Disentangled Representation Learning in Non-Markovian Causal Systems

W#1 “The practicality of the proposed method is questionable…”

We address this question directly in Appendix I. FAQ Q2 (pg 46) and provide some details in author’s rebuttal section II.

Firstly, expert knowledge can help infer causal structures. For instance, in Appendix D.1 (pg 31), it's known that "Age" directly affects "Hair Color" in faces.

Beyond the CRL setting, the causal inference literature has decades of work demonstrating the utility of imposing causal assumptions in the form of a causal graph, which can be traced back to its origins (e.g., see Pearl’s original paper in 1995 [2]). We highlight a few [1-4]. [1-2] demonstrates that it is possible to identify causal effects given only observational data and knowledge of the causal graph. [3-4] shows how to estimate counterfactual quantities. [5] discusses external validity and the transportability of causal effects across domains. Similar to these prior works, our paper demonstrates how to leverage the knowledge of the latent causal structure (assumptions), along with the data distributions (input) to achieve a generalized notion of identifiability (output).

Finally, our work complements methods that learn causal graphs. One can leverage our work, and propose further assumptions that enable simultaneously learning the latent causal structure.

Further insights are available in FAQ Q1 (pg 46).

[1] Pearl. Causality: Models, Reasoning, and Inference. 2000.

[2] Pearl, Judea. “Causal Diagrams for Empirical Research.”1995.

[3] Pearl, "Probabilities of causation: three counterfactual interpretations and their identification," 1999.

[4] Pearl. "Direct and indirect effects." 2001.

[5] Bareinboim and Pearl, "Causal inference and the data-fusion problem," 2016.

W#2 “...what new value…disentanglement notation”

First, we argue that relaxing full disentanglement to achieve causal disentanglement has practical value. Often, full disentanglement of all variables isn't necessary. For example, in the medical scenario with EEG data (L65-82), clinicians may only need to disentangle drug treatment from seizures, not all variables. Achieving full disentanglement typically requires specific distributions, such as single-node interventions for each latent variable, which are often unavailable or hard to obtain. Thus, allowing arbitrary combinations of distributions and focusing on partial disentanglement is more practical in real-world settings.

To help readers outside the specific research topic understand the impact of learning partially disentangled representations, we have added to the text:

“As another example to motivate learning a representation that only disentangles a subset of variables, consider a marketing company generating face images for a female product. The relevant latent factors are Gender $\leftrightarrow$ Age $\rightarrow$ Hair Color. If Gender and Age are entangled, altering Age could also change Gender, which is undesirable. The company needs a model where Age is disentangled from Gender but can remain correlated with Hair Color. Thus, the company doesn’t need to ensure disentanglement between Age and Hair Color. Our paper addresses the generalized identifiability problem of determining the sufficient training data for learning such a disentangled representation.”

Q#1 “How can the relaxed disentangled estimation be used?”

Consider a motivating example as described in Appendix D.1 pg 32, where Gender $\leftrightarrow$ Age $\rightarrow$ Haircolor. If Age is disentangled from Gender, but is entangled with Haircolor, then after editing the representation of Age, we would expect that Gender does not arbitrarily change, while Haircolor may change. This flexibility may be useful in the context where the user does not care about the entanglement of Haircolor and Age, but does care a lot about keeping faces the same gender during image generation (i.e. not spuriously turning a woman's face into a male face, and vice-versa).

In the context of your example, if we have V1 -> V2 -> V3 -> V4, and V2 and V3 can be disentangled from Vtar = {V1, V4}, then one can predict that variables in Vtar will not arbitrarily change when intervening on V2 and/or V3, unless they are causal descendants of V2 and/or V3. For example, perturbing V2 will translate into changes downstream of V2, such as V4 (and V3 since we assume they are still entangled). However, perturbing V2 will not spuriously change V1 since we assume in this example that the representation of V2 is disentangled from V1.

Q#2 “advantage…unobserved confounding factors”

The non-Markovianity applies to the underlying SCM over V, not the ASCM including X. Unobserved confounders (U) are not part of V and do not generate X. We expand on this in the author’s rebuttal IV.

Unobserved confounders are crucial as they allow for causal relationships within V that cannot be encoded by a Markovian model but exist in the real world. For concreteness, consider V1 and V2 generating X. In Markovian settings, three causal relationships are considered: (1) V1 and V2 are independent; (2) V1 -> V2; (3) V1 <- V2. Now consider that there exists a common cause, U, which both affects V1 and V2 (i.e., V1 <- U -> V2). This cannot be encoded through the Markovian language. Here, U is not a parent of X and does not mix with V1 and V2 to generate X. Introducing unobserved confounders is necessary for modeling general causal relationships in CRL.

For concreteness, consider the discussion in Q1, where Gender and Age are spuriously correlated because there are many old males and young females in the dataset due to face images being collected in a non-diverse setting. This unobserved confounding though does not manifest itself in the face image, except through the causal factors Gender and Age. The causal relationships between generative factors Gender and Age (V) are assumed from domain knowledge, but they cannot be assumed independent or to have a direct effect on each other.

Thank you for your answer. For W1, I still think the applicability is fairly limited. For W2 and Qs, the authors' response makes sense, thus I raised my score.

Gender and Age are spuriously correlated because there are many old males and young females in the dataset due to face images being collected in a non-diverse setting. This unobserved confounding though does not manifest itself in the face image, except through the causal factors Gender and Age.

This is an example of selection bias rather than confounding.

New Experiment Details

The images are a modification of the popular MNIST dataset with a bar and color added (to both the digit and the bar). Each digit in the original database contains many examples with different writing styles. The goal is to demonstrate that a disentangled representation can robustly modify the color of the bar and digit without spuriously changing other latent factors.

Exp details: Data is generated according to Fig. 2 (in author’s rebuttal pdf), where the digit ($V_1$) causes the color of the digit ($V_2$), which in turn causes the color of the bar ($V_3$). Each digit is written in its own style ($V_4$), and is confounded by the setting in which the writer wrote the digit. With observational data from two domains, soft intervention on the distribution of color-bar, and a hard intervention on the color-digit, CRID outputs that $V_2,V_3$ are disentangled from $V_1,V_4$, a new identifiability result compared to the prior literature. Fig. 3 verifies this quantitatively since the correlation of the representations w.r.t. disentangled variables are lower than w.r.t. entangled variables. Since this evaluation requires the ground-truth of latent factors while we do not have access to the truth writing style ($V_4$), and thus only show correlations w.r.t. the others. However, this can be verified qualitatively through Fig. 4. Upon modification of $V_2$ and $V_3$’s representation, we note that robust changes in only the color-digit/color-bar are observed, while no spurious changes in the digit or writing style take place.

W#1 “Experiments are limited...”

We want to clarify that the proposed work’s goal is to identify the presence of disentanglement in fully learned representations. The experiments in this work are used for verifying our identifiability results. Compared with similar identifiability literature, there are many works that focus on the theoretical side (identifiability) and only use pure simulation data to verify their theoretical results (e.g., see [1-8]).

Having said that and based on your suggestion, we have included an additional experiment on hand-written digit images to highlight the potential impact of learning disentangled representations.

[1] Hyvarinen et al., "Nonlinear ICA using auxiliary variables and generalized contrastive learning," 2019.

[2] Khemakhem et al., "Variational autoencoders and nonlinear ICA: A unifying framework," 2020.

[3] Liang et al., "Causal component analysis," 2024.

[4] Zhang et al., "Causal representation learning from multiple distributions: A general setting," 2024.

[5] Varici et al., "General identifiability and achievability for causal representation learning," 2024.

[6] von Kügelgen et al., "Nonparametric identifiability of causal representations from unknown interventions," 2024.

[7] Varici et al., "Score-based causal representation learning with interventions," 2023.

[8] Sturma et al., "Unpaired multi-domain causal representation learning," 2024.

W#2 “The text is very dense”

Thank you for noting the many results we try to introduce in this work. We also agree that there are numerous concepts that must be introduced in order to arrive at the theoretical contributions. The submission includes a table of the various notations in Appendix A.1 (pg 16), and also Ex. 1 to introduce the notation (L198-203).

To further help the introduction of the theory, we will add a paragraph describing the assumptions informally to help visualize the critical technical assumptions in a conceptual manner. We anticipate this will help readers conceptualize the technical content better. In addition, we will add additional examples that conceptualize the notation and definitions.

Q#1 “What do you think about the scalability of CRID to real-world data…”

CRID Algorithm Applicability: Yes, the CRID algorithm produces a causal disentanglement map (CDM) that indicates what can be disentangled based on data and latent causal structure assumptions.

Let us elaborate. CRID answers: “Given my distributions and causal assumptions, what is identifiable?” It generates a Causal Disentanglement Map from the selection diagram and interventional targets but does not estimate or learn representations, thus handling complex confounders among latent variables, and any mixing function. Note the identifiability of variables may change with different confounders.

Weaknesses: CRID determines identifiability but does not estimate representations with complex confounders and mixing functions. Estimating representations involves building a proxy ASCM model compatible with the selection diagram and observed distributions.

Existing prior work (and our work Def. 1) suggests that the mixing function is a diffeomorphism, making normalizing flows a common approach. However, these require matching dimensions between latent variables and data, which can be problematic with high-dimensional data. Future research could focus on models bridging low-dimensional latent factors and high-dimensional data.

Summary: Our paper focuses on identifiability, not on developing new models for learning representations. In the author's rebuttal with pdf, we show a disentangled learning experiment on a “causal” MNIST dataset to inspire further research into learning disentangled representations using our theoretical framework.

W#1 “There is a gap …experiments”

We want to clarify that the proposed work aims to identify if a learned representation will exhibit disentanglement. This is a separate question from estimating the disentangled representations, and using said representations in a downstream task, such as realistic data simulations.

The contributions on identification are of a theoretical nature and work for both low and high-dimensional mixture X. To illustrate, the CRID algorithm is proposed to identify the disentanglement of learned representations. CRID is a pure symbolic method that takes the diagram and intervention targets as input but does not require any training procedure. Consider the EEG example in the introduction (L65-82). With the input $G^S$ and intervention targets described in Ex.1, we can obtain the causal disentanglement map as shown in Figure 4 (please see Ex. 6 for details.)

The experiments in the paper are used for verifying the identifiability results. In the literature, there are many causal disentangled representation learning works that focus mainly on the theoretical problem of identifiability and also only use simulation data to verify their theoretical results [1-9].

We understand that seeing more than pure simulated data would be compelling, which led us to include an additional experiment on hand-written digit images. We hope this can illustrate the potential impact of learning disentangled representations. In particular, the images are a modification of the popular MNIST dataset with a bar and color added (to both the digit and the bar). Each digit in the original database contains many examples with different writing styles. The goal is to demonstrate that a disentangled representation can robustly modify the color of the bar and digit without spuriously changing other latent factors.

Exp details: Data is generated according to Fig. 2 (in author’s rebuttal pdf), where the digit ($V_1$) causes the color of the digit ($V_2$), which in turn causes the color of the bar ($V_3$). Each digit is written in its own style ($V_4$), and is confounded by the setting in which the writer wrote the digit. With observational data from two domains, soft intervention on the distribution of color-bar, and a hard intervention on the color-digit, CRID outputs that $V_2,V_3$ are disentangled from $V_1,V_4$, a new identifiability result compared to the prior literature. Fig. 3 verifies this quantitatively since the correlation of the representations w.r.t. disentangled variables are lower than w.r.t. entangled variables. Since this evaluation requires the ground-truth of latent factors while we do not have access to the truth writing style ($V_4$), and thus only show correlations w.r.t. the others. However, this can be verified qualitatively through Fig. 4. Upon modification of $V_2$ and $V_3$’s representation, we note that robust changes in only the color-digit/color-bar are observed, while no spurious changes in the digit or writing style take place.

[1] Hyvarinen et al., "Nonlinear ICA using auxiliary variables and generalized contrastive learning," 2019.

[2] Khemakhem et al., "Variational autoencoders and nonlinear ICA: A unifying framework," 2020.

[3] Liang et al., "Causal component analysis," 2024.

[4] Zhang et al., "Causal representation learning from multiple distributions.." 2024.

[5] Varici et al., "General identifiability and achievability for causal representation learning," 2024.

[6] von Kügelgen et al., "Nonparametric identifiability of causal representations from unknown interventions," 2024.

[7] Varici et al., "Score-based causal representation learning with interventions," 2023.

[8] Sturma et al., "Unpaired multi-domain causal representation learning," 2024.

[9] Lachapelle et al., "Disentanglement via Mechanism Sparsity Regularization…" 2022.

W#2 “The simulation result, … not supported with detailed discussion…”

The simulation results are obtained over 10 different random seed runs with a fixed set of hyperparameters, such as learning rate, MLP normalizing flow architecture, and max epochs. Note: disentangled representation learning theory (in general and in our paper) holds in the setting of infinite data and perfect optimization. However, given finite data and numerical optimization, this will generally not be perfect. Thus we ran multiple trials to verify that on average disentanglement was achieved. We also point out that our paper does not claim to propose a robust method for learning the representations, which is a separate problem.

Re MCC (correlation) values and how to interpret: According to ID definition (Def. 2.3, pg 5), if there exists a function $\tau$ such that $\widehat{V}_j = \tau(V^{en})$, then $V_j$ is disentangled from $\mathbf{V} \backslash V^{en}$. Thus the simulation results show MCC values between $V_k$ and $\widehat{V}_j$ are relatively lower when $V_j$ is disentangled from $V_k$. In contrast, MCC values will be higher on average when variables are entangled. The MCC values in Fig. 5 show on average that variables that were expected to be disentangled by CRID were in fact relatively less correlated to their entangled counterparts.

For instance, Fig. 5(a) (in manuscript) considers a causal chain $V_1\rightarrow V_2\rightarrow V_3$. Based on prior work [1], with a single-node soft intervention on each variable, one can achieve disentanglement up to ancestors; i.e. $V_3$ would still be entangled with $V_1$. However, in line with the motivation of this paper, what if one does not have a single-node intervention per latent variable? Figure 5(a) demonstrates that with two $V_3$ soft interventions, one can disentangle $V_3$ from $V_1$ as evidenced by the average correlation of $\widehat{V}_3$ with V1 (MCC=0.3) being lower than the correlation of $\widehat{V}_3$ and $V_3$ (MCC=0.8).

We have added text to the Experiments section describing the above setting to provide more discussion.

We thank the reviewer for the feedback. We feel that a few misreadings of our work make the evaluation overly harsh, and hope you can reconsider the paper based on the clarifications provided below.

W#1

W#1 (cont) “I'm not completely sure about the significance of the theoretical results, especially compared to previous work. Since the hidden causal structure is already provided, the problem is easier than most existing works in causal representation learning.”

We respectfully disagree on this point. The latent causal structure is assumed in many prior causal inference literature and CRL works. Compared to existing works, our paper is not easier since we consider a more general setting. Please see section II in the author's rebuttal for further discussion about assuming the graph.

W#1 (cont) “Compared to causal component analysis [1], the extension from Markovian to non-Markovian settings is introduced.”

Thanks for highlighting the Markovianity extension! However, our work extends causal component analysis across all three axes we mentioned in the global review: input-assumptions, input-data, and output-disentanglement goals, not just in terms of non-Markovianity.

While we generalize to the non-Markovian setting, we also improve results within the Markovian context. The CRID algorithm encompasses all disentanglement results from [1]. For instance, in the Markovian diagram $V_1 \rightarrow V_2 \rightarrow V_3$, [1] requires one hard intervention per node plus observational data for full disentanglement, ensuring $V_i$ is disentangled from $V_j$ for all $i \neq j$. [1] shows ancestral disentanglement with soft interventions per node plus observational data, achieving disentanglement as: $V_1$ from $V_2, V_3$, and $V_2$ from $V_3$. In contrast, Appendix F.5 (pg 37) shows how we can disentangle a $V_3$ from its ancestors using only two soft interventions on $V_3$.

W#1 (cont) “However, given that the hidden structure is already known (we can precisely locate those hidden confounders that make the system non-Markovian), this extension seems somewhat compromised.”

That’s a misconception, to see why the non-Markovian setting is reasonable and valuable, refer to the author’s rebuttal Section IV.

W#2

W#2 (and W#1 cont.) “The combination of assumptions 1 and 2 seems unrealistic ...”

Please see why the non-Markovain setting in our work is reasonable and valuable in author’s rebuttal Section IV.

W#3

W#3 “The organization of the paper could be improved …”

Thank you for pointing this out. We added a paragraph in Section 2 to introduce the assumptions and their conceptual ideas informally early on.

Q#1-3

Q#1 “In the FAQ part, it is mentioned that domains and interventions are distinct.”

We added a response to this issue in Section II. of author rebuttal. In addition, we provide an example illustrating the differences and possibly the issues that may arise in conflating the two.

Q#2 “In Table 1, it is stated that the proposed method could deal with general distributions, while previous works focus either on one intervention per node or multiple distributions. ”

We want to clarify what we discuss in the manuscript: the current literature focuses on distributions from interventions per node or observational distributions from multiple domains. We handle the general case since our distributions come from arbitrary combinations of interventions and domains.

“However, it seems that the identifiability result in this paper also needs changing distributions (sufficient changes) and soft interventions, …, especially when compared with previous work on causal representation learning with multiple distributions.”

Thank you for pointing out we leverage different distributions for disentanglement. However, requiring different distributions does not mean we do not generalize the input distributions. In fact, the challenge here is to determine how distributions change given arbitrary combinations of intervention targets and domains encoded in the selection diagram. Under Markovianity, determining how distributions change given soft interventions per node is relatively easier and less complex compared to interventions in different domains under non-Markobanity settings. This challenge is answered by Prop 2. In addition, our setting does not require observational data necessarily either. Finally, note that existing work explicitly specifies the required distributions beforehand, whereas CRID accepts as input whatever distributions a user has, and determines the identifiability of a learned representation. Taken together, the proposed work generalizes the input distributions assumed compared to prior work.

Q#3 “Are assumptions strictly weaker than those required in causal component analysis (CCA)?”

First, expanding on our earlier point in Q2, CCA requires an intervention per node, whereas our work does not impose such distribution requirements for disentanglement. We offer a more general approach with the causal disentanglement map (CDM) and provide additional graphical criteria (Props. 3, 4, 5) for identifiability, even in cases where CCA may not apply. Second, our non-Markovanity cases are strictly weaker than Markovanity in CCA.

In addition, our theory does not specify exact conditions for disentanglement because the goal is user-defined. Instead, we use graphical criteria to link assumptions in the latent causal graph with data distributions to determine expected disentanglement.

Q#3 (cont.) “If so, are there any trade-offs?”

There is indeed a tradeoff in terms of how much disentanglement can be achieved, which we highlight in L65-82. Specifically, the more diverse distributions are given, the more disentanglement can be generally achieved. However, if they only care about disentanglement among a subset of variables, they may be able to design more targeted experiments, or data collection procedures to achieve a subset of full disentanglement.

Thank you for your response and follow-up questions.

1. Does this imply that CCA addresses a special case of the problem in the paper under stronger assumptions?

Yes, our problem setting is strictly more general in terms of the assumptions and input data (as discussed in Q3 of the rebuttal). Even though CCA is also nonparametric, it focuses on the Markovian setting, which is a special case of the non-Markovian setting we study in our paper. Also, the results in CCA rely on input data that contains interventional data for every node. On the other hand, our result accepts arbitrary combinations of distributions from heterogeneous domains, where not all nodes need to be intervened on. Thus, the required input distributions in CCA are a special case of the proposed work.

Summary: The disentanglement results in CCA can be derived using CRID. For further discussion on how CCA and our work relates, please refer to L1256-1290 (p. 39) in the submitted manuscript.

2. Can we view the intervention as a specific form of domain change? If so, I am still not entirely convinced of the need to treat them separately.

The answer is no, not in generality.

Section III of the author rebuttal discusses this issue. As another example, consider a modification of the Ex., where one treats a domain as an intervention. Consider the causal chain $S \rightarrow V1 \rightarrow V2 \rightarrow V3$. Assume a hypothetical setup, where one has observations in humans ($P^{\Pi_1}(X)$), and observations in bonobos ($P^{\Pi_2}(X)$), and two soft interventions on hair color in bonobos ($P^{\Pi_2}(X;\sigma_{V_3}), P^{\Pi_2}(X;\sigma_{V_3})$). Leveraging CRID, one can disentangle hair color ($V_3$) from sun exposure ($V_1$).

If we ignore the domain index and consider any bonobo distribution as an intervention w.r.t. the observational distribution in humans, then the available distributions would be the following: $P(X), P(X;\sigma_{V_1}), P(X;\sigma_{V_1, V_3}), P(X;\sigma_{V_1, V_3})$. Interestingly, the original result that $V_3$ is ID w.r.t. $V_1$ cannot be obtained in this case since $V_1, V_2, V_3$ are all in \deltaQ set (Def. 3.1) ($P(V_1)$ and $P(V_3 \mid V_2)$ can both change) when comparing $P(X;\sigma_{V_1, V_3})$ with any other distribution. The lesson here is that it would be naive to ignore that distributions within the same domain contain invariances that can be leveraged.

In addition, further discussion about this matter is provided in Q5 (FAQ) in p. 47 of the submitted manuscript. After all, the domain index is not only useful mathematically, but it formally encodes important information about the world. Finally, the distinction of interventions and domains is also useful and has far-reaching consequences in the context of causal discovery (structure learning), as noted in [1]. We would be happy to provide further clarifications.

[1] Li, Adam, Amin Jaber, and Elias Bareinboim. "Causal discovery from observational and interventional data across multiple environments." Advances in Neural Information Processing Systems 36 (2023).

3. Additionally, I couldn't find the updated part concerning W3. Did I overlook it?

Since we could not upload the new manuscript, we copy the text here, which was added at the end of Section 2.

“Assumptions (Informal) and Modeling Concepts

Before discussing the main theoretical contributions, we informally state our assumptions and provide remarks to ground the ASCM model.

Assumptions

1. Soft interventions without altering the causal structure

Hard interventions cut all incoming parent edges, and soft interventions preserve them. However, more general interventions may arbitrarily change the parent set for any given node. We do not consider such interventions and leave this general case for future work.

2. Known-target interventions

All interventions occur with known targets, reducing permutation indeterminacy for intervened variables.

3. Sufficiently different distributions Each pair of distributions $P^{(j)}, P^{(k)} \in \mathcal{P}$ are sufficiently different, unless stated otherwise. This is naturally satisfied if ASCMs and interventions are randomly chosen. Similar assumptions include the "genericity", "interventional discrepancy", and "sufficient changes" assumptions.

Remarks

1. Mixing is invertible

As a consequence of Def. 2.1, the mixing function $f_X$​ is invertible since it is defined as a diffeomorphism, ensuring that latent variables are uniquely learnable.

2. Confounders are not part of the mixing function

According to Def. 2.1, latent exogenous variables U influence the high-dimensional mixture X only through latent causal variables V, so unobserved confounding does not directly affect the mixing function. For instance, in EEG data, sleep quality and drug treatment may influence EEG appearance, while socioeconomic status may confound sleep and drug treatment but not directly affect EEG. This idea is also present in prior work, such as nonlinear ICA, where independent exogenous variables $U_i$​ each point to a single $V_i$.

3. Mixing function is shared across all domains

According to Def. 2.1, the mixing function $f_X$​ is the same for all ASCMs $M_i \in M$, enabling cross-domain analysis. If the mixing function varied across distributions, the latent representations would not be identifiable from iid data alone.

4. Shared causal structure

As a consequence of Def. 2.1, each environment's ASCM shares the same latent causal graph, with no structural changes among latent variables. The assumption that there are no structural changes between domains can be relaxed and is considered in the context of inference when causal variables are fully observed, as discussed in (Bareinboim 2011). This is an interesting topic for future explorations, and we do not consider this avenue here.“

W#1 “The contributions seem a bit overstated…”

Thank you for pointing out additional relevant works. Broadly, please refer to Section I in the author’s rebuttal and Fig.1 in the pdf for a global view of this setting. Specifically, we have added a comparison of these papers [1-4] in the Appendix and discuss them here. Input-Assumptions: Our paper operates in a nonparametric, non-Markovian setting. Among [1-4], only [1] considers a nonparametric approach but assumes sparsity in the graph and mechanism, unlike our work. All papers assume Markovianity. Input-Data: We accept various combinations of distributions (observational, soft, or hard interventions) across domains. [2] uses observational data across domains, [3] focuses on hard interventions in one domain, [4] considers interventions in one domain, and [1] includes interventions in one domain (and time series, which we do not discuss).

Output: We target partial disentanglement, whereas [2-4] aim for full or structured disentanglement (e.g., up to ancestors). Our approach encompasses these as special cases, with notable differences from [1] as described above.

W#1 (cont.) “lines 63-64, 172-197”

We reworded the sentences:

L63: “In terms of the expected output, rather than aiming for a full disentanglement, or structured disentanglement (such as from ancestors, or surrounding nodes), we consider a more relaxed type of disentanglement, such as in Ex. 6”.

L172-197 to compare to prior work: “Recent work has also considered a similar goal of generalized disentanglement [1]. Our work still differs from theirs in the following ways: i) [assumptions] we model a completely nonparametric non-Markovian ASCM, whereas [1] assumes sparsity and a Markovian ASCM … (see Appendix F for more details).”

[1] https://arxiv.org/pdf/2401.04890

[2] https://arxiv.org/pdf/2311.12267

[3] https://proceedings.mlr.press/v236/bing24a

[4] https://arxiv.org/abs/2406.05937

W#2 and L#1 “i) Knowledge of the causal structure…”

We recognize causal discovery/structure learning is an important problem within CRL and causal inference more broadly. Please check the author's rebuttal Section II for why we assume the causal graph and the justification about this assumption. To summarize, the causal diagram is a common assumption made in both causal inference and CRL literature since its inception. Causal inference as we study in AI arguably started in its modern form with the work of Judea Pearl in Biometrika 1995 where he introduced the causal graph assumption and developed the do-calculus, which highlights the tradeoff between assumptions and the types of conclusions one can draw. Also note that achieving identifiability given the diagram, as discussed in this work, is essential for learning the diagram. We note that inferential and learning components already appear together in the literature, albeit not yet in the context of representations, such as [1-4]. Adding a representation layer on top of these works is certainly a challenging but very compelling research direction.

[1] Zhang. "Causal Reasoning with Ancestral Graphs" (2008).

[2] Jaber et al. “Causal Identification under Markov equivalence: Calculus, Algorithm, and Completeness” (2022).

[3] Jaber et al. Identification of Conditional Causal Effects under Markov Equivalence (2019).

[4] Jaber et al. Causal Identification under Markov Equivalence: Completeness Results. (2019).

W#2 (cont.), Q#3 and L#1 “ii) …relies somewhat on the availability of “do” interventions…”

Thanks for mentioning this point and giving us the opportunity to clarify. To be clear, “do” interventions are not required by our work. In the text, “arbitrary distributions” implies that neither hard nor soft are necessarily required. Formally, do(T) = {} means there is no hard intervention. We have reworded L163-164 to be more clear: “When $\mathbf{I}^{(k)} = \{V_i^{\Pi^{(k)}, \{b\}}\}$, where $t$ is omitted, then the intervention is assumed to be soft. Thus when $do[I^{(k)}] = do[\{\}]$, it implies there are no variables with hard interventions in $\sigma^{(k)}$.”

In summary, CRID accepts as input any arbitrary combination of “do” and “soft” interventions, including “soft” interventions alone. For concreteness, Ex. 4 in the paper provides an example precisely that does not rely on any hard intervention.

W#3 “I think the authors tried to convey some outlook...”

The granularity discussion in the Introduction is to motivate why learning disentangled representations is important to the broader puzzle. The goal of this paper is foundational, and we show a more general setting where it is possible to learn disentangled representations. The simulation experiments corroborate the theory in settings that previously were thought that disentanglement could not be achieved. We also added a new image experiments shown in author's rebuttal V.

Q#1: It would be nice if the example in lines 24-25 could be expanded/elaborated on.

We include in the introduction the following elaboration:

"For example, images may capture a natural scene, where the causal variables are the objects within the scene while the pixels themselves are not the causal variables. Thus AI must disentangle the representations of latent causal variables given the pixels to capture correct causal effects for downstream tasks. Similarly, given written text describing this natural scene, the characters in the text are not the causal variables, but rather a low-level mixture of the underlying causal variables."

Also please refer to a motivating example in Appendix D.1 (pg 32).

Q#3

See response to W2(ii).

L#1

See W#1 and W#2 responses.

L#2 “This brings us one step closer to building robust AI…”

When we say “one step closer”, it does not mean we claim to solve the problem of “robust AI”, but rather the disentanglement results in the paper are a critical ingredient towards developing AI with causal awareness. We will reword the sentence.