A Sample Efficient Conditional Independence Test in the Presence of Discretization

Multivariate Gaussian distribution.

Thank you for raising the concern. We fully agree that the assumption of Gaussianity will limit the generality of the proposed test. At the same time, please allow us to share a few points regarding its reasonableness:

1. Challenges in Conditional Independence in the Presence of Discretization: Inferring the conditional independence of latent variables based on their discretized values is indeed challenging, despite being a common practical issue. Discretization significantly reduces the available information. Without introducing mild assumptions, it is particularly difficult—if not overly ambitious—to construct statistics that correctly reflect conditional independence of latent continuous variables, let alone develop valid inference procedures. In this work, we rely on two key properties:
   (1) The Gaussian assumption enables consistent estimation of the latent covariance matrix from discretized observations thanks to its parametric structure. (2) Under the Gaussian model, conditional independence can be inferred solely from the covariance matrix.
2. Popularity of nonparanormal Model: The assumption of latent variables following multivariate Gaussian, also called the nonparanormal model, is well-studied and widely accepted in the community. There is a substantial body of work demonstrating the effectiveness of the nonparanormal model in various scenarios [1,2].
3. Empirical Performance: To alleviate your concern and thanks to the insightul suggestion from Reviewer Nwpp, we conducted experiments investigating the Type I and Type II error of the proposed test where the data generation process violates our assumption. Specifically, the data are generated as either linear or nonlinear non-Gaussian, where the linear parameters follow the same setting of the main experiment and the nonlinear functions are randomly choose from $(a) f(x) = sin(x), (b) f(x) = x^3, (c) f(x) = tanh(x), (d) f(x) = ReLu(x)$. Figure~1 in the link https://anonymous.4open.science/r/DCT-GMM-0D6D shows the comparison. From the experiment result, DCT-GMM demonstrates comparable or superior Type I error control relative to DCT. In terms of Type II errors, it also outperforms DCT under most distributions. Overall, the discretization-aware tests clearly outperform other baselines.

Slight size inflation. What might be causing this phenomenon?

We appreciate the reviewer for the valuable question. The most plausible explanation towards this phenomenon is the neglect of the second order and higher order terms in the deriving distribution of $\hat{\sigma} - \sigma^*$ (Theorem 3.1 and Lemma 3.2 ). Specifically, our derivation relies on a Taylor expansion of the form (kindly refer to line 1045-1048): $\hat{g}(\mathbf{\theta}^*) = \hat{g}(\hat{\mathbf{\theta}}) + \hat{\mathbf{G}}(\mathbf{\theta}^* - \hat{\mathbf{\theta}}) + \dots,$ In our derivation, we omit higher order terms, which might influence the accuracy of the derived distribution. To validate the hypothesis, we conducted the experiment where we further increase the sample size, and strengthen the influence from the conditioning set to the observed pairs, thereby reducing the impact of higher-order terms. As shown in Figure 4 (link: https://anonymous.4open.science/r/DCT-GMM-0D6D), the proposed test effectively controls the Type I error rate under these conditions, supporting the validity of our explanation. We will acknowledge this approximation as a limitation of our approach in the revised version of the paper.

How does the number of discretization levels impact the results?

Thank you for the insightful question. Towards this question, we conducted additional experiments shown in Figure2 and Figure3 in the anonymous link https://anonymous.4open.science/r/DCT-GMM-0D6D.

Figure 2 compares Type I and II errors of DCT-GMM and baselines, using discretization level $M=5$, the cardinality of conditioning set $D=1$ and change the sample size $n=(100,500,1000,2000)$. Similar to the main experiment, both DCT and DCT-GMM control Type I error well, while DCT-GMM achieves higher power.

Figure 3 varies $M=(4,5,6,7,8)$ and fix $D=1$, and $n=2000$ in Figure3. DCT-GMM consistently maintains Type I error control and high power. Notably, the two-step DCT-GMM outperforms the one-step version, supporting our theoretical results.

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[1] Fan, J., Liu, H., Ning, Y., and Zou, H. High dimensional semiparametric latent graphical model for mixed data. Journal of the Royal Statistical Society Series B: Statistical Methodology, 79(2):405–421, 2017.

[2] Zhang A, Fang J, Hu W, et al. A latent Gaussian copula model for mixed data analysis in brain imaging genetics[J]. IEEE/ACM transactions on computational biology and bioinformatics, 2019, 18(4): 1350-1360.

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Thank you again for your thoughtful question. We hope the additional results clarify our findings. Please feel free to reach out if you have further questions or feedback.

The multivariate normal distribution assumption.

Thank you for raising the insightful concern. Please kindly refer to our response~1 to Reviewer 9WLL for a detailed explanation. Due to the 5000-character limit, we were unable to include the full response here. We apologize for the inconvenience.

The technical contribution is rather limited since the main results use standard asymptotic tools.

Thank you for this important comment. We believe this comment arises from a lack of sufficient discussion in our paper. We have included additional discussion to clarify our use of asymptotic tools.

1. Standard Practice in CI Testing: Asymptotic approaches are a cornerstone of CI testing. Many well-established tests, including the classical Chi-square test [2], Fisher-z test [3], kernel-based tests [4] and more recent approaches [5], rely on asymptotic theory. While permutation-type tests may be constructed controlling Type I error in some special cases [1], there are currently no non-asymptotic methods available for CI tests involving hidden variables under nonparanormal models.

2. Finite Sample Solution: Recognizing the limitations inherent in asymptotic analysis, our framework can be readily extended with resampling techniques, such as the bootstrap. These methods offer a viable route to improve finite-sample performance without deviating from the core asymptotic framework.

3. Empirical Support from Simulation Studies: Extensive simulation studies in the literature demonstrate that asymptotic approximations yield accurate results even at moderate sample sizes (e.g., n>100 [4, 6]), supporting their practical applicability.

results from DCT

We are sorry for the confusion. Both DCT and our method involve inferring the precision matrix, for which the standard technique of nodewise regression is employed in both methods [7,8]. This is why the derived CI test in Theorem 3.4 has the same form (though the analytical solutions $\xi$ involved are entirely different). We have properly acknowledged it.

Apart from the use of nodewise regression, all other components employ entirely different techniques. Our aim lies in making the CI test sample-efficient in the presence of discretization. To achieve this, we cannot follow DCT, which further binarizes observations and conducts estimation and inference based on the binarized data—a limitation explicitly acknowledged in DCT (Appendix G in [9]). Instead, we reformulate the problem as an overparameterized one and leverage GMM to provide a principled solution with theoretical guarantees. The rationale and techniques are fundamentally different.

We will carefully discuss the above points in the paper to avoid further confusion. Thanks for the valuable feedback.

In Equation (1), it should be clarified that m ranges from 2 to M−1,

Thanks for your great advice. We have included "$m$ is an integer ranging from $2$ to $M-1$" in our revised paper.

definitions for Σ−jj and Σ−j−j−1 should be stated explicitly

Thanks for your great suggestion. Kindly note that we define the notation in right part of line 99-102: "Similarly, $\mathbf X_{-j-j}$ is the submatrix of $\mathbf X$ without $j$th column and $j$th row, and...". However, your suggestion made us realize that it might be not sufficiently clear. We have included the specific definition of $\mathbf \Sigma_{-j-j}$ and $\mathbf \Sigma_{-jj}$ in Lemma 3.4 to improve the clarity.

In the two-step procedure, is there a specific reason for limiting the process to only two steps, rather than iterating further until some form of convergence in the weight matrix is achieved? or the convergence itself is not guaranteed?

Thanks for your insightful question. Yes, we can iterate the procedure alternately until convergence, and convergence is indeed guaranteed. we don't adopt the iterative update since the two-step estimator is already consistent and variance-efficient, additional iterations would offer limited benefit while incurring higher computational cost.

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[1] Berrett et al., The conditional permutation test for independence while controlling for confounders.

[2] Pearson, on the criterion that a given system of deviations from the probable in the case of a correlated system ...

[3] Fisher, On the "Probable Error" of a Coefficient of Correlation Deduced from a Small Sample.

[4] Zhang et al., Kernel-based conditional independence test and application in causal discovery

[5] Azadkia & Chatterjee, A Simple Measure of Conditional Dependence

[6] Gretton et al., A kernel two-sample test

[7] Qiu et al., Inference on multi-level partial correlations based on multi-subject time series data

[8] Chang et al., Confidence regions for entries of a large precision matrix

[9] Sun et al., A conditional independence test in the presence of discretization

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Please feel free to let us know if any part remains unclear or if you have further questions. Thank you again for your time and valuable feedback.

Most of the claims presented are clear, except for Theorem 3.5.

Thank you for this construction comments. We followed your comment and carefully revised Theorem 3.5 to make it more intutive and clear. Note that the formal theorem demonstrating the superiority of DCT-GMM over DCT is provided in Appendix G. We avoid to include it in the main body of the submission because presenting the claim requires a detailed introduction to DCT, which may detract from the overall flow of the main text.

It would be beneficial if the authors could provide additional details about Theorem 3.3 in the main content

Thank you for your thoughtful feedback. We believe you are referring to Lemma 3.3, which is intended to illustrate a property of nodewise regression that supports the derivation of the CI test. Inspired by your comments, we have added sentences to clarify the motivation behind using nodewise regression in this context:

" While Theorem 3.1 and Lemma 3.2 effectively address the independence test, they do not directly resolve the CI testing problem. A seemingly straightforward approach is to invert the estimated covariance matrix; however, this does not provide a valid solution for inference."

The parameters described in lines 407 to 410 on the left differ from those used in the original DCT work.

Thanks for your careful review. Since DCT is the only method specifically designed to handle discretization scenario, we generally followed their experimental setup for fair comparison. However, our parameter setting for the experiment is more challenging, involving fewer samples and more variables to validate the superiority of DCT-GMM. We summarize the comparison of causal discovery of DCT paper and our own:

overlap between lines 91 to 102 and lines 297 to 304 on the right-hand column with the corresponding sections of the DCT work.

We appreciate the reviewer for the careful review. We have rephrased the related work part (line 91 to 102) with citing additional references [1, 2, 3] (thank again to reviewer NWpp for his great suggestion).

For the experiment part (line 297 to 304), our intention is to follow the previous work DCT. Given your concern, we have rephrased it as " In the first part, we evaluate the Type I and Type II errors of DCT-GMM across various scenarios, comparing it with baseline methods including DCT (Sun et al., 2024), the Fisher-z test (Fisher, 1921), and the Chi-square test (F.R.S., 2009). In the second part, we assess the performance of DCT-GMM on causal discovery tasks and compare with same baseline methods. In the third part, we directly compare DCT-GMM and DCT to empirically validate Theorem 3.5. "

Benefits of using nodewise regression rather than inverting a covariance matrix?

Thanks for your insightful question. The problem with inverting the covariance matrix is that it does not address the issue of inference, i.e., deriving the distribution of $\hat \omega_{jk} - \omega_{jk}^*$.

While the GMM solves the estimation of covariance $\mathbf{\hat{\Sigma}}$ effectively, and we can directly invert it and obtain $\mathbf{\hat{\Omega}}$ , whose entry $\hat \omega_{jk}$ captures the relation of $X_j$ with $X_k$ conditioning on all other variables. The problem is, it does not provide a tractable way to infer the distribution $\hat \omega_{jk} - \omega_{jk}^*$. To address this, we adopt a standard technique---nodewise regression [4, 5]. This allows us to: 1. Show $\beta_{j,k}$ acts as an effective surrogate of $\omega_{jk}$, thereby the wanted distribution transfers from $\hat \omega_{jk} - \omega_{jk}^*$ to $\hat \beta_{j,k} - \beta_{j,k}^*$; 2. Express $\hat \beta_{j,k}-\beta_{j,k}^*$ as linear combination of the known distribution $\hat{\mathbf{\Sigma}} - \mathbf{\Sigma}^*$ as equation~(11).

[1] Li S, et al. K-nearest-neighbor local sampling based conditional independence testing[J]. Advances in Neural Information Processing Systems, 2023.

[2] Cai Z, et al. A distribution free conditional independence test with applications to causal discovery[J]. Journal of Machine Learning Research, 2022.

[3] Kim I, et al. Local permutation tests for conditional independence[J]. The Annals of Statistics, 2022.

[4] Qiu, Yumou, et al. "Inference on multi-level partial correlations based on multi-subject time series data." Journal of the American Statistical Association, 2022.

[5] Chang J, et al. Confidence regions for entries of a large precision matrix[J]. Journal of Econometrics, 2018.

Some practical scenarios

Thank you for your valuable question. We appreciate the chance to highlight the common and often unavoidable discretization due to practical measurement constraints. In principle, any variables measuring "degree" or "intensity"(e.g., happiness, severity) are inherently continuous but often recorded in discrete form. For example

• In medical diagnostics, the severity of cancer is commonly categorized into stages like "Stage I" or "Stage II", not because the disease progresses in discrete jumps, but because there is no equipment that can exactly quantify its severity on a truly continuous scale.

• Similarly, in questionnaires—including the real-world dataset used in our paper—latent continuous variables such as level of depression are often measured using Likert scales (e.g., 1 to 5) for fast assessment and clinical usability.

We hope those practical scenarios can clarifies its practical significance of DCT-GMM.

Groundtruth of real-world dataset

Thanks for your question. You're right—there is no ground truth in the Big Five Personality Dataset. We followed the setup from DCT. Despite the absence of ground truth for reference, the results of the discretization-aware CI tests (DCT and DCT-GMM) appear more reasonable. Specifically, the conclusion that $N4 \perp N10 \mid N3$, is intuitively plausible---since N10 represents "I often feel blue" is a stronger indicator of mood and should naturally subsume the information in N4 ("I seldom feel blue").

At the same time, we would greatly appreciate it if the reviewer could suggest any datasets that align well with our setting.

Normal assumption

Thank you for this very insightful question. Please kindly refer to our response~1 to Reviewer 9WLL for a detailed explanation. Due to the 5000-character limit, we were unable to include the full response here. We sincerely apologize for the inconvenience.

Citing CI tests that work with discrete data...

Thanks for your great suggestion. We have included the suggested reference [1] and other recent CI tests for discrete variables in our revised version [2,3].

Sentences are difficult to understand. e.g, the proportion...

Thank you for the helpful comment. We have followed your comment and revised it to: “The empirical probability of observed discrete pairs reflects the covariance of the underlying continuous variables.”

Use heavily from DCT of nodewise regression

Thank you for discussing the connection between DCT and ours. Nodewise regression is a standard technique for addressing the inference problem of the precision matrix [4,5]. Both DCT and our method involve the inference of the precision matrix. By using the nodewise regression in nonparanormal model, the derived CI test in Thoerem 3.4 would have the same form (but the analytical solutions involved are entirely different). We have properly acknowledged this in the main text and references.

Moreover, it is worth noting that apart from the use of nodewise regression, all other components employ entirely different techniques. Our unique contribution lies in making the CI test sample-efficient in the presence of discretization. To achieve this, we cannot follow DCT, which further binarizes observations and conducts estimation and inference based on the binarized data—a limitation explicitly acknowledged in DCT (Appendix G in [6]). Instead, we reformulate the problem as an overparameterized one and leverage GMM to provide a principled solution with theoretical guarantees. The rationale and techniques are fundamentally different.

• DAG generation

Kindly refer to lines 380–382, where we state that the DAG is generated using the BP model.

"Fisherz" meaning..

Thank you for your insightful question. Yes, "Fisherz" refers to directly applying the Fisher z test to discretized data. Our goal is twofold:

1. To compare it with the Fisher z test on the original continuous data and highlight how discretization distorts causal discovery.
2. To show that even if users know the variables are inherently continuous, directly applying a CI test designed for continuous data—such as the Fisher z test—on discretized values can significantly degrade the resulting causal graph.

[1] Li et al., K-nearest-neighbor local sampling based conditional independence testing

[2] Cai et al., A distribution free conditional independence test with applications to causal discovery

[3] Kim et al., Local permutation tests for conditional independence

[4] Qiu et al., Inference on multi-level partial correlations based on multi-subject time series data

[5] Chang et al., Confidence regions for entries of a large precision matrix

[6] Sun et al., A conditional independence test in the presence of discretization

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We sincerely thank the reviewer for the thorough and constructive review, which has greatly improved our paper. Please let us know if there are any further questions.