Coupling Generative Modeling and an Autoencoder with the Causal Bridge

We would like to thank the reviewer for their constructive feedback. Below we address the three concerns raised by the reviewer.

1. 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.

2. 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.

3. 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:

4. 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.

5. 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.

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.

1. 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.

2. 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.

3. 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).

4. 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.

5. 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.

6. 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.

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.

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.

1. 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.

2. 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.

3. 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).