Response to Reviewer 2XuD

We thank the reviewer for the valuable comments and the pointers to Caroline Uhler’s lab related research. We would like to address some of the reviewer’s concerns.

On the baselines: we agree with the reviewer that using the purely observational data the pooled data might not be a fair comparison due to their simplicity. However, using joint interventional data is the closest one gets to an “oracle” version, since then the problem is reduced to estimating a conditional function $P(Y \mid do(A))$. Moreover we believe it would be unfair to use the methods of Bravo-Hermsdorff et.al., Jung et.al. and Saengkyongam & Silva as benchmarks, as their assumptions usually do not match ours.

In particular, we cannot apply Saengkyongam & Silva’s method because their method requires the noise to be Gaussian (indeed, our method is also applicable for discrete variables). Likewise, our setting with the additive outcome mechanism is not identifiable in the interventional factor graph model by Bravo-Hermsdorff, since the additive structure of the causal mechanism does not translate to an extra factorization of the joint probability in the different interventional settings.

About Caroline Uhler’s lab related research: We thank the reviewer for the pointers. These are indeed quite related to our work.

• In particular Ribot et.al. is also interested in causal effect estimation in novel scenarios when some action-context pairs have been observed, which in the language of our paper would amount to estimating joint causal effects from observing only certain subsets of the interventions. Likewise, their identification strategy relies on making a linearity assumption, which shows that a latent factor model is induced.
• On the other hand, Bijan et.al. cluster variables with respect to interventions that come from a mixture distribution and then use the clustering to produce Synthetic Potential Outcomes to compute a previously unseen intervention. Their setting differ from ours in that we have several possible actions as opposed to a single treatment and we make an assumption on the functional form of the SCM in order to make the joint interventions identifiable.

We will add an extended comparison in the camera-ready version of the paper.

On the running example: We believe that having a running example is a good idea. However, there are two reasons why we don’t include it. First, lack of space in the initial submission, and second, including a running example on the camera-ready version would potentially require large changes to the paper, which would be challenging to do given the time constraints and difficult to review.

We appreciate the time and effort you have invested in providing constructive feedback, which will help us improve our paper.

Thank you for taking the time to review our paper and for your thoughtful comments. We appreciate your feedback and would like to address your concerns below.

On the need for parametric identification

From g-ID theory [1], we know that our problem setting is non-identifiable in the non-parametric case (i.e., only considering the causal structure in Fig. 1). I.e., given single-interventional and observational data, the joint-interventional effect cannot be uniquely determined in general. The causal structure, together with the intervention sets, forms what's known as a "thicket" (Def. 6 in [1]), which precludes identifiability. We believe that such multi-cause settings are interesting across many domains (see citations in Introduction). Unfortunately, these settings are limited by non-identifiability in the intervention generalization setting without additional assumptions.

On the novelty of the additivity assumption

We don't claim the additivity assumption itself is novel. There is indeed a rich literature on additivity in statistics [2,3]. In causal inference, additivity of exogenous noise has been quite common. However, functional additivity, where the function is composed of a sum of nonlinear functions, to the best of our knowledge, has not been explored in causal inference. In particular, it has not been investigated as a parametric assumption in the intervention generalization setting.

This function class is particularly interesting here since it limits the complexity of interaction between actions in the outcome mechanism. This effectively prevents two distinct causal models from agreeing across single-interventional regimes while differing on the joint-interventional.

On the limitations of Assumption 2

We agree that Ass. 2 limits the generality of our results. However, as discussed above, the problem is hard and not solvable in the most general case. For many if not most real-world settings with confounding between outcome and actions, intervention generalization is impossible without additional assumptions.

The key question becomes: "What parametric and structural assumptions do we have to make to achieve identifiability in the intervention generalization setting?" In our work, we have shown that under our set of assumptions using the additive outcome mechanism, intervention generalization is possible, and we have provided a practical estimator for it.

On the practical applicability

While intervention generalization may not be achievable in all real-world cases, the benefits for the cases where it is applicable are immense. We can reduce the required experimental conditions from growing exponentially with the No. of actions to a linear regime. This makes a significant difference in many real-world settings, where obtaining experimental data is often very hard or expensive.

Additionally, Cor. 1 provides a way to trade-off assumptions on additivity for additional experimental data. If for practical applications it is unclear whether a subset of actions contributes additively, we can choose drop the assumption on this subset. This then comes at the cost of having to collect joint interventional data for this subset. We think that allowing for this trade-off between additivity and joint interventional data makes our method applicable to a broader range of real-world cases than full additivity would allow.

On future directions

Our set of assumptions may not be the most parsimonious, and we hope that future work will find more general solutions. One motivation for choosing the additive model class was its relation to more general model classes, namely Generalized Additive Models (GAMs) [4], which in turn are related to the general functions of the Kolmogorov-Arnold Theorem [5]. We hope that our work can serve as a stepping stone toward proving identifiability results for the GAM case.

On extensions to more general confounding between actions

If we add confounding between actions, Lem. 1 would not hold. However, we believe this is not a complete blocker for the problem. Our identifiability proof is agnostic to the causal structure between the actions (assuming it has a DAG structure). We suppose that addressing more general action confounding would require modeling the dependencies between the actions as well as the outcome mechanism to achieve intervention generalization. We hope to make progress in this direction in future work.

References

[1] Lee, S. et al. "General identifiability with arbitrary surrogate experiments." 2020.

[2] Friedman, J. H. et al. "Projection pursuit regression." 1981.

[3] Breiman, L. et al. "Estimating optimal transformations for multiple regression and correlation." 1985.

[4] Hastie, T. J. et al. "Generalized Additive Models." 1990.

[5] Kolmogorov, A.N. "On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables." 1956.

Response to Reviewer 9nyV

Thank you for your thoughtful review and for engaging with our work. We appreciate your careful reading and feedback.

We would like to address your primary concern regarding the novelty of our approach and the necessity of our assumptions: You mentioned that Assumption 1 might be sufficient for identifying the joint interventional effect using only observational data, referring to Definition 5 and Theorem 3 in "On the validity of covariate adjustment for estimating causal effects" (UAI 2010).

Critical clarification: In our problem setting, the confounders ${C_1, \ldots, C_K}$ are unobserved. This is a fundamental aspect of our work, explicitly stated in Section 2.2 where we define $C = {C_1, \ldots, C_K}$ as "a set of unobserved confounders."

The covariate adjustment approach you reference would indeed be applicable if these confounders were observed. However, when confounders are unobserved (as in our setting), covariate adjustment is not possible, and identification becomes significantly more challenging. This is precisely why we need both Assumption 1 (Intervention Support) and Assumption 2 (Additive Outcome Mechanism), along with single-variable interventional data, to achieve identification.

We believe this clarification addresses your main concern about the novelty and necessity of our approach. Thank you again for your review, and we hope this clarifies the contribution of our work.

Response to Reviewer dotZ

Thank you for your thoughtful review and constructive feedback on our paper. We appreciate the time you've invested and would like to address your key concerns.

On the strong assumptions of our method

We acknowledge that our model makes strong assumptions, particularly regarding the additive outcome mechanism and the single-cause confounding structure. These assumptions indeed limit the generalizability to certain real-world scenarios where shared confounding or more complex outcome mechanisms might be present.

From general identifiability theory [1,2], we know that the intervention generalization problem is non-identifiable in the non-parametric case (considering only the causal structure). Without additional assumptions, given single-interventional and observational data, the joint-interventional effect cannot be uniquely determined in general.

The key question becomes: "What minimal set of assumptions enables identification in this setting?" Our work proposes one such set of assumptions that makes the problem identifiable. We view this as a starting point rather than a complete solution to all real-world scenarios with confounding.

Importantly, our Corollary 1 provides a way to trade-off assumptions on additivity for additional experimental data. If for practical applications it is unclear whether a subset of actions contributes additively, we can choose not to make that assumption on this subset. This then comes at the cost of having to collect joint interventional data for this subset. We think that allowing for this trade-off between additivity and joint interventional data makes our method applicable to a broader range of real-world cases than full additivity would allow.

In many domains where additive models are already commonly used (e.g., marketing analytics [3], certain pharmacological interactions [4]), our approach could provide practical value despite its limitations. We hope future work will find more general solutions with weaker assumptions that cover a broader range of real-world scenarios. (See also our response to Reviewer tbRD.)

On the multi-treatment causal inference literature

You make an excellent point about the connection to multi-treatment causal inference literature. We thank you for suggesting these important references. Our work is indeed related to this literature, and we will add a discussion of these connections in our Related Works section. The papers you suggested (Miao et al., 2023; Wang and Blei, 2020; Zheng et al., 2021) provide valuable context for our work and would strengthen our paper's positioning within the broader causal inference literature.

On the limitations of synthetic experiments

Our main goal in the experimental section was to show that, while we have shown that the problem setting is identifiable, the joint-interventional effect is practically estimatable from data with our proposed estimator.

We agree that our empirical evaluation using synthetic data has limitations in representing the full complexity of real-world causal structures. While we've tried to incorporate some level of complexity through randomly generated SCMs with varying parameters, this approach cannot fully capture all real-world nuances.

Thank you again for highlighting these important limitations. Your feedback will help us improve our research and better understand the practical applicability of our method.

References

[1] Lee, S., Correa, J. D., and Bareinboim, E. "General identifiability with arbitrary surrogate experiments." Uncertainty in artificial intelligence. PMLR, 2020.

[2] Kivva, Y., et al. "Revisiting the general identifiability problem." Uncertainty in Artificial Intelligence. PMLR, 2022.

[3] Chan, D., & Perry, M. "Challenges and opportunities in media mix modeling." Google Inc, 16, 2017.

[4] Pearson, R. A., Wicha, S. G., & Okour, M. "Drug combination modeling: methods and applications in drug development." The Journal of Clinical Pharmacology, 63(2), 151-165, 2023.