Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors study causal discovery of linear models in the presence of measurement errors and unobserved confounders, for the situation where the mixing matrix can be identified up to column scaling and permutation. They show that there is a one-to-many relationship between models subject to measurement error and models with unobserved parentless causes. Given this mapping, they analyze the identifiability of both models and propose additional faithfulness assumptions for the two classes of models leading to identifiable structures, and in some cases, to identifiable parameters. Finally, they define graphical partitions (ordered groupings) that are invariant for the identified models.
strengths_and_weaknesses: The analogy between linear causal models in the presence of measurement error and linear causal models with (top-level) latent variables is, to best of my knowledge, novel and constitutes an interesting perspective on causal discovery without causal sufficiency. The technical quality of the paper is high, the text is clearly written for the most part, and the results seem to bridge an important gap in the causal discovery literature. The simulation section is brief, but comprehensive enough.

On the negative side, confidence bars are missing from the simulation results in Figure 4, even though the authors mention in the checklist that they have been reported. The figure is a bit hard to read, and some minor writing errors hinder clarity.

Minor comments:
- p1 (page 1), l34 (line 34): "application" $\to$ 'applications'
- p1, l56: "casual" $\to$ 'causal'
- p3, l95: "latter" $\to$ 'later', to be consistent with p6, l289 
- p6, l263: "passes" $\to$ 'pass'
- p7, l341: "exist" $\to$ 'exists'
- p8, l356: "parent" $\to$ 'parents'
- p8, l359: "child" $\to$ 'children'
- p8, l371: "the edge in" $\to$ 'edges in the'
- p8, l372: "are" $\to$ 'is'
- p8, l373: "group" $\to$ 'groups'
- p9, l409: add 'the' before "Reconstruction"
questions: On page 7, line 300, why does the structure defined in Definition 4 lead to a star graph? This does not seem immediately obvious to me and perhaps should be explained in a few sentences. I would also briefly describe what a star graph is.

How many simulations were performed in Figure 4? If the experiments were replicated multiple times, as they should be, then why are there no confidence bars reported?
limitations: The authors have not discussed any potential negative societal impacts of their work, which can be excused for such a highly theoretical work. I concur with the authors' claim that causal structure learning from observational data is less prone to these impacts than randomized experiments.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The paper shows a correspondence between linear SEMs with measurement error and linear SEMs with hidden variables.

The paper then characterizes the equivalence class for these cases which can be learned when variables are linear and non-Gaussian under two faithfulness assumptions.

A recovery algorithm is provided along with simulation results, which is shown to outperform LiNGAM.
strengths_and_weaknesses: Strengths
- Rigorous theoretical results that contribute to the literature on identifiability of linear non-Gaussian models
- Very detailed and generally well written paper

Weaknesses
- Very limited discussion of motivation and impact of the work
- Limited empirical results and extremely limited discussion of empirical results
questions: What is the primary impact of these results on causal discovery broadly?

What are some applications that these results contribute to?
limitations: Some discussion of limitations.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The article studies causal discovery in linear systems where some of the variables are observed with measurement error. The identification is based on the mixing matrix (which maps independent exogenous noise to the observations). The article first shows that the model with measurement error is equivalent to a linear latent variable model where the latent variables don't have any parents, so any identifiability results based on the mixing matrix are applicable to both cases (Thm 1).

Thm 2 states that under the regular faithfulness assumption the models can be identified up to an equivalence class consisting of systems with the same ancestral ordered grouping (AOG), but the edges between the groups can't be identified. Then the article presents an additional condition (Condition 1) such that iff every edge from a non-leaf node satisfies the condition, then even the edges between the groups can be identified, in other words, the system can be identified up to the Directed Ordered Grouping (DOG).

Furthermore, the assumptions imply that any ground-truth generating model has fewer edges than any model that has the same mixing matrix but does not belong to the same DOG as the true model. This insight can be used to devise an algorithm for structure recovery: 1) find mixing matrix, 2) find AOG of the true model, 3) find the center for each group that leads to the graph with fewest edges.

A simple simulation study shows that using the algorithm with DOG identifies the correct model better than recovery based on just AOG or another published baseline: ICA-LiNGAM.

strengths_and_weaknesses: Strenghts:

The presentation is clear for this kind of work that is somewhat heavy on technical detail. All the math is presented accurately (though I only glanced through the supplement) and the theorems are associated with illustrative figures and examples that demonstrate the concepts in concrete setups.

The main novelty are the theoretical results connecting the two types of graphs as well as the new results on identifiability that are stronger than the previous ones. I find these results original and significant in the sub-field machine learning studying causal identifiability with latent variable models.

Weaknesses: 

There are no major weaknesses. Some real-world example would have strengthened the paper further, but that might not be necessary in this kind of paper where the main contribution are the new theoretical results.

One major assumption for identifiability is that the noise terms are non-Gaussian. Whereas this might be true in many real-world situations, it is equally true that Gaussian noise is the default assumption in the majority of applications. Hence, the assumption may be limiting.

questions: Figure 4 could benefit from additional annotation of the rows/columns and a more detailed caption.
limitations: Yes.
review_title: null

Meta-review: metareview: There is a consensus among all expert reviewers that the paper provides solid and rigorous theoretical results that contribute to the sub-field of causal inference, especially the identifiability of linear non-Gaussian models **when there is a measurement noise**, although the authors could have done better at providing concrete motivation and demonstrating the real-world impact of the results. Some of the issues however have been addressed during the discussion phase. I do hope the authors will take reviewers' comments into consideration when preparing the camera-ready version of this paper.

The main contribution of this work is to show the analogy between linear causal models in the presence of measurement error and linear causal models **with parentless latent variables**. This constitutes an interesting perspective on causal discovery without causal sufficiency and enables the authors to obtain identifiability results based on the mixing matrix in both cases. The results, therefore, bridge an important gap in the causal discovery literature.
confidence: Less certain
Decision: Accept