Learning Mixtures of Unknown Causal Interventions

Summary

This paper explores the challenge of disentangling and identifying causal relationships in situations where interventional data is noisy and mixed with both intended and unintended effects. It focuses on applying interventions within linear SEMs with Gaussian noise without prior knowledge of the true causal graph. It presents an efficient algorithm that can learn and separate individual components of a mixture of unknown interventions, allowing for the recovery of true interventional distributions even when intervention targets are unknown.

Strengths

This paper addresses a relatively unexplored problem of disentangling mixed interventional and observational data within linear SEMs without prior knowledge of the causal graph. This is a challenging problem with significant implications for causal inference, making the study innovative and important for the field. The authors propose an efficient algorithm that leverages the properties of linear SEMs with Gaussian noise to recover individual components of the mixture. The methodological approach is rigorous, with detailed theoretical support including proofs of identifiability and sample complexity. This thorough theoretical treatment provides a solid foundation for the claims made in the paper. The paper is well-structured with clear explanations of the problem, methodology, and results.

Weaknesses

The methods developed in the paper are specific to linear SEMs with Gaussian noise, which might limit their applicability in scenarios where these assumptions do not hold. Non-linear relationships or non-Gaussian noise structures, which are common in many real-world datasets, may not be adequately addressed by the proposed approach.

Questions

1. I suggest improving readability by properly describing insights related to theorems, even though some of the proofs focus on matrix computations. 2. In the experimental setup, what would be the effect of different variances in Gaussian noise? 3.There might be some symbols used incorrectly: In line 104, "the edge U" should be checked. In line 128, the definition of M_I needs verification. In line 134, the letter \(\kappa\) is used when mentioning shift interventions, but \(\gamma\) is used below. I know that do-interventions also change \(\gamma\), but I suggest keeping the notation consistent. In line 279, "A_{ij} > 0" should possibly be "|A_{ij}| > 0".

Rating

8

Confidence

5

Soundness

4

Presentation

3

Contribution

3

Limitations

The author has explained the limitations of the study. This paper has no possible negative social impact.

Summary

The paper proposes a method to disentangle the observational and interventional data under linear SEM with Gaussian noise.

Strengths

The algorithm proposed in this paper efficiently disentangles components of mixtures arising from unknown interventions, accommodating both soft and hard interventions. The proposed method is supported by thorough theoretical analysis. Additionally, the paper is well-written and easy to follow.

Weaknesses

The experiments are not comprehensive enough to thoroughly assess the effectiveness of the proposed method. More experiments on real-world datasets are required. This is especially necessary since the problem setup for the mixture of interventions is a bit restrictive. Also, it would be interesting to analyze how the method empirically performs on datasets that are not generated from Linear SEM or do not have Gaussian noises.

Questions

1) Could you please elaborate on how realistic the problem setting is? Especially concerning the noisy intervention setting [6, 28], how compatible is the paper's problem setup with these real-world scenarios? 2) Could you please elaborate on the connection between your proposed method and existing literature on learning mixture of Gaussians?

Rating

6

Confidence

5

Soundness

3

Presentation

3

Contribution

3

Limitations

Yes

Summary

The paper proposes linear structural equation models with additive Gaussian noise to address the challenge of disentangling mixed interventional and observational data. The problem is highly relevant to real-world applications with mixed data.

Strengths

* The theoretical framework is robust, with clear assumptions and derivations. The paper provides theoretical guarantees on the identifiability of mixture parameters. * The key idea is clearly explained and easy to follow.

Weaknesses

* The use of linear structural equation models is well-established in recent literature. Implementing SEMs with unknown interventions is not a novel approach. * The identifiability guarantees rely on the assumption of soft interventions. This assumption may restrict the broader applicability of the model. * The proposed method lacks sufficient experimental support. And it seems that the proposed method is primarily validated through simulations in the experiments, is it possible to provide real-world data examples?

Questions

See Weaknesses.

Rating

5

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

* The method focuses on linear structural equation models with additive Gaussian noise only, and the theoretical guarantees rely on the linear- SEM assumptions, which may limit the method's applicability in practice.

Summary

This paper considers a setup where an intervention results in obtaining iid data from a mixture of multiple interventional data and observational data and the goal is to learn the mixing weights and the resulting interventional distributions. Particular focus is on the linear SEM setting with Gaussian noise where interventions are allowed to be soft with some restrictions. The goal is to then learn the mixing weights and the interventional distribution (all multivariate Gaussians) parameters without knowledge of the causal graph. Given the large amount of literature on learning mixtures of Gaussians, it only remains to be shown that the individual components are well-separated in terms of changes in the interventional distributions.

Strengths

Interventions in the real world turn out to be messy and this paper continues on the thread of research that deals with unknown interventions. The paper is well-written with a clear flow of ideas. The numerical evaluations are also thorough.

Weaknesses

While I understand that this is the first step in learning mixtures of interventions, I believe that it has limited novelty both conceptually and technically since it largely follows from existing work on learning Gaussian mixtures. The considered setup also assumes unconfoundedness on top of linear SEMs and Gaussian noise. Few more detailed questions follow in the next section. The experimental evaluation is also limited to simulation data which don't completely endorse the validation of the algorithm. See next section on suggestions/questions. The writing can be more careful. There are typos in the main assumption (4.1) and there's some confusing notation about the high probability delta and the difference in variance delta_i.

Questions

1. Theorem 4.1 is an existence theorem and in the experimental section, EM is used to learn the parameters. Given that the polynomial time guarantee includes information about A, how would we claim to not requiring knowledge of the causal graph? 2. While misspecifying k for the 'half' interventions case is indeed a feature, is there an improvement if k is correctly specified. Currently, it's not completely clear to me that the error goes down in (d). 3. Is there any intuition for why when SHD and Jaccard similarity metrics improve, the parameter estimation error still does not? See nodes = 8 in half interventions case.

Rating

6

Confidence

4

Soundness

3

Presentation

3

Contribution

2

Limitations

The authors have addressed limitations in a separate section.

Response to the Author

Thank you for your response. The authors have partially addressed my concerns, thus I will keep my rating.

Response to rebuttal

Thanks for the response and apologies for my delayed response. Regarding limited novelty, I am not convinced yet mainly because I don't see non-trivial cases where separation wouldn't hold. Perhaps it would be more constructive to consider a non-trivial example of intervention in your model where separation does not hold? I understand this request comes late. Regarding Question 1 - Yes, I meant to write sample complexity. This is convincing since you're not really claiming polynomial sample complexity but just identifiability so the mismatch in the theorem statement and experimental algorithm is ok. Experimental Evaluation - The authors have responded with a thorough evaluation including experiments on a real-world dataset. A question regarding the automatic component selection algorithm - how are the cutoff percentages decided? I still find the novelty issue a minus but the experimental evaluation is now thorough for me to increase my score.