Paper: On the Recoverability of Causal Relations from Temporally Aggregated I.I.D Data (ICLR-2024)

Reviewer comment (Writing):
Much of the paper is poorly explained, leading to issues with understandability. There are sentences in important sections that are hard to follow, particularly in the description of the intuition behind Definition 1 and Theorem 1.

Author response:
Sorry to cause vagueness. We modified Definition 1 and Theorem 1 and rewrite the explanation for Definition 1 in revised version.



Definition 1:

Functional Consistency means that there exists a function $\hat{f}$ such that the temporally aggregated data \(\overline{X}\) is described by $\overline{X} = \hat{f}(\overline{X}, e)$, where $e$ is a noise vector with independent components that are only related to the independent noises encountered in the process, denoted as $e_{2:k+1}$.

Explanation:

This definition implies that if functional consistency holds, then the aggregated data can at least be represented as a Structural Causal Model (SCM) in vector form, and the source of independent noise aligns with the underlying process. Please note that we allow the generative mechanism $\hat{f}$ to differ from the underlying causal function $f$. However, even with this allowance, achieving functional consistency in the nonlinear case remains challenging, as we will demonstrate in this section.

Theorem 1:

Consider a function $f(X, e)$ that is differentiable with respect to $X$. Define the following:

Statement 1: The function $f$ is of the form $f(x, e) = Ax + f_2(e)$ for some function $f_2$.

Statement 2: For any positive integer $k$, there exists a function $\hat{f}$ such that the functional equation 
$\frac{\sum_{i=1}^k f(X_{i}, e_{i+1})}{g(k)} = \hat{f}(\overline{X}, e)$
holds for any $X_i$, $e_i$, and any normalization factor $g(k)$, where $e$ is related only to $e_i$ for $i = 2, \dots, k+1$.

Statement 1 is a necessary condition for Statement 2.

Reviewer comment (Writing):
In the paragraph after Definition 1, the first sentence is a run-on (“Intuitively, functional consistency might suggest X=f(X,e), here we introduce a different function f, which allows for the possibility”), and the sentence after that sounds very strange (“This is because actually existing such f is difficult for nonlinear case”).

Author response:
We have corrected this in the revised version. We do need $\lim_{k \to \infty} g(k) = +\infty$ for the approximation part. However, when our analysis begins with the aggregation of the instantaneous model and considers any finite $k$, the choice of $g(k)$ does not matter at all.

For how large $k$ needs to be, please note that in our paper, we only require a large $k$ to diminish unaligned terms for approximation/alignment. We need to approximate the aggregation of the time-delay model as the aggregation of the instantaneous model. Once we start the analysis from an aligned model, the value of $k$ is no longer a concern. $k$ can be any finite integer larger or equal to 2. In the experiment in Appendix F, we discuss how large a $k$ value is sufficient for approximation. The experiments show that when $k = 30$, the results of the aggregation of the time-delay model and the aggregation of the instantaneous model are very similar, which is sufficient to make the approximation reasonable.

Reviewer comment (Writing):
Theorem 2 could benefit from more discussion around intuition. While Theorems 1 and 2 seem to be the main results of the paper (conditions for consistency for functional and CI-based methods), Theorem 1 suffers from the grammatical lack of clarity previously described, and Theorem 2 introduces new variables (alpha, beta, and gamma) that do not appear to be defined anywhere. These theorems would benefit from intuitive discussion and an editing pass for clarity.

Author response:
The definitions of the notations alpha, beta, and gamma have been provided within the content of Theorem 2 in the original manuscript.

Reviewer comment (Theory):
In Section 3.1, after equation 1, it is stated that g(k) can be any normalizing function such as g(k)=1 or g(k)=k. However, later in that section it is said that as k becomes large, g(k) approaches 0, allowing removal of that term under large k. This is unclear because g(k)=1 will not tend to 0, and g(k)=k will only tend to 0 for very large k, without any discussion about how large k must be for the paper's results to hold. The experiments use a maximum k of 50, which may not be large enough to assume g(k) → 0.

Author response:
We have corrected this in the revised version. We do need $\lim_{k \to \infty} g(k) = +\infty$ for the approximation part. However, when our analysis begins with the aggregation of the instantaneous model and considers any finite $k$, the choice of $g(k)$ does not matter at all.

For how large $k$ needs to be, please note that in our paper, we only require a large $k$ to diminish unaligned terms for approximation/alignment. We need to approximate the aggregation of the time-delay model as the aggregation of the instantaneous model. Once we start the analysis from an aligned model, the value of $k$ is no longer a concern. $k$ can be any finite integer larger or equal to 2. In the experiment in Appendix F, we discuss how large a $k$ value is sufficient for approximation. The experiments show that when $k = 30$, the results of the aggregation of the time-delay model and the aggregation of the instantaneous model are very similar, which is sufficient to make the approximation reasonable.

Reviewer comment (Writing):
While the dangers of temporal aggregation are discussed at a high level in the introduction, the only example given (effect of temperature on ice cream sales aggregated daily) does not seem to be problematic. This example does not illustrate spurious instantaneous relationships from aggregation. An example where such relationships can arise would help motivate the paper.

Author response:
“While the effects become instantaneous at the aggregate level, it's an effect that's consistent with the dynamics at the de-aggregated level and one that seems reasonable for causal discovery methods to detect.” No. The aggregation of an instantaneous causal model still suffers from the distortions caused by aggregation.

All theoretical results in our paper are about the aggregation of instantaneous effects. The damage to causal discovery performance is due not only to time-delay causal models but also to the aggregation itself. The primary focus of our paper is to investigate the effects caused by aggregation itself, underscoring how it can potentially distort causal relationships.

Reviewer comment (Experiments):
The experiments are weak. The first set is a simulation with a single two-variable model showing Direct LiNGAM issues as k increases (possibly due to the non-Gaussianity assumption). The second set uses a three-variable chain, fork, or collider structure to show how CI tests perform poorly without linearity, but does not contain any aggregation (using only two time steps) or a specific causal discovery algorithm, and focuses only on CI test performance. The absence of realistic data, models with more than 2–3 variables, more than 2 time steps, or any complex structures makes the evaluation feel very weak.

Author response:
Due to the page limit, most of the pages are dedicated to presenting theoretical results, leading to only a limited number of experimental results included in the main text. The methods we have tested encompass PC, FCI, GES, LiNGAM, and ANM. The settings cover different aggregation levels and scenarios, including linear/non-linear scenarios, with and without prior knowledge, and the aggregation of both time-delay and instantaneous models.

We have included 5 experiments in our original submission:

* applied widely-used causal discovery methods to aggregation data(Appendix C)
* functional consistency(Appendix D)
* conditional independence consistency(Appendix E)
* the impact of the k value(Appendix F)
* preliminary solution to the aggregation problem(Appendix G)

"doesn't actually employ a specific causal discovery algorithm": Please see Appendix C: we applied widely-used causal discovery methods(PC, FCI, GES) to aggregation data with 4 variables.

"data with more than 2-3 variables": Please see Appendix C,G

"across more than 2 time steps": Please see Appendix D,F

In the revised version, we have rewritten the first paragraph in the experiment section of the main text to clearly outline the experiments included, making it easier for readers to understand what experiments were conducted.

For the statement 'the second set of experiments doesn't actually contain any aggregation.': If the data represents the aggregation of an instantaneous model, the experimental results for different aggregated levels k large or equal to 2 will not show much difference. Moreover, an aggregation level of k=2 still qualifies as aggregation.

Reviewer comment (Experiments):
In Section 4, only chain-like, fork-like, and collider-like models are considered. Given the focus on temporal aggregation, cycle structures are likely (e.g., X{t-1} → Y{t}, Y{t-1} → Z{t}, Z{t-1} → X{t}), and these cause problems upon aggregation if the model cannot handle cyclic models. Such cases seem especially relevant to a discussion about temporal aggregation.

Author response:
We agree that the cyclic model is important in discussions about aggregation, and the analysis in Section 3 includes the cyclic model. However, Section 4, which focuses on conditional independence consistency, does not discuss the cyclic model. This is because, to our knowledge, the methods for discovering cyclic models are mainly FCM-based (discussed in Section 3) or score-based methods.

Regarding your example, when structure is cyclic, there is no conditional independence involving $\overline{X}, \overline{Y}, \overline{Z}$. Aggregation generally makes conditional independence conditionally dependent; it does not change a dependent relationship into an independent one. Therefore, we believe that aggregation has no influence on conditional independence in your example.

Reviewer comment (Theory):
I doubt that there is a technical flaw in the proof of Theorem 3, which is described in the Questions section.

Author response:
In the formula derivation, $s_Y$ is not a function of $y_1, \ldots, y_k$. $s_Y$ is a value given at the beginning of the derivation: $p_{S_{Z}|S_{Y},S_{X}}(s_{Z}|s_{Y},s_{X})$, which is completely independent of $y_1, \ldots, y_k$. For the relationship $S_Y = \sum_{i=1}^k Y_i$, yes, it does hold. It is naturally guaranteed by the integrands: $p_{S_{Y},S_{X},Y_{1:k}}(s_{Y},s_{X},y_{1:k})$ (the third equation) and $p_{Y_{1:k}|S_{Y},S_{X}}(y_{1:k}|s_{Y},s_{X})$ (the fourth equation): when $\sum_{i=1}^k y_i \neq s_Y$, both $p_{S_{Y},S_{X},Y_{1:k}}(s_{Y},s_{X},y_{1:k})$ and $p_{Y_{1:k}|S_{Y},S_{X}}(y_{1:k}|s_{Y},s_{X})$ are equal to $0$.

Reviewer comment (Theory):
The credibility of Corollary 1 is also in question, as it follows Theorem 3. This is mentioned in the Questions section.

Author response:
> Given Fig.2, how can $S_X$ independent with $Y_{1:k}$ given $S_Y$?

Without the causal faithfulness assumption, it is possible to have conditional independence that does not correspond to d-separation in the DAG. For examples of violations of faithfulness, you can refer to [1].

> In what situation does the "if" statement hold?

Corollary 2 then informs the reader under what conditions the 'if' statement holds.

The logic of Section 4.2 is as follows:

Remark 1: Given the structure of the DAG as shown in Figure 2, chain and fork structures cannot have conditional independence consistency unless there are coincidental circumstances (violations of faithfulness) in the causal functions within the DAG.

Then, where should the violations of faithfulness occur to gain consistency?

Theorem 2 and Corollary 1: The violations of faithfulness should appear in $\{S_X \perp\perp Y_{1:k}\mid S_Y\}$ or $\{S_Z \perp \perp Y_{1:k}\mid S_Y\}$ to gain the consistency.

Regarding the causal functions, how can such conditional independence be established?

Corollary 2: If the causal function between X and Y or the one between Y and Z is linear, then the corresponding conditional independence holds.

Reviewer comment (Theory):
While the name of Theorem 3 is "Necessary and Sufficient Condition for Conditional Independence Consistency," the content of "Theorem 3" only presents three equivalent statements that appear to just utilize the marginalization of the joint distribution. Please briefly explain how the three equivalent statements are necessary and sufficient conditions for conditional independence consistency if I misunderstand the point.

Author response:
You may be referring to Theorem 2. Sorry to cause vagueness. The full name of Theorem 2 should be "Necessary and Sufficient Condition for Conditional Independence Consistency of Chain and Fork Structure." We have modified the name to avoid misunderstanding in the revised version. The logic related to this is as follows:



* We consider three fundamental cases: chain, fork, and collider structures.
* Under typical assumptions of constraint-based causal discovery, collider structures naturally have conditional independence consistency, but chain and fork structures do not. (Remark 1)
* Then, under what conditions do chain and fork structures have conditional independence consistency? (See Necessary and Sufficient Condition Theorem 1, Sufficient Conditions Corollaries 1 and 2)

According to the definition of conditional independence consistency, chain and fork structures are supposed to satisfy $S_{X} \perp\perp S_{Z} \mid S_{Y}$(condition (i)), and this is equivalent to conditions (ii) and (iii).




The significance of conditions (ii) and (iii): Although conditions (ii) and (iii) are derived from condition (i) through some simple transformations, they are more specific requirements compared to condition (i). The integrands in (ii) and (iii) are always the multiplication of two parts, one part only related to components X and Y, and the other part only related to components Y and Z. This motivates us to conclude that only by meeting requirements for the substructure can we obtain consistency for the full structure. From this, we claim: partial linearity is sufficient.

Reviewer comment (Novelty):
As mentioned in the limitation section, the paper discussed the impact of temporal aggregation but did not offer any resolution. Consequently, it is challenging to assess the significance of this paper's contribution.

Author response:
Our contributions are summarized as follows:

What is the problem?

'Is the aggregation of causality recoverable?' This is a fundamental question because almost every real-world dataset can be seen to have some degree of aggregation. If recoverability cannot be guaranteed, then the practical application of nonlinear causal discovery would be largely doubtful. However, this issue is largely overlooked. Our paper aims to bring more attention to this issue within the causality community.

When and why does this issue occur?

Most of our work focuses on this question. We provide both theoretical analysis and simulation experiments to enhance the community's understanding of the nature of this issue.

How to solve it?

The theoretical results in our paper can motivate potential ways to solve this problem. We make an initial attempt in the original manuscript: if we divide the process of discovering the Markov equivalence class into two steps - skeleton discovery and determining v-structure - Remark 1 in our paper implies that we can always discover the correct v-structure with the skeleton prior given. The experimental results (see Appendix G) show that with a skeleton prior, the results of causal discovery are not damaged by aggregation. Therefore, we believe future work should focus on skeleton discovery or how to obtain the skeleton prior.

Reviewer comment (Evaluation):
Given Fig.2 and the discussion between 4.2 and Remark 1, the experimental results appear to illustrate that discussion rather than provide evidence for any significant claims in the main paper.

Author response:
Roughly speaking, the main results of Section 4.2 are: 'collider structures always maintain consistency,' and 'partial linearity is sufficient for chain and fork structures to achieve consistency.' Our experimental results regarding conditional independence consistency (Appendix E) do support these claims. The results show that chain and fork structures yield correct results in Linear+Linear, Linear+Nonlinear, and Nonlinear+Linear cases, but produce incorrect results in Nonlinear+Nonlinear cases. For colliders, correct results are obtained in all cases. It is important to note that our experiment randomly selects combinations of different nonlinear functions in each repetition, ensuring our experiments cover a wide range of nonlinear scenarios.

Reviewer comment (Theory):
In the proof of Theorem 3, the derivation from the third equation and the fourth equation below the sentence "On the left hand side (LHS):" is in question. The denominator of the third equation is \( p_{S_Y, S_X}(S_Y, S_X) \) and \( S_Y \) is a function of \( Y_1,\cdots, Y_k \). Given this, it appears inappropriate to directly insert \( p_{S_Y, S_X}(S_Y, S_X) \) into the integrand, as it is a function dependent on the integrated variables.

Author response:
In the formula derivation, $s_Y$ is not a function of $y_1, \ldots, y_k$. $s_Y$ is a value given at the beginning of the derivation: $p_{S_{Z}|S_{Y},S_{X}}(s_{Z}|s_{Y},s_{X})$, which is completely independent of $y_1, \ldots, y_k$. For the relationship $S_Y = \sum_{i=1}^k Y_i$, yes, it does hold. It is naturally guaranteed by the integrands: $p_{S_{Y},S_{X},Y_{1:k}}(s_{Y},s_{X},y_{1:k})$ (the third equation) and $p_{Y_{1:k}|S_{Y},S_{X}}(y_{1:k}|s_{Y},s_{X})$ (the fourth equation): when $\sum_{i=1}^k y_i \neq s_Y$, both $p_{S_{Y},S_{X},Y_{1:k}}(s_{Y},s_{X},y_{1:k})$ and $p_{Y_{1:k}|S_{Y},S_{X}}(y_{1:k}|s_{Y},s_{X})$ are equal to $0$.

Reviewer comment (Theory):
Even after ignoring the doubts in the proof of Theorem 3, Corollary 1 is questionable. Given Fig.2, how can \( S_X \) be independent with \( Y_{1:k} \) given \( S_Y \)? In what situation does the "if" statement hold?

Author response:
> Given Fig.2, how can $S_X$ independent with $Y_{1:k}$ given $S_Y$?

Without the causal faithfulness assumption, it is possible to have conditional independence that does not correspond to d-separation in the DAG. For examples of violations of faithfulness, you can refer to [1].

> In what situation does the "if" statement hold?

Corollary 2 then informs the reader under what conditions the 'if' statement holds.

The logic of Section 4.2 is as follows:

Remark 1: Given the structure of the DAG as shown in Figure 2, chain and fork structures cannot have conditional independence consistency unless there are coincidental circumstances (violations of faithfulness) in the causal functions within the DAG.

Then, where should the violations of faithfulness occur to gain consistency?

Theorem 2 and Corollary 1: The violations of faithfulness should appear in $\{S_X \perp\perp Y_{1:k}\mid S_Y\}$ or $\{S_Z \perp \perp Y_{1:k}\mid S_Y\}$ to gain the consistency.

Regarding the causal functions, how can such conditional independence be established?

Corollary 2: If the causal function between X and Y or the one between Y and Z is linear, then the corresponding conditional independence holds.

Reviewer comment (Theory):
Could you state the necessary and sufficient conditions for conditional independence consistency in words, given the three equivalent statements in Theorem 3?

Author response:
You may be referring to Theorem 2. Sorry to cause vagueness. The full name of Theorem 2 should be "Necessary and Sufficient Condition for Conditional Independence Consistency of Chain and Fork Structure." We have modified the name to avoid misunderstanding in the revised version. The logic related to this is as follows:



* We consider three fundamental cases: chain, fork, and collider structures.
* Under typical assumptions of constraint-based causal discovery, collider structures naturally have conditional independence consistency, but chain and fork structures do not. (Remark 1)
* Then, under what conditions do chain and fork structures have conditional independence consistency? (See Necessary and Sufficient Condition Theorem 1, Sufficient Conditions Corollaries 1 and 2)

According to the definition of conditional independence consistency, chain and fork structures are supposed to satisfy $S_{X} \perp\perp S_{Z} \mid S_{Y}$(condition (i)), and this is equivalent to conditions (ii) and (iii).




The significance of conditions (ii) and (iii): Although conditions (ii) and (iii) are derived from condition (i) through some simple transformations, they are more specific requirements compared to condition (i). The integrands in (ii) and (iii) are always the multiplication of two parts, one part only related to components X and Y, and the other part only related to components Y and Z. This motivates us to conclude that only by meeting requirements for the substructure can we obtain consistency for the full structure. From this, we claim: partial linearity is sufficient.

Reviewer comment (Writing):
What does the \( n \) mean in Theorem 1? Should it be \( k \)?

Author response:
Yes. Corrected in revised version.

Reviewer comment (Writing):
In definition 1, it said that "Moreover, the function \( \hat{f} \) has some degree of consistency with the underlying causal function \( f \)." The phrase "some degree of consistency" is ambiguous; could it be explained more precisely?

Author response:
Sorry to cause vagueness. We've made it clear in the revised version and rewrite the explanation for Def 1. 

Modified Def 1: 

Functional Consistency means that there exists a function $\hat{f}$ such that the temporally aggregated data \(\overline{X}\) is described by $\overline{X} = \hat{f}(\overline{X}, e)$, where $e$ is a noise vector with independent components that are only related to the independent noises encountered in the process, denoted as $e_{2:k+1}$.

Explanation:

This definition implies that if functional consistency holds, then the aggregated data can at least be represented as a Structural Causal Model (SCM) in vector form, and the source of independent noise aligns with the underlying process. Please note that we allow the generative mechanism $\hat{f}$ to differ from the underlying causal function $f$. However, even with this allowance, achieving functional consistency in the nonlinear case remains challenging, as we will demonstrate in this section.

Reviewer comment (Theory):
Could you explain how to get the equation \( f(X, e_1) + f(0, e_2) = f(0, e_1) + f(X, e_2) \) in the appendix?

Author response:
According to the definition of functional consistency:$f(X, e_1) + f(0, e_2)=\hat{f}(X,e)$,$f(0, e_1) + f(X, e_2)=\hat{f}(X,e)$, where e is only related to $e_1$, $e_2$.

Therefore, $f(X, e_1) + f(0, e_2) = f(0, e_1) + f(X, e_2)=\hat{f}(X,e_1+e_2)$

Reviewer comment (Theory):
Can you provide an explanation of how the second equality in \( f(X_1,e_1)+f(X_2,e_2) = f_1(X_1)+f_1(X_2)+f_2(e_1)+f_2(e_2) = f_1(X_1 +X_2)+f_2(e_1)+f_2(e_2) \), is derived in the appendix?

Author response:
$f(X_1,e_1)+f(X_2,e_2) = f_1(X_1)+f_1(X_2)+f_2(e_1)+f_2(e_2)$ is directly derived from the result we obtin before: $f(X,e)=f_1(X)+f_2(e)$

And $f(X_1,e_1)+f(X_2,e_2)=f_1(X_1 +X_2)+f_2(e_1)+f_2(e_2)$ is derived from 
the definition of functional consistency.

Reviewer comment (Theory):
There is no \( \hat{f} \) appearing in the proof of Theorem 1; is it unrelated?

Author response:
It appears in the detailed proof.

Reviewer comment (Writing):
Section 3 is confusing, mainly due to the fact that some terms are a bit vague. While "functional consistency" is defined, it uses words like "compatible" and "degree of consistency" that are vague and confuse the definition. For example, does compatible seem to mean that the aggregated data is a function of the aggregated data and a noise term? Furthermore, does $\hat{f}$ having some degree of consistency with underlying $f$ mean that they are the same function? As a result of this vagueness, the rest of this section is unclear. A clearer connection to why this consistency is needed would also help the reader a lot.

Author response:
Sorry to cause vagueness. We modified Definition 1 and Theorem 1 and rewrite the explanation for Definition 1 in revised version.



Definition 1:

Functional Consistency means that there exists a function $\hat{f}$ such that the temporally aggregated data \(\overline{X}\) is described by $\overline{X} = \hat{f}(\overline{X}, e)$, where $e$ is a noise vector with independent components that are only related to the independent noises encountered in the process, denoted as $e_{2:k+1}$.

Explanation:

This definition implies that if functional consistency holds, then the aggregated data can at least be represented as a Structural Causal Model (SCM) in vector form, and the source of independent noise aligns with the underlying process. Please note that we allow the generative mechanism $\hat{f}$ to differ from the underlying causal function $f$. However, even with this allowance, achieving functional consistency in the nonlinear case remains challenging, as we will demonstrate in this section.

Theorem 1:

Consider a function $f(X, e)$ that is differentiable with respect to $X$. Define the following:

Statement 1: The function $f$ is of the form $f(x, e) = Ax + f_2(e)$ for some function $f_2$.

Statement 2: For any positive integer $k$, there exists a function $\hat{f}$ such that the functional equation 
$\frac{\sum_{i=1}^k f(X_{i}, e_{i+1})}{g(k)} = \hat{f}(\overline{X}, e)$
holds for any $X_i$, $e_i$, and any normalization factor $g(k)$, where $e$ is related only to $e_i$ for $i = 2, \dots, k+1$.

Statement 1 is a necessary condition for Statement 2.

Reviewer comment (Experiments):
I would have liked to see more experiments with various models. For example, only the LiNGAM algorithm is tested, what about ANM? Does the performance degrade for methods that do not make any functional assumptions (e.g. kernel based methods like KCDC [1])?

Author response:
We included experiments on Additive Noise Models (ANM) and methods that do not impose any functional assumptions, such as PC with Kernel-based Conditional Independence (KCI), in our original submission. Please refer to Appendices C, D, and E for details. Figure 8 presents the experimental results for ANM. In our experiments with non-linear models, we consistently employed general score functions for score-based methods and the Kernel Conditional Independence test for constraint-based methods, which are designed to be effective for any non-linear functions.

Reviewer comment (Theory):
Why does the infinite K only consider the linear model? Is this due to the previous section? This is not clear.

Author response:
Why not consider infinite k for nonlinear model?

An infinite k introduces several mathematical challenges in terms of assumption or definition. For instance, the convergence of infinitely aggregated data heavily depends on functions f and g. If f is linear and g(k) = k, then the aggregated data loses randomness due to the law of large numbers. We require $g(k) \sim O(\sqrt{k})$ to maintain a converged variance. Additionally, we must assume data centralization to avoid infinite expectations. If f is nonlinear,(a) an appropriate $g$ that fits $f$ is necessary to maintain a well-defined variance; (b) different variables in the aggregated data may require different $g$ functions; and (c) for some nonlinear $f$, the variance of aggregated data may never converge. In practical scenarios, it is unrealistic to expect that observed data will perfectly align with these requirements of suitable $g$ and centralization. Furthermore, even if we overcome these assumption and definition issues, analyzing the central limit of non-independent and non-identically distributed variables in nonlinear case remains a complex problem in probability theory.

In summary, an infinite k is unnecessary and impractical in real-world scenarios, and addressing convergence issues detracts from our work's fundamental question: "Is aggregation of causality recoverable?". 

Why consider an infinite k for the linear model?

The main purpose is to alert the reader to a special issue caused by aggregation: it transforms any distribution into a Gaussian. This issue causes non-Gaussian based causal discovery methods to collapse, even when the functional form itself is not disrupted by aggregation.

Reviewer comment (Theory):
Is there a way to validate the assumptions in Theorem 2?

Author response:
Theorem 2 itself does not introduce additional assumptions. If you are referring to the assumptions mentioned in Remark 1, the causal Markov condition and faithfulness assumption are common in causal discovery. To validate the faithfulness assumption, please refer to [1]. In our paper, this means that, given the structure of the DAG as shown in Figure 2, chain and fork structures cannot have conditional independence consistency unless there are coincidental circumstances in the causal functions within the DAG. Remarks 2 and 3 further explain under which specific coincidental conditions the faithfulness assumption might be violated and how chain and fork structures could become conditionally independence consistent.


[1] Zhang, Jiji, and Peter Spirtes. "Detection of unfaithfulness and robust causal inference." Minds and Machines 18 (2008): 239-271.

Reviewer comment (Writing):
VAR (1) is typically accepted as a linear model. It is not clear if f() covers another model. The terminology seems inconsistent to the notation.

Author response:
In Section 3.1, our model definition is similar to the definition of a Nonlinear Vector Autoregressive (NVAR) model as described in [1]. However, in our paper, the function f can be either linear or nonlinear. Therefore, we refer to it as a 'general VAR(1)' in Section 3.1. To clarify, we have added some text in the revised version to emphasize that the VAR discussed in our paper encompasses both linear and nonlinear VAR models.

Reviewer comment (Writing):
The number of variables seems to be s but is not clearly defined.

Author response:
Your understanding is correct. We have added additional text in the revised version to clearly define the number of variables as s.

Reviewer comment (Writing):
What "k is large" means is not clear.

Author response:
In our paper, we present the sufficient and necessary conditions for two types of consistency in three steps:

*   Define the original model as a time-delay causal model (VAR) and define its temporal aggregated version.
*   Approximate the aggregation of the time-lag causal model as the aggregation of the instantaneous causal model (at the end of section 3.1 for functional consistency as you mentioned, and section 4.1 for conditional independence consistency).
*   Present the sufficient and necessary conditions of consistency for the aggregation of the instantaneous model, for any finite positive k, and any choice of g(k).

Among these steps, we only apply the large k condition in the approximation step. Yes, from a mathematical perspective, "k is infinity" would imply a perfect approximation. However, we write "k is large" for the following reasons:


1. According to our experiments, a large but not extremely large(infinity) k is enough to make sure the approximation reasonable. Please see Figure 9 in Appendix F. When k=30, the results of aggregation of time-delay model and the one of instantaneous model become very similar. 

2. An infinite k introduces several mathematical challenges in terms of assumption or definition. For instance, the convergence of infinitely aggregated data heavily depends on functions f and g. If f is linear and g(k) = k, then the aggregated data loses randomness due to the law of large numbers. We require $g(k) \sim O(\sqrt{k})$ to maintain a converged variance. Additionally, we must assume data centralization to avoid infinite expectations. If f is nonlinear,(a) an appropriate 
g that fits f is necessary to maintain a well-defined variance; (b) different variables in the aggregated data may require different g functions; and (c) for some nonlinear f, the variance of aggregated data may never converge. In practical scenarios, it is unrealistic to expect that observed data will perfectly align with these requirements of suitable g and centralization. Furthermore, even if we overcome these assumption and definition issues, analyzing the central limit of non-independent and non-identically distributed variables remains a complex problem in probability theory.

In summary, an infinite k is unnecessary and impractical in real-world scenarios, and addressing convergence issues detracts from our work's fundamental question: "Is aggregation of causality recoverable?". Therefore, we use "large k" in the paper. In the revised version, we have included references to experimental results when mentioning "large k" for additional clarity.

Reviewer comment (Theory):
$g(k)$ is defined as "any normalization function like g(k)=1". Then, how do you guarantee that $(X_1 -X_{k+1})/g(k)$ vanishes?

Author response:
We have corrected this in the revised version. Our intent is to convey that in Step 3, when we begin with the aggregation of the instantaneous model and consider any finite k, the choice of g(k) does not matter at all.