same models and hyperparameters are used for gey (z, W, heg (W, x e)) (this allows consideration of the impact of the autoencoder, without anything else changed). We also note that model selection for all approaches is based only on the loss for Y in (1))
In all places where we sample from p[W|x, 2). 100 samples are drawn. For the model hap (W, xx, c). for each draw of W. 5 unique e are drawn from p(c). These values were selected based on empirical validation to ensure stable learning and accurate estimation.
Results Figure shows that for experiments with synthetic data, Demand and dSprites, using our generator for p W|r, 2) significantly improves performance relative to DFPV (52), which is likely due the advantages of sampling (we draw 100 samples from p(W (>,2) in these experiments). Consequently, we have the flexibility to generate many samples of W for each combination of
[(X x, Z }. enabling better learning and estimation of causal effects. Moreover, utilizing ha, (W, xx c) in the generalized bridge model go,, (a, W. ho,, (W. x., c)), and thus moving beyond the DFPV representation of the bridge, further improves performance. The gains of this approach using [22 are most noticeable on the Demand dataset. Incorporating the autoencoder for (X, Z) using both
1 and [3) aids with the learning of the shared he, (W, x, t), particularly when the sample size small, which is of particular interest in real-world scenarios. Finally, when comparing the proposed
CB and CB+ + AE with NMMR-U, NMMR-V and KAP, we see that our approaches are significantly better, especially the dSprites dataset. Results for KAP on Demand and CEVAE on both datasets
8

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Haaard hadoo 55% Confieece Haree
Figure 3: sam-ll MSE results for (Left) Demand and (Middle) dSprite data. (Right) Hazard-ratio (HR) results with 95%CIs for Framingham data. The different methods are listed along the axis, including results from the RCT, to which CB AE agrees best. The red and green dashed lines correspond to the null HR= 1 and the reference (mean RCT estimate), respectively.
are not shown because they are out of range, however, they are presented in Appendix Table along with all results in Tigure3 completeness.
In Appendices [ and J we present two ablation studies. In the former, we explore the dimensionality of U and noise for the shared autoencoder heg (W, 2 e), by showing in Figure 12 that our model is insensitive to them. In Appendix U we explore the generalization in (IO). but instead of using hew <W,
xr e) to sample from ((U] we do directly via MCMC (we can do this when we assume the unrealistic case of knowing the underlying model - we do this as test of the utility of the model hay (W, x. e)). We show in Figure that learning g(x, W. U) with samples from p(U (U, x)) produces slightly more accurate causal estimates compared to using hay (W, xt c), which is good indication that the latter is a good approximation for p(U[W,,) x)) for the specific purpose of estimating causal effects.
7.2 Real-World data
Data Framingham is an observational longitudinal study designed to learn about the incidence and prevalence of cardiovascular discase (CVD) and its risk factors [5). The data used here are the Offspring cohort, which consists of 3435 subjects split into 2404, 516, and 515 training, validation, and test samples, respectively. For this experiment, we are interested in estimating the average causal effect of taking statins (a cholesterol-lowing timing Y of future CVD events. These data are in the public domain, and therefore these experiments can be replicated.
medication), i.e, the treatment X {0, 1}, on the
Constructing proxies for Framingham The 32 covariates available for this dataset are not split in terms of treatment and outcome controls, Z and W, respectively. Consequently, we adopt the proxy bucketing strategy of (47), which divides covariates into Z and W according their association with treatments and outcomes using effect sizes estimated from linear models on the training data. Details
are provided in Appendix E.3
Models considered We compare the proposed framework with three strong baselines: variants of balancing weighting schemes for the Cox proportional hazards (CoxPH) model for survival analysis [42. Balancing weights are obtained from a linear logistic regression model built to estimate propensity scores P(X 1 w, Z = 2). Then, these weights are used to fit a CoxPH model where the input is only the treatment and the objective uses the balancing weights. We consider three weighting schemes: (CoxPH-Uniform) where all weights are 1; (CoxPH-IPWW using inverse probability weighting, w = r// + (1 x1 - s) IO:: (CoxPH-OW) using overlapping weighting, w x s) + (1 - I)ss (27). We also consider the two proposed variants of the causal bridge, i.e. CB and CB AE. Additional details about the CoxPH model and overlapping weights are provided in Appendix G The sampler p( 2), the bridge and autoencoder architectures, and model selection details are provided in Appendix F
Randomized control trial We also report the HR estimated from separate randomized control trial (RCT) conducted specifically to assess the effects of statins on CVD outcomes [56]. These RCT results provide a powerful reference against which causal estimates can be compared.
Metries We estimate the causal effect of the treatment using the hazard ratio (HR). For the CoxPH model, the HR is obtained simply as exp( where 3 is the coefficient for the treatment obtained
9

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from fitting the weighted model. For the proposed bridge model we use the strategy described in Section 6 HR values are interpreted as positive (HR< 1). negative (HR> 1) or neutral (HR> 1) in relation to the effect of the treatment on the outcome. Confidence intervals (95% Cls) for the HR are obtained using asymptotic estimates for CoxPH model [3] and empirical quantile estimates (from multiple model runs and samples of W) for the proposed model. Additional details are provided in Appendix G For completeness, in Figure [3 in Appendix K we also report the concordance index (C-Index) on the test set, which is widely used metric to assess the predictive power of survival models. Note that we cannot obtain a C-Index for CoxPH variants because these are built only using treatment (X) and outcome (Y), thus unable to produce predictions.
Results Figure shows HR estimates with 95%CIs indicating that the CB approaches outperform the CoxPH variants. We note that CoxPH-
and CoxPH-IPW both result in HR> indicating that the treatment increases the risk of CVD events, which occurs because subjects at high risk of CVD are the ones treated with statins (see Figure 14 in Appendix K thus effectively confounding its effect on the outcome when using observational data. The other three estimates (CoxPH-OW, CB and CB AE) result in HR 1, however, notably, CB + AR results in tighter estimates with 95% CI away from HR= 1. Importantly, the causal-bridge results are consistent with expectations from the RCT. For completeness, Figure 15 in Appendix K show results for CB and CB + AE as boxplots.
8 Conclusions
We have introduced a framework for causal inference with control variables by re-examining core assumptions connected to the causal bridge. This yielded a new bound on the error of the causal bridge, providing insights into the estimation of causal effects with unobserved confounders. Our approach employed a conditional generative model for W, and motivated a new framing of the model by the inclusion of an autoencoder. We also extended the causal- framework to survival analysis, broadening its applicability to a wider range of real-world problems. Empirical validation on both synthetic and real- real-world datasets confirms the superior performance of our proposed approach compared to state-of-the ahrt methods. As an interesting direction for future work, we recognize that it is possible to derive similar result presented here for the causal (outcome) bridge but for the treatment confounding bridge using ideas from [14], although noting that such a treatment bridge is most useful for binary treatments (interventions), while most of our experiments considered continuous (real-valued) treatments. Moreover, we can also extend the proposed methodology to scenarios where post-treatment variables are available like in [49].
Limitations This work has several limitations, i) analogous to other causal inference frameworks, verifying the assumptions is difficult; ii) defining splitting covariates into proxies, W and Z, may be challenging in practical scenarios; iii) we acknowledge that the proportional hazard assumption used for the time-to-event model can be seen as limitation; however, one can alternatively consider a specific formulation for it like in [54], however, such an approach has also issues related to model misspecification that are difficult to address in practice; and iv) understand that having additional real-world datasets will make the experimental results stronger; however, it is difficult to find real- world datasets for which the ground-truth causal effects are known, such as it is for the case shown in our experiments using the Framingham dataset.
Acknowledgments
This work was supported by ONR grant number 313000130.
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Summary of Appendix Content
This appendix provides extensive set of details associated with the material in the paper. To aid the reader in navigating and using this Appendix, we summarize what is provided and in what section (with hyperlink it).
Section
Broader Impact Statement.
Section
Proofs (Theorem 3 and Corollary 1, and a Concentration Inequality).
Section
Relative Error under Structural Equations Model (SEM).
Section
SEM Experiments for Gaussian Unobserved Confounder.
Section
Experimental Details.
Section
Network Structures and Hyper-parameters.
Section
CoxPH Loss with Weighting,
Section
Additional results for Demand and dSprite.
Section
Ablation Study.
Section
Bridge Generalization from Proposition
Section
Additional Survival Analysis Results.
A Broader Impact Statement
This research involves the estimation of causal treatment effects, using proxy variables for the real-world survival dataset. Even though the proposed approach is not a replacement for direct measurements of drug effects, more accurate causal inference from real-world data can improve clinical decision making by supplementing it. Although this work has the potential to enhance drug impact assessment, careful application is essential to avoid unexpected outcomes. From theoretical perspective, this study seeks to advance the generalization of causal inference using proxy data, which is increasingly vital for machine learning models, partieularly in high-stakes decision-making domains such as healthcare, finance, and policy. Thus, it includes the societal consequences, ranging from ethical to environmental considerations linked to the field of machine learning.
B Proofs (Theorem 3, Corollary 1, and Concentration Inequality)
B.1 Proof of Theorem 3
We first consider the case Z z, and then will average over Fix and z. For each W, define
8(W)= := Eo=p((]].,)(((U))
where f(U)- E|Y|æ, W, U]. Since f is C'Lipschitz and U is supported on set of radius R.
follows from Pinsker's ss inequality and the standard variational form of KL divergence (see (50 37 that:
[0(W)| s CR. 22 (P(U W.,, 2) (U a))).
This inequality is related to result the proof of Lemma in 50 using here that a C-Lipschitz function f()), with U supported on ball of radius R. is Cu--sbGaussiann 48
Now take the expectation over W PWW[z, 2);
[Ew>p(Wle.e) [b x) bo bo x, =)]l S (W.p(W1e.e]((W)))
Apply Jensen's inequality to the concave square root function, and the series of inequalities continues:
Ew=p(W1z.e) ((W) < CR Ew Ew-p(W (
V2Drr. / p(U|W,,))
< CRR v Ewwxp(W)..)) [KL(p(U |UU
From above, have result for fixed z. We now take expectation over Z p(Z[zz) on both sides:
Ep(Z(a) [|Ep(W]r. (6(,,)) (6W,,
bo W, x, 2)]]
S CR (p(z)w) [v
14

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Now apply Jensen's 's inequality (the square root is concave) to move the outer expectation inside the square root, the subsequent inequality follows:
SCR- 2 | DDDGHN/I
The right-hand side becomes:
CR- 2 (() (PRi,p(U)W, 2) ())) = CR 221 x). which is the statement of Theorem 3, recalling that E(Y|,, 2) Bw- (o, W, U)]. Providing more detail, the final step states:
E (xw. ( =
To show that, consider that the conditional mutual information [UU 2IW, x)) is defined as:
[ p(u, w/x) log
p(u, 2 ww,) x)
2U;
du dz dw
(17)
[p(ulw x x))((z)uu (1,
f [ plu, 2, worr) log
p(u 2,
du dz dw
(18)
p(u ((ulw,2))
This can be rewritten as:
/ / p(u, 2|0)
[f p(uluw 2, 2)g log
p(ulw x)
I(U; x),
du dw dz
p(ulus w. x))
The inner integral is precisely the KL divergence
z)), so:
/ p(W, Z(x) Dw..b(UW...))))llluu
I(U; Z))
dW dZ
(19)
-Ep.w.zee//DeebbtU/.. 2, Z,)
(20)
B.2 Complete Statement and Proof of Corollary 1
Corollary 1: Suppose that the random variables (U, X, W) satisfy the following conditions:
(i) W = =U) + c, where V RR + Bddw is an invertible, continuously differentiable (C*)
function,
(ii) is independent of (U, X)X,, with zero mean and covariance matrix o.
(iii) The latent confounder U has finite differential entropy h(U) < cc,
(iv) The conditional distribution pl p(Z(U,,) x)) Lipschitz in U; specifically, there exists L2, such that
log p( (Z((,,,,))) 2) log pl (12 S Lz 11 well, Vug, u2-
(v) Z T W (U, (vi) Quin min P(U W,x) p(Z W,x) > 0.
Then, there exists a constant Co 0 depending on the Lipschitz constants of v and p(Z(U) such that, for sufficiently small o..
I(U; Z),,x) < Coa
In particular,
lim I(U; WW x) 0. 20-
Proof:
Since W V (U') and V is invertible and Cº,, can locally linearize v around any U. For small noise e, we have the Taylor expansion:
W (U))
U 1 - -)).
15

---

Expanding & VW - -c) around W gives:
V
Vy vw- o(le]]);
where vu- is the Jacobian of v at W.
Thus, conditioned on W, the distribution of U is centered around vp w'W(W,, with fluctuations proportional to To be more specific, as - 0, we consider following approximation
PUU W2x) " (U)
(21)
where "> refers to weak convergence, μr (W) L (W) and (w-(w) is Dirac function centered at jr" (W). Moreover, the conditional covariance of p(U W, can be approximated as
Var(U|W, x V (W) Var(c) (v (ow'((w) (W)) O
(22)
because Var(e) = 1, and VW- is bounded by the invertibility and smoothness of V. With above approximation, we have
p(Z W, / p(z UU w.
(Using T. W (U,
~ p(Z (W))
(Using 21
r(Z n² x)
(23)
Then, by Lipschitz assumption log p( 7 U), derive
12(Z W,xx p(2 1 (W), = / [p(z U,,) p(Z k² (W), p(U W, z))U
(Using 1 W (U,x))
$ $ f/It2 |P( U2x) -p(z 10 ('(W), x)) p(U W,
< LZu /
[IV 10 (w-N((W)V(0 W,
LZ BUJW, (W)]]-
(24)
Notice that
Eujw, 110 / (W)]] B VVU 1 (W)
V Belwaell 1 (W))
(Using Jensen's inequality)
Var(UlW, = O(oz).
(Using 22
(25)
Hence, we have
Io(Z W,x) p(2 14~ (WWW,, S LZ O(a.).
(26)
Now, observe that the pUU Z1W, x dZ is close 2 because
the factorized distribution
p(U, 2 do = x(U W. W,x)p(2 dU
(Using Z JL W (U,x))
(w) (U)p(Z U,z)dUdZ
(Using 21
(Lr((W(('))( μ²
P(U W, W. d.
(Using 23
(27)
16

---

We consider P = p(U, Z W,x), Q - p(U W. p(Z] W, The total variation distance would
be
TV( - / 2 W,x) - pPU W, p(Z dZ
2 / [l(Z U,,) 22 W. pLU W. dd
1 / [M( U,z) p( W. (U)
(Using (21)
/ 1M(2 ('(W), p(2 W, (w)
(w) (U)
2
/ 1p(Z (w()), pp W, (U) dU)
<
/ Lz,U, O(o,) (w) (U)
(Using (26))
Lz,U. O
(28)
2
Applying reverse Pinsker's inequality] | which states
DRl((P)))) - 2log 2loge TV((P Q)²
(29)
Qmin
where Qmin ( p(U W,x) p(Z W, and we assume Quin O. Combining 28 and 29 we can derive
L3 u. loge O(o²).
Dkr. -
(30)
2Q
Thus, for the mutual information I(U; Z W, = Ew DKL. (PI(),, we have
L u loge 0(o2)) 12, log
I(U Z W,x) S Ewizl
-(a²).
(31)
2Q
2Q
Finally, by the definition O(-), we know that there exists there exists Co depending on Lz such that
[UU S Cooe for sufficiently small o.. Hence, Z + 0 as o² - 0, completing the proof.
C Relative Error for Structural Equation Model (SEM)
Consider causal graph in Figure induced by the following structural equations model (SEM)
7
Y = ayxX + HywWW W + +ayuU +
(32)
W awuU +Ew
(33)
Z = azul +ez
(34)
X axzz + axuU +EX.
(35)
where oyx, ayw, ayg, azu are nonzero real coefficients, and ey *O o. ~ NO ), ~ NO 0 N/(0, (x), with distinct variance.
)x are all drawn from mean Gaussian distributions
Consider the SEM in 35) with the unobserved confounder U also assumed to be Gaussian. i.e. ~ (N0, ) This is done for convenience, because such a case, W, and X being linear U, thus (W, 2, X.U) jointly follow multivariate Gaussian distribution. More specifically, since we
know that E[U] = O,
E W] E + Ew] U] EU 0
(36)
E | E/azu + Ez] = U = azu = 0.
(37)
This inequality is stated in Section 7.6 of 37
17

---

W
azu
OWU
Z
W
XXUU
αYU
azx
dyw
EX
X
QYX
Y
Figure 4: Graphical model the structural equation model (SEM) in (32 represent unobserved random variables.
35
Grayed out nodes
and
[[X] = ElaxzZ + axuU + = Blaxz + axvU] axz + axx axx:BB 0, (38)
hence (W, 7, X) can be expressed as (0, Ewzxx,wzxu). where EwZxe,wZXU is the corre- sponding covariance matrix whose entries are stated in the following proposition.
Proposition 3 Assume that the unobserved confounder U, treatment variable X, outcome variable Y, treatment-related proxy variable and outcome-releee rreated variable W satisfy the structural equation model and U (0, ) The entries of the covariance matrix Ew 2xXX would be
Cov oū:
2 Cov(W, W) awyot ow
Cov(Z, Z) azzot +0z
Cov(X,X) (axzazu + axu) 22202 + ox
Cov(W, U) awoot:
Cov (Z, U)) azuoy,
Cov(X, U' = (axzazu + axu)ot
Cov(W, Z) awvazvot
Cov(W, = awo fax
Cov(Z, X) azulaxzazu + axu) + axzo2
Proof: In the derivations that follow, we will make use of the linearity of expectation and the formula for the variance of linear combination random variables.
Further, the derivations will repeatedly use the fact that, since, U, Sw, EZ, and ex are mutually independent,
E[Ue x] E[U]E Ex| 0,
E[Uxx] E[U]E[[x]] 0,
E[ewU] E[(w] EE[ O,
B[Ewez] E[Ew [[Ezl = O,
E[Ewex] Elew |E[xx] 0.
-Cov(U,U). Following the definition, we have Cov(U,U) Var(U) oi.
18

---

. Cov(W, W). Note that W = awoU Ew. We can derive
Cov(W, () = Var(W) = VarlawoU sw)
aivv Var(U) + Var(Ew)
(39)
2 awvot tow.
(40)
39 holds because U IL Sw.
Cov(Z, Z). Note that Z uzuU + 8z We can derive
Cov = Var(Z) = Var(azuU + =zz)
u Var(U) + Var(ez)
(41)
azzot + +
(42)
uses the fact that U 1L Ez.
Cov(X,X) X). Note that X = -xxz + axuU + We have
Cov(X, X) = -Vr(X) Varlaxz + axuu +ex)
Var(axz|azuu E2) exxU+Ez
Var((oxzazz +axu aazez +
(axzazu + ( + ax Var(ez) Var(ex)]
(43)
(axzaze t xx o
(44)
43
follows from the fact that ez and sx mutually independent.
Cov(W, U). Note that Cov(W, = B[W U]
We have
E(W U] E[(owo aw) U] Elawou + EWUU
awr +
awo = awe (Var(U) + B(U?)) = awuot.
(45)
and,
E[W] B[U] =
Thus, we can infer that Cov(WUU) owrot.
Cov(Z, U). Note that Cov E|Z U] - E E[U]. We have
B[Z U] [ + +z) U] BlazvU2 Uezl
D EUu
(46)
a = azu (Var(U) + B(U azuot
(47)
and,
E[Z] E[U] = 0- 0
infer that Cov (Ov(Z o
Cov(X,U). Note that Cov(X, = E|X U] E[X] ELU]. Since we have
E[X U = El(axz axu + +xx) U]
+ (xx ( + +xx) U]
(axzazu +oxu [| U]
(axzazu +oxu [U
+ ax [U E[Ex [E[]]
(48)
(axzazu 1axu)EUU**
(axz0zu + (Var()))
(axzazu +axu)o
(49)
and,
E E(U] 0 0=
we can infer that Cov (ov(XXU)= =X + axu )a?
488 uses the fact that U 1 Ez and U ex.
19

---

. Cov(W, Z). Note that Cov(W. = E|W [[[] B[Z]. Since we can derive,
E|W- Z] Bl(awoU (lawoU+eww) (azuU +
B((awuazu Ewazuu +wu =wEz)
Elawvazo + UlswazuU] + BlawwUezz + [[swez]
Elawwazu U azu [BU] awr + E|e2]
(50)
ElawvazuU?!
awcazvBBUu awuazu E awrazuot
(51)
and
B [ = 0
we can infer that Cov(W, Z) awoazool
(50) follows from the fact that using U, Ew and Ez are mutually independent.
Cov(W, X). We can derive
R[W x] BlawoU + &ww) + +xx)]
E((wwUU + tew) (axz(azuU + az) + axxU + +xx]
B((awUU +ew) +xxxzaz++xxx0 aazzez EX)]
Elawwlaxza2z =lavolo1z22x+xxx0000= +
tawueex
+ (axzazu + // +
axx 1o aaw
+ ( +oox:/oottooooooooooooooooooaa E[swex]
El(awo(axaazu
awelaxzazu a
awelaxzazu + axu (Va E(U?²)
awe(axzazu ax.
(52)
Besides, since [[W] [[X] = 0, we can infer that Cov(W, = awu(axzazu + axu)ot.
Cov((, X). Finally, we consider that
B|Z x]
El(azu +82) (axzž+axol) +
E[(azuU +zz) (axz(azzl + + axxu +ex)]
El(azuú + (laxzazu + a +axzez +ex)
+ (xtp://
azu
+ +axzazuu +xx txxxl
+ oo azu B(uxx
(axzazu ( [||zex]
+
(53)
(Vxz(Var())) +B(E2))
azu o taxz03
(54)
Moreover, since B[2] [[[] = 0, we can infer that Cov(Z, = azu +axu) axzoz.
+
Closed-from Expression for Relative Approximation Error With the help of Proposition 3, we can start derive the closed-from expression for the relative error r(n.). Given pair of Gaussian random vectors.
(), ~N -N(1,
EA.A
ΣA,B EB,B
EB.a
20

---

the conditional A B = b can be expressed
A B = ((AA, CAAA EA,BLi n, B, B.,A),
(55)
stated in section 2.3.1 of [6].
With this expression and Proposition 3, we can express p[W|X, 2). p(U] X) and p(U|W as Gaussian distributions using components of Ewzxu follows.
p(W 2) Let W. B (X, Z) and (X x. Z 2). Then, have
(w](X== Ew E X2,X2
(56)
2 WI o. Ew.x www.x2Xxz.x X2,X2 Exz,w.
(57)
p(U W, Let U, (W,X and (W = w, X = x, 2 =2). 2). Then,
N= = SX
I
(58)
i W -.w. = ot. St.wx2Wxxz.xx Xw EwxzU
(59)
p(U X). Let A = U.B B (W,X)" and (W w, X = =x). Similarly, we have
(ol(W=u.X=o) = Ev.wx W
WXWX
(60)
a (e = Eu,
D
W.W.X EW
(61)
Recall that al for the SEM, before the expectation wrt p( X) and p(X) can be expressed (in slight abuse of notation)
E 23] Ew ( ((W
E 2 2|l
Y [2,, 24]
aav Ew 0i.:
B B U1]
(62)
x + Ew [arw W + aav E B
With 56 61
we can further express By =| as
MUl(W=W' X=xxill
(63)
Y Fi 2:]
l ayww" + Mo](W= Xxx;
where w (X=n,
Thus, can expressed r(T))
r(n) = E Empp((((E=s)
E 2d
Jayu ju (W= (=m, Pul X=nii]
=Barrp(Z(X==)
(64)
|ayx aY + PUI(W= ( X=x;
Analytic Solution for r()) = 0 From 64 we note that if we can show X==
[W][Ww..e- for any (W = w, X = x, Z = z), then r(n) would be zero. It is clear that when U Z (W, X). then (U](W.X.2) = Mul (W.X) holds, and hence r(w) would zero. Following this idea, we can derive sufficient condition such that r(n)) =
Lemma 1 Assume that the unobserved confounder U. treatment variable X, outcome variable Y, ceatment--elltted proxy variable Z and outcome-related variable W satisfy the SEM and U (O ). Then, if
xxUaxz
2
azU
21

---

Corretation
Correlation Mean
3.022
3.00
3,115
2.02
1.92
0p-6..
2402
1--0
0p-1.
0p-12
5.0
5.4 Minia interodionn 20:
6.0
5.2
5.4
0.6
0.8
1.2
14
Minus Wnnmmddonn N02.W
o 0.5
(b) xx 1.0
Figure 5: Relative approximation error (r(1)) mutual information (1(U; WW, 2)) both averaged over X. Each line represents value of for increasing values of ow = {0.1, 0.25, 0.5, 0.75, 1). which are consistent with TU; a.)
then we can infer that 1 Z(WW,X) and hence r(n) =
Proof: By 155 we can derive
EuZWxXZZXX = EUZ2 - X Ewx,U2,
(65)
which implies that
Cov 2 XXX)= = Cov(U, Eu,
E X x
(66)
Finally, we plug-in the value Proposition 3 into (66). Thus, we can derive
Cov(U, W)
of ( (xe axz + azu 6
(67)
aiveot2 t80 + otv ooo axw obot 2oxuoxzazr ofai 032020 ob 0x2 oi +
Assume 3 expoxz Then, (- azuo) = and hence Cov(U, Z X) which suggests that U and uncorrelated conditioned on W,
Further, since (U, (W, X) are jointly Gaussian, uncorrelatedness implies independence, as stated in section 11.5 of (26).
Hence, we can conclude that if of (-axvaxzo azuoz) 0, U 1 XIWW and, thus r(%:) = O.
Lemma 1 seems indicate that once the treatment variable X and treatment-relate proxy variable satisfy some relation, then the information of U in Z would be contained in the information of U in (W, X). In other words, we can have an approximation of BUU X, Z] with EEU ie.. BUU X, Z] Re EIUIW, X], which implies that r(n) would be close to zero.
D SEM Experiments for Gaussian Unobserved Confounder
To analyze the association between the relative error r(n) [= [x, Z]] vs. T(U; Z), both averaged over X, we present Figure 2 showing the results for the mean of the relative error r(z) Vs. T(U; Z (. averaged over X, for au = 10, ax = 0.1 and oz.ow (0.1,0.25, 0.5, 0.75, 1}, all weights xy ayw, ayu, azz set 1. Moreover, to show the effect of changing respectively. For these experiments, we generate A 10, 000 samples, D {[4,, wi, 2 drawn using U (0, of) and (34 (35). and the corresponding noise variances to compute the mean the relative error r(n) and mutual information [(U; Z x)) averaged over X.
we set ox 0.5 and σx = 1.0 in Figure 5]a) and Figure 5Jb).
We observe that across ox (0.1, 0.5, 1.0). increasing ow drives up both the mutual information T(U; ZW, æ) and the relative error r(n) tandem. This shows that our mutual information-ben -ased bound reliably captures the growth of the error with increasing ow. Furthermore, present the
22

---

0w-0.11
0p- 0.25
0.5
0g
0.-1.0
9.063
0.032
9.092
6.03
6.004
6.005
000
666
0.031
0.055
6.009
0.005
6.005
0.000
0.000
0.009
001
6,20
0.003
6.034
0.060
5.09
640
0.023
6.011
6.09
6.024
6.013
0.035
000
5.000
6.020
0.003
500
0.031
0.020
5.099
620
001
0.020
0.013
5:0
0.020
0,015
0.011
0.06
5.00%
0.030
0.002
0.
0.21
0.5
0.79
1.0
21
0.5
05
0.15
1.5
0.1
0.25
05
0.75
6.1
0.25
6.5
0.75
1.
0.1
0.25
0.5
0.75
1.0
0.000
0x
Figure 6: Heatmap of Mean of
21
0wm0.1
0.25
0.5
0u-00775
0,71.0
0.000
3.002
0.001 0.000
0.003
0.03
3:005
5,000
0.003
5:015
0.007
0.001
9.093
0.007
55.03
5:008
0.20
0.00
0.2
0.22
055
0.25
6.75
01
025
05
5.25
0.72
0.25
0.5
0.75
1.0
0.500
42
4x
0%
Figure 7: Heatmap of Standard Deviation of |
heatmap of the mean and standard deviation of 2]] in Figure and Figure respectively. It worth mentioning that the mean of |_]||z.z] is equal to the mean of r(n), and that the standard deviation of 71 =, is actually the upper bound of the standard deviation of the relative error r(n). which is induced by the Jensen's inequality. We observe that the mean of r(1) and standard deviation of 7V [. monotonically increase as ow increases. Notably, when ox = oz. both r(1) = 0 and
I(U; Z x) O, which is special case formalized in Lemma in Appendix C
Note that when computing the statistics of the relative error, we use trimmed mean and standard deviation, i..., we remove 10% of the outlying relative error values to mitigate the influence of large outliers occurring when the denominator of r(n) is too small, thus causing precision errors.
E Experimental Details
In both experiments, we generate M 1000 5000 data for both stage Dj {1T, Ris Wa W1). and D: = {(I, 2i. ys h=M.M to share.
E.1 Details of the Generation of the Demand Dataset
The observations are generated using the following model:
V (V-P A5 55)))5gg)))))
(68)
Y=
exp
+
10
where Y represents sales, P is the treatment variable (price), and these are influenced by potential demand D. Here, a A b denotes min(a. b), and the function is defined as:
g(d) = (d ()
+ ((d-5)) 5)
5)21) + 10 2.
600
23

---

As negative control (proxy) data, cost-shifters C1 and C2 are introduced as treatment weatment-inducing proxies, and the views V of the reservation page are used as the negative control outcome proxy data. The data is generated as follows:
D ~ Unif 10 10]
C1 ~ 2sin(2Dt/10) +1
C2 2 cos((200
6
V ~ 7g( +5
P 35 + (C1 + 3)g(D) C2 + c4.
where ei, E2, <3, E4 (O, 1). These specifications and notation are as given in 52 Concerning the notation in this paper, we make the following correspondences: D ++ U, (C1, G2) + Z, + W, and P + X. The outcome is Y, consistent with the notation of the main body of the paper. We select 10 evenly spaced treatment values within the range [10, 30) as the test data, following the same setup introduced by [52]]
E.2 Details of the Generation of the dSprite dataset
This dataset consists of images parameterized by five variables (shape, scale, rotation, posX, and Y)s The images are 64 64 pixels, resulting in 4096-dimensional vector. The shape parameter fixed to a "heart," hence using only heart-shaped images (see below). The other parameters take values within the following ranges: scale E (0.5, I], rotation [0, 2T], posX E [0, 1]. and posY [0, 1].
The treatment variable A and the outcome Y are generated as follows:
1. Uniformly sample latent parameters (scale, rotation, posX, p)s
2. Generate the treatment variable as
= Fig( rec rotation, posX, p)sY) + 17A-
3. Generate the outcome variable as
0.0,
1
(posY 0.5) (vec(B) A) 3000
Y
+
12
500
The function Fig returns the corresponding image for the latent parameters, and 1)л and are noise variables generated from 7) (O000 0.17) and (0, 0.5). The matrix B E R11x61 is defined as B, = |32 jl/32. From this data generation process, we observe that and Y are confounded by Y.sY. While the treatment variable A is given figure corrupted with Gaussian random noise, the variable posY Y is not revealed to the model, and hence there is no observable confounder.
The structural function fstrutt is defined as:
A)2 (vec( TA)2 3000
Farue((A)
500
The negative control treatment is given by the tuple (scale, rotation, posX) IR. and the negative control outcome is
V/ ~ N (0.0.0..1)
W Fig 0.8, 0.5. 0...0.sss.s. posY) + nw.
We set aside 588 test points to quantify the validation error (out of sample), which is generated from grid of points on the latent variable space. The grids consist of 7 evenly spaced values for posX and pos Y, 3 evenly spaced values for scale, and 4 evenly spaced values for orientation. The above settings in [53].
Note that in 52 the dSprite data generation process and results are slightly different to those presented here. The experiment process was refined in [53] and is the one used in our experiments.

---

True W
True vs. C1(Z)
True W C2(Z)
35
10
-4
20
25
30
35
40
45
20
25
30
35
40
45
20
25
30
35
40
45
Sampled W vs. x
Sampled W vs. C1(Z)
Sampled W vs. C2(Z)
20
-2
20
30
40
50
Figure 8: Samples drawn from the model and ground-truth distribution of p(W]=, x) where (Z, X) drawn from the true distribution, and (C1, C2). From right in a row, samples are shown as (w,, æz), (w. Cl;) and (wa, C2:). The top row is from the ground truth distribution and the bottom is from the model.
E.3 Proxy Bucketing Strategy
Since the covariates in the Framingham dataset are not segregated into outcome-inducing proxy W and the treatment-inducing proxy Z, follow 47 to group the 32 covariates, namely: age6, age61, age62, ascvd_hxé bmi6, bmi61, bmi62. bpmeds6, cho15, cho151, cho152, dbp6, dbp61, dbp62, diab6, female, gluc5, gluc51. gluc52, hd15, hd151, hd152, pad_hxt hx6, sbp6, sbp61, sbp62, smoke6, stk_ mi mi_hx6, trigly5, trigly51, trigly52.
As do not have the covariate groupings beforehand, together. We begin by ranking the proxies based on their strength of association with: i) The outcome from the coefficients obtained using CoxPH regression of (Y, E) given (X, C), and ii) The treatment from the coefficients obtained using logistic regression of X given C. Then, we select proxies in decreasing order of strength of association, first choosing the proxy having the strongest correlation with the outcome as an outcome-inducing proxy. and we do the same for the treatment-ine sraatenn-innuucig proxy, The algorithm halts upon allocation of all the proxies. We obtain the following proxy allocation (with 16 variables in both W and 2);
C' to denote all covariates (W, Z)
W: smoke6, female, ascvd_hx6, diab6, pad_hx6, gluc52, stk_ age62, sbp62, gluc51, age61, age6, hd15, gluc5, trigly52, trigly51
Z: hx6, dbp62, bpmeds6, hd152, dbp61, bmi62, hd151, bmi61, bmi6, cho152, dbp6, sbp61, chol51, chol5, sbp6, trigly5
E.4 Model of pW|,, x) for Demand data
In Figure we compare samples drawn from the ground-truth distribution of p( p(W[2, a) to those of the model P.i Z 2, X .x), which is represented by Gaussian distribution with mean and variance modeled via simple neural networks. We observe close agreement between the true samples and our conditional distribution model.
25

---

Figure 9: Examples of true dSprite proxy data W (images).
Figure 10: Examples of synthesized samples from p(W] x, for dSprite, manifested manifested via conditional GAN 32
E.5 Model of pWW|,, .)) for dSprite data
The capacity to sample from pW|2,, x) for the dSprite data is implemented via conditional GAN Here, the proxy variable W and the treatment X are 64 x 64 images, and the proxy variable is a three-dimensional vector, representing image scale, rotation and position in the horizontal direction (posX). The unobserved confounder is the vertical position of the image (posY). Figures TO true and synthesized images from the dSprite data, respectively, and Figure I1 shows the true treatments X connected to Figure TO emphasized in the caption Figure TT important to note that the synthesized W is able to infer the proper PosY from X, which is inferred from the image X).
32
Network Structures and Hyperparameters
In both synthetic data experiments, we use k-fold cross-validation (k: 5) to select the learning rate from the range [10-3, 3, 10 10-51 The learning rate is the same for all model components, but one can try to make the learning rate different across different components. We can also use k-fold cross-validation for synthetic data and the validation split for Framingham data to determine the weights for each autoencoder loss accordingly. Once the hyperparameters are selected, the model trained with train/validation split of 0.8/0.2 Early stopping is implemented using the validation loss of the causal bridge Lay only, even for the autoencoder version of model. Framingham experiments, we use the validation split provided in the dataset to measure the concordance index (C-Index) and select the batch size from the range [384, 512, 768), keeping the learning rate and weight decay fixed 10-1 and 10-4 respectively. The test split in Framingham is used only to
26

---

Figure 11: True treatments X associated with the generative model used to draw from p(W x, x) in Figure 10 Note that the generator is given Z (scale, rotation, posX). but must infer the latent U posY from which shares here independent of Z), the model is able to correctly extract posY from X. The subfigures here correspond to those in Figure 10
Note while treatments have different than that used for W
evaluate the C-Index of the trained model. To ensure fair comparison, we utilize the training and validation split learn the baseline CoxPH models with three different weighting schemes.
The details of the architectures used for Demand, dSprite and Framingham are shown in Tables 2 and 4 respectively.
F1 Demand Experiments
For the demand experiment, we use the AdamW optimizer with weight decay 10-b and learning rate of 10-4 for all models. The autoencoder loss Lax is weighted with 1, and Cag is weighted with W2
F.2 dSprite Experiments
For the dSprite experiment, we use the Adam with learning rate of 10-3 for all models. The autoencoder loss Lax is weighted with wx 100, and Lez is weighted with wN 0.01.
F.3 Framingham Experiments
For the Framingham dataset, the model structures are linear bridges. We have shown 0, = (6x, Φ2) for ease of explanation in Table separateell To model p(W|X, X, 2), we utilize simple MLP-based conditional diffusion probabilistic model (DDPM) [20]. To train the conditional DDPM, we use the AdamW optimizer with weight decay of 10 learning rate of 10 and the number of diffusion timesteps 1000. For the bridge and autoencoder modules, we use the Adam W optimizer with weight decay of 10 and a learning rate of 10-1, The autoencoder loss Lox is weighted with W. = 0.5, and La,, is weighted with W₂ 0.1.
G CoxPH Loss with Weighting
For learning the Cox proportional hazards (CoxPH) model with various weighting schemes, maximize the propensity-weighte partial likelihood 42 46) to obtain the parameters, 7;
wm
exp(xm
[(>) = II Engy. Un exp(x.>) micm m1
(69)
Here, for th example, yn denotes the observed time yn- = min(kn, Cn), is the time which the event of interest occurs and en is the follow-up time. yn = In < Cn. it is said that the event of interest is observed and en 1, otherwise, In Cn. tn. en 0 and the event is reght-censored.
27

---

60
39
[4]-)00000 1000
[i 228, (1 2000
10
1000 5000
Figure 12: Ablation results with the + AE model when varying the dimensionality of (Left) on Demand dataset and (Right) U and e on the dSprite dataset.
Table Out-of-sample MSE results Demand and dSprite data. Figures are medians interquartile ranges
(QI and Q3). These results match those in Figure 3 but also include results for CEVAE and KAP.
Demand
dSprite
Method
Sample Size
Median (Q1, Q3)
Median (Q1, Q3)
CEVAE
1000
360.05 (205.15, 602.59)
52.92 (51,45, 45, 53.29)
NMMR-V V
1000
23.07 (17.27, 28.12)
24.48 (21.32, 25.73)
NMMR-U
1000
21.6 (20.41, 25.38)
22.22 (18.37, 25.57)
KAP
1000
109.86 (79.01, 176.86)
22.4 (21.75, 23.03)
DFPV
1000
43.93 (38.38, 48.34)
22.08 (19.96, 24.38)
DFPV (our generator)
1000
34.15 (29.65, 40.88)
15.25 (13.97, 18.59)
CB (our method)
1000
20.5 (17.52, 24.78)
14.06 (13.21, 14.94)
CB + AE (our method)
1000
16.66 (14.59, 20.32)
13.68 (13.39, 14.49)
CEVAE
5000
262.24 (161,96, 438.63)
53.89 (53.07, 54.18)
NMMR-V V
5000
31.03 (24.07, 41.42)
11.62 (8.77, 14.44)
NMMR-U
5000
17.12 (13.19, 24.1)
19.11 (18.11, 20.3)
KAP
5000
153.19 (96.4, 221.03)
26.53 (25.85, 27.1)
DFPV
5000
43.35 (32.65, 53.2)
16.33 (15.23, 17.44)
DFPV (our generator)
5000
27.84 (24.73, 30.71)
8.79 (8.2, 9.49)
CB (our method)
5000
15,84 (14, 17.28)
7.45 (6.92, 7.96)
CB + AE (our method)
5000
14.77 (12.9, 16.89)
7.27 (7.04,
Thus, the weighting schemes estimate weights for each subject using the entire dataset and these weights to fit CoxPH model. The unknown propensity score P(X W Wn, Z = 2n, is modeled using a linear logistic regression model: šn oW wn. 2n). The weight for each mit example is computed as follows:
CoxPH-OW: overlapping weights,
Wm 2i 8m) + (1 xmn 8m
CoxPH-IPW: inverse probability weighting,
Fm
1 Fm
Wyn
8
5m
CoxPH-Uniffrm: uniform weights (standard RCT uniform assumption).
H Additional Results for Demand and dSprite
Table I present all the results for Demand and dSprite like in Figure 3 but also including CEVAE and KAP.
28

---

I Ablation Study
We explore the impact of the dimensionality of the latent U and noise model for the shared encoder
hog (W. x, <). For the Demand dataset we consider | = 1 and |c| = {1,3, 5). where [0] indicates the cardinality (dimensionality) of U. For the dSprite dataset,
consider [0]
{32, 64, 128)
and |c| = {500, 2000). Note that in the main results shown in Figure 3 | = |c| = 1, and
JU| = 32 and |e| 500, for the Demand and dSprites datasets, respectively. All other parameters
of the model are fixed and consistent with those used for the main results, as shown in Appendix P The optimal range for the dimensions of U and can be determined by domain knowledge or cross-validation. For instance, the dimensionalities of W in Demand and dSprite are |W] 1 and [ 4096, respectively, although the latter is contained on a lower dimensional manifold. which explain the differences in U above. The difference between [U] and |e] for the dSprite dataset necessary because smaller values of |c| tend to cause the learning to ignore the variation of hog (W, I, c), which can cause overfitting issues.
The ablation results for Demand and dSprite shown in Figure 12 indicate that the proposed CB + AE model is fairly insensitive to the dimensionality of the latent U and e. These are a subset of all ablation studies have considered, consistent with our experience that the proposed models train well and are not particularly sensitive to "reasonable" parameter settings.
J Bridge Generalization from Proposition 2
This document describes the implementation and analysis of Structural Equation Modeling (SEM) experiment using Bayesian methods. The experiment involves estimating latent variables from observed data while accounting for confounding factors.
The observed variables are generated from the SEM in 32 35 with all structural coefficients
{azu, aWU, axzz QXU, xy aYw, ayv) = 1, treatment and outcome variances [x) = 1, and the latent confounder is set to U Uniform (0, 10). We generate n 100 observations of (2r, 2, w), Using these observations we can train model p[W|æ, 2), after the model is trained, we can sample J 50 W using p( [W[|a: 2) for every (æ. Using these generated W we can then use MCMC to get M = 100 samples of U from p(U [w, x, 2) and p(U w. x)) for every {w,x,2).
The details of experiment steps are the follows:
Sampling W We model p(W] x, 2) as a Gaussian with parameters (mean and variance) given by neural network with inputs 2x. With samples (w,, 24, 24), we train network to generate samples from p(W| X, Z) using maximum likelihood estimation with gradient descent. Then for each (I,, 2)) we draw
Wi, ~ (W 24, j = I.....hh
(70)
where J = 50.
Sampling U We compile two NUTS 21] samplers in PyMC 45): one targeting p(U (X) and one targeting p(U|W J/W,,, X, 2). For each proxy sample Wi, we generate
".i.. p(U (w),,,))),
(1)
uigon p(U [u Ti. 21,
(2)
n= I,....,, M.
(71)
where M = 100.
Learning the Bridge Function Define a feedforward network go (u, w). We approximate the bridge using Proposition 2 as
1 M M
bo (W x)
2
go Σ
(w Uugg.n Wir, Wk).)
(72)
n=1
Then we learn in by (W, x) by minimizing
i = arg min Σ
(ELY
I. 2,]
Ebbo x,, 2:1
(73)
Evaluation Finally, we compute the
EE [ on the causal effect per data point as
E Wi zi.
Blbb (W 2)),, 2:]l
(74)
Y [1, 2:]
B[ ( 2:]
29

---

Comparison of
Statistics for Different ow and oz
FYX 2
10a
0==100, 0=1.0
0.=011, 0=0.11
0==1.0, 0.
0w=0.1, 0,=100
Value (log scale)
10
10
10
Mean min glu.w.x)
Std काiंत glu.W.x)
Mean uा E(Yj.,W.x)
Std काों E(YjU.W.))
Figure 13: The statistics (mean and standard deviation) of | for learning the bridge function compared to using known U, W, X).
and report the empirical mean and standard deviation of { [Ely x.. H for 100. Note that this mean is an approximation the mean of r(n).
2
Results E[Y [r, 2;] is calculated using samples from p(U/w, x, 2), p( [x, 2) and observed (x, z). using the true expectation E(Y)U, U XX) samples from p(U [W,, 2). p( W
+ W + U, where E[bo pa, 24] is calculated using xx, 2) and observed (x, 2) using:
The true expectation [UY W,X)
W +U.
The learned function gU, W,x))
Results for the mean and standard deviation of { [2]] [ Mi=1 in Figure
13
K
Additional Survival Analysis Results
For the Framingham dataset, we plot the KM curves, which provide the survival probability estimates without considering the confounding factors. F4 the Kaplan-Meier (KM) survival curves for the two groups with treatment X = and treatment X 0. Notably, the KM curves indicate has decreased survival probabilities compared to the group that received the treatment (X = 0), even though previous large-scale longitudinal RCT trials indicated hazard ratio (HR) of 0.75. This emphasizes the necessity of modeling the Framingham survival analysis using a framework that precisely captures the causal relationship.
that the group of patients that received the treatment (X
For the Framingham dataset, we report the concordance index (C-Index) to show the effectiveness of proposed approach to correctly rank the survival times. The interpretation of C-Index is similar to the area under the ROC curve (AUC), but it is a generalization for censored data. CI ranges from 0 to 1, such that score of 1 implies a perfect ranking, 0.5 implies random predictions and < 0.5 indicates performance worse than random.
Figure 15 shows the box plots for the hazard ratios and concordance index (CI) the test split over runs. Our proposed framework of the causal bridge along with the autoencoder (CB AE) outperforms only using the causal bridge (AE) in terms of both the hazard ratio and concordance
30

---

Kaplan-Mtier Ourves Trestment==1 Trestment=00
Teeimenee Tresieeee
Suneea Ploveddyy
Figure 14: Framingham dataset: Kaplan Meier Curves.
0.78
1.0
Concaraanc dance Indoo (C0
0.75
Haaadd FaltON (HNN)
Patrod
0.74
0.73
0.8
0.8
0.70
Figure 15: Framingham dataset: (Left) Hazard ratio and (Right) Concordance Index (CI).
index, demonstrating that using AutoEncoder helps not only in learning the causal relationship but also improves the predictive capability of the model.
31

---

Table 2: Models for the Demand Experiment.
Model
p(W|Z 2, =x)
Model
h(w, x, c)
Input(2,,))
Input(w:,)) e)
FC(3,32) ReLU
FC(3, (2). ReLU FC(32, 64), ReLU FC(64, 16), ReLU FC(16, 1)
FC(32, 64), ReLU
FC(64, 16), ReLU
Mean: FC(16, 1) Std: FC(16, 1), Softplus
Model
0y
Model
Φx Input(æ:) F2),,32). ReLU FC(32, 64), ReLU FC(64, 16), ReLU FC(16,
Input(ar, u, w) FC(3, ReLU FC(32, 64), ReLU FC(64, 16), ReLU FC(16.
3
5
Model
0,
Model
02 Input(u) FC( (2)), ReLU FC(32, 32), ReLU FC(32, 2)
Input(2, u) FC(3,32) ReLU FC(32, 64), ReLU FC(64, 16), ReLU FC(16, 1)
Table Models the dSprite Experiment.
Model
pW = x)) G
Model
p(WIZ = 2, X x) D
Input(rr, z, noise)
Inp
FC(4299, 2048), ReLUU
FC(8195, 4096), ReLU
(0(2048,4996)
FC(4096, 1). Sigmoid
Model
h(W,z,e)
Model
Φx Input(æ:)
Input(w, e) FC(8692, 1024), ReLU (C0144, 256), ReLU FC 64), ReLU FC( 32)
2
FC(4096, 64), ReLU FC(64, 32)
4
Model
O,
Model
0y
Input(2, u) FC(35, 64), ReLU FC(64, 128), ReLU FC(128, 256), ReLU FC((56, 4096)
Input(#, u, w) FC(4160, 1024), ReLU FC(1024,256) ReLU FC(256, 1)
2
Model
02
1
Input(u)
2
FC(32, 512), ReLU
3
FC(512,
32

---

Table 4: Models for the Framingham Experiment.
Embedding Layer
Input(2; =) FC(17, 64), Linear
2
Time Step Embedding Layers
FC(64, 64), Linear
2
SiLU Activation FC(64, 64), Linear
Projection Layer
FC(16, 64), Linear
DDPM Structure
FC(64, 256), ReLU, Dropout(O.) I)
2
FC(256, 512), ReLU, (ropout(O.1)
FC(512, 256), ReLU, Or0))
4
FC(256, 16), Linear
Model
e,W Input(w,, x, c) FC(18, 1)
2
Model
0g Input(æ, w)
2
FC(18, 1)
Model
6. (part of (,,))
Input(2) FC(16, 16)
2
Model
6n (part of 0..)
Input(((2)), FC(17,
2
Model
Input(w)
2
FC(1, 16)
33

---

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Reviews:
[{"review_id": "3BVFsqT3jW", "reviewer": "Reviewer_2mQh", "summary": "This paper addresses the challenge of estimating treatment effects under unobserved confounding by leveraging a causal bridge function within a proxy variable setting. It reformulates the bridge in a way that enables an intuitive deep learning implementation that improves the interpretability and statistical performance. The authors present theoretical conditions under which treatment effects can be identified using two proxy variables and introduce a novel set of assumptions that maintain the same inferential soundness. They further contribute a sensitivity analysis framework to assess robustness to assumption violations and demonstrate how their method enables statistical strength-sharing across variables. The framework consists of a two-step modeling pipeline: training a generative model to sample from proxy variables and then estimating treatment effects using these samples along with simulated unobserved confounders. A joint modeling strategy is employed to refine the confounder sampler. The approach is also extended to survival outcomes and is shown empirically to outperform state-of-the-art methods.", "questions": "Full disclosure: I have previously reviewed this paper and gave it an accepting score. In the rebuttal, my questions were answered, and my opinion of the paper remains largely unchanged.", "limitations": "Limitations are stated.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. The main contribution of the new bridge formulation is a novel result that cleverly allows a new practical implementation for estimating treatment effects.\n2. Assumptions are very clearly stated.\n3. Both the causal bridge approach and the joint objective are clearly demonstrated to lower estimate errors in out-of-sample settings in the experiments.\n\n**Weaknesses:**\n\n4. The paper is dense and is difficult to read. It would benefit from having figures and examples explaining the intuition of the bridge function, as well as the proven results.", "was_revised": true, "final_justification": "I maintain my score and recommend the paper for acceptance.", "rebuttal": "We would like to thank the reviewer for their positive assessment of our paper and general feedback. We understand that the paper is somewhat dense, however, necessarily so due to space constraints. We have done our best to balance its contents with extensive supplementary information in the Appendix containing an illustrative example of the bridge function as well as experiments using an intuitive structural equation model (SEM) in Appendices C and D. As a complement, we will include a new figure in Appendix C illustrating the SEM model in equations (27)-(30), which can further provide intuition about the bridge function."}, {"review_id": "xrhH7Cnt5C", "reviewer": "Reviewer_5YZq", "summary": "The paper presents theoretical insights into the causal bridge method, a causal inference approach based on proxy variables, designed to estimate the causal effect of a treatment variable on an outcome in the presence of unobserved confounders. Building on these theoretical analyses, the paper proposes an architecture aimed at improving the quality of causal bridge estimates.", "questions": "Please refer to Strengths And Weaknesses.", "limitations": "Yes.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is well-motivated and addresses an important problem. Causal effect estimation in the presence of unobserved confounding is a challenging yet critical issue, and proposing a framework to handle this is valuable.\n2. The paper is interesting, particularly the theoretical analysis and how it is used to justify the proposed method. The comparison with CEVAE is also compelling.\n3. The paper is well-structured and easy to follow and the proposed method demonstrates good performance, outperforming the baselines.\n\n\nWeaknesses:\n\n1. Some contributions of the paper appear incremental. In particular, much of Section 2, e.g., including Propositions 1 and 2, are direct derivations from Equation (1). Additionally, some findings, such as W and Z are noisy mappings of U have already been discussed in prior work [20].\n2. Some of the language in the paper needs to be more precise. For example, the introduction should explain more clearly the advantages of proxy variable methods, especially causal bridge, compared to others, such as instrumental variable approaches.\n3. The paper only uses one real-world dataset in the experiments. Including more real-world datasets would help better demonstrate the performance of the proposed method. Also, given the similarities between the proposed approach and the CEVAE, the paper should also include that as baseline. Also, I highly suggest that the paper includes more ablation studies in the main paper, rather than the appendix.", "was_revised": true, "final_justification": "I believe the paper addresses an interesting problem and proposes both theoretical analysis an a well-justified methodology. However, the concerns regarding the clarity of the paper remains.Overall, I'd like to keep my positive score.", "rebuttal": "We would like to thank the reviewer for their positive assessment of our paper and constructive feedback. Below we address the three concerns raised by the reviewer.\n\n1. Some of the contents of the paper including Propositions 1 and 2 though can be derived from (1) as the reviewer point out have not been introduced elsewhere. However, the reviewer is correct in that the assumption that $W$ and $Z$ are noisy mappings of $U$ have been discussed elsewhere. We would like to point out that beyond re-examining the assumptions underlying the causal bridge in Section 4, which includes new theoretical results in Theorem 3 and Corollary 1, we also introduce a new formulation for the causal bridge leveraging generative models for the outcome proxy ($W$), an autoencoder architecture that enables sharing of statistical strength between observed quantities ($W$, $Z$, $X$ and $Y$), an extension to survival outcomes and extensive experimental results with synthetic and artificial data demonstrating the effectiveness of our generative, autoencoding and survival approaches to model the causal bridge.\n\n2. The suggestion about clarifying the advantages of proxy variable methods is well taken. In the revision, we will clarify the need and advantages of proxy variable methods in the unobserved confounding setting. Importantly, we will emphasize that it requires weaker assumptions than an instrumental variable approach, *i.e.*, that the instrumental variable and the unobserved confounder are independent and the latter influences the treatment but not the outcome.\n\n3. We agree that having additional real-world datasets will make the experimental results stronger; however, it is difficult to find real-world datasets for which the ground-truth causal effects are known, such as it is for the case shown in our experiments using the Framingham dataset. This will be discussed as a limitation of the proposed approach in the Conclusions section. We did not include results for CEVAE because they are not competitive (they did not fit within the range of the vertical axes we used in our figure); however, we will include them in the revision and are presented below (in the revision these will be presented graphically as in Figure 3), where Q1 and Q3 are the first and third quartiles presented along with the median out-of-sample MSE results. We also include results from two additional competitive baselines suggested by **Reviewer rsP2**, namely NMMR (Kompa et al., 2022) and Kernel Alternative Proxy (Bozkurt et al., 2025). We see that our methods (CB and CB + AE, namely with and without the autoencoder formulation, respectively) outperform all the baselines considered. We agree with the reviewer that including the ablation studies in the main paper is desirable, and we plan to do so in the revision, considering that we will have additional space (1 page).\n\n\n| Method |\tSample Size |\tDemand Median (Q1, Q3) |\tdSprite Median (Q1, Q3) |\n| :--- | :----: | :----: | :----: | \n| CEVAE |\t1000 |\t360.05 (205.15, 602.59) |\t52.92 (51.45, 53.29) |\n| NMMR-V |\t1000 |\t23.07 (17.27, 28.12) |\t24.48 (21.32, 25.73) |\n| NMMR-U |\t1000 |\t21.6 (20.41, 25.38) |\t22.22 (18.37, 25.57) |\n| Kernel Alternative Proxy |\t1000 |\t109.86 (79.01, 176.86) |\t22.4 (21.75, 23.03) |\n| DFPV |\t1000 |\t43.93 (38.38, 48.34) |\t22.08 (19.96, 24.38) |\n| DFPV $\\sim p(W\\|x,z)$ (our generator) |\t1000 |\t34.15 (29.65, 40.88) |\t15.25 (13.97, 18.59) |\n| CB (our method) |\t1000 |\t20.5 (17.52, 24.78) |\t14.06 (13.21, 14.94) |\n| CB + AE (our method) |\t1000 |\t**16.66 (14.59, 20.32)** |\t**13.68 (13.39, 14.49)** |\n| CEVAE |\t5000 |\t262.24 (161.96, 438.63) |\t53.89 (53.07, 54.18) |\n| NMMR-V |\t5000 |\t31.03 (24.07, 41.42) |\t11.62 (8.77, 14.44) |\n| NMMR-U |\t5000 |\t17.12 (13.19, 24.1) |\t19.11 (18.11, 20.3) |\n| Kernel Alternative Proxy |\t5000 |\t153.19 (96.4, 221.03) |\t26.53 (25.85, 27.1) |\n| DFPV |\t5000 |\t43.35 (32.65, 53.2) |\t16.33 (15.23, 17.44) |\n| DFPV $\\sim p(W\\|x,z)$ (our generator) |\t5000 |\t27.84 (24.73, 30.71) |\t8.79 (8.2, 9.49) |\n| CB (our method) |\t5000 |\t15.84 (14, 17.28) |\t7.45 (6.92, 7.96) |\n| CB + AE (our method) |\t5000 |\t**14.77 (12.9, 16.89)** |\t**7.27 (7.04, 8)** |"}, {"review_id": "yJsiC0KQu8", "reviewer": "Reviewer_vSpr", "summary": "This paper focuses on the effect estimation with latent variables and two additional observed proxies,  a framework commonly referred to as proximal methods. The paper derives a new error bound on the causal bridge function (more specifically,  the outcome confounding bridge function), highlighting that the proxies should serve as low-noise measurements of the latent confounders. Further, the paper proposes a generative model based on an autoencoder architecture to learn the bridge function. Experimental results demonstrate the effectiveness of the proposed approach.", "questions": "see above", "limitations": "see above", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### strengths\n\n1. The paper is clearly written and well-organized.\n\n1. It proposes an autoencoder-based architecture for learning the causal bridge function within the proximal framework, which performs well empirically.\n\n   \n\n### Weaknesses / Questions \n\n1. Corollary is not a surprising result, since $\\Psi$ is a non-information-loss function. Therefore, the information loss stems solely from the exogenous noise $\\epsilon_w$.  As $\\sigma_{\\epsilon_w} \\rightarrow 0$, the proxy $W$ become equivalent to $U$, leading to $I(U;Z|W) \\approx I(\\Psi(W);Z|W)=I(W;Z|W)=0$. \n1. The authors discuss solely the case of outcome confounding bridge function $b$ (denoted $h$ in [1]). Is it possible to drive similar result for treatment  confounding bridge function $q$​ [1]? \n1. Relatedly, is it possible to learn the treatment bridge function $q$ using the proposed autoencoder architecture?\n1. It would be better to clarify why we should access $D_1$. What is the implication of using a separate dataset versus the same dataset?\n1. The experimental evaluation could be strengthened by comparing the proposed method with more state-of-the-art proximal baselines.\n1.  The paper discusses how the additional loss terms in Equation (13) improve the learned bridge function hhh, but an ablation study on these terms is missing.\n\n\n\n[1] Semiparametric proximal causal inference. JASA 2024.", "was_revised": true, "final_justification": "During the rebuttal, all my concerns were well addressed. Thus I increase my score from 3 to 4.", "rebuttal": "We would like to thank the reviewer for their positive assessment of our paper and constructive feedback. Below we address the concerns raised by the reviewer.\n\n1. We completely agree with the reviewer that the Corollary is not surprising, but we decided to include it because it is an interesting special case for structural equation models. To our knowledge Theorem 3 is new, and we hoped to aid understanding of it by demonstrating that it implies the expected result in the Corollary.\n\n2. It is certainly possible to derive a similar result for the treatment confounding bridge; this is an excellent question. We have results (experimental and theoretical) for the treatment bridge, which we have obtained since we submitted to NeurIPS, however, we consider them a separate contribution. Nevertheless, we will discuss the treatment bridge as a natural extension that will be addressed in subsequent work. Additionally, please note that the treatment bridge is most useful for binary treatments (interventions), and most of our experiments considered real treatments.\n\n3. The reviewer is also correct in that it is possible to learn the treatment bridge in an autoencoder setting, so both the treatment and outcome bridge share statistical strength when learning the function that is used to implicitly sample from $u_j$ using $u_j = h(w_j,x,\\epsilon_j)$, which for the treatment bridge will be a function of $z_j$, matched in distribution to that using $w_j$ via an integral probability measure (IPM) or minimum mean discrepancy (MMD).\n\n4. We should clarify that $D_1$ is not necessary; however, it is done to weaken the requirement of having a common dataset for which ($W$,$Z$,$X$,$Y$) are available. This distinction is especially useful in situations where the dataset for which the outcome is available $D_2=\\{(X,Z,Y)\\}$ is not the same or is (much) smaller than $D_1=\\{(X,Z,W)\\}$. We will clarify it in the revision to avoid confusion.\n\n5. Following your suggestion and that of **Reviewer rsP2** we have produced results for CEVAE, NNMR (Kompa et al., 2022) and Kernel Alternative Proxy (Bozkurt et al., 2025) reproduced below, where Q1 and Q3 are the first and third quartiles presented along with the median out-of-sample MSE results. In the revision these will be presented graphically as in Figure 3. We see that our methods (CB and CB + AE, namely with and without the autoencoder formulation, respectively) outperform all the baselines considered.\n\n| Method |\tSample Size |\tDemand Median (Q1, Q3) |\tdSprite Median (Q1, Q3) |\n| :--- | :----: | :----: | :----: | \n| CEVAE |\t1000 |\t360.05 (205.15, 602.59) |\t52.92 (51.45, 53.29) |\n| NMMR-V |\t1000 |\t23.07 (17.27, 28.12) |\t24.48 (21.32, 25.73) |\n| NMMR-U |\t1000 |\t21.6 (20.41, 25.38) |\t22.22 (18.37, 25.57) |\n| Kernel Alternative Proxy |\t1000 |\t109.86 (79.01, 176.86) |\t22.4 (21.75, 23.03) |\n| DFPV |\t1000 |\t43.93 (38.38, 48.34) |\t22.08 (19.96, 24.38) |\n| DFPV $\\sim p(W\\|x,z)$ (our generator) |\t1000 |\t34.15 (29.65, 40.88) |\t15.25 (13.97, 18.59) |\n| CB (our method) |\t1000 |\t20.5 (17.52, 24.78) |\t14.06 (13.21, 14.94) |\n| CB + AE (our method) |\t1000 |\t**16.66 (14.59, 20.32)** |\t**13.68 (13.39, 14.49)** |\n| CEVAE |\t5000 |\t262.24 (161.96, 438.63) |\t53.89 (53.07, 54.18) |\n| NMMR-V |\t5000 |\t31.03 (24.07, 41.42) |\t11.62 (8.77, 14.44) |\n| NMMR-U |\t5000 |\t17.12 (13.19, 24.1) |\t19.11 (18.11, 20.3) |\n| Kernel Alternative Proxy |\t5000 |\t153.19 (96.4, 221.03) |\t26.53 (25.85, 27.1) |\n| DFPV |\t5000 |\t43.35 (32.65, 53.2) |\t16.33 (15.23, 17.44) |\n| DFPV $\\sim p(W\\|x,z)$ (our generator) |\t5000 |\t27.84 (24.73, 30.71) |\t8.79 (8.2, 9.49) |\n| CB (our method) |\t5000 |\t15.84 (14, 17.28) |\t7.45 (6.92, 7.96) |\n| CB + AE (our method) |\t5000 |\t**14.77 (12.9, 16.89)** |\t**7.27 (7.04, 8)** |\n\n6. We apologize for the confusion regarding equation (13). We do have an ablation study in the sense that we present results both using (12), *i.e.*, just the bridge (CB in Figure 3), and (12) + (13), *i.e.*, the proposed autoencoding setup (CB + AE in Figure 3). These are reproduced above for convenience. Moreover, the weights of the three loss terms ($L_{\\theta_Y}$, $L_{\\theta_X}$, $L_{\\theta_Z}$) are also optimized as described in Appendix F. Moreover, we do not present ablation results only using $L_{\\theta_X}$ and $L_{\\theta_Z}$ because the autoencoding setup is intended to model the joint distribution $p(X,Z|U)$ following the graphical model in Figure 1."}, {"review_id": "G4RDmUmSa1", "reviewer": "Reviewer_rsP2", "summary": "This paper derives theoretical bounds on the average error in treatment effect estimation under proximal causal inference when the causal bridge (CB) assumption is violated. The authors also propose an autoencoder-based architecture for estimating the causal bridge function, providing both theoretical guarantees and a practical implementation framework.", "questions": "I might increase or decrease my score based on authors' responses\n- What motivated the choice of autoencoders over diffusion models for generating samples from the conditional distribution, particularly given diffusion models' demonstrated effectiveness in generative tasks?\n- Could you include comparisons with the missing baselines identified in the weakness section (CAVAE, Kompa et al. 2022, and Ying et al. 2022) to provide a more comprehensive experimental evaluation?", "limitations": "The authors fail to acknowledge that their experimental evaluation omits several important baselines from recent literature, and do not discuss why they chose auto-encoders instead of diffusion models.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 1, "strengths_and_weaknesses": "*Strengths*:\n- The paper extends existing theory for proximal causal inference by providing theoretical analysis for the scenario where the bridge function independence assumption (relative to the proxy variable) is violated.\n- The experimental evaluation includes both synthetic datasets and real-world data with ground truth established through randomized controlled trials, providing comprehensive empirical validation.\n\n*Weaknesses*:\n- The paper fails to discuss or compare against recent neural network approaches for unmeasured confounding, including key works such as Deep Learning Methods for Proximal Inference via Maximum Moment Restriction (Kompa et al., 2022), Causal Effect Estimation using Identifiable Variational AutoEncoder with Latent Confounders and Post-Treatment Variables (Xie et al., 2024), and Density Ratio-based Proxy Causal Learning Without Density Ratios (Bozkurt et al., 2025).\n- The synthetic data experiments omit important baselines: (1) CAVAE: the authors' claim that results are not numerically stable does not justify excluding this established method, and (2) the approach from Kompa et al. (2022).\n- The real-world data evaluation lacks comparison with relevant baselines, particularly Ying et al. (2022).\n- The authors present their method's proportional hazard assumptions as an advantage over Ying et al. (2022) in survival outcomes, but fail to acknowledge that requiring this assumption could be viewed as a limitation of their own approach.\n- The paper provides no justification for choosing autoencoders over diffusion models for sampling, despite diffusion models' growing prominence in generative modeling", "was_revised": true, "final_justification": "The authors responses addressed my concerns and questions. As such I increased the score.", "rebuttal": "We would like to thank the reviewer for their constructive feedback. Below we address the three concerns raised by the reviewer.\n\n1. Thanks for pointing us to the three related papers. Below we show results for Kompa et al., 2022 and Bozkurt et al., 2025, showing that the proposed approaches (CB and CB + AE, namely with and without the autoencoder formulation, respectively) outperform both of their settings NNMR-U and NNMR-V, and Kernel Alternative Proxy, respectively. In the table below, Q1 and Q3 are the first and third quartiles presented along with the median out-of-sample MSE results. The other paper will be discussed in the revision; however, it is not considered in the experiments because Xie et al., 2024 is for a different setting that requires post-treatment variables.\n\n| Method |\tSample Size |\tDemand Median (Q1, Q3) |\tdSprite Median (Q1, Q3) |\n| :--- | :----: | :----: | :----: | \n| CEVAE |\t1000 |\t360.05 (205.15, 602.59) |\t52.92 (51.45, 53.29) |\n| NMMR-V |\t1000 |\t23.07 (17.27, 28.12) |\t24.48 (21.32, 25.73) |\n| NMMR-U |\t1000 |\t21.6 (20.41, 25.38) |\t22.22 (18.37, 25.57) |\n| Kernel Alternative Proxy |\t1000 |\t109.86 (79.01, 176.86) |\t22.4 (21.75, 23.03) |\n| DFPV |\t1000 |\t43.93 (38.38, 48.34) |\t22.08 (19.96, 24.38) |\n| DFPV $\\sim p(W\\|x,z)$ (our generator) |\t1000 |\t34.15 (29.65, 40.88) |\t15.25 (13.97, 18.59) |\n| CB (our method) |\t1000 |\t20.5 (17.52, 24.78) |\t14.06 (13.21, 14.94) |\n| CB + AE (our method) |\t1000 |\t**16.66 (14.59, 20.32)** |\t**13.68 (13.39, 14.49)** |\n| CEVAE |\t5000 |\t262.24 (161.96, 438.63) |\t53.89 (53.07, 54.18) |\n| NMMR-V |\t5000 |\t31.03 (24.07, 41.42) |\t11.62 (8.77, 14.44) |\n| NMMR-U |\t5000 |\t17.12 (13.19, 24.1) |\t19.11 (18.11, 20.3) |\n| Kernel Alternative Proxy |\t5000 |\t153.19 (96.4, 221.03) |\t26.53 (25.85, 27.1) |\n| DFPV |\t5000 |\t43.35 (32.65, 53.2) |\t16.33 (15.23, 17.44) |\n| DFPV $\\sim p(W\\|x,z)$ (our generator) |\t5000 |\t27.84 (24.73, 30.71) |\t8.79 (8.2, 9.49) |\n| CB (our method) |\t5000 |\t15.84 (14, 17.28) |\t7.45 (6.92, 7.96) |\n| CB + AE (our method) |\t5000 |\t**14.77 (12.9, 16.89)** |\t**7.27 (7.04, 8)** |\n\n2. Above we show results for CEVAE, NNMR-U, NNMR-V (Kompa et al., 2022) and Kernel Alternative Proxy (Bozkurt et al., 2025), from which we see that the proposed approach outperforms the alternatives in both datasets (Demand and dSprite) and dataset sizes (1000 and 5000). These will be shown graphically in the revision as in Figure 3.\n\n3. We produced results using the method in Ying et al. 2022 using the same data settings in our experiment with the Framingham dataset shown in Figure 3 (Right). We use their formulation for the hazard function (described below as $h(t,W,X,V)$) and estimate hazard ratios (HRs) for three distinct event quantile time points $t=(0.25,0.5,0.75)$ which correspond to event horizons at 1924, 3709, and 5494 days, respectively, consistent with the experiments in Ying et al. 2022. Note that unlike in our approach, they estimate causal effects for specific points in time, while we consider a time-independent causal estimate. Results below indicate that the approach in Ying et al. 2022 fails to capture the correct direction of the hazard ratio (HR$< 1$) and magnitude of the ground-truth HR=0.75. Summaries using the method of Ying et al. 2022 (mean and standard deviation for the HR) are calculated over 30 runs:\n\n| **t**         | **0.25** | **0.5**  | **0.75** |\n|:--------------|---------:|---------:|---------:|\n| **Mean**      | 1.0454   | 1.1414   | 1.2103   |\n| **Standard Deviation** | 0.0011   | 0.0077   | 0.0007   | \n\n4. We understand that the proportional hazard assumption can be seen as a limitation; however, in Ying et al., 2022, they consider a very specific formulation for the hazard function that can be similarly perceived as a limitation. Specifically, they use $h(t,W,X,V) = exp( - b_0 t - b_1 t^2 - b_a t X - b_w t W - b_v V )$, *i.e.*, an exponentiated linear function of time ($t$), time $\\times$ time ($t^2$), time $\\times$ treatment ($t X$), time $\\times$ outcome proxy ($t W$) and additional covariates denoted as $V$. In the revision, we will acknowledge the limitations of both approaches, *i.e.*, using a proportional hazard assumption *vs.* choosing a specific form of the hazard function a priori.\n\n4. We understand the confusion about choosing the autoencoder for sampling, which we clarify as follows. We are not using the autoencoding formulation for sampling, but to learn functions $g_{\\theta_Y}$, $g_{\\theta_Z}$, $g_{\\theta_X}$, and $h_{\\theta_U}$. Sampling is done through a separate model trained to sample from $w \\sim p(W|x,z)$, which is not implemented with an autoencoder. In fact, we use a conditional GAN and a conditional diffusion probabilistic model for the dSprite and Framingham experiments as described in Appendices E.5 and F.3, respectively. The Demand data is very simple; thus, a simple MLP modeling the (Gaussian) mean and variance of W is sufficient as described in Appendix E.4. In answer to the question from the reviewer, we are effectively using a diffusion model when there is a need for sampling."}]

Author rebuttal:

Meta-review: The paper provides a new error bound for the causal bridge under violated assumptions, introduces a generative + autoencoder formulation that improves estimation, extends to survival outcomes, and shows empirical gains over strong baselines on synthetic and real-world data. In addition, the authors have effectively addressed the reviewers’ concerns. Therefore, I believe this paper is suitable for publication.
Decision: Accept (poster)