Causal Abstraction Inference under Lossy Representations

We appreciate the reviewer's feedback and believe some misunderstandings may have led to a harsh evaluation. We respectfully ask for reconsideration based on the clarifications below.

In general the paper is very difficult to read [...]

Response: We recognize that the reviewer’s primary concern is the paper’s density. While we aimed for clarity through diagrams and examples, some content had to be relegated to Apps. B and C, preventing us from elaborating as thoroughly as we would have liked. The density arises from building upon several works that already involve complex semantic systems, including:

1. Pearl Causal Hierarchy (PCH): Our approach is general, encompassing all 3 layers of the PCH. While we chose to present examples using simpler interventional quantities from Layer 2, we present all definitions generally to convey the full scope of our contribution (Layer 3).
2. Abstractions: We follow established abstraction theory, using concepts like inter/intravariable clusters, $\tau$-abstractions, AIC, and C-DAGs, with notation aligned to prior literature.
3. Causal Generative Modeling: Our work extends neural causal models (NCMs), forming the basis for Sec. 4 experiments.

In the main paper, we prioritized our new contributions over reviewing prior work, referencing established concepts rather than restating them.

That said, we are committed to improving clarity by:

• adding a notation summary table,
• expanding discussion on related works,
• using the extra page to elaborate on each definition. We appreciate your willingness to work with us in evaluating the contributions presented.

I am unsure what significant claims this paper wants to make.

Response: We emphasize in our paper that our primary goal is to develop a causal abstraction framework robust to violations of the abstract invariance condition (AIC). When two low-level interventions produce different effects but map to the same high-level intervention, existing frameworks fail to address the resulting ambiguity. This problem is particularly prevalent when working in the representation space after performing representation learning, as a learned, lossy representation is unlikely to avoid such ambiguity. Working in a representation space offers substantial advantages for high-dimensional causal inference, a direction we aim to advance in this paper. We hope this motivation is clearly conveyed in the introduction (lines 58–75, left), supplemented by the example on HDL and LDL cholesterol, as is our proposed solution (discussed from lines 76-95 (left)). We will refine the introduction for clarity and welcome any specific suggestions.

the approach starts from the premise to have a low-level SCM already available [...]

Response: The relaxation of this requirement is the entire contribution of Sec. 3. Recognizing that we are unlikely to have the whole low-level SCM, we introduce identification results (Def. 9, Thm. 3) that specifically aim to infer causal quantities given limited data from the low-level SCM, such as observational data.

if I already had a low-level representation (input of your algorithm) [...]

Response: The benefits of working in the abstraction space are similar to those motivating the use of representation learning in non-causal contexts: reducing dimensionality for improved tractability, transforming data into spaces with desirable mathematical properties (e.g., the linearity of Word2Vec), and better interpretability.

The AIC challenge is unique to the causal setting, as non-causal contexts do not require consideration of how causal relationships between variables are preserved or altered when the representation is lossy. Previously, the trade-off of working in the abstracted space, as you mentioned, meant accepting AIC violation risks for the benefits of representation learning. Our work resolves this dilemma by enabling the advantages of representation learning without incurring any drawbacks from AIC violations.

even if I am interested in summarizing variables [...]

Response: Certainly, but doing so would prevent leveraging the computational benefits of representation learning in the causal modeling process.

in a similar line: If you admit dependency between U-variables [...]

Response: This is a good question. Recognizing that AIC violations can be reinterpreted as projections into the exogenous space is one of the key insights of our paper. Previously, the prevailing view was that if a high-level intervention was ambiguous, performing the abstraction was futile since it meant ignoring details that were relevant to the setting (resulting in mathematical contradictions).

The natural connection to SCM partial projections enabled a generalization of previous frameworks, accommodating AIC violations by interpreting the lost details as exogenous variables. Establishing this formal connection and generalization constitutes a key and non-trivial set of contributions presented in Sec. 2.

We thank the reviewer for the positive feedback and review. We are happy that the work was understood. To answer your concerns:

The experimental section seems to be good to me, but it will be difficult for readers not familiar with NCMs. I would recommend trying to give the experimental section some more room to breath.

Response: Thank you for the suggestion. We will expand the experimental section to contextualize NCMs for unfamiliar readers, and we will include a section in the appendix discussing NCMs and their relevant results to the discussion in this paper.

However, given that causal abstraction has been applied to mechanistic interpretability, it might be good to situate the paper relative to that literature and explain to readers from that field how this relates.

Response: Thank you for these citation suggestions; we will be sure to include them. Mechanistic interpretability is a prominent issue in the field of causal abstractions, and we will certainly add an appendix section to discuss it in more detail. Although this work focuses more on the foundational aspects of abstraction theory, particularly performing cross-layer inferences, we find that many of the results are strongly applicable to both problem settings. Relating this work to the mechanistic interpretability problem is a clear next step.

How does this relate to causal feature learning? I don't think you cite any work from that area, but it seems deeply related to me.

Response: Actually, the problem of causal feature learning is highly relevant. Papers such as “Visual Causal Feature Learning” (Chalupka et al., 2014) leverage data-identifiable forms of the AIC to learn ideal representations and variables that best preserve the causal relationships between variables. This connection is briefly discussed in “Neural Causal Abstractions” (Xia & Bareinboim, 2024) Appendix D.2. We will also include a section in our paper complete with these citations.

We thank the reviewer for the positive review and valuable insight in our experimental analysis. We address your comments below.

I would like to see a discussion of the limitations of the proposed theoretical framework, as well as well as future perspectives.

Response: Indeed, one of the main limitations of the theoretical framework is that, as a tradeoff for violating the AIC, it is possible that more causal dependencies are introduced, as illustrated through the definition of the projected C-DAG in Def. 8 (more edges are added when there are AIC violators). We will emphasize this point further in the paper. This naturally implies that there is a challenge of determining how to balance the tradeoff between lossy representations and loss of causal constraints, and we will also make a point of this as a potential future direction of work.

"The abstractionless model has higher error than the projected C-DAG model since it operates in a higher-dimensional space." I don't find this explanation fully satisfying [...]

Response: In this particular experiment, the dimensionality reduction is quite extreme. In the $\mathcal{G}$-NCM, the model is trying to sample the entire image $X$ as an intermediate step, while in the $\mathcal{G}_{\mathbb{C}}^{\dagger}$-NCM, the variable $X$ has been abstracted into a binary variable. Since the high-level model has much less information in the input dataset, its only serious advantage is the dimensionality reduction. That said, the error of the $\mathcal{G}$-NCM still trends downward with higher data, since it is theoretically still a sound method of estimating the query. With much more parameters and compute, it is possible that the approach could achieve similar errors. We will include more discussion on this in Appendix D, as well as discussion on the tradeoff of performing such a large dimensionality reduction.

In the coloured MNIST experiment one thing is not clear to me [...]

Response: Indeed, we chose the binary representation as the most egregious example of an AIC violation, and we illustrate one of its clear flaws – one should not be able to achieve any good reconstruction of the image with such a lossy representation. This highlights the strength of our approach of projected sampling – whatever we lose in the representation, we can represent it in the exogenous space. An example analogy is this: suppose we have an image of 10 bits, but we represent it with only 1 bit. We certainly cannot reconstruct a 10-bit image with only 1 bit, but we can reconstruct samples by sampling the other 9 bits. The particular distribution of 9 bits is based on the theory discussed earlier in Sec. 2: we sample conditioned on the 1 available bit and the parents of the variable as if we are translating a high-level intervention to the low-level. We will include this discussion in the paper.

We see an improvement with respect to the non-causal model, but the digits themselves don't look right compared to the 16 dimension RNCM model [...]

Response: Thank you for the suggestion, we will run the two methods at different representation sizes to see how the performance scales. Our expectation is that while the original RNCM degrades in performance as the AIC is more and more violated (translating to worse performance with lower dimensionality), the RNCM with projected sampling does not suffer from this issue. To continue the bit analogy from above, the reconstruction of a 10 bit image would look better if we had 9 of the bits compared to only having 1 bit. However, with projected sampling, we are sampling the remaining bits anyway, so we always have the full 10 bits. Generally speaking, performance of the projected sampling approach could potentially be improved by simply improving the base architecture of the model, although this is out of the scope of our work.

I suggest to nuance/precise this type of statement: "Combining these two modes of reasoning is vital for building more advanced AI systems." [...]

Response: Thank you for the suggestion, we will change it to “Combining these two modes of reasoning unlocks great potential for building more advanced AI systems.”

In definition 7, is there a consistent generalization of the PCH to soft interventions? [...]

Response: Yes, this is indeed an interesting point. We frame Def. 7 in the context where the high-level model is performing hard interventions to emphasize where the noise in the corresponding low-level soft interventions arises as a response to AIC violations. That said, the method could certainly be generalized to handle cases where the high-level model is performing soft interventions as well. Then, the corresponding low-level interventions would be soft interventions that aggregate the results of all of the possible sampled interventions in their high-level counterparts. We will add a discussion on this in the appendix, as you suggested. Thank you!

We thank the reviewer for the positive review and detailed suggestions. Addressing the reviewers points:

The Abstraction Function, as defined in Definition 4, is required to satisfy certain assumptions [...]

Response: Yes, broadly, the paper is focused on a specific family of abstractions called constructive abstractions. Most works found in this literature, including ours, focus on this family because there is a natural mapping between low- and high-level interventions and distributions, allowing for well-defined inferences. These abstractions are most easily realizable in practice as well, since the construction of the high-level model naturally follows from the declaration of the clusters, which are interpretable and can also sometimes be learned (see “Neural Causal Abstractions” (Xia & Bareinboim, 2024).) If one would prefer to allow unrestricted abstraction functions, the results of the paper may still be applicable, but one would have to define how low-level interventions correspond to high-level interventions on a case-by-case basis (i.e., it is not necessarily the case that an intervention on $do(\mathbf{X}_L = \mathbf{x}_L))$ on the low-level is equivalent to an intervention on $do(\tau(\mathbf{x}_L))$ on the high-level, or even well-defined).

The three rules in Definition 8 are key factors in the proposed method. I am curious about the underlying motivation [...]

Response: Indeed, that’s a cool question, thank you. The three rules demonstrate the core insight of allowing lossy abstractions – by losing some information in the variables, we must make up for it by removing causal constraints (i.e., adding more edges). These rules are necessary for guaranteeing the validity of the downstream causal inferences, as shown through Thm. 2. Without adding these edges, one risks claiming that a causal query is identifiable when it is not. For example, in Fig. 1(a), if the edge from $Z$ to $Y$ is not included, one may mistakenly assume that $P(Y \mid do(X), Z) = P(Y \mid do(X))$, which is the mistake made in Ex. 2. The proof of Thm. 2 shows the math behind what could go wrong when each of these rules are ignored.

Overall, this work mainly addresses AIC violations, but I am concerned whether the results hold regardless of the degree of AIC violations [...]

Response: This is a great question. In fact, this is precisely the issue that we are solving. In the past, approaches that rely on the AIC would deteriorate in performance as the AIC becomes more and more violated. In contrast, the projected abstraction approach does not care how much the AIC is violated and performs identically regardless. Note, for example, in Fig. 4 of our MNIST experiment, reducing the representation size to a mere binary variable strongly violates the AIC and therefore results in very poor performance from the RNCM (3rd row), but with projected sampling, we can produce images as usual (4th row).

My final concern is regarding the experiment settings. [...]

Response: Our goal with the MNIST experiment was to provide a proof-of-concept of how one could potentially learn a causal generative model without worrying about AIC violations. More generally, this would allow for a unification of out-of-the-box representation learning methods and causal generative modeling methods since it would not matter how the representation is learned once we move to the causal generative modeling phase. With this unification understood, scaling to more complex datasets is a matter of scaling the architecture and representation learning methods using state-of-the-art techniques from general deep learning research, which can be done separately from the causal results guaranteed by our work.

While I can understand the overall story, I feel that the writing style may not be very accessible [...]

Response: We aimed for a more rigorous analysis of this highly technical topic for the sake of concreteness, so its more mathematical nature is somewhat to be expected. We do appreciate the suggestion and will add some clarifying sentences in the final report and include an appendix section discussing some of the key prior works to help bring readers up to speed with the contents of the paper.

It would be better to introduce, from an intuitive (high-level) standpoint, why consistency estimation is still possible when faced with lossy representations, and how the proposed method addresses this issue.

Response: Yes, thank you for the suggestion. We will include the discussion mentioned above on the tradeoff between the lossiness in the abstracted variables and the loss of constraints in the projected C-DAG and why inferences in the new model still apply.

We thank the reviewer for the insightful questions and positive review. We appreciate the multiple suggestions and will address them as follows:

L110-118 could be put into Def 2, as I got confused by undefined notations when reading Def 2.

Response: We will adjust Def. 2 to accommodate this and make it clearer; thanks!

Def 5: $\widetilde{\mathbf{pa}}_V$ is not used.

Response: Thanks for pointing this out. It was originally used in settings in which the AIC was not violated, but we do not consider such cases in the main body, so we will omit this from the definition to improve legibility.

Def 6: Definition of $\tau$-abstraction is missing in the main text (please move or point to Def 10 in Appendix A).

Response: We will include this as suggested.

The paper is notational heavy [...]

Response: Thank you for the feedback. We do assume some background of the previous literature, but indeed, our efforts to fit all of our content into the page requirement have unfortunately led to the paper being a bit dense. To remedy this problem, we have included Appendix B for further discussion and Appendix C for clarifying examples. If accepted, we will use the extra page to clarify each definition and make the theory easier to follow. We will also add a table in the appendix to clarify all of the notation we use throughout the paper.

Finally, to answer your questions:

1. The details of the theoretical contribution are unclear. It seems like projection has been proposed in Lee & Bareinboim, 2019 [...]

Answer: Yes, SCM projections were proposed in Lee & Bareinboim (2019). We draw inspiration from this result to generalize to partial SCM projections (one of our contributions). In contrast to Lee & Bareinboim (2019), which solves a causal RL problem to find optimal intervention sets in a causal bandits setting, we work in an entirely different space, applying projections to define a causal abstraction formalism. Def. 6 and Thm. 1 (both contributions of our paper) establish this intriguing duality between partial SCM projections and causal abstractions, which is surprising and may otherwise seem unrelated. The idea that a lossy abstraction that violates the AIC can be thought of as partially projecting away the abstracted variables is the main insight that allows us to generalize previous abstraction inference results to new settings where the AIC does not necessarily have to hold. This insight leads to further results about how to perform more general causal inferences across abstractions (Sec. 3) and a new type of sampling procedure for high-dimensional causal generative modeling (Sec. 4). In short, the generalization of SCM projections to the partial form is indeed a contribution of the paper, but the main motivation for this contribution is to generalize challenging instances of causal abstraction inference. We thank the reviewer for pointing this out, and we will include this discussion in the paper's appendix.

2. For experiments, what does the variance in each method (Fig 5) come from? [...]

Answer: The variance for each approach arises from the sampling of the training data and the randomness in the training process (e.g., parameter initialization). One explanation for the seemingly higher variance in the projected C-DAG approach may simply be that the plot somewhat exaggerates it due to the log-log formatting. Another deeper reason may be that there is a tradeoff between granularity and constraints when we convert from the original C-DAG to the projected C-DAG (as we are potentially adding more edges). With more edges, there are fewer constraints; therefore, converging may be more challenging despite the higher accuracy. Of course, this more nuanced view is just brought up by the new machinery developed in this paper. This paper is the first to study relaxed abstractions under violations of AIC, which are very likely to occur in almost any scenario.

3. What is the scalability of the proposed method in a more complicated, real-world scenario? [...]

Answer: One issue that is resolved by our paper is that the AIC can be a challenging thorn when attempting to apply out-of-the-box representation learning tools in causal generative modeling contexts. We would like to be able to learn an NCM on top of any representation, but the AIC prevents us from doing so. The goal of the MNIST experiment is to show a proof of concept that, regardless of how the representation arises, we can still perform causal inferences and sample causal images using the new projected-sampling approach. This is powerful as it means that the method can achieve robust performance regardless of the method of representation learning used. Scaling to even higher-dimensional settings may require larger architectures, but fortunately, one can leverage the vast research of the general deep learning community to accomplish this while still enjoying the causal benefits from this work.