Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The authors investigate the connection between structural causal models and neural networks from an learning perspective. Their theoretical results emphasize, notably, that the impossibility to learn neural networks that are interventionally equivalent to the true SCM based on observations only. The paper provides an interesting perspective on the connection between causal inference and learning with neural network, although clarity could be improved, notably regarding the limitations and the contribution with respect to previous work.
main_review: While the overall effort of this paper is clearly valuable, I struggled with the presentation of the theoretical results. It might be partly due to my partial understanding of the framework, although I think the main text lacks some key information.
1/ First, only by going through the appendix one can notice that the theory applies to variables in countable numerical domains (lemma 4). Besides, I tend to think this is not fully accurate, as the proof of lemma 4 seems to assume *finite* domains.
While one can argue that binary/finite domains can approximate continuous ones with arbitrary precision, it is important to specify it in the main text, as results would appear very confusing to readers familiar with technical issues, while unfamiliar readers who see the use of neural networks would naturally assume every functions defined can act on, say, Euclidean domains.
2/ It is confusing to have theorem 1 on L3 consistency while this property is not defined in main text. An informal presentation of Li would help.
3/ Corollary 1 conveys an important point, but only to those not familiar with causal inference (pointing to structural identifiability issues), for whom the level of abstraction will be difficult to parse. On the other hand, I tend to think that the setting of this result is too “coarse” to capture when causal inference is useful (and I assume the authors do agree with me that sometimes it is). If I did not miss anything, if we restrict ourselves to an SCM with two nodes, and we add additional information on the causal graph (e.g. uncounfounded, and the direction of causation in known), then we are in good shape to be able to collapse the interventional level to the observational one. This kind of positive result is not captured by the corollary one because a) we consider “all SCMs” at once, thus ignoring the informativeness of certain graph structures, b) we do not include the possibility to incorporate assumptions on the graph structure at inference, which is key in practice. I do get that the next sections address this issue, but I question the benefit this side result. Overall, I tend to think the target readers would benefit more from a high-level discussion, possibly in relation to the examples in supplemental, that from such abstract result (besides largely not self-contained in the paper). 
4/ Overall, it is difficult to disentangle the contribution of the paper with respect to what is already known regarding learnability of SCMs. Once we have established the ability of neural networks to perfectly fit functions with finite domains (which can be found in the appendix, and are typically not too challenging to show), classical result from causality seem to apply as usual. I certainly may have missed key novel results that are fundamentally different from previous work, but they are not really emphasized in main text. It looks like the lemmas in appendix reflect better the gist of the contribution than the main text results. 

limitations_and_societal_impact: Assumptions of finite domain seems to be central for the proofs, but does not seem to be mentioned in main text, while only a countability assumption is mentioned in only one lemma in supplemental, although finite domain seems to be exploited in the proof.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper studies the expressivity, identification, estimation of causality with neural models. For expressiveness, it shows that Neural Causal Model (NCM) is expressive enough to represent any structural causal model (SCM), but in general, given observation data alone, without further constraints, the NCM is not able to identify higher layers of Pearl Causal Hierarchy (PCH). The authors then develop a necessary and sufficient condition to determine whether a causal effect can be learned from data (i.e., causal identifiability); if identifiability holds, it also estimates the effect from data (causal estimation). Empirical experiments on a simple synthetic dataset support the theoretical results.
main_review: In terms of novelty, to my knowledge, the theory and the algorithms introduced are novel. 

In terms of quality, I did not check through the theoretical proof in the Appendix, but I believe they are correct. For the experiments, they are evaluate on simple scenarios which demonstrate the effectiveness of the algorithm's capability of causal identification and estimation. I wonder how well the algorithm is able to scale to more complex scenarios where the observed number of variables and their edges are larger, and its run-time compared to training a standard network of similar structure. Furthermore, for the neural architecture, it would offer more insight if the authors can perform experiments that test if and how the expressivity of the neural models (number of layers, number of neurons per hidden layer, etc.) affects the results, and how many examples are needed for confident identification and estimation.

In terms of clarity, the paper is very theory-heavy. To improve intuition, if possible, it may be useful to use a running example, starting from the beginning, to accompany the proved theorems, so as to make it more friendly for readers not in the causal community. For example, the examples in Appendix C may be good examples.

Overall, I think this paper is well-written and of high-quality, and makes useful contribution to causal identification and estimation with neural methods. If the more thorough experiments are performed, it will further strengthen the paper.
limitations_and_societal_impact: The authors did not address the limitation of the paper.
review_title: null

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: A new class of models called Neural Causal Models is proposed which retains causal reasoning properties of Structural Causal Models but can be learned through neural net based methods. contributions include formally proving connections between SCMs and NCMs, disentangling expressiveness with learnability of NCMs and deriving constrained classes of NCMs, a practical learning algorithm for NCMs.
main_review: The paper proposes a new model called Neural Causal model that has the properties similar to a structural causal model but can be learned through gradient-based methods. NCMs are a subset of SCMs but have causal reasoning power like SCMs. Formal properties are proved about the expressive power of NCMs, i.e., they can represent distributions as an SCM. However, this expressiveness does not mean that NCMs are learnable and require additional constraints imposed on the causal diagram. The constraints give rise to a learnable class of NCMs. It is shown that in this class it is possible to perform causal reasoning using observational data. Soundness and completeness results are presented for neural inference in NCMs and it is shown that it is equivalent to symbolic inference in SCMs. A gradient-based training method is presented for approximately learning NCMs.

Experiments are performed that show if an effect can be identified from a cause accurately through NCMs for different causal settings. For non-identifiable effects it is shown that the effects cannot be accurately identified through NCM learning. A generative model through NCM is compared with a generative model that does not use the causal graph (only data-driven) and also WERM, a state-of-the-art causal learning model.

The paper is well written and covers a lot given the space limits. The clarity of writing makes it accessible even to non-experts in this area I think. The significance of this work seems quite remarkable since it proves several fundamental connections between SCMs and neural nets. I can see that this work can have impact in many other domains such as explainability in AI. A case study of a real-world application would have been beneficial for understanding the impact in a practical setting. Overall, though this seems to be a highly significant paper connecting Neural and Symbolic AI.


limitations_and_societal_impact: Societal impact not directly addressed, but since the paper addresses fundamental connections between symbolic and neural models, I cannot think of any negative societal impact or limitation to be concerned about.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper considers a restricted class of SCMs (structural causal models) where each variable in a graph is generated by applying a neural-network-parameterized function of the variable's parents, called Neural Causal Models (NCMs). The paper shows a wealth of results mapping the relationship between the full class of SCMs and the smaller class of NCMs in terms of when and how identification holds and how estimation of the interventional distribution can be achieved. The method proposed to identify and estimate effects takes a graph as an input, computes the maximum and minimum of the effect of a particular intervention within the class of NCMs that are consistent with the observed data distribution (at the lowest level in the causal hierarchy, observational); if the maximum and minimum match, the effect is identified. The method is evaluated on a few experiments.
main_review: The paper is well-written and puts forth an interesting perspective of estimating interventional distributions. However, it is unclear why the notion of neural networks is central here in theory. I understand that practically there are appealing properties in terms of optimization and techniques used to quickly build good predictors using neural networks. However, all of the theory about NCMs being an interesting class of causal models, i.e. with identification and estimation, holds for any sufficiently flexible class of functions (lemmas 4,5 are constructive and nice but any large enough function class contains the constructed functions). 

I think the distinction above is important because under this perspective, the main claim of the paper is that neural networks help learn interventional distributions better than other existing causal effect estimation methods that do not rely on the machinery of the NCMs; for example, one could check causal identifiability based on the given graph and then apply causal effect estimation methods like the following: like https://arxiv.org/abs/2102.12034 and https://arxiv.org/abs/2001.07426. 

As the theory holds for any flexible class of functions, it seems to me that the experiments should show that the proposed estimation method is better than other baseline methods as suggested above.  As these baselines are not included, the experiments, while encouraging, are insufficient to establish the improvement that the proposed method (and NCM-based methodology) brings to causal effect estimation.


--- UPDATES after rebuttal

score updated after clarifying the position of the paper.
limitations_and_societal_impact: I am concerned about the applicability of the proposed method to estimation of interventional distributions in general seeing that the experiments are limited. It is unclear why WERM is the only (non-naive) method compared against and how it is state-of-art. If the authors could explain why no other baselines were considered, that would help clarify the standing of the paper; for example, compare against interventional distributions using knowledge of the graph and effect estimation methods like like https://arxiv.org/abs/2102.12034 and https://arxiv.org/abs/2001.07426.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: In this paper a subclass of SCMs whose structural equations are based on feed-forward NNs are considered.
The authors show the expressiveness of this class and address the problem of training those model and evaluated identifiability of a causal task. Experiments on models (with few variables) are promising.
main_review: Causal machine learning is definitely an important field of research for the NeurIPS community. This paper offers an interesting class of models to practically achieve that. Considering that NNs are universal approximators, the results in the technical part are not surprising, but I think the paper might really help researchers in this area. The fact that the experiments are only focused on very small models leave some doubts about the practical scalability of these techniques.
limitations_and_societal_impact: Nothing.
review_title: null

Meta-review: The paper considers the Pearl causal hierarchy layers, with $L_1$ representing the observations, $L_2$ the interventions and $L_3$ the counter-factuals. 

The contribution establishes that the fact that a powerful model (neural nets in the following (1)) fits the true causal model 
${\cal M}^*$ regarding $L_1$, does not imply that it fits ${\cal M}^*$ regarding $L_2$. The proof, following from [6], is based on that fact that the set of models fitting ${\cal M}^*$ on (at least two) layers has measure 0.


In order to tackle the second layer $L_2$ of the Pearl causal hierarchy, one must consider the causal identification task (predicting the causal diagram (2)) and the causal estimation (predicting the effects of intervention). 

The second contribution is to establish identifiability conditions (including the knowledge of the causal diagram) for an intervention query. 

The algorithmic condition for this identifiability is stated as the fact that the neural models fitting the interventional data and maximizing / minimizing the outcome of the intervention, converge toward each other. 

The significance of these contributions has been challenged and very convincingly argued by the authors. 

Some limitations regarding the domain of the observed variables (discrete and finite) have been discussed and will be stated in the revised version. 

(1) The neural net class is considered in the paper, though the discussion suggests that any class with sufficient expressivity could be considered instead.

(2) The paper claims that there exist no neural methods today focused on solving the causal identification task, which is not exact (though restricted to Markovian), consider https://arxiv.org/abs/1803.04929. 

Decision: Accept (Poster)