We are deeply grateful for the time you devoted to reviewing our manuscript. We appreciate that you think our proposal is a very solid paper. We hope that the following responses adequately address your concerns.

Q1. ``Additional details regarding the learning of $\mathcal{L}_{V_i}$ should be provided.''

A1. Thanks for being thoughtful. Specifically, the learning process of $\mathcal{L}_{V_i}$ first learns the Markov blanket of $V_i$, then performs skeleton learning based on the variables in $MB^{+}(V_i)$, and finally determines the V-structures. As you suggested, we have incorporated this clarification in the revised version.

Q2. ``Detailed technical comparisons between the proposed and compared methods (e.g., IDA, LV-IDA) are not explicitly stated in the manuscript, which should be clarified.''

A2. Thanks for the helpful suggestion. After a PAG (or CPDAG) is learned, LV-IDA（or IDA）enumerates all corresponding MAGs (or DAGs), identifies adjustment sets that satisfy the generalized back-door criterion, and estimates causal effects. If all these estimated effects are consistently nonzero (or consistently zero), an invariant ancestor (or non-ancestor) is determined. By contrast, our method does not require global graph learning or enumeration. We will include the above discussion.

Q3. ``More details about what is the specific difference between the MMB-by-MMB algorithm and Algorithm 1.''

A3. Thank you for raising this point. The MMB-by-MMB algorithm focuses solely on learning the local structure involving the target variable and its adjacent variables. To identify local characterizations (Theorems 1, 2, 3, 4), we extend the MMB-by-MMB algorithm to learn the $\mathcal{P}_{MB^+(X)}$. As a result, we can identify causal relationships between any pair of variables, no longer limited to the target variable and its adjacent variables.

Q4 ``The edges between G and H, C and F in Fig. 7 (b) are correct, why aren't they preserved in $\mathcal{P}$ ?''

A4. Thank you for your careful observation. We would like to clarify that in each iteration, from the learned $\mathcal{L}_{V_i}$, we utilize Propositions 1 and 2 to identify valid edges and colliders and store them in $\mathcal{P}$. Since the edges between G and H, as well as C and F, are not validated by Propositions 1 and 2, we do not include them in $\mathcal{P}$. In other words, we only retain the information that we can confirm to be correct.

We sincerely appreciate the time you dedicated to reviewing our paper, as well as your insightful and encouraging comments. Below, we provide our responses to your comments.

Q1. ``I encourage the authors to compare against some of more recent baselines''

A1. Thank you for the suggestion. Within the limited time, we have added a performance comparison with M3HC [1] and ICD [2] under the same settings as in our main paper. After learning the graph using ICD and M3HC, we applied Zhang's causal identification criteria (similar to RFCI-Zhang). As shown in the table below, LocICR outperforms other methods on all evaluation metrics and significantly reduces the number of CI tests compared to ICD-Zhang, which focuses on global structure learning with latent variables. Since M3HC is a hybrid method rather than fully CI-based, we did not compare the number of CI tests.

Q2. ``What prevents the authors from using the default setting instead of adopting a linear Gaussian parameterization?''

A2. Following existing studies—specifically the setups of RFCI, LV-IDA, ITC, and IDA—we adopt a linear Gaussian parameterization, which is also used by M3HC and ICD. Additionally, we tested our method on the ALARM network with its default parameters in bnlearn, keeping other settings unchanged. The results below demonstrate that our method continues to perform well.

Q3. ``A more thorough review of the state of the art in global causal discovery with latent confounders is missing.''

A3. Thank you for the valuable suggestion. In the revised version, we have included a more comprehensive review of state-of-the-art methods in global causal discovery with latent confounders, highlighting recent advancements (e.g., [1–4]).

[1] On scoring maximal ancestral graphs with the max–min hill climbing algorithm International Journal of Approximate Reasoning 2018

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

[3] Differentiable causal discovery under unmeasured confounding AISTATS 2021

[4] Greedy equivalence search in the presence of latent confounders UAI 2022

We sincerely appreciate your thorough review and insightful comments. We hope the following response properly addresses your concerns.

Q1 Regarding ``...the theoretical contribution...'':

A1. We would like to clarify that one of our paper’s main theoretical contributions is proposing necessary and sufficient local characterizations (Theorems 1–4) that account for latent variables—an aspect not found in Xie et al. (2024). To learn these local characterizations, we extend the method proposed by Xie et al., 2024 to learn $P_{MB^+(X)}$. Unlike their method, which is restricted to the target variable and its adjacent variables, we generalize it to apply to any two variables (see lines 280–285).

Q2 Regarding Proposition 1:

A2. We would like to clarify that we have double-checked Theorem 1 in Xie et al. (2024) and confirmed its validity. A key reason is the important fact stated in Zhang (2008): given any DAG $\mathcal{G}$ over $\mathbf{V} = \mathbf{O} \cup \mathbf{L}$—where $\mathbf{O}$ denotes the set of observed variables, and $\mathbf{L}$ denotes the set of latent variables—there is a MAG over $\mathbf{O}$ alone such that for any disjoint $\mathbf{X}, \mathbf{Y}, \mathbf{Z} \subseteq \mathbf{O}$, $\mathbf{X}$ and $\mathbf{Y}$ are d-separated by $\mathbf{Z}$ in $\mathcal{G}$ if and only if they are m-separated by $\mathbf{Z}$ in the MAG. We also found similar conclusions in related works [see page 6 in Akbari et al., 2021; page 6 in Pellet & Elisseeff, 2008a].

• Zhang J. Causal Reasoning with Ancestral Graphs. JMLR, 2008.

Q3 Regarding Proposition 2:

A3. Sorry for the confusion. We would like to clarify that if $\mathbf{S}$ contains some colliders that open inactive paths, there will exist vertices belonging to $MB(X)$ similar to $A$ that can be added to $\mathbf{S}$ to block these newly activated paths. According to your suggestion, we can define $\mathbf{S}$ as the specific set $Pa^*(V_1)\cup S_{X,V_2}$. Note that when we verify how $\mathbf{S}$ blocks the three types of active paths, any collider variables introduced are also included in $\mathbf{S}$. Here, all $A_i$ belong to $Pa^*(V_1)$ and $Pa^*(V_1)\subseteq MB^+(X)$, which is observable.

More specifically, the process is as follows:

If there exists an active path of the form $V_1 \leftarrow A_1\dots V_2$, then $A_1\in MB(X)$ can block that path. If $A_1$ is a collider on a path $p1:$ $V_1 \dots * \rightarrow A_1 \leftarrow * \dots V_2$, due to the graph being ancestral and $V_1 \leftarrow A_1$, there must exist $V_1 \leftarrow A_2$ on $p1$. Thus $A_2\in MB(X)$ also blocks $p1$, and if $A_2$ is also a collider on a path, it falls back to the case where $A_1$ is a collider.

If there exists an active path of the form $V_1 \leftrightarrow A_3 \rightarrow\dots V_2$, then $A_3\in MB(X)$ can block that path. If $A_3$ is a collider on a path $p2:$ $V_1 \dots*\rightarrow A_3 \leftarrow * \dots V_2$, due to the graph being ancestral and $A_3 \rightarrow\dots V_2$, there must exist a $A_3 \leftarrow* A_4$ on $p2$ . Thus $A_4\in MB(X)$ also blocks $p2$, and if $A_4$ is also a collider on a path, it falls back to the case where $A_3$ or $A_1$ is a collider.

If there exists an active path of the form $V_1 \rightarrow\dots V_2$, then $S_{X,V_2}\subseteq MB(X)$ can block that path due to $X*\rightarrow V_1$ and $V_1 \notin S_{X,V_2}$. Thus, the edge $V_1-V_2$ is true.

We will incorporate the above discussion to make the proof clearer.

Q4 Regarding Proposition 3:

A4. First, if we do not stop, $M_L$ will indeed become an empty graph. We would like to clarify that lines 1055–1061 illustrate the following fact: by suitably removing leaf nodes, we can derive the local subgraph $M_L$ of interest from the global MAG $M$.

Next, the reason two identical marginal distributions imply that continuing the algorithm will not further orient the undirected edges is that newly added leaf nodes do not affect the joint distribution of the existing variables in $M_L$. In other words, intuitively speaking, these leaf nodes do not introduce new directional information (e.g., V-structures) that could help further orient the undirected edges in the local structure.

Q5 Does PAG in this paper actually refer to maximally informative PAG?

A5. Yes, PAG refers to the maximally informative PAG.

We sincerely appreciate your constructive and thoughtful feedback, as well as your recognition of the importance, clarity, and empirical rigor of our work in the field of causal inference. We hope the following responses properly address your concerns.

Q1 ``introducing the correct definition of Markov Blanket for MAGs in the main paper''

A1. Following your suggestion, we have moved the definition of the Markov Blanket for MAGs from the Appendix into the main text.

Q2 ``citing the seminal paper of Chekering when they first mention PAG''

A2. Thank you for noting this. It was indeed a citation error, which we have now corrected in the revised version. We have properly cited works specifically addressing PAGs, including [Spirtes and Richardson, 1996], [Ali et al., 2004], [Zhang and Spirtes, 2005], and [Zhao et al., 2005].

Q3 ``... without requiring the learning of a full PAG, the enumeration of MAGs, or the assumption of causal sufficiency.'' Having a redundancy.

A3. Thank you for your thoughts. We have removed this redundancy in the revised version. Our initial aim was to emphasize that the proposed approach does not rely on the assumption of causal sufficiency.

Q4 ``Why is the performance of the proposed algorithm on MILDEW is better at small sample sizes (in WR and WP)?''

A4. Because our algorithm reduces the number of conditional independence (CI) tests through localization, it helps avoid bias introduced by excessive testing. Consequently, even with smaller sample sizes, our method outperforms other global approaches. Notably, existing local methods often perform worse because they require the assumption of causal sufficiency, while latent variables may be present in the system.

Q5 ``Enumerating all MAGs within a class and determining that X is an invariant ancestor across all equivalent MAGs is equivalent to saying that X is an invariant ancestor in the PAG?''

A5. You are correct! Stating that X is an invariant ancestor in all equivalent MAGs is equivalent to saying that X is an invariant ancestor in the PAG.

Q6 ``My intuition is that PAGs represents uncertainty within a class and since in all MAGs X is an ancestor then there is no uncertainty and so this information should be visible in the PAG, assuming you are using the complete rules of FCI. ... However finding a set satisfying the back-door criterion is not the same as finding ancestors. ''

A6. First, we would like to clarify that although $X$ may be an ancestor in all equivalent MAGs, this fact might not always be explicitly visible(i.e., by checking the directed path) in the PAG. This discrepancy arises because PAGs represent uncertainty within a class of MAGs and do not explicitly encode all ancestral relationships. For instance, as illustrated in Fig. 6, there is no directed path from $A$ to $E$ in the PAG shown in Fig. 6(b), yet $A$ is an ancestor of $E$ in all equivalent MAGs (Fig. 6(c)-(g)).

Furthermore, LV-IDA enumerates all (local) MAGs, identifies adjustment sets that satisfy the generalized back-door criterion, and estimates the possible causal effects accordingly. If all such causal effects are non-zero, the method concludes an invariant ancestor (Fang et al., 2022). For instance, in Fig. 6, LV-IDA calculates the causal effect from $A$ to $E$ for every equivalent MAG (Fig. 6(c)-(g)) as non-zero, thereby concluding that $A$ is an invariant ancestor of $E$.

Q7 ``is it possible to reduce the faithfulness assumption into a "local" faithfulness?''

A7. Thank you for your thoughtful consideration. The local learning—such as our approach—can reduce unnecessary CI tests, which may help mitigate minor violations of the faithfulness assumption [Isozaki, 2014]. We will discuss this point in our paper and plan to explore it more thoroughly in future work.

Isozaki T. A robust causal discovery algorithm against faithfulness violation[J]. Information and Media Technologies, 2014, 9(1): 121-131.

Thank you again for your careful reading. We have corrected the typos in the citations and simulation table in the revised version.