SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 6: Classification performance by pre- and postprocessing strategy for XGBoost across all datasets using Cleanlab as the data-centric method.
C.2 Performance by dataset
As shown in Figures 6 to 8 the relative ranking of the five generative models remains largely consistent across datasets. Specifically, the classification performance results shown in Figure 6 indicate that ddpm produces synthetic data that best retains the ability to train classification models.
The model selection results shown in Figure 7 display higher variance across datasets and pre- and postprocessing methods. This variance can likely be attributed to the findings presented in Table 5, which demonstrate that the performance of several classification models is relatively similar. Consequently, minor performance variations caused by data-centric processing can significantly change the models' relative ranking.
As shown in Figure 8, data generated with the ddpm model tends to lead to the best ranking of important features. Consistently with the findings in classification performance and model selection, the bayesian_network model tends to perform the worst across datasets.
Figure 7: Model selection performance by pre- and postprocessing strategy across all datasets using Cleanlab as the data-centric method.
C.3 Identifying optimal thresholds for the data-centric methods
To identify the optimal threshold  $\tau$  for the scoring function  $S$  for each of the data-centric methods, we constructed four simulated datasets with varying levels of label noise. In particular, we generated datasets with 10 features and 10 samples, 10 features and 50 samples, 50 features and 10 samples, and 50 features and 50 samples. The correlation of each feature with the outcome label was sampled from a uniform distribution ranging from -0.7 to 0.7.
For each dataset, we randomly flipped a proportion (0, 0.02, 0.04, 0.06, 0.08, 0.1) of the outcome labels and recorded the indices of the flipped indices.
Data-IQ. Following the Data-IQ implementation, we explored two methods for setting  $\tau$ : one based on whether the aleatoric uncertainty of the sample was below a certain percentile, and the other based on whether the aleatoric uncertainty was below a certain value. In both cases, to categorize a sample as 'hard', the average predicted probability of the correct label (throughout the model training) had to be  $\leq 0.25$  and the aleatoric uncertainty had to be below  $\tau$ . 'Easy' samples were defined as samples where the average predicted probability of the correct label was  $\geq 0.75$  and the aleatoric uncertainty was below  $\tau$ . Any remaining samples were categorized as 'ambiguous'. We tested setting  $\tau$  at the [20, 30, 40, 50, 60, 70, 80] percentile, and using raw values of [0.1, 0.125, 0.15, 0.175, 0.2].
Data Maps. The data-profiling procedure for Data Maps followed the same approach as Data-IQ, with the exception that the epistemic uncertainty was used instead of the aleatoric uncertainty. We tested the same values of  $\tau$  as used for Data-IQ.
Figure 8: Feature selection performance by pre- and postprocessing strategy for XGBoost across all datasets using Cleanlab as the data-centric method.
Cleanlab. The Cleanlab package provides a 'label quality' score which was used to set  $\tau$  to categorize samples as 'easy' or 'hard'. We tested using the default score as well as the values [0.1, 0.125, 0.15, 0.175, 0.2]. For details of the theoretical basis of Cleanlab's thresholds, we refer to [34], Section 3.1.
Experiment. For each augmented dataset (varying levels of label noise), we applied each of the data-centric methods and tested all the specified values of  $\tau$ . We recorded the number of samples identified as 'hard',  $n_{hard}$  as well as the number of samples that were correctly identified as hard (i.e., a flipped label)  $n_{hard}^{correct}$ . Recall was calculated as  $\frac{n_{hard}^{correct}}{n_{flipped}}$ , precision as  $\frac{n_{hard}^{correct}}{n_{hard}}$  and the F1 score as  $2 \cdot \frac{\text{precision - recall}}{\text{recall + precision}}$
Results. As shown in Table 6, Cleanlab achieves markedly higher F1 scores and recall compared to Data Maps and Data-IQ, albeit with slightly lower precision. Given the substantially higher F1 and recall scores, we chose to focus the main analysis on Cleanlab. We selected the threshold that maximised F1 for the main experiments, resulting in a  $\tau$  of 0.2 for all methods.
Table 6: Average results across all simulated datasets and levels of noise. Numbers in parentheses indicate the standard deviation. The highest value per data-centric method and metric is highlighted in bold. The value in bold for  $\tau$  indicates our chosen threshold for the main experiments.
Method	Type	τ	F1	Recall	Precision
Cleanlab	cutoff	Default	0.407 (0.29)	0.633 (0.30)	0.332 (0.28)
0.100	0.384 (0.30)	0.354 (0.30)	0.5 (0.33)
0.125	0.403 (0.30)	0.393 (0.30)	0.478 (0.33)
0.150	0.41 (0.30)	0.42 (0.31)	0.45 (0.32)
0.175	0.417 (0.30)	0.45 (0.32)	0.428 (0.31)
0.200	0.42 (0.29)	0.478 (0.32)	0.408 (0.30)
Data-IQ	cutoff	0.100	0.083 (0.15)	0.051 (0.10)	0.507 (0.42)
0.125	0.118 (0.19)	0.076 (0.13)	0.562 (0.40)
0.150	0.16 (0.22)	0.11 (0.16)	0.547 (0.40)
0.175	0.206 (0.24)	0.153 (0.18)	0.523 (0.40)
0.200	0.221 (0.24)	0.17 (0.19)	0.513 (0.40)
percentile	20	0.033 (0.05)	0.019 (0.03)	0.279 (0.41)
30	0.057 (0.07)	0.038 (0.05)	0.433 (0.43)
40	0.075 (0.08)	0.052 (0.06)	0.371 (0.39)
50	0.099 (0.10)	0.067 (0.07)	0.373 (0.39)
60	0.138 (0.12)	0.093 (0.08)	0.407 (0.40)
70	0.17 (0.16)	0.12 (0.11)	0.4 (0.40)
80	0.199 (0.21)	0.146 (0.15)	0.438 (0.40)
Data Maps	cutoff	0.100	0.21 (0.24)	0.16 (0.18)	0.426 (0.39)
0.125	0.216 (0.24)	0.166 (0.19)	0.47 (0.40)
0.150	0.219 (0.24)	0.169 (0.19)	0.513 (0.40)
0.175	0.221 (0.24)	0.17 (0.19)	0.513 (0.40)
0.200	0.221 (0.24)	0.17 (0.19)	0.513 (0.40)
percentile	20	0.058 (0.07)	0.033 (0.04)	0.327 (0.37)
30	0.082 (0.10)	0.048 (0.06)	0.331 (0.37)
40	0.103 (0.12)	0.064 (0.07)	0.334 (0.37)
50	0.124 (0.14)	0.08 (0.09)	0.336 (0.37)
60	0.144 (0.17)	0.097 (0.11)	0.376 (0.38)
70	0.163 (0.19)	0.114 (0.13)	0.419 (0.39)
80	0.183 (0.21)	0.133 (0.15)	0.423 (0.39)
C.4 Development of Figure 1
The results shown in Figure 1 were obtained performing the following steps: First, the Adult dataset was split into an  $80\%$  training set and a  $20\%$  test set. Second, each of the five generative models (with the hyperparameters specified in Table 3) was trained on the training set. Third, A synthetic dataset,  $D_{synth}$ , with the same dimensions as the training set was generated using each generative model. Fourth, the inverse KL divergence between  $D_{synth}$  and the training data was calculated, an XGBoost model was trained on  $D_{synth}$  and evaluated on the test set to measure the performance, and Cleanlab was applied to  $D_{synth}$  to extract the proportion of easy and hard examples from each synthetic dataset.
Table 7: Percentage increase in performance from baseline, i.e., no data-centric pre- or postprocessing (as seen in Table 1), per generative model for each pre- and postprocessing strategy, averaged across all datasets and seeds, using Data-IQ as the data-centric method.
Generative Model	Preprocessing Strategy	Postprocessing Strategy	Classification	Model Selection	Feature Selection
bayesian_network	baseline	no-hard	0.27 (-6.25, 6.66)↑	-3.49 (-98.06, 92.25)↓	-9.71 (-61.47, 44.94)↓
easy_ambi-hard	baseline	-0.92 (-7.92, 6.3)↓	2.33 (-91.86, 115.12)↑	-27.86 (-90.12, 33.36)↓
no-hard	-0.64 (-7.8, 7.64)↓	63.18 (-35.66, 164.73)↑	-20.85 (-76.47, 35.22)↓
easy-hard	baseline	-0.33 (-7.34, 6.76)↓	61.24 (-9.69, 131.4)↑	-64.55 (-128.52, -4.18)↓
no-hard	-0.1 (-6.77, 6.99)↓	29.84 (-33.33, 93.02)↑	-81.7 (-135.04, -23.46)↓
ctgan	baseline	no-hard	0.29 (-2.26, 2.75)↑	2.13 (-9.95, 13.86)↑	0.43 (-7.16, 7.78)↑
easy_ambi-hard	baseline	-0.3 (-3.24, 2.37)↓	8.51 (-5.0, 21.51)↑	-0.56 (-9.67, 8.99)↓
no-hard	-0.17 (-2.7, 2.38)↓	6.96 (-5.12, 18.29)↑	0.56 (-9.08, 10.0)↑
easy-hard	baseline	-0.08 (-2.83, 2.44)↓	-12.02 (-25.19, 1.38)↓	-4.02 (-10.59, 2.31)↓
no-hard	0.13 (-2.51, 2.83)↑	-13.11 (-25.82, 0.12)↓	-3.97 (-11.06, 2.72)↓
ddpm	baseline	no-hard	0.2 (-3.48, 4.02)↑	-0.23 (-16.27, 15.02)↓	1.39 (-11.55, 13.66)↑
easy_ambi-hard	baseline	-5.08 (-9.82, 0.18)↓	-28.29 (-47.34, -9.98)↓	-4.45 (-18.54, 9.27)↓
no-hard	-5.06 (-9.84, -0.29)↓	-18.2 (-34.64, -0.06)↓	-5.43 (-20.07, 8.01)↓
easy-hard	baseline	-3.19 (-7.5, 0.98)↓	-11.51 (-26.59, 3.12)↓	-10.6 (-22.83, 0.19)↓
no-hard	-2.99 (-7.3, 1.19)↓	-17.46 (-34.07, -1.47)↓	-11.44 (-23.96, 0.79)↓
nflow	baseline	no-hard	0.35 (-2.39, 3.07)↑	1.6 (-15.26, 19.65)↑	4.66 (-10.09, 19.15)↑
easy_ambi-hard	baseline	0.16 (-2.58, 2.79)↑	6.32 (-15.85, 27.4)↑	-1.57 (-17.11, 15.53)↓
no-hard	0.5 (-2.14, 3.24)↑	6.16 (-14.67, 25.8)↑	0.21 (-15.45, 16.06)↑
easy-hard	baseline	1.02 (-1.91, 3.62)↑	5.65 (-13.41, 25.13)↑	6.18 (-9.79, 21.52)↑
no-hard	1.03 (-2.06, 3.67)↑	3.46 (-15.85, 22.85)↑	4.82 (-10.25, 18.61)↑
tvae	baseline	no-hard	0.19 (-2.63, 3.11)↑	3.27 (-11.38, 16.1)↑	0.97 (-4.51, 6.27)↑
easy_ambi-hard	baseline	-0.43 (-3.47, 2.35)↓	-6.48 (-22.11, 7.53)↓	-1.28 (-9.16, 6.16)↓
no-hard	-0.27 (-3.37, 2.71)↓	-4.03 (-20.01, 10.97)↓	-0.06 (-7.57, 6.89)↓
easy-hard	baseline	1.02 (-1.93, 3.63)↑	-18.03 (-32.56, -3.15)↓	-0.8 (-5.49, 3.83)↓
no-hard	1.07 (-1.86, 3.97)↑	-19.84 (-34.66, -5.19)↓	0.9 (-4.16, 5.73)↑
C.5 Results with Data-IQ and Datamaps
The following subsections show the main results using Data-IQ and Data Maps as the data-centric method. Note, that statistical fidelity metrics for Data-IQ and Data Maps are not included in the table to reduce the complexity of the table, as they do not demonstrate any correspondence with task-related performance.
C.5.1 Data-IQ
As shown in Table 7 and Figure 9, the results from using Data-IQ are relatively similar to what's reported in Table 2. However, the impact of data-centric processing is weaker and more variable when using Data-IQ. This effect be attributed to Data-IQ's tendency to identify fewer 'hard' examples (as shown in Table 6 by markedly lower recall) and thereby being a less strong intervention than Cleanlab.
C.5.2 Data Maps
The main results using Data Maps as the data-centric method are presented in Table 8 and Figure 10. Similarly to Data-IQ, the variation across performance metrics is generally higher when using Data Maps compared to Cleanlab. The benefits of Data Maps compared to no processing varies to some extent, with models like nflow seeming to draw some benefit from it, while other models do not see major gains. This might again be attributed to the relatively low recall and the small number of identified 'hard' samples with Data Maps, which reduces the strength of the intervention.
Table 8: Percentage increase in performance from baseline, i.e., no data-centric pre- or postprocessing (as seen in Table 1), per generative model for each pre- and postprocessing strategy, averaged across all datasets and seeds, using Data-IQ as the data-centric method.
Generative Model	Preprocessing Strategy	Postprocessing Strategy	Classification	Model Selection	Feature Selection
bayesian_network	baseline	no-hard	0.14 (-5.08, 6.41)↑	-14.67 (-179.33, 148.67)↓	-70.0 (-243.76, 116.63)↓
easy_ambi-hard	baseline	0.87 (-5.18, 7.4)↑	190.67 (39.33, 340.67)↑	135.71 (-33.95, 301.53)↑
no-hard	1.2 (-4.46, 8.79)↑	199.33 (52.67, 341.33)↑	104.23 (-63.28, 272.74)↑
easy-hard	baseline	2.18 (-3.58, 8.95)↑	262.0 (74.67, 443.33)↑	-51.7 (-223.73, 137.24)↓
no-hard	2.46 (-3.46, 9.15)↑	213.33 (26.0, 400.0)↑	-80.31 (-249.14, 118.38)↓
ctgan	baseline	no-hard	0.17 (-2.67, 2.72)↑	3.63 (-7.26, 14.19)↑	0.33 (-6.6, 6.91)↑
easy_ambi-hard	baseline	-0.57 (-3.39, 2.17)↓	7.54 (-4.08, 18.38)↑	-2.52 (-11.64, 5.99)↓
no-hard	-0.32 (-3.11, 2.47)↓	13.13 (1.34, 24.25)↑	-3.6 (-13.19, 5.62)↓
easy-hard	baseline	-0.75 (-3.8, 2.14)↓	-15.2 (-28.21, -1.96)↓	-2.47 (-9.4, 4.63)↓
no-hard	-0.65 (-3.62, 2.23)↓	-10.34 (-22.79, 1.9)↓	-3.13 (-9.47, 3.35)↓
ddpm	baseline	no-hard	0.13 (-3.82, 3.73)↑	-10.19 (-24.66, 4.66)↓	1.34 (-12.26, 13.9)↑
easy_ambi-hard	baseline	-3.98 (-8.9, 0.5)↓	-11.78 (-25.59, 1.7)↓	-4.52 (-18.29, 8.67)↓
no-hard	-4.03 (-8.96, 0.77)↓	-19.23 (-35.29, -4.05)↓	-4.14 (-17.14, 9.47)↓
easy-hard	baseline	-2.53 (-7.05, 1.58)↓	-20.93 (-36.11, -5.92)↓	-4.42 (-16.55, 6.67)↓
no-hard	-2.82 (-7.49, 1.12)↓	-16.27 (-28.99, -3.18)↓	-1.53 (-12.95, 9.81)↓
nflow	baseline	no-hard	0.22 (-2.48, 2.75)↑	3.83 (-13.8, 21.77)↑	-0.4 (-13.37, 13.22)↓
easy_ambi-hard	baseline	0.77 (-1.89, 3.22)↑	8.13 (-10.21, 25.68)↑	4.43 (-10.51, 19.46)↑
no-hard	1.13 (-1.53, 3.75)↑	-0.72 (-16.59, 14.75)↓	5.48 (-9.09, 19.53)↑
easy-hard	baseline	1.23 (-1.5, 4.15)↑	-3.03 (-18.74, 12.84)↓	9.02 (-5.26, 22.42)↑
no-hard	1.46 (-1.02, 4.2)↑	6.7 (-10.93, 22.49)↑	9.53 (-4.33, 21.96)↑
tvae	baseline	no-hard	0.21 (-2.65, 2.92)↑	4.5 (-10.54, 18.35)↑	-1.23 (-6.94, 4.55)↓
easy_ambi-hard	baseline	-0.45 (-3.26, 2.43)↓	-14.39 (-28.0, 0.3)↓	-1.25 (-7.76, 5.62)↓
no-hard	-0.26 (-3.14, 2.68)↓	-14.68 (-30.55, 0.71)↓	-0.67 (-8.08, 5.57)↓
easy-hard	baseline	0.8 (-2.13, 3.5)↑	-9.47 (-24.33, 6.34)↓	-0.61 (-5.74, 4.4)↓
no-hard	0.88 (-1.97, 3.55)↑	-10.3 (-24.33, 4.32)↓	0.31 (-4.84, 5.71)↑
C.6 Performance on data with added label noise across all tasks and generative models
The performance of all generative models across all tasks (of which a subset is presented in Figure 4) is shown in Figure 11. The overall trend is consistent across generative models, although not all models are affected equally hard by varying levels of noise. For instance, tvae and ctgan seem to be more robust to noise than e.g. ddpm.
Figure 9: Average performance across all datasets for each generative model by pre- and postprocessing method using Data-IQ as data-centric method.
Table 9: Feature selection performance by classification model, averaged across all datasets and generative models using Cleanlab as the data-centric method. Numbers in parentheses indicate  $95\%$  bootstrapped CI.
Preprocessing Strategy	Postprocessing Strategy	Classification Model	Spearman's Rank Correlation
baseline	baseline	DecisionTreeClassifier	0.68 (0.65, 0.71)
RandomForestClassifier	0.71 (0.68, 0.74)
XGBClassifier	0.51 (0.47, 0.55)
no-hard	DecisionTreeClassifier	0.67 (0.63, 0.69)
RandomForestClassifier	0.69 (0.66, 0.73)
XGBClassifier	0.51 (0.48, 0.56)
easy-hard	baseline	DecisionTreeClassifier	0.69 (0.66, 0.72)
RandomForestClassifier	0.73 (0.7, 0.76)
XGBClassifier	0.52 (0.49, 0.56)
no-hard	DecisionTreeClassifier	0.67 (0.63, 0.7)
RandomForestClassifier	0.71 (0.68, 0.74)
XGBClassifier	0.53 (0.49, 0.56)
C.7 Feature selection results with other models
As indicated in Table 9 and Figure 12, the feature selection results reported in the main manuscript are stable across generative models and classification models. Across all comparisons, Spearman's rank correlation is calculated between the feature rankings of the same model type trained on  $\mathbb{D}_{\mathrm{train}}$  and  $\mathbb{D}_{\mathrm{synth}}^{\mathrm{post}}$ . In general, random forest and decision trees models tend to display higher concordance in their feature rankings between models trained on synthetic data and models trained on real data compared to xgboost models.
Figure 10: Average performance across all datasets for each generative model by pre- and postprocessing method using Data-IQ as the data-centric method.
Figure 11: Performance across all models and tasks on the Covid mortality data with added label noise. Using Cleanlab as the data-centric method.
C.8 Causal Discovery
As a proof of concept, we investigated whether data-centric processing might be useful for causal discovery. For this purpose, we used DAGMA [43] to estimate a causal graph,  $DAG_{train}$  based on  $\mathbb{D}_{\mathrm{train}}$ .  $DAG_{train}$  was used as the reference graph, and compared to a causal graph estimated from  $\mathbb{D}_{\mathrm{synth}}^{\mathrm{post}}$ . Results for a single dataset are shown in Figure 13 in terms of Structural Hamming Distance (SHD). The results appear promising in this particular example, however, further work is required.
Figure 12: Mean feature selection performance across all datasets by classification model and generative model, using Cleanlab as the data-centric method.
Figure 13: Causal graph discovery performance using Cleanlab as the data-centric method on a single dataset (id=361055).
C.9 Distribution of Scores
C.9.1 Main Experiment
To investigate the variability of the performance across the three main tasks, we show the distribution of performance metrics across all generative models, datasets, and processing strategies. The findings for classification performance, feature selection, and model selection are shown in Figure 14, Figure 15, and Figure 16, respectively. All figures are shown using Cleanlab as the data-centric method. Overall, the difficulty of the datasets is highly variable, and the variance by generative model is heterogeneous. In general, the best performing models, such as TabDDPM, exhibit comparatively lower variability across different seeds. In contrast, models with poorer performance, like Bayesian networks and normalizing flows, display higher variability across seeds.
The variance of the performance metrics is higher for the tasks of feature selection and model selection than classification. This discrepancy can likely be partly attributed to the inherent smoothness of the performance metrics. The model ranking score is established by accurately ranking the eight supervised classification models listed in Appendix B.3. Similarly, the feature selection score is determined by correctly ranking the metrics for feature importance across all features within the dataset - a quantity that naturally varies by dataset.
In contrast, classification performance relies on AUROC values computed on the test dataset, which encompasses a relatively large number of samples (1,141 in the smallest dataset). Due to the dissimilarity in the number of samples, models, and features, the metrics for model and feature ranking are inevitably more susceptible to larger variance. This occurs because disparities in a single element can lead to more substantial effects on the performance metrics in comparison to the impact of single elements for classification metric.
Figure 14: Distribution of classification performance across the random seeds by generative model (rows) and dataset ids (columns).
Figure 15: Distribution of feature selection performance across the random seeds by generative model (rows) and dataset ids (columns).
Figure 16: Distribution of model selection performance across the random seeds by generative model (rows) and dataset ids (columns).
C.9.2 Label noise
To investigate the variability in the label noise experiments, we show the distribution of performance metrics across all generative models, proportions of label noise, and processing strategies. The findings for classification performance, feature selection, and model selection are shown in Figure 17, Figure 18, and Figure 19, respectively. Further, we show the performance across the generative models, summarized over the levels of label noise, in Table 10 and show the performance by the level of label noise, summarized over all generative models, in Table 11. All figures and tables are shown using Cleanlab as the data-centric method.
Figure 17: Distribution of classification performance across the random seeds by generative model (rows) and percent label noise (columns).
Preprocessing - postprocessing
Figure 18: Distribution of feature selection performance across the random seeds by generative model (rows) and percent label noise (columns).
baseline - baseline
baseline - no_hard
easy-hard - baseline
easy-hard - no-hard
Figure 19: Distribution of model selection performance across the random seeds by generative model (rows) and percent label noise (columns).
Table 10: Summarised performance for the label noise experiments across all levels of label noise. Classification is measured by AUROC, feature selection and model selection by Spearman's Rank Correlation, and statistical fidelity by inverse KL divergence. Numbers show bootstrapped mean and  $95\%$  CI.
Generative Model	Preprocessing Strategy	Postprocessing Strategy	Classification	Feature Selection	Model Selection	Statistical Fidelity
Real data	None	None	0.766 (0.764, 0.767)	-	-	-
bayesian_network	baseline	baseline	0.69 (0.678, 0.704)	0.393 (0.349, 0.435)	0.376 (0.284, 0.46)	1.0 (1.0, 1.0)
no-hard	0.697 (0.684, 0.71)	0.417 (0.371, 0.462)	0.45 (0.355, 0.539)	1.0 (1.0, 1.0)
easy-hard	baseline	0.742 (0.737, 0.747)	0.535 (0.505, 0.566)	0.44 (0.336, 0.526)	1.0 (1.0, 1.0)
no-hard	0.751 (0.744, 0.756)	0.522 (0.478, 0.563)	0.608 (0.536, 0.683)	1.0 (1.0, 1.0)
ctgan	baseline	baseline	0.741 (0.734, 0.746)	0.662 (0.62, 0.7)	0.451 (0.346, 0.559)	0.995 (0.994, 0.996)
no-hard	0.752 (0.747, 0.757)	0.63 (0.601, 0.661)	0.56 (0.467, 0.651)	0.995 (0.994, 0.996)
easy-hard	baseline	0.734 (0.73, 0.738)	0.62 (0.581, 0.666)	0.366 (0.292, 0.442)	0.998 (0.997, 0.998)
no-hard	0.746 (0.74, 0.751)	0.58 (0.54, 0.625)	0.544 (0.46, 0.626)	0.998 (0.997, 0.998)
ddpm	baseline	baseline	0.685 (0.677, 0.693)	0.272 (0.208, 0.338)	0.294 (0.23, 0.36)	1.0 (1.0, 1.0)
no-hard	0.69 (0.682, 0.699)	0.28 (0.212, 0.356)	0.298 (0.229, 0.371)	1.0 (1.0, 1.0)
easy-hard	baseline	0.72 (0.71, 0.729)	0.43 (0.347, 0.511)	0.309 (0.217, 0.398)	0.999 (0.999, 0.999)
no-hard	0.728 (0.717, 0.739)	0.439 (0.365, 0.51)	0.518 (0.421, 0.626)	0.999 (0.999, 0.999)
nflow	baseline	baseline	0.686 (0.674, 0.697)	0.325 (0.268, 0.378)	0.393 (0.328, 0.457)	0.995 (0.995, 0.996)
no-hard	0.699 (0.686, 0.711)	0.344 (0.294, 0.396)	0.467 (0.37, 0.556)	0.995 (0.995, 0.996)
easy-hard	baseline	0.707 (0.699, 0.716)	0.342 (0.286, 0.395)	0.521 (0.45, 0.586)	0.995 (0.995, 0.996)
no-hard	0.717 (0.71, 0.724)	0.324 (0.26, 0.383)	0.577 (0.488, 0.663)	0.995 (0.995, 0.996)
tvae	baseline	baseline	0.735 (0.731, 0.74)	0.64 (0.605, 0.671)	0.41 (0.327, 0.498)	0.987 (0.986, 0.989)
no-hard	0.747 (0.743, 0.752)	0.643 (0.606, 0.679)	0.537 (0.45, 0.619)	0.987 (0.985, 0.988)
easy-hard	baseline	0.722 (0.715, 0.729)	0.608 (0.558, 0.663)	0.437 (0.356, 0.519)	0.988 (0.987, 0.99)
no-hard	0.732 (0.725, 0.738)	0.601 (0.547, 0.658)	0.415 (0.315, 0.506)	0.987 (0.986, 0.989)
C.10 Other Measures of Statistical Fidelity
Table 12 shows additional measures of statistical fidelity (Wasserstein Distance (WD) and Maximum Mean Discrepancy (MMD)) for the main experiment.