Venue: CLeaR 2022 Conference

Reviews:

final_score: rating: 8: Top 50% of accepted papers, clear accept
review_content: Summary: The main contribution of the paper to the literature is to provide identifiability results for causal discovery under the assumption of linearity and allowing for both latent confounding and deterministic relationships.
Main review: Originality: The identifiability results under determinism are new and relevant. The novelty compared to existing works is well justified.

Significance: The paper addresses an important problem, which is relevant for the CLeaR community. Determinism is for example implicit in constitutive relationships. Complex systems typically mix up constitutive and causal relationships. The paper for sure provides new insights. The insight of using blind source separation to find out deterministic relationships is interesting and full of potentiality. The paper is likely to have broad impacts also outside the CLeaR community, in all complex settings where causal and constitutive relationships are mixed up.

Technical quality: The proposed approach is technically sound, with claims substantiated by sound theoretical results. However, some the implications of the so-called "unique components condition" should be better spelled out. It seems to me that this condition implies that one observes all the variables that enter as argument in a deterministic function of the structural causal model. Suppose that, in the SCM, X=f(Y,Z). It seems to me that the unique component condition (and the proposed identification method to function) needs that one observed Y and Z. Latent confounding is allowed only when is nondeterministic. I think this should be spelled out. Moreover, at p. 6 (bottom) it is written: "we can use this unique component to compute the total causal effect from $x_i$ to $x_k$ (by dividing the coefficient of this unique component in $x_k$ by its coefficient in $x_i$)". What is claimed between brackets should be rigorously proved.

The paper is clearly written and well organised. However, in the introduction section some illustrative examples of the different SCMs would help.
review_title: null

final_score: rating: 8: Top 50% of accepted papers, clear accept
review_content: Summary: The paper generalizes the linear SCM setting to the case where simultaneously latent variables and deterministic relationships between endogenous variables are existent. The main result of the paper provides necessary and sufficient conditions for unique identifiability of causal effects among the variables. For the sufficiency direction, an algorithm to learn the causal structure is provided. It is shown that this algorithm recovers the true causal structure better than some of the alternative algorithms when dealing with some different subclasses of this general SCMs.
Main review: I personally like the paper. It is clearly written and contains a clear result. The result and the algorithm, although marginally significant in my opinion, are novel and worth publishing. The technicalities seem correct to me although I did not check the Appendix carefully. A problem with the paper in current from is that a large number of examples, explanations, and numerical experiments that fit in the main body of the paper and help to understand the content are put in the Appendix. I think the authors could have chosen what goes into the main text and what goes to the Appendix better. I have just a few comments below:

At first, the paper sounds as though they have solutions for all possible deterministic relationships, but as it turns out later there are rather strong combinatorial conditions that need to be satisfied. I think the authors should be more clear about this from the beginning.

The argument about the faithfulness assumption being satisfied almost surely, although formally correct, is misleading. It is true that any additional conditional independence is of order zero, but any SCM assumes many of these order-zero constraints, so it is not totally unreasonable to wonder whether more of these independencies not lie around. In addition, even after looking at the related Appendix section, I could not see that it is shown formally that this condition is equivalent to the faithfulness assumption.   

There should be more discussion on the main two assumption required in the main result. What are the causal interpretation and implications of these assumptions, and also how restrictive do they become? 

It seems to me that the conditions could be used beyond the linear setting, as, in principle they do not use the assumption of linearity. It would be worthwhile to have a discussion on whether they generalize the known results on identifiability of non-linear SCMs.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: Summary: This paper considers the causal discovery problem in linear structural causal models, where the observed variables are of a deterministic function of other observed variables or latent confounders, but not with an exogenous source. 
Main review: This paper considers the causal discovery problem in linear structural causal models, where the observed variables are of a deterministic function of other observed variables or latent confounders, but not with an exogenous source. The authors give necessary and sufficient conditions for the unique identifiability of the model and the relative estimation method. Overall, this paper is written well, with complete theoretical identifiability analysis for the model and sufficient supplementary materials. However, I have some confuses, listed below.

1. About the motivation of the introduced deterministic linear model, I agree that “G-SCMs without distinct sources can be used to model scenarios with less factors than observed nodes;”, but why deterministic relations are implied? For example, I mean in Fig.1(a), s_1 and s_2 can be removed while x_3 could have a distinct exogenous variable. I do not think the motivation is applicable for the proposed model. Btw, if it is indeed deterministic relation, the model in Eq. (2) cannot characterize this determinism.

2. I am confused about Assumption 2, which I think is contradictory with the proposed “deterministic relations”. For instance, if we consider a two-variable case and suppose x_i -> x_j, then according to Assumption 2, x_j has at least one source in x_j that is not in x_i, which implies x_j does not deterministically depend on x_i or latent confounders. Please explain it.

3. For the nonlinear causal discovery methods with deterministic relations and with identifiability, please also refer to the following papers.
[1] Chen Z, Zhang K, and Chan L, "Nonlinear Causal Discovery for High Dimensional Data: A Kernelized Trace Method," 2013 IEEE 13th International Conference on Data Mining, 2013, pp. 1003-1008, doi: 10.1109/ICDM.2013.103
[2] Zeng Y, Hao Z, Cai R, et al. Nonlinear Causal Discovery for High-Dimensional Deterministic Data[J]. IEEE Transactions on Neural Networks and Learning Systems, 2021.
[3] Janzing D, Mooij J, Zhang K, et al. Information-geometric approach to inferring causal directions[J]. Artificial Intelligence, 2012, 182: 1-31.

4. The sentence in the paper “when the sources are non-Gaussian random variables, the mixing matrix W can be recovered using Independent Component Analysis (ICA) or overcomplete ICA methods” is not correct, but using the under-complete ICA. 

5. It might be better to use the assumptions in ICA or BSS to derive Assumption 4, rather than “separability”, and the input of the algorithm is the data X, but not the mixing matrix. 

6. How could we know the number of latent sources or the size of the mixing matrix? 

I would raise my score if the authors could resolve all my concerns.
review_title: null

Meta-review: The paper studies causal structure learning of linear SCMs under confounding, while allowing for deterministic relationships in the structural equations. The authors derive necessary and sufficient conditions for identifiability and provide an algorithm to recover the structure. The reviewers agree that the results are an interesting contribution to the literature. That being said, there are minor issues in the exposition that should be addressed in the final revision (see reviewer fNIS’s suggestions). In addition, it would be great if the authors could provide more intuition on the two major assumptions in the main paper (e.g., by providing examples).
Decision: Accept (Poster)