Causal Inference with Conditional Front-Door Adjustment and Identifiable Variational Autoencoder

Thanks for your review and insights.

Regarding the first weakness, we think there might be some misunderstanding by the reviewer. We do not assume Z_CFD is observed. In fact, we consider Z_CFD to be unobserved and it is why CFDiVAE learns the representation of Z_CFD that satisfies the CFD criterion such that we can use the learned representation for CFD adjustment by following Theorem 3 (specifically Eqn. (3)). In other words, Theorem 3 states how to do CFD adjustment when a CFD adjustment variable is given, but this does not mean that our work assumes a CFD adjustment variable is known.

For the second question, firstly we would like to clarify that in the paper, we used ground truth to refer to the ground truth adjustment "variable" or its distribution/density function (which is known since we generated the data, including the data of the adjustment variable), instead of the ground truth representation or its distribution. Then the purpose of the experiments in Section 6.2 is to compare the distribution of the learned representation of the adjustment variable with the distribution of the ground truth adjustment variable. In this way we can validate the correctness of the learned representation, if the representation is correctly learned, its distribution should be similar to that of the distribution of the ground truth adjustment variable.

Regarding dimensionality, the proxy variables' dimensions we investigated are 4, 8, and 16. Following your comments, we conducted additional experiments on dimensionality analysis. The results show that CFDiVAE achieves its best performance in most sample sizes when $D_{\text{L}} = D_{\text{X}}$ (in this setting $D_{\text{L}} > D_{\text{R}}$, i.e., a larger setting than the correct dimension of the latent variable).

About Section 5, CFDiVAE is for learning the representation of the CFD adjustment variable, then after Z_CFD has been learned, we can use it (as the adjustment variable) with any ATE estimation method and the choice of the ATE estimation method is based on the data generation (e.g., linear, or non-linear). For linear models, we propose a basic solution in the main paper. For nonlinear models, we are aware that semi-parametric methods could be applicable, but delving into these methods is outside the scope of our contributions.

Following your suggestions, we addressed the inconsistencies in the notation and added two comparison methods based on the standard front door adjustment.

Please find the latest version in the Rebuttal Revision.

Thanks for the comment. Now we understand your concern, that is, the learned representation Z_CFD is a transformed version of the true conditional front door adjustment variable. However, since the transformation is linear, i.e., a linear invertible transformation (based on the assumption of iVAE [1]), this transformed variable is still a proper conditional front door adjustment variable because the causal relationships are maintained in this case.

Hope this addresses your concern. Thanks a lot.

[1] Khemakhem, Ilyes, et al. "Variational autoencoders and nonlinear ica: A unifying framework." International Conference on Artificial Intelligence and Statistics. PMLR, 2020.

Thanks for your review and insights.

Regarding your first weakness about the difficulty of recovering the adjustment variable from its proxies, we have expanded the discussion in the latest version of the paper and included references [3,4] as you suggested.

For your second weakness, we partially concur. CFDiVAE is for learning the representation of the CFD adjustment variable, then after Z_CFD has been learned, we can use it (as the adjustment variable) with any ATE estimation method and the choice of the ATE estimation method is based on the data generation (e.g., linear, or non-linear). For linear models, we propose a basic solution in the main paper. For nonlinear models, we are aware that semi-parametric methods could be applicable, but delving into these methods is outside the scope of our contributions. The initial implementation of our front-door adjustment formula was inspired by reference [1], but we acknowledge your point about its suitability. In the latest version, we now cite both [1] and [2], the latter of which details a semi-parametric method for Natural Indirect Effect (NIE) estimation. It is highly relevant because the causal effect via the front door equates to the NIE in the absence of a direct causal path between T and Y. We have emulated the procedure from [1] and adapted the semi-parametric NIE method to front-door adjustment scenarios. To clarify our approach, we utilize the example causal graph "T->M->Y, T->Y" from mediation analysis, where the NIE represents the effect via T->M->Y, and the Natural Direct Effect (NDE) represents the effect via T->Y. When no causal link exists between T and Y, the NDE becomes zero, and the NIE equates to the Total Effect (TE), making M the front-door adjustment set.

Concerning the Adult dataset, we concur that there is no causal path from W to T, as illustrated in Figure 8. We retain the causal path W to Y based on the comprehensive DAG. In our experimental setup, we categorize all mediating factors under the umbrella of stereotypes, which are regarded as proxies for stereotypes. Thus, the findings on indirect discrimination and discrimination through stereotypes are in alignment. Owing to the scarcity of real-world examples, we can only substantiate the applicability of CFDiVAE in this manner. Following your advice, we will include a discussion on direct and indirect causal effects in the main body of the text.

Following your recommendations, we have incorporated FindFDSet and ListFDsets into our experimental comparison and have stated the limitations of our model, particularly its diminished performance with small sample sizes. We have also revised any unsupported or misleading statements, ensuring that our contributions and conclusions are articulated more clearly.

Please find the latest version in the Rebuttal Revision.

[1] Iain Barr. Causal inference with python part 3 - frontdoor adjustment, Sep 2018. URL http://www.degeneratestate.org/posts/2018/Sep/03/causal-inference-with-python-part-3-frontdoor-adjustment/.

[2] Eric J Tchetgen Tchetgen and Ilya Shpitser. Semiparametric theory for causal mediation analysis: efficiency bounds, multiple robustness, and sensitivity analysis. Annals of statistics, 40(3):1816, 2012.

[3] Wu, Pengzhou Abel, and Kenji Fukumizu. beta-Intact-VAE: Identifying and Estimating Causal Effects under Limited Overlap. International Conference on Learning Representations, 2022.

[4] Miao, Wang, Zhi Geng, and Eric J. Tchetgen Tchetgen. Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika 105.4: 987-993, 2018.

Thank you for the follow-up discussion and all your suggestions. Please find our responses to your latest comments below. We have updated the paper based on your latest comment, and the updated contents are highlighted in blue.

Difficulty of the proxy setting (and others): We have revised all the assumptions as you suggested in the latest version. To clarify these assumptions, we summarize them and our problem setting in Section 4.1.

Outcome estimation: As you recommended, we have now provided more detailed explanations to bridge the gap between Equation (3) and the ATE estimation method. In the latest version, we use the discrete data case as an example to demonstrate that Algorithm 1 in Appendix C aligns with Equation (3). Although the solution for continuous data with the linear model does not mirror Equation (3), it is still consistent with the (conditional) front-door adjustment shown in Theorem 3. As we know, the front-door adjustment consists of two back-door adjustments. In our case with continuous variables, these two back-door adjustments are implemented by regression adjustments, as shown in Equation (30) and Equation (31) in Appendix C. Specifically, for the first backdoor adjustment, we use regression adjustment to get the causal effect of $T$ on ${Z_{\text{\rm CFD}}}$ (i.e., $\beta_{T,{Z_{\text{\rm CFD}}}}$) adjusting on $W$, as shown in Equation (30) and Step 1 of the process in Appendix C; for the second backdoor adjustment, we also use regression adjustment as shown in Equation (31) in Appendix C to obtain the causal effect of ${Z_{\text{\rm CFD}}}$ on $Y$ adjusting on $W$ and $U$. However, for this regression adjustment, since $U$ is unobserved, we follow the approach [1-2] to firstly obtain the error $e_{Z_{\text{\rm CFD}}}$, which is independent of $U$, and regress $e_{Z_{\text{\rm CFD}}}$ on $Y$ instead. In this way, we can eliminate the confounding bias caused by $U$. This is shown in Step 2 of the process in Appendix C.

Experiments: Regarding your question about the Adult dataset, we performed discretization on the dataset. After learning the CFD variable via CFDiVAE, we applied Algorithm 1 (as presented in Appendix C) to estimate the ATE. The solution outlined in Algorithm 1 is consistent with Equation (3).

Limitations: Following the comment, we have updated the conclusion to discuss the limitations.

Please let us know if you have any further comments, concerns, or suggestions. We greatly value your feedback and discussion.

[1] Iain Barr. Causal inference with python part 3 - frontdoor adjustment, Sep 2018. URL http://www.degeneratestate.org/posts/2018/Sep/03/causal-inference-with-python-part-3-frontdoor-adjustment/.

[2] Eric J Tchetgen Tchetgen and Ilya Shpitser. Semiparametric theory for causal mediation analysis: efficiency bounds, multiple robustness, and sensitivity analysis. Annals of statistics, 40(3):1816, 2012.

Thanks for your review and insights. We revised the typo in the latest version. Following your comments, we updated more explanations for the real-world application, added two comparison methods based on the standard front door adjustment, and conducted additional experiments on dimensionality analysis. Please find the latest version in the Rebuttal Revision.

Thanks for your review and insights.

Regarding your concern on the main contribution, while Pearl's book has informally mentioned the relaxation of the front-door criterion, it does not offer a formalisation or solution to the problem. Our paper not only clearly defines the conditional front-door criterion (CFD) but also provides comprehensive proof to support CFD adjustment, thus making a novel and non-trivial theoretical contribution to causal effect identification in the presence of latent confounders. Additionally, it is important to note that the scarcity of literature on the front-door criterion has resulted in less attention to this potent technique for obtaining unbiased causal effect estimation in the presence of unobserved confounders, and the strict requirements of the standard front-door criterion have restricted its practical use, our paper, therefore, makes a very useful contribution by considering a more realistic scenario to enable the practical use of the technique.

As for the second contribution, a main novelty of our work, as you mentioned, lies in the combination of CFD and the VAE based model, especially the employment of the VAE technique makes it possible to learn a CFD adjustment variable directly from data, enabling data-driven causal effect estimation. Moreover, we employ an identifiable VAE, which theoretically ensures the identifiability of the representation learning process. This represents a novel contribution to the practical application of deep generative models for causal inference.

For the question on the changes to our model when considering the example in Pearl’s book, we agree with you that the inference model would be different given the difference in the causal structure.

Thank you for the follow-up discussion and all your suggestions.

We really appreciate your valuable comments.

We have updated the paper based on the follow-up discussion, and the updated contents are highlighted in blue.