SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The running time of CAREFL-H is slightly longer than CAREFL-M, due to the additional independence tests, i.e.  $HSIC(X_{test}, \widehat{Z}_Y)$  and  $HSIC(Y_{test}, \widehat{Z}_X)$ . Please see Table 6. The running time is measured on a computer running Ubuntu 20.04.5 LTS with Intel Core i7-6850K 3.60GHz CPU and 32 GB memory. No GPUs are used.
Table 6: Accuracy running time in seconds over 10 datasets generated by SCM LSNM-sine-tanh (definition in Appendix M) with  $N = 10,000$  samples.  $\alpha = 1$ . Ground-truth noise distribution is uniform(-1,1). Model prior distribution is Gaussian(0,1).
Method	Average Running Time in Seconds
CAREFL-M	33.23
CAREFL-H	39.38
P Additive noise models under misleading conditional variances
We show the accuracy of ANM ML methods with ANM data under misleading CVs, and compare with LSNM ML methods. The ANM data is generated by the following SCMs:
$$
\text {A N M - s i n e :} \quad Y := \sin (X \cdot \theta_ {1}) \cdot \theta_ {2} + Z _ {Y}
$$
$$
\text {A N M - t a n h :} \quad Y := \tanh  \left(X \cdot \theta_ {1}\right) \cdot \theta_ {2} + Z _ {Y} \tag {11}
$$
$$
\text {A N M - s i g m o i d :} \quad Y := \sigma (X \cdot \theta_ {1}) \cdot \theta_ {2} + Z _ {Y}
$$
where  $Z_{Y}$  is the ground-truth noise sampled from one of the following distributions: uniform  $(-1, 1)$  and Gaussian  $(0, 1)$ .  $\sigma$  is the sigmoid function. Following [33, 34], each  $\theta$  is sampled uniformly from range  $[-2, -0.5] \cup [0.5, 2]$ .
CAREFL-ANM-M is the ANM counterpart of CAREFL-M (see Appendix I). The LSNM flows (see Equation (6)) in CAREFL-M are changed to model ANMs:
$$
X = t _ {1} + Z _ {X}
$$
$$
Y = t _ {2} (X) + Z _ {Y}
$$
Similar to LSNM ML methods, the accuracy of ANM ML methods is also negatively related to misleading CVs, as shown in Figure 5. Unlike LSNM ML methods, which suffer from misleading CVs under noise misspecification (see Table 1 and Figures 2 and 4), an ANM ML method suffers from misleading CVs with either noise misspecification or correct specification, as illustrated in Figure 5. Figure 1 summarizes the differences between ANM and LSNM ML methods.
(a)LSNM-sine-tanh and uniform(-1,1) noise
(b) LSNM-sine-tanh and Gaussian(0,1) noise
(c) LSNM-tanh-exp-cosine and uniform(-1,1) noise
(e) LSNM-sigmoid-sigmoid and uniform(-1,1) noise
Figure 4: We rewrite Equations (10) as  $Y \coloneqq f(X) + \alpha \cdot g(X) \cdot Z_Y$ . Altering  $\alpha$  changes the variance of noise, and creates datasets with difference CVs. 10 datasets are generated with each  $\alpha$ . CVs are computed by binning the true cause. All the methods in the figure assume Gaussian(0,1) whereas the ground-truth noise is either uniform(-1,1) or Gaussian(0,1). All the datasets are standardized to have mean 0 and variance 1.
(d) LSNM-tanh-exp-cosine and Gaussian(0,1) noise
(f) LSNM-sigmoid-sigmoid and Gaussian(0,1) noise
Figure 5: We rewrite Equations (11) as  $Y \coloneqq f(X) + \alpha \cdot Z_Y$ . Altering  $\alpha$  changes the variance of noise, and creates datasets with difference CVs. 10 datasets are generated with each  $\alpha$ . CVs are computed by binning the true cause. All the methods in the figure assume Gaussian(0,1) whereas the ground-truth noise is either uniform(-1,1) or Gaussian(0,1). All the datasets are standardized to have mean 0 and variance 1.
(f) ANM-sigmoid and Gaussian(0,1) noise
Table 7: (Best viewed in color) Summary of datasets. Settings in Section 7.1 are color-coded. (1) Blue: with noise misspecification; (2) Red: with more than  $50\%$  of datasets having misleading CVs; (3) Brown: both (1) and (2). The datasets in Section 7.1 are generated by LSNMs and the datasets in Sections 7.2 and 7.3 are not.
Type	Name	Noise	Number of Datasets	Percentage of Datasets With Misleading CVs (%)
Synthetic (Section 7.1)	LSNM-tanh-exp-cosine	Uniform(-1,1)	10	60
Beta(0.5,0.5)	10	80
ContinuousBernoulli(0.9)	10	70
Exponential(1)	10	20
Gaussian(0,1)	10	40
Laplace(0,1)	10	40
LSNM-sine-tanh	Uniform(-1,1)	10	100
Beta(0.5,0.5)	10	100
ContinuousBernoulli(0.9)	10	60
Exponential(1)	10	20
Gaussian(0,1)	10	60
Laplace(0,1)	10	80
LSNM-sigmoid-sigmoid	Uniform(-1,1)	10	90
Beta(0.5,0.5)	10	100
ContinuousBernoulli(0.9)	10	80
Exponential(1)	10	70
Gaussian(0,1)	10	90
Laplace(0,1)	10	100
Synthetic (Section 7.2)	SIM	N/A	100	40
SIM-c	100	46
SIM-ln	100	23
SIM-G	100	29
Real-World (Section 7.3)	Tübingen Cause-Effect Pairs	N/A	99	40
(a)  $N = 500$
(b)  $N = 5000$
Figure 6: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and uniform  $(-1,1)$  noise.
CAREFL-M (0.8)
CAREFL-H (0.8)
CAREFL-M (1.0)
CAREFL-H (1.0)
Figure 7: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and beta(0.5, 0.5) noise.
Figure 8: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and continuousBernoulli(0.9) noise.
LSNM-tanh-exp-cosine and exponential(1) noise.
Figure 10: Accuracy over 10 datasets: LSNM-sine-tanh and uniform  $(-1,1)$  noise.
Figure 11: Accuracy over 10 datasets: LSNM-sine-tanh and beta(0.5, 0.5) noise.
Figure 12: Accuracy over 10 datasets: LSNM-sine-tanh and continuousBernoulli(0.9) noise.
Figure 13: Accuracy over 10 datasets: LSNM-sine-tanh and exponential(1) noise.
Figure 14: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and uniform  $(-1,1)$  noise.
Figure 15: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and beta(0.5, 0.5) noise.
Figure 16: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and continuousBernoulli(0.9) noise.
Figure 17: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and exponential(1) noise.
Figure 18: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Gaussian(0,1) noise.
Figure 19: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Laplace(0,1) noise.
Figure 20: Accuracy over 10 datasets: LSNM-sine-tanh and Gaussian(0,1) noise.
Figure 21: Accuracy over 10 datasets: LSNM-sine-tanh and Laplace(0, 1) noise.
Figure 22: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Gaussian(0,1) noise.
Figure 23: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Laplace(0,1) noise.
(a) Accuracy over 100 datasets from SIM sub-benchmark.
(b) Accuracy over 100 datasets from SIM-c sub-benchmark.
(c) Accuracy over 100 datasets from SIM-ln sub-benchmark.
CAREFL-M (0.8)  
CAREFL-H (0.8)
(d) Accuracy over 100 datasets from SIM-G sub-benchmark.
CAREFL-M (1.0)  
CAREFL-H (1.0)
Figure 24: Results with SIM benchmarks.
CAREFL-M (1.0)  
CAREFL-H (1.0)