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: ### Problem setting 
The paper addresses the causal representation learning task, i.e., inferring high-level causal latent variables from low-level observations. Specifically, it considers a weakly-supervised setting in which pairs $(x,x’)$ of pre- and post-intervention observations are available. This can be seen as a generalization of Locatello et al. [5] to a setting in which the latents are not mutually independent but causally related. The paper formalizes this setting via latent causal models (LCMs) which consist of a structural causal model (SCM) over latent variables $z$ and a decoder $g$ mapping $z$ to observations $x$ (as also considered in previous works).

### Theory
The main theoretical contribution (Thm. 1) is an identifiability result stating that, given an infinite number of pairs $(x,x’)$ resulting from all perfect stochastic single-node interventions (those that set a causal variable $z_i$ to a new noise variable, thus completely removing any influence from its causal parents), the true LCM is identified up to a permutation and element-wise invertible reparametrisation of the causal variables. 

### Algorithm/Method
Based on this insight, the paper then proceeds to investigate algorithmic approaches to learning such LCMs from data. Specifically, two VAE-based approaches are investigated which either represent the causal variables and graph explicitly or only implicitly (ELCMs and ILCMs, respectively). ELCMs are only briefly discussed at a high level and are dismissed due optimization challenges. ILCMs, in which the latent space corresponds to the exogenous noise variables in the LCM, are portrayed as a more promising practical approach. ILCMs consist of an intervention encoder, a noise encoder and decoder, and a solution function (a map between exogenous noise and endogenous causal variables) and are trained by maximizing the corresponding ELBO. Two methods are proposed for post-hoc extracting causal structure from a trained ILCM. 

### Experiments
In experiments on three synthetic and image datasets, ILCMs are compared with other VAE architectures and SlotAttention w.r.t. DCI score, accuracy in inferring intervention targets, and structural Hamming distance and are shown to compare favourably. A new Mujoco-based dataset called CausalCircuit is also introduced in the process.  
strengths_and_weaknesses: ## Post-Rebuttal Update
After discussion with the authors, some of my doubts and questions (particularly regarding the use of solution functions) have been resolved and addressed, and I have decided to increase my score as a result. I believe this is a solid paper that would benefit the NeurIPS community and recommend acceptance. 

I encourage the authors to take serious the suggestions brought up during the review phase to further improve the paper, particularly its accessibility and presentation, as they already indicated they would.
____ 

*Disclaimer / review context: I have read the main paper carefully, but only skimmed the Appendix. Due to lack of familiarity with category theory, I did not check the proof of the main theorem and can thus not judge its soundness. I am quite familiar with the setting and related literature, but was not able to follow certain aspects in detail---specifically, the exact use of different solution functions---which I hope to be clarified during the discussion period. My score reflects my current impression of the paper, but I remain open to adjusting my score based on the authors' response.* 

### Strengths
- The paper is generally well-written and well-motivated.
- The problem setting of learning causal representation from weak supervision is novel and highly relevant.
- The paper makes contributions to both theory (identifiability result) and algorithms (ELCMs and ILCMs) for causal representation learning.
- The paper is well positioned in the related literature. 
- The paper is honest and transparent in discussing limitations, both regarding theoretical assumptions and implementation.
- The paper introduces useful concepts such as isomorphisms between LCMs, which are of interest to the field of causal representation learning beyond the present work. 

### Weaknesses
- Due to addressing both theory and algorithmic approaches, many technical details are deferred to the Appendix, which makes it hard to understand the proposed method in sufficient detail based on just the main text. 
- I have some doubts/reservations regarding the soundness of the proposed ILCMs (specifically, regarding the representation of SCMs through the noise variables and solution functions) and the evalution in terms of the DCI score (see below for more details).

questions: ### Main questions and comments
My main concerns and questions relate (i) to the concept of representing SCMs through the combination of exogenous variables and a solution function (which is also known as the reduced form SCM in the literature), as well as to (ii) the soundness of evaluation based on DCI scores.

(i) To my understanding, this representation of SCMs is not equivalent to the original SCM, but strictly less expressive in that the reduced form SCM cannot model arbitrary interventions to the true causal mechanisms---see, e.g., Sec. 10 of (Schölkopf, B. & von Kügelgen, J. From Statistical to Causal Learning. 2022). Would the authors agree with this statement?

I would like to better understand how the new solution functions $\tilde{s}_I$ are defined:
- Does this not require knowledge of the true mechanisms?  
- How does this relate to the assumption of perfect, stochastic interventions? 
- What aspect of the theory and proposed method would break down, e.g., for hard interventions (which set $z_i$ to a constant) or for soft interventions that preserve some dependence on the parents?

Some more specific questions regarding the solution function(s):
- Before Defn. 1, the new mechanism $\tilde{f}_i$ is defined over the same exogenous domain $\mathcal{E}_i$, but in Defn. 3, $\tilde{\epsilon}_i$ seems to be a new noise variable with different domain. Can you clarify?
- Regarding footnote 3, do you have a formal argument or reference to back up this claim? 
- In Sec. 4.2, paragraph “Latents”, the noise variables are defined as $e=s^{-1}(z)$ and $\tilde{e}=s^{-1}(\tilde{z})$ and it is correctly stated that the latter corresponds to noise value that would have generated  $\tilde{z}$ under the *unintervened* SCM mechanisms. In the next paragraph, it is then stated that only those components of $\epsilon$ corresponding to intervened nodes change. Where exactly in Appendix A is this proven?  While I found this result surprising at first, after looking at some toy examples, it seems to hold. However, the way in which $\epsilon_I$ needs to change will depend on the value of the other noise terms $\epsilon_{-I}$, thus rendering them dependent. Could you comment on the relevance of this? 

(ii) Based on my understanding, the DCI score of Eastwood and Williams is tailored to a setting with mutually independent ground truth factors since it is based on predicting different ground truth latents from the learnt representation. It would seem that this is no longer necessarily a sound evaluation when the $z_i$ are dependent.
- Did you consider this aspect? 
- What was your choice for the feature importance matrix?
- Do you report only the disentanglement (D) score or the whole DCI score (average of D, completeness C and informativeness I)? Since the abbreviation D was used, this was not completely clear. 

### Other more minor comments, questions, and suggestions
- l.4: “this requires…” makes it sound like a necessary condition, whereas what is shown is that the considered setting is sufficient. Consider rewording, e.g., “this involves…” 
- the restriction to perfect stochastic interventions seems important enough to be mentioned in the abstract and introduction
- Fig. 1: consider adding an explanation of what orange nodes represent (changed post-intervention values?)
- Related work: I would consider adding the following causal representation learning references:
- - Adams, J., Hansen, N., & Zhang, K. Identification of partially observed linear causal models: Graphical conditions for the non-gaussian and heterogeneous cases. Advances in Neural Information Processing Systems 34, 2021.
- - Xie, F., Cai, R., Huang, B., Glymour, C., Hao, Z., & Zhang, K. Generalized independent noise condition for estimating latent variable causal graphs. Advances in Neural Information Processing Systems 33, 2020.
- - Kivva, B., Rajendran, G., Ravikumar, P., & Aragam, B. Learning latent causal graphs via mixture oracles. Advances in Neural Information Processing Systems 34, 2021.
- - Chalupka, K., Perona, P., & Eberhardt, F. Visual causal feature learning. Uncertainty in Artificial Intelligence, 2015.
- - Beckers, S., & Halpern, J. Y. Abstracting causal models. AAAI Conference on Artificial Intelligence, 2019.
- l.104: including a intervention distribution in defining SCMs relates to (Rubenstein, P., Weichwald, S., Bongers, S., Mooij, J., Janzing, D., Grosse-Wentrup, M., & Schölkopf, B. Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence, 2017.)
- Defn. 2: What does “compatible” mean here? What implications does this have for the (obs./int./cf.) distributions implied by the two SCMs, will they match?
- Thm. 1: consider adding “in the sense of Defn. 2” to statement 2.
- l. 151: AFAIK, Pearl defines hard interventions as those which set a variable to a constant; in this sense, the considered stochastic interventions are not hard (in general)
- Sec. 4.1: some more details on ELCMs (e.g., form of the prior $p(z,z’)$) would be helpful
- Sec. 4.2: the idea of ILCMs where structure is embedded implicitly seems related to: Leeb, F., Lanzillotta, G., Annadani, Y., Besserve, M., Bauer, S., & Schölkopf, B. Structure by architecture: Disentangled representations without regularization. 2020.
- l. 214: what if the true post-intervention distribution is non-Gaussian? Why is Gaussianity needed here, or could a more flexible density also be used? I understand this is only the base density used to encode a more complex distribution over the noise variable, but this seems somewhat counterintuitive: do we not mostly care about the distribution of causal variables in the end? In this sense, forcing it to be Gaussian seems restrictive. 
- L. 215 ff.: I was not able to really follow this and the next paragraph in satisfying detail; perhaps consider expanding/clarifying.
- L. 238: since $q(I|x, x’)$ involves $\mu_e$, does this not depend on the noise encoder? Is $\mu_e$ shared between both?
- L. 296: is the graph of dVAE-E not always the empty graph by construction? 
- L. 300: are objects/latents learned by SlotAttention assumed/constructed to be independent? 
- Evaluation: it could be interesting to also evaluate how well different approaches capture the true interventional distribution, e.g., by looking at an average KL between different true and inferred interventional distributions. 	
- Fig.5: which ground truth variables are intervened upon here? Also, some more details on how the ILCM intervention is generated exactly would be helpful.
- L.369: any intuition on what the failure mode is here? 
- (L.378) Generally, I think the pre- and post-intervention terminology is slightly misleading: Typically, interventions are associated with layer 2 queries in which the exogenous variables are not shared but randomly resampled. In the considered setting, the exogenous variables are shared which makes it a layer 3 / counterfactual query. This point has been made in a similar multi-view/weakly-supervised setting by von Kügelgen et al. [14, Sec. 3] and I think this could be articulated more clearly throughout. In particular, “pre-“ and “post-“ suggest a temporal succession, which is not the same as the considered *hypothetical* intervention under the same context/background condition/noise values; the analogy to agents observing the effect of actions may thus only hold approximately, as there some noise variables may change (naturally, i.e., not as the result of the intervention).
limitations: N/A
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This paper aims to identify the latent causal models from the observational data, where data can be intervened by some pre-defined distributions. To achieve this goal, the authors firstly demonstrate that if the observation data is generated according to equation 1, then the causal model can be identified. Based on this theory, the authors design two specific models to learn the causal variable and causal structure. The first model is based on VAE and the second one infers the intervention from the noisy variable. In addition, the authors also discuss the potential functions of their proposed models. In the experiments, the authors demonstrate the effectiveness of their model on the tasks of pedagogy, Causal3DIdent and CausalCircuit.
strengths_and_weaknesses: 
In general, I believe the studied problem is interesting. However, I find the paper quite hard to read. There are so many contents referred to the appendix or other papers. For example, the formal definition of diffeomorphic function, at least, there should be some intuitive explanations. What is faithful in the causal domain? What is p_M^X? Ideally, the reader should clearly understand what the authors would like to deliver. However, I think there are a lot of concepts that should not be regarded as common knowledge (it is not necessary for the reviewers to read the appendix). Because of these unclear notations, I have trouble in reading this paper. 

There should be an explicit algorithm on how to train the ELCM and ILCMs. I cannot capture how to relate the solution of ELCM and ILCMs with the theoretical results in section 3. Whether the results learned from ELCM and ILCMs can lead to a optimal result of theory 1?

I'm wondering how accurate are the inferred interventions, and how the error of the inference influence the following causal prediction? Whether there are some strategies to overcome the inference error?

Considering that the paper seems to study a very interesting problem, I tend to be slightly positive. However, I can lower my rating if the other reviewers do not support this paper.

questions: See the Strengths And Weaknesses
limitations: The authors have discussed the limitations of their model.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This paper has contribution in two parts: Identifiability of causal models of high level (latent) variables from low-level data like that of pixels of images based on weak supervision, and two practical algorithms to learn causal models based on the weak supervision. The identifiability result is based on the assumption that pairs of datum where one of them is not intervened upon and the other is, after an atomic hard intervention to one of the high level variables, is available for training. The latent variables need to be continuous for the identifiability result to hold. 
In addition, there are two algorithms, Explicit Latent Causal Models which are based on explicit modelling of causal graphs as well as Implicit Latent Causal Models which do not model the causal graph directly but rather parameterise the noise encodings. Experiments are performed on very simple datasets to demonstrate the practical algorithms.
strengths_and_weaknesses: **Strengths**:

The identifiability result follows from the work of Locatello et al. 2020 where the identifiability of disentangled representations are presented. The authors in this paper do a good job of extending it to incorporate the notion of graphs in latent variable models and hence it enables the identifiability of arbitrary causal models from unknown interventions.

In addition, the idea of implicitly parameterising the causal model and then using the latent variable data to learn a causal model from an off-the-shelf algorithm is very interesting as this is something I believe has not been done before in causal representation learning.

**Weaknesses**:

The identifiability result is based on some assumptions which are hard to verify. For any given dataset of samples, it is hard to verify, let alone know, if the data samples were produced from a single hard intervention on one of its latent variables or if there was a soft intervention or a multi-target intervention. So in this regard, I am a bit unsure in which real world image data does these assumptions hold. The presented results are on very simple and hand-designed datasets where the assumptions are made to hold. Also see below in limitations regarding other assumptions which are not addressed.

The main weakness of the paper is that the experiments are very weak, to say the least. The datasets are already simple in the sense that all the assumptions of the identifiability result are respected, and moreover, the number of high level variables on which the causal graph is defined is utmost 4, which is a very small number. For reference, for up to 4 variables, all the DAGs can even be enumerated and is a much simpler task. It is disappointing to see that even the implicit LCM algorithm which does not model a causal graph explicitly is not shown to work on more number of variables (10 or 20 for example). For the ELCM, the authors mention that it takes a performance hit when the number of variables increases due to optimisation issues. If it is the same case with ILCM as well, then I would argue that the presented algorithms are far from being practical. This could be a case of identifiability does not imply learnability, unless the authors convincingly demonstrate in higher dimensions that this works reasonably.

Adding to the above point, I suggest the authors to consider a synthetic dataset (maybe with a random nonlinear SCM plus normalizing flow like in the pedagogical experiment) where they can create data samples such that number of high level variables are 10 (or more). Multiple such random datasets could be produced and trained on with an ILCM. This would at least highlight how the ILCM performs in higher dimensions. Given that ENCO can handle variables upto 1000 variables, I do not see any limitations in performing these experiments.

The proposed algorithm achieves both high disentanglement scores as well as SHD. But this seems a bit counterintuitive. Disentanglement measures how much each of the factors of variation are disentangled from each other while SHD measures how well the ground truth graph is recovered. If the ground truth graph is not extremely sparse, then both cannot be high. Maybe this is only a behaviour in low dimensions which the authors consider. In addition, it could be that DCI score maybe captures something similar to a structural metric. I personally do not think that it is important to measure disentanglement if the focus is to get the correct graph. If it is still important for some tasks, I encourage the authors to measure other disentanglement metrics as well.

In summary, I feel that the theoretical contribution is very interesting and can be a stepping stone for identifiability results with more relaxed assumptions. However, given that the experiments are weak and the practical algorithms need more realistic settings, I think that more work needs to be done in this aspect before the paper can be accepted.
questions: Here are some of the questions I have for the authors:

- For Causal3Dindent dataset, only 6 graphs are considered, why not more? And are the considered six graphs random? Also please give the confidence intervals.

- I do not get the intuition behind using a beta term for the ELBO. Is it not the case that the goal of the current setup is to get the correct causal graph? Encouraging disentanglement through the beta term might be encouraging more independent factors, however, that might not be necessarily the case with hierarchical latent spaces with causal graphs.

- Does the ILCM also suffer from optimisation issues in higher dimensions, and if so, what advantages then practically speaking does it offer as against ELCM.

 
limitations: One of the main limitations of the proposed approach is that the causal variables are assumed to be causally sufficient (i.e. they are apriori known or given in some way through slots etc and they alone define a causal model). However, this assumption is notoriously hard to be true in any realistic dataset. For any image, the cause of a particular variable could be outside the image itself, and hence hidden. This point should be discussed and highlighted. In my opinion, a true practical algorithm should be able to handle latent variables in causal models as well. Nevertheless, I feel that this aspect should be discussed as it is an important aspect of causal representation learning.
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: This paper provides identifiability results for causal discovery under a weakly supervised setting where we have access to data generated from perfect atomic interventions. Along with the theoretical results, it also proposes two algorithms for learning, one requiring an explicit DAG search and the other can learn the graph implicitly.

strengths_and_weaknesses: This is a nice result and I enjoyed reading the paper! The strength of the paper are:

- The theoretical result is novel and allows identifiability of causal graphs with interventional data. This is important and future algorithms can be made based on this.

- The ILCM description is nice as it allows learning of the graph without explicit graph searching.

- The experiments are non trivial and shows off the implication of the result.

- The paper is well presented and written clearly. It was an enjoyable read.


questions: As it stands, I think it is a very good paper and it should be accepted. A few questions and (minor) criticisms:

- This is the question most directly related to the current paper - I didn't find it clear in the paper as to why ILCM should perform better than ELCM. I can see that ILCM doesn't require explicit graph search which is nice, but can it not also be stuck in local minima just like ELCMs do, if it has found an (implicit) incorrect graph? Perhaps you could clarify this.

- The situation where we have interventional data from perfect and atomic interventions is not very applicable. Although the current work is a good starting point, further investigation into how to relax the interventional data assumption is needed.
limitations: The authors should be commended for calling out the limitation of their assumptions. I am quite interested in understanding what happens to the identifiability results if we still have data from heterogeneous regimes, but possibly from imperfect/non-atomic interventions? Can the results be extended? In the paper, the authors give a counter example for when the intervention is imperfect, but what about the case when we don't have atomic interventions?
review_title: null

Meta-review: metareview: The reviewers were split about this paper: on one hand they would have liked to see better experimental results, particularly for larger graphs, on the other they appreciated the identifiability results and the ILCM algorithm. After going through the paper and discussion I have voted to accept for the following reason: even though the experimental results could be strengthened, papers with novel approaches to long-standing problems are the kind that make NeurIPS an uniquely interesting conference, particularly if those paper have strong theoretical guarantees. I urge the authors to take all of the reviewers changes into account (if not already done so). Once done this paper will be a nice addition to the conference!
confidence: Certain
Decision: Accept