SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We demonstrate the effectiveness of our approach and compare it with alternative methods in Scenario (13). The results are shown in Figure 4, which pertains to high-dimensional  $Z$ , and Figure 5, which focuses on low-dimensional  $Z$ . The results consistently demonstrate that our test achieves favorable performance in terms of the type I error and power under  $H_{1}$ . Although LPCIT, CMIknn, and NNSCIT effectively control the type I error, they exhibit noticeably lower power compared to our method, when the dimension exceeds 60, often by a substantial margin. Furthermore, KCIT, GCIT, and CCIT all yield high power under  $H_{1}$ , but they either always or sometimes suffer from inflated type I errors.
Figure 3: Comparison of the type I error (lower is better) and power under  $H_{1}$  (higher is better) for our test in Scenario (13) across different values of  $k$ .
Figure 4: Comparison of the type I error (lower is better) and power under  $H_{1}$  (higher is better) of our test with six SOTA tests in Scenario (13). Left: The results when varying the dimension of  $Z$ . Right: The results when varying the sample size.
B.3 Additional empirical results for Scenario I
In Figure 6, we present the type I error and power under  $H_{1}$  in low dimensions of  $Z$  ranging from 5 to 30 for Scenario I (Eq. (9)) with Gaussian or Laplace noises. It can be observed that our test and LPCIT consistently achieve good and stable performance in terms of type I error and power under  $H_{1}$ , when the dimensionality of  $Z$  is lower than 30. On the other hand, GCIT, CCIT, and KCIT exhibit high power under  $H_{1}$  but fail to control the type I error. NNSCIT and CMIknn demonstrate relatively good control of type I errors but lack sufficient power under  $H_{1}$ .
B.4 Computational efficiency analysis
Figure 7 shows the timing performance of all methods for a single test under Scenario I with Laplace noises. Our test is found to be highly computationally efficient even when dealing with large
Figure 5: Comparison of the type I error (lower is better) and power under  $H_{1}$  (higher is better) of our test with six SOTA tests in Scenario (13).
Figure 6: Comparison of the type I error (lower is better) and power under  $H_{1}$  (higher is better) of our method with six SOTA methods on the post-nonlinear model under Gaussian or Laplace distributions in Scenario I. Left: The results under Gaussian distribution. Right: The results under Laplace distribution.
sample sizes and high-dimensional conditioning sets. In contrast, CMIknn and CCIT for sample sizes exceeding 1000, and LPCIT for dimension of Z higher than 50 are impractical due to their prohibitively long running time.
B.5 The detailed experimental setup for Scenario III
For the chain structure  $Y \to Z \to X$ , we generate synthetic data as follows:
$$
H _ {0}: Y \sim N (1, 1), Z = Y a + \epsilon_ {1}, X = Z ^ {T} b + \epsilon_ {2},
$$
$$
H _ {1}: Y \sim N (1, 1), Z = Y a + \epsilon_ {1}, X = Z ^ {T} b + Y + \epsilon_ {2},
$$
Figure 7: Running times in seconds as a function of sample size or dimension of  $Z$  on the post-nonlinear model under Laplace distribution in Scenario I. Left: The results when varying the dimension of  $Z$ . Right: The results when varying the sample size.
where  $a$  and  $b$  are both  $d_{z}$ -dimensional, the entries of  $a$  and  $b$  are both randomly and uniformly sampled from  $[0, 0.3]$ ,  $\epsilon_{1}$  is generated from a  $d_{z}$ -dimensional standard multivariate Gaussian distribution, and  $\epsilon_{2}$  is sampled from a standard Gaussian distribution.
C Real Data Analysis
In order to showcase the superior performance of our test, we conduct a comparative evaluation against other state-of-the-art (SOTA) CI tests using real datasets. We assess the effectiveness of our method along with six SOTA approaches on two specific datasets: the ABALONE dataset [9] and the Flow-Cytometry dataset [39].
C.1 Real ABALONE dataset
The ABALONE dataset [9] comprises measurements obtained from a study conducted to predict the age of abalones based on their physical characteristics. The dataset is publicly available at the UCI Machine Learning Repository and can be downloaded from https://archive.ics.uci.edu/ml/datasets/abalone. In our evaluation, we consider the graph structure recovered by [31] as the ground truth, as depicted in Figure 4 of their paper. This graph represents the causal relationships among the 8 variables in the dataset. We specifically select 35 CI relations and 35 non-  $\mathrm{CI}$  relations from this graph. The philosophy used is that a node  $X$  is independent of all other nodes  $Y$  in the graph when conditioned on its parents, children, and parents of children [26, 41]. Additionally, if there exists a direct edge between node  $X$  and node  $Y$  in the graph, they are never conditionally independent given any other set of variables. As a result, the conditioning set  $Z$  can be arbitrarily selected from the remaining nodes. The dataset consists of 4177 samples, and  $d_{z}$  varies from 1 to 6.
In order to evaluate the performance of various tests, we utilize precision, recall, and F-score as evaluation metrics. Precision is calculated as  $\mathrm{TP / (TP + FP)}$ , where TP represents the number of true CI instances correctly identified, and FP represents the number of non-CI instances incorrectly identified as CI. Recall is calculated as  $\mathrm{TP / (TP + FN)}$ , where FN represents the number of CI instances not identified. The F-score is then computed as the harmonic mean of precision and recall, given by  $2\times$  precision  $\times$  recall / (precision + recall) [29]. TN represents the number of correctly identified true
non-CI instances. Table 1 presents the results for all methods. It should be noted that we do not record the results for GCIT as it does not correctly identify any CI relations. Our approach successfully identifies 31 CI relations and 32 non-CI relations, achieving the highest F-score among the testing methods, while maintaining high precision and recall.
Table 1: The TP, TN, precision (pre), recall (rec) and F-score of our test and six SOTA methods for the real ABALONE dataset.
Method	TP	TN	Pre	Rec	F-score
KCIT	5	35	1	0.1429	0.2501
CCIT	12	34	0.9231	0.3429	0.5
CMIknn	22	35	1	0.6286	0.7720
LPCIT	5	35	1	0.1429	0.2501
NNSCIT	33	6	0.5323	0.9429	0.6805
Ours	31	32	0.9118	0.8857	0.8986
C.2 Real Flow-Cytometry dataset
The Flow-Cytometry dataset is a widely used benchmark in the field of causal structure learning [36, 48]. This dataset captures the expression levels of proteins and phospholipids in human cells [39]. The data can be obtained from the website https://www.science.org/doi/10.1126/science.1105809. In our evaluation, we consider the consensus graph proposed in [33] as the ground truth, which has also been adopted by [41] for verifying CI relations. Figure 5(a) in [33] illustrates the causal relationships among the 11 proteins in the dataset. Following the philosophy outlined in Section C.1, we select 50 CI relations and 40 non-Cl relations from this graph. The number of samples is 1755 and  $d_{z}$  varies from 1 to 9.
Table 2 presents the results for all tests. Our method outperforms other approaches by correctly identifying 47 CI relations and achieving the highest recall and F-score.
Table 2: The TP, TN, precision (pre), recall (rec) and F-score of our test and six SOTA methods for the real Flow-Cytometry dataset.
Method	TP	TN	Pre	Rec	F-score
KCIT	32	30	0.7619	0.64	0.6957
CCIT	33	29	0.75	0.66	0.7021
CMIknn	41	26	0.7455	0.82	0.7810
GCIT	40	24	0.7143	0.8	0.7547
LPCIT	38	25	0.7170	0.76	0.7379
NNSCIT	33	26	0.7021	0.66	0.6804
Ours	47	23	0.7344	0.94	0.8246