Firstly, we would like to sincerely thank you for the considerable time and effort invested in reviewing our paper. Your insightful comments and suggestions have significantly helped us enhance the clarity and readability of our manuscript. In response to your valuable feedback, we have carefully revised and clarified our paper:

Q1: What do the symbols and reference table mean in Figure 3?

We add a detailed explanation for Figure 3 to clarify the notation.

The set of symbols (up/down triangles, diamonds, and circles) represents the four types of conditional independence (CI) relationships listed in the first column of the reference table in Figure 3(f). Each symbol corresponds to a specific CI pattern. The green check indicates that the CI relationship is satisfied and is marked with corresponding black symbols, while the red X indicates that the CI relationship is not satisfied and is marked with white symbols. For example, a black upward triangle indicates that $I_X \perp Y \mid S$ holds (i.e., I_X and Y are conditionally independent given S), whereas a white upward triangle indicates that $I_X \not\perp Y \mid S$ (i.e., they are dependent). We hope this clarification makes the figure more accessible and easier to interpret.

Taking Figure 3(a) as an example, the structure exhibits conditional dependence $Y \not\perp I_X \mid S$, which is represented by the white upward triangles (△) in the reference table. The conditional independence $Y \perp I_X \mid X, S$ is shown in (a) when $X$ is given and is represented by black downward triangles (▼). These CI patterns for (a) are summarized using the corresponding symbols displayed below the causal graph.

Q2: What do the grey-shaded columns (c) and (e) mean in Figure 3?

The grey shading in columns (c) and (e) indicates that both cases yield identical conditional independence (CI) patterns. Specifically, as discussed in Remark 3.4, the selection process introduces additional conditional dependencies into the observed data, resulting in indistinguishable CI patterns for these two scenarios. Consequently, GISL is unable to differentiate a scenario involving only selection bias (c) from one involving both selection bias and a causal relationship (e). However, this limitation does not impact GISL's capability to identify the presence of selection bias itself, even though the precise causal structure remains ambiguous. This point is formalized in Theorem 3.7. The grey shading thus serves to distinguish scenarios that GISL can uniquely identify (white shading: cases a, b, and d) from those it cannot (grey shading: cases c and e, which are indistinguishable).

Please let us know if you have any questions about Figure 3. We would be delighted to discuss this further and deeply appreciate your thoughtful feedback.

Q3: Technical terminologies & terms are Undefined: mutilated DAG, knockout, knockup, and knockdown.

We sincerely appreciate your careful reading and thoughtful comments. We are grateful for the opportunity to clarify these terms and improve the clarity of our presentation.

To enhance readability, we have added explicit definitions for commonly used terms introduced in the preliminaries, including mutilated DAG (widely used in interventional causal discovery), perturbation indicator (Definition 1 on page 6 of [1]), intervention targets (third paragraph on page 4 of [2]), and hard/soft interventions (illustrated in the figures on pages 4 and 6 of [3]).

Why introduce Mutilated DAG? Mutilated DAG represents the graphical structure for hard intervention (illustrated in Figure 2(a) and described in Lines 102–104), while Augmented DAG is introduced in Lines 105–108 to represent soft interventions. Since our framework accommodates both types of interventions, it is necessary to present both forms of DAG representations.

Logical progression in introducing terms. We have aimed to reduce cognitive load for the reader by introducing terminology in a gradual, structured manner. In our preliminaries (Lines 100–102), we first introduce gene perturbation methodologies—gene knockout and transcriptional modulation (gene knockup or knockdown)—which correspond to hard and soft interventions, respectively, in the causal inference framework. We then describe how these two types of interventions are represented graphically. Specifically, a mutilated DAG refers to the graph obtained from a causal DAG by removing all incoming edges to the intervened variables, effectively breaking the causal dependence of those nodes on their parents. Similarly, the augmented DAG is introduced in Lines 105–108 to represent soft interventions, and is illustrated in Figure 2(b).

[1] Varieties of causal intervention. PRICAI 2004.

[2] Characterization and greedy learning of interventional Markov equivalence classes of directed acyclic graphs. JMLR 2012.

[3] Interventions and causal inference. PSA 2006.

Q4: $T_k$ in the sentences is Unparsable.

Each $T_k$ denotes the target variable set for the k-th intervention. We have added the definition of $\mathcal{T} = \{T_1, \ldots, T_K\}$ in the revised version.

Q5: Why GISL performs not so well regarding the recall?

As you pointed out, GISL shows a little bit lower recall in identifying regulatory relationships under selection bias. This is because, as discussed in Q2 and Figure 3, in theory, it is challenging to distinguish (c): selection bias and (e): causal + selection bias -- they share the same CI patterns (all conditionally dependent). As a result, our GISL categorizes both cases (c) and (e) as selection bias and thus shows lower recall when identifying regulatory relationships (causal relations).

On the other hand, the baseline algorithm, e.g., GIES, exhibits higher recall, but this does not necessarily indicate better performance. The key issue is that GIES is unable to account for selection bias, which introduces spurious dependencies into the data. When GIES relies on a score function to detect conditional dependencies, it may mistakenly interpret these spurious associations as causal relationships. As a result, the algorithm tends to include additional edges regardless of whether true causal relationships exist, leading to inflated recall but low accuracy.

We hope our responses have clarified the key points and improved the overall clarity of the paper. If any aspects remain unclear, we would be happy to provide further explanation. Thank you again for your thoughtful review and the considerable effort you invested.

Thank you for the follow-up. Our core idea is that different causal graphs lead to different patterns of which variables are independent or dependent. We can use conditional independence (CI) tests to check these patterns. By testing the CI relations between gene-perturbation indicators ($I_X$ and $I_Y$ in Figure 3) and the observed variables ($X$ and $Y$ in Figure 3), some fundamental causal structures in the presence of selection bias and latent confounders can be uniquely identified.

In prose:

In Step 3 of Algorithm 1, for any two variables ($X_i, X_j$) (corresponding to ($X, Y$) in Figure 3), we apply conditional independence (CI) tests to see how each perturbation variable ($I_i, I_j$) relates to the other observational variable given appropriate conditioning. For example, we can observe the following CI relations hold from the data $\mathcal{J} =$ {$I_{i} \not\perp X_{j} \mid \mathcal{S}$, $I_{i} \perp X_j\mid X_i, \mathcal{S}$, $I_{j} \perp X_i \mid \mathcal{S}$, $I_{j} \not\perp X_i \mid X_j, \mathcal{S}$}. We then compare these four independence/dependence results to the CI relations revealed by the causal graph of Figure 3(a). If and only if those four outcomes line up perfectly with what Figure 3(a) prescribes, we trigger the “If $\mathcal{J}==Figure 3(a)$” condition and conclude $X_i \to X_j$.

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I thank the authors for their responses. I am still confused about how to parse the main algorithm (Algorithm 1) using Figure 3. For example in step 3, can the authors explain in prose exactly when the condition "If $\mathcal{J}== \operatorname{Figure 3a}$" is triggered?

Edit: correcting LaTeX formatting.

Re: "To further improve the figure, we would be grateful if you could elaborate on which aspects remain difficult to parse." In general, I think the current way the figure is used to define the method leaves too many steps for the reader to reconstruct on their own. For example, in their algorithm, the authors define $\mathcal{I}$ as a set of 4 independence tests, and then reference logical conditions like "If $\mathcal{I}==\operatorname{Figure}\ 3(a)$," where Figure 3(a) displays two in/dependence relationships. As the authors explain in their reply above, what they intend is for the reader to use symmetry to deduce that Figure 3(a) IMPLICITLY references 4 in/dependence relationships obtained by exchanging $X_i$ and $X_j$. This is the kind of detail that is obvious to the authors, but unclear to readers, especially given the fact that even understanding that Figure 3(a) encodes 2 in/dependence relationships is difficult because it is communicated with black/white triangles, diamonds, and circles rather than explained anywhere in prose. The novel method in Algorithm 1 is a key contribution of the paper. Given that readers struggle to understand Algorithm 1's statement without the ability to ask specific questions of the authors (as evidenced by multiple reviewers' confusion), I think the paper offers limited benefits to the community in its current form.

I suggest the authors either restructure Figure 3 so that it is correct and readable or provide precise pseudocode for defining Algorithm 1.

Edit: formatting quotations in markdown.

Dear Reviewer 7xy2,

We appreciate your time and effort. Your patience and feedback on Figure 3 were invaluable in guiding revisions that substantially improve the paper’s clarity. We have updated Figure 3 with an extended caption and an illustrative example to enhance readability, and we hope these changes address your concerns. Thank you again for your engagement and constructive input.

Sincerely,

Authors of #17977

Thank you for the time dedicated to reviewing our paper. Your insightful comments and valuable feedback have greatly improved its clarity. Following your suggestions, we have carefully revised the manuscript. Please see our point-by-point responses below.

Q1: "I don't fully understand the symbols in Figure 3. Could you please explain that in more detail with the reference table?"

We have added a detailed explanation of Figure 3 with the reference table to improve clarity.

We begin by explaining the reference table in Figure 3(f). The first column lists key conditional independence (CI) patterns used to distinguish different causal structures. Each pattern is represented by a specific symbol—such as upward/downward triangles, diamonds, and circles—corresponding to different types of CI relations (e.g., between $I_X$ and $Y$).

A green check indicates that a CI relation holds, shown by a black symbol. Conversely, a red X indicates that the CI relation does not hold (i.e., conditional dependence), represented by a white symbol. This symbolic representation helps visually summarize the CI patterns associated with each structure in Figure 3.

Taking Figure 3(a) as an example, the structure exhibits conditional dependence $Y \not\perp I_X \mid S$, which is represented by the white upward triangles (△) in the reference table. The conditional independence $Y \perp I_X \mid X, S$ is shown in (a) when $X$ is given and is represented by black downward triangles (▼). These CI patterns for (a) are summarized using the corresponding symbols displayed below the causal graph.

Q2: "The paper does not compare with some existing work with a very similar setting (such as Dai et al. 2025)."

Thank you for pointing out this related work. Following your suggestion, we have added a dedicated discussion in the revised manuscript. In short, while the two methods appear similar at a high level, the key assumptions and problem formulations are in fact quite different. Specifically,

Assumptions of CDIS (Dai et al.):

CDIS models selection as a one-time process that occurs only before intervention and becomes inactive afterward, using an interventional twin graph. Under this assumption, the distribution satisfies $P(Y \mid \text{do}(X) = x, S) = P(Y \mid S)$, meaning that after an intervention on $X$ is applied, the selection variable S no longer influences $Y$.

Assumptions of GISL:

In contrast, our work focuses on biological constraints in genes—a form of inherent selection bias that exists both before and after intervention. These constraints come from basic biological rules, such as essential gene functions or conditions needed for a cell to survive, which do not disappear even when a gene is perturbed. As a result, selection continues to affect the system even after $\text{do}(X)$, violating the CDIS assumption. In our setting, the correct relation is $P(Y \mid \text{do}(X) = x, S) \neq P(Y \mid S)$. This fundamental difference means that CDIS cannot account for such persistent selection effects, leading to poor performance in settings where biological constraints are present.

Moreover, in light of your suggestion, we have conducted additional experimental comparing GISL and CDIS (Dai et al) on synthetic datasets (number of variables $|\mathcal{X}|$=15, sample size n = 1500, soft intervention). The results are as below:

We demonstrate that, within the context of specific biological constraints, our method achieves superior performance due to the correct identification results.

Q3: The problem setting of the selection variable is vague. The interpretation of the selection variable and perturbation indicator is unclear.

Thanks for your feedback, which has certainly improved the readability of our paper.

Problem Setting:

• Goal: Identify regulatory relationships, latent confounders, and selection bias (biological constraints) in gene expression data.

• Assumption: Gene expression is affected by latent confounders and persistent selection bias, as supported by biological evidence (see Lines 29–31 and 33–39).

• Solution: We formulate this as a causal discovery problem under latent confounding and selection bias. By leveraging both observational and gene perturbation (interventional) data, GISL identifies regulatory relationships, latent confounders, and selection bias by capturing differences in perturbation symmetry and intervention effects (see Section 3.1).

Clarification on the Selection Variable and Perturbation Indicator:

As described in Lines 85–88, the selection variable S imposes a constraint where only samples with S = 1 are observed; data with S = 0 is unobservable. This formulation is formally discussed in the "M Bias" section on page 5 in [1] and has been widely adopted in subsequent works [2–5]. Thus, our analysis is restricted to the selected subpopulation.

Importantly, selection bias and intervention are independent—selection can affect both observational and interventional data. The intervention is designed and classified in Sections 4 and 5 (Pages 5-7) in [6]. The GISL is designed to identify selection bias, not eliminate it. To this end, we introduce a perturbation indicator into an augmented causal graph, which enables joint modeling of observational and interventional distributions. This allows CI tests to capture both perturbation symmetry and perturbation effects, facilitating the identification of regulatory relationships, latent confounders, and selection bias.

[1] Quantifying Biases in Causal Models: Classical Confounding vs Collider-Stratification Bias. Epidemiology, 2003.

[2] Causal discovery from data in the presence of selection bias. PMLR, 1995.

[3] Structural approach to selection bias. Epidemiology, 2004.

[4] Sample selection bias as a specification error. Econometrica: Journal of the Econometric Society, 1979.

[5] Endogenous Selection Bias: The Problem of Conditioning on a Collider Variable. Annual Review of Sociology, 2014.

[6] Variables of causal intervention. UAI, 2004.

Q4: "What are the terms like $[I_i, X_j \mid S]$? Are those indicator variables for the conditional independence?"

Thanks for pointing this out. $[I_i, X_j \mid S]$ denotes that we need to capture the CI relation between the intervention indicator $I_i$ and the variable $X_j$, given the selection variable $S$.

Q5: "How scalable is the proposed method? How sensitive is the proposed method to the size of datasets?"

Scalability is a key consideration in the design of GISL. GISL is built to handle large-scale gene expression datasets, as demonstrated in our real-world experiments (Section 4.3 and Appendix F.1, Lines 760–786), where GISL was applied to datasets with over 5,000 observed genes (variables) and more than 100 perturbed genes. In comparison, the original nonparametric CI test KCI [8] was only evaluated on graphs with 5 variables, and baseline methods such as [7] consider at most 20 variables. These comparisons highlight the practical scalability of GISL beyond prior work.

In terms of sensitivity to dataset size, our experiments on synthetic data (10–25 variables, 500–2,000 samples) and real data show that GISL maintains stable performance across varying sample sizes. This is enabled by two design choices: (1) the use of skeleton discovery via FGES, and (2) the application of pairwise CI tests based on limited conditioning sets, both of which help ensure computational efficiency as dataset size grows.

[7] Permutation-Based Causal Structure Learning with Unknown Intervention Targets. UAI, 2020.

[8] Kernel-based Conditional Independence Test and Application in Causal Discovery. UAI, 2011.

Q6: "The option in Algorithm 1 $\mathcal{J}==(c)$ and $\mathcal{J}==(d)$ seems to be flipped and inconsistent with the caption of Figure 3."

Thanks for pointing this out. The options for $\mathcal{J}==(c)$ and $\mathcal{J}==(d)$ in Algorithm 1 were inadvertently flipped. We have corrected this error to ensure consistency between the algorithm and the figure.

Q7: "Do the terms regulatory relations and causal relations mean the same thing in this context? The paper seems to use these terms interchangeably."

Yes, in this context, regulatory relation and causal relation refer to the same concept. We primarily use the term regulatory relationship to align with the gene regulatory network inference (GRNI) context, while causal relationship is used when introducing the formal model to remain consistent with standard causal discovery terminology. This interchangeable use is clarified in Lines 79–81 and is consistent with the GRNI definition in Line 22, where regulatory interactions are modeled as causal effects.

If you have any questions, please do not hesitate to let us know. We would greatly appreciate your feedback and are happy to discuss further.

Dear Reviewer 7UVK,

Thank you very much for your time and effort in reviewing our manuscript. We have incorporated your suggestions to clarify the problem setting and the Figures, and demonstrate GISL’s scalability.

As the discussion deadline approaches, please let us know if you have any further concerns, and we will be happy to discuss further.

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Yours sincerely,

Authors of Submission 17977

Dear Reviewer 7UVK,

Thank you very much for your valuable feedback, which has greatly contributed to improving the clarity and readability of our paper.

As the discussion deadline approaches (with only 48 hours remaining), we would be grateful for the opportunity to address any remaining concerns you may have. If there are any outstanding questions or points that require further clarification, we would be more than happy to respond.

With sincere appreciation,

Authors of Submission 17977

Thank you very much for your encouragement. We are pleased that many of your concerns have been addressed and that Figure 3 is now clearer. We sincerely appreciate your recognition of the additional experiments and revisions, and we are grateful for your willingness to improve the score. Your feedback has been invaluable in strengthening the paper.

We first sincerely appreciate your insightful suggestions and valuable feedback, which have enabled us to strengthen our method through a more comprehensive evaluation. We have carefully considered the concerns you raised and provide our detailed responses below.

Q1: Robustness of Simulation Experiments.

In light of your suggestion, we have conducted comprehensive experiments to evaluate the robustness of GISL as follows:

Noise level

We conducted experiments to evaluate the robustness of GISL under varying levels of Gaussian noise variance, where the noise mean was randomly sampled from a uniform distribution U(0, 4). Experimental results demonstrate that GISL’s performance improves as Gaussian noise variance increases, because larger distributional changes are more easily detected by conditional independence (CI) tests, yielding more accurate inference.

Experimental setting: hard intervention, 15 variables, 1500 samples.

Experimental setting: hard intervention, 15 variables, 500 samples.

SCM complexity

We conducted experiments to evaluate the robustness of GISL under varying levels of structural complexity, quantified by the number of edges in the graph. Experimental results indicate that GISL’s performance experiences a slight decline as graph structural complexity increases.

Experimental setting: hard intervention, 15 variables, 1500 samples.

Experimental setting: hard intervention, 15 variables, 500 samples.

The number of genes

We conducted experiments to evaluate the robustness of GISL under a varying number of genes. Experimental results indicate that GISL’s performance remains stable and even improves as the gene count increases, owing to the greater sparsity of the corresponding graphs.

Experimental setting: hard intervention, 1500 samples.

Experimental setting: hard intervention, 15 variables, 500 samples.

For larger datasets, please see Figures 11 and 13, which present real-world experiments on gene expression data encompassing over 5,000 observed genes and more than 100 perturbed genes.

Q2: Realism of the Synthetic Data Generator.

In light of your suggestions, we consider biological simulators, which simulate the regulatory relationship by using the normalised-Hill differential equations and gene expression is a truncated distribution. Experimental results are as follows:

Experimental setting: hard intervention, 15 variables, 1500 samples.

Experimental setting: hard intervention, 15 variables, 500 samples.

Experimental results demonstrate stable performance on data generated by SCM-driven biological simulators.

Q3: Need for More Real-World Evidence.

According to your suggestion, we have added more real-world evidence to support the results of GISL as follows:

Human Lung Epithelial Cells

GISL reports that there exist biological constraints between gene CNN1 and CDKN1A. CNN1 (Calponin 1) is normally low in healthy lung epithelium but markedly increases during epithelial–mesenchymal transition or fibrotic remodeling. [1] CDKN1A (p21) is a cell-cycle inhibitor that surges under DNA damage or stress (e.g., p53 activation), causing growth arrest or senescence; indeed, lung injury models and fibrotic lungs (IPF) show abnormally high p21 in alveolar epithelial cells [2]. These observations imply that when lung epithelial cells are pushed toward a damaged or transitioning state, both CNN1 and p21 tend to rise, which suggests that the cell may need to keep its combined activity in check to maintain normal function. Excessive co-elevation of CNN1 (indicating mesenchymal/fibrotic shift) and CDKN1A (indicating cell-cycle arrest) could drive cells into an unsustainable state (senescence or fibrosis), so a balance in their expression is likely required to preserve viability and tissue homeostasis. This suggests a form of biological constraint whereby lung epithelial cells limit the concurrent upregulation of CNN1 and CDKN1A to avoid tipping into pathological remodeling or loss of proliferative capacity.

K562 Lukemia Cells

GISL reports that there exist biological constraints between gene ELF1 and gene GABPA. ELF1, an ETS-family transcription factor, regulates hematopoietic differentiation and immune genes; in K562 cells, it controls the G₁/S transition via CDKN1A (p21), with overexpression inducing apoptosis and underexpression promoting unchecked proliferation [3]. GABPA, a nuclear respiratory factor, orchestrates mitochondrial biogenesis and metabolism through the PI3K/Akt/mTOR axis; its depletion triggers G₀/G₁ arrest, and its binding affinity (pKD) and phosphorylation response (pEC₅₀) underscore a requirement for dosage matching downstream signaling intensity [4]. Both factors converge on stress and apoptotic pathways—GABPA targets (Caspase-9, Bcl-2) overlap with ELF1-regulated DNA-damage genes—and respond coordinately to PI3K/Akt perturbations (e.g., LY294002) and autophagy induction (e.g., in K562/ADM cells), suggesting K562 cells impose a constraint on ELF1 and GABPA expression to maintain proliferative homeostasis and survival [5,6].

[1] Alveolar epithelial cells express mesenchymal proteins in patients with idiopathic pulmonary fibrosis. American Journal of Physiology-Lung Cellular and Molecular Physiology 2011.

[2] P21Waf1/Cip1/Sdi1 and p53 expression in association with DNA strand breaks in idiopathic pulmonary fibrosis. American journal of respiratory and critical care medicine 1996.

[3] SIRT1 is a critical regulator of K562 cell growth, survival, and differentiation. Expression Cell Research 2016.

[4] Decrypting drug actions and protein modifications by dose-and time-resolved proteomics. Science 2023.

[5] A genetic map of the response to DNA damage in human cells, Cell 2020.

[6] Elf1 promotes transcription-coupled repair in yeast by using its C-terminal domain to bind TFIIH. Nature Communication 2024.

Dear Reviewer yzJf,

Thank you for taking the time to review our paper and for your helpful suggestions. Your feedback on evaluating the robustness of GISL and real-world extension was constructive in improving our paper. We hope that our responses have addressed your concerns.

Due to the limited time for rebuttal discussion, we look forward to receiving your feedback at your earliest convenience and the opportunity to respond to it.

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Yours sincerely,

Authors of Submission 17977

We sincerely thank you for the time dedicated to reviewing our paper and valuable feedback. Please find the response to your comments and questions below.

Q1: Why does the paper explicitly look at gene regulatory networks when the method is more generally for causal discovery with interventional data?

Thank you for this insightful suggestion. Indeed, our method is broadly applicable to general causal discovery, but we chose the gene regulatory network inference (GRNI) framework because it was inspired by a real‐world challenge—the pervasive yet often overlooked selection (survival) bias in biological data—and directly addressing this issue offers immediate value for GRNI and beyond.

Specifically, we focus on GRNI because:

1. GRN inference is a classical causal discovery task with strong biological significance, supported by well-established gene perturbation technologies that provide abundant interventional data.
2. Despite being a classical problem, modeling gene expression from a causal perspective remains challenging due to latent confounding and overlooked selection bias, making it a meaningful testbed for our method.

Q2: Related works should be in the main paper and could be more complete. The difference between GISL and related works should be clear.

In the revised version, we will move the discussion of related work into the main paper and expand it accordingly. Due to limited space, we list representative causal discovery methods for comparison in settings and highlight the novelty and difference compared with related works as follows:

In comparison, GISL is novel in utilizing interventional data to identify relationships, latent confounders, and selection bias in a general setting without relying on parametric assumptions, and is specifically tailored to handle the biological selection constraints that arise in gene regulatory networks.

[1]. Permutation-Based Causal Structure Learning with Unknown Intervention Targets. UAI 2020.

[2]. Iterative causal discovery in the possible presence of latent confounders and selection bias. NeurIPS 2021.

[3]. A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables. ICLR 2023.

[4]. Characterization and Learning of Causal Graphs with Latent Variables from Soft Interventions. NeurIPS 2019.

[5]. Identifying selection bias from observational data. AAAI 2018.

Q3: The statements about the identifiability in Theorem 3.7 could be more accurate. There are certain cases that are not identifiable (like figure 7(d) as mentioned in Appendix D).

We agree with you. We have revised the statement in Theorem 3.7 to make it more accurate as follows:

The causal relationships, selection processes, and latent confounders are identifiable up to the equivalence classes of four types of CI patterns in Figure 3 among the variables that are subject to intervention. The presence of selection and latent confounders (existing or not) is identifiable among these variables.

Additionally, as shown in Figure 7(d) in Appendix D, certain structures such as $X \rightarrow L \leftarrow Y$ differ in causal structure but yield the same Conditional Independence (CI) patterns, placing them in the same Markov equivalence class under PAG representation. While the exact structure may not be distinguishable, the presence of latent confounders between X and Y can still be inferred. Thus, GISL can reliably identify whether selection bias or latent confounders exist among the intervened variables and identify causal structures up to the equivalence class.

Q4: Lacking the accuracy of identifying latent confounders.

We report GISL’s ability to identify latent confounders on synthetic datasets with 1500 samples as follows:

We focus on selection bias in our experiments for the following reasons:

1. One of the main contributions of this work is to highlight the often-overlooked role of selection bias in gene expression data, and to propose GISL specifically to identify it.
2. Selection bias can be systematically evaluated on both synthetic and real-world datasets, as the selection processes (e.g., biological constraints) are either known or can be reasonably assumed. In contrast, latent confounders can only be evaluated on synthetic datasets, since there is no ground truth or systematic benchmark for latent confounding in real-world gene expression data.

Q5: The SID (structural intervention distance) metric might also be a more natural measure of how good the recovered graph is.

Thank you for the suggestion. GISL focuses on recovering the graphical structure, and SHD, which directly measures structural differences globally, is a commonly used and standard evaluation metric in this context. While SID provides a meaningful assessment of local interventional reasoning, it focuses on ancestral relationships and is generally more tolerant of structural discrepancies. As a result, SID may overlook structural inaccuracies that do not impact interventional distributions.

Nevertheless, we appreciate your comment and are happy to incorporate it for a more comprehensive analysis. According to your suggestion, we evaluate the SID under varying sample size n for hard intervention as follows:

Q6: Figure 2 is not decipherable.

We will revise and reformat Figure 2 to improve its clarity and readability in the final version of the paper.

Q7: Figure 3 is quite hard to understand, and (b) top is incorrect.

We thank you for your patience in interpreting Figure 3. We have added a detailed description for better clarity and corrected the mistake in panel (b). Additionally, we have added a detailed explanation to help readers interpret the figure more easily.

Specifically, Figure 3 (f) serves as a reference table summarizing the CI patterns for each target gene pair. In this table, black symbols indicate that the corresponding CI relation holds (i.e., the variables are conditionally independent, while white symbols indicate that the CI relation does not hold (i.e., the variables are conditionally dependent).

Taking Figure 3(a) as an example, the structure exhibits conditional dependence $Y \not\perp I_X \mid S$, which is represented by the white upward triangles (△) in the reference table. The conditional independence $Y \perp I_X \mid X, S$ is shown in (a) when $X$ is given and is represented by black downward triangles (▼). These CI patterns for (a) are summarized using the corresponding symbols displayed below the causal graph.

Q8: Incorrect reference to example 2.

Thanks for pointing this out. We have corrected it.

Q9: L271: The function in nonparametric simulation is simple. Would functions drawn from a GP or an NN not be more appropriate?

According to your suggestion, we have expanded the function pool to include a neural network (NN) with a hidden dimension of [5, 5] and ReLU activation. The regulatory function is now randomly sampled from a set that includes linear, quadratic, sin, log, and NN functions. The results of number of variables $\mathcal{X}$ = 15, number of samples n = 1500 for soft intervention are as follows:

Q10: Baselines for identifying colliders as upper bound performance.

Following your advice, we conducted experiments on synthetic data comprising 15 variables and 2000 samples under soft interventions to evaluate the identifiability of GISL in detecting colliders, as detailed below:

Q11: Lacking the actual conditional independence test used or other experimental details.

We have added detailed descriptions of the experimental setup as follows: Specifically, following Algorithm 1, the first step—skeleton discovery—is performed using FGES with the BIC score to identify the conditional dependence. In Step 3, we capture CI patterns from both observational and interventional data. For the nonparametric CI testing, we use the Kernel-based Conditional Independence (KCI) test.

If any aspect remains unclear, please do not hesitate to let us know; we would be more than happy to discuss it further with you.

Dear Reviewer TNYS,

Thank you again for your valuable time dedicated to reviewing our paper and for your helpful suggestions. We particularly appreciate your questions regarding the complexity of the simulation, theorem, and related works. So we are eager to see whether our responses properly addressed your concerns and would be grateful for your feedback.

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With best wishes,

Authors of Submission 17977

We are so happy that your other concerns have been addressed, and we sincerely thank you for your recognition of our work.

Regarding the statement on SID, thank you for pointing this out. What we intended to convey is that SID and SHD capture complementary aspects of model evaluation: SHD penalizes all structural inaccuracies, while SID quantifies differences in the intervention distributions (SID counts the number of falsely inferred interventional distances, as you correctly pointed out). For a more comprehensive evaluation, we will revise the manuscript to clarify this more carefully to avoid any confusion and will put results on both metrics in the updated version.

We sincerely appreciate your thoughtful and constructive feedback, which has helped us present a more comprehensive evaluation. Thank you.