Nonparametric Expert DAG Learning with Accurate Edge Strengths and Realistic Knowledge Incorporation

1. The methodological innovation behind NEDAG-GP is limited. Specifically, the literature review indicates that GP-based DAG methods are already available, though NEDAG-GP sets up the weighted adjacency matrix differently. Moreover, incorporating expert knowledge is seemingly straightforward, though Section 4.2 does not actually explain how the knowledge-based constraints are enforced.
2. The paper's primary focus is unclear as it attempts to address two distinct problems simultaneously: nonparametric DAG learning and expert knowledge incorporation. Is there any reason expert knowledge cannot be included in linear DAGs, MLP-based DAGs, or spline-based DAGs? Or is there something particular about GP-based DAGs that makes them more amenable to integrating expert knowledge?
3. The experimental evidence in favor of NEDAG-GP (without expert knowledge) is limited. Figure 3 suggests that its good performance depends on whether the ground truth is a GP, so evaluations on a wider range of functions would be helpful. Also, DAGMA should be included as a baseline since it has superseded NOTEARS as the de facto DAG learning method in this area.
4. The paper does not provide a discussion or results about NEDAG-GP's uncertainty quantification performance, which is odd since it uses GPs.

1. Although I selected “good” for presentation, it would be better if the authors could include a pseudocode of their algorithm for a clearer presentation.

2. How to select the parameters of the Gaussian Processes? In supplementary B.1, the authors described the objective function, and it seems that the notation $\theta$ is unexplained. Does $\theta$ refer to the parameters of the Gaussian Processes? Also, It is still unclear to me how the expert knowledge is incorporated. Is it formulated as constraints of the optimization problem?

3. It seems that using non-parametric estimation method and incorporating expert knowledge make NEDAG-GP outperform NOTEARS-MLP. What if we compare NEDAG-GP with NOTEARS that is augmented with non-parametric estimation methods or expert knowledge incorporated? i.e. an ablation study.

• Unfortunately, the paper contain many imprecise statements (see below).

• only one method is compared, also ignoring many literatures on GP based causal models.

• motivation on using GP is not fully justified.

Other comments:

• " local decisions about adding, removing,or reversing edges are made without clear visibility into their global impact": this is not true, global consistency (and in some extend local consistency) properties of scores have been proven to show the optimality in these operations
• L65: is it true that a single number that can reveal the full caual relationships, esp they often come with specific distribution assumptions? In addition, score-based approach produce specific distribution scores, constraint-based approaches offer test stats, which all represent edge weights.
• L144: The knowledge on edge weights can be easily be via regularization, such as the L1 sparsity coefficient to achieve confidence in forbidden edges. The objective itself is data fitting + prior as regularizations. Topological order itself can be expressed by a set of forbidden edges.
• Section 4.2: I don't see how these W constraints can not be expressed by existing continuos learning approaches. In addition, exppressing prior knowledge as an exact numerical value seems harder

(1) While the paper aims to address DAG learning for modeling causal structures and complex dependencies, the purpose could be clarified. It seems that causal structures inherently involve complex dependencies, so it would help to clarify how these terms are being distinguished in the context of this work. If the intent is to use a DAG for causal reasoning, some discussion on the identifiability of the learned DAG would strengthen the contribution. Specifically, it would be helpful to know if the learned DAG represents a unique solution given the data or if it belongs to an equivalence class that includes the ground truth. Reviewing classic works on causal discovery algorithms, such as PC, GES, or PNL, could help refine the objectives and theoretical foundation of the approach.

(2) The paper introduces the idea of incorporating Gaussian Processes into DAG learning, leveraging their nonparametric properties. While this is an interesting direction, the novelty may be somewhat limited, as Gaussian Processes are a known approach for handling nonparametric modeling. Given an adjacency matrix with binary indicators, there are many established methods for estimating associated parameters, so it would be valuable to see a discussion on how this approach contributes uniquely to the field.

(3) Some aspects of the writing could be more clear. For instance, in the introduction, two bolded statements emphasize the importance of incorporating expert knowledge while minimizing reliance on expert-specified parameters and distributional assumptions. Since expert knowledge can encompass information on edges, parameters, and distributions, it would help to clarify the intended balance between these elements. Addressing this and similar points throughout the paper would enhance readability and help readers better understand the author’s perspective and familiarity with the field.