## 8 Experiments and Results
Further, we posit that when model performance on an arbitrary MSE dataset \(\mathcal{X}_{x}\) meets or exceeds the performance of the same model when trained on \(\mathcal{X}_{gt}\), then it is likely that the distribution learned by the model, presumably \(\sim\)\(P(x_{k}^{n}|\mathcal{X}_{k}^{n})\), is in some meaningful way similar to the distribution learned by the model trained on ground truth--presumably \(P(x_{gt}^{n}|\mathcal{X}_{gt}^{n})\). This line of thinking suggests that the NDM and ODM, in so far as they are able to predict aspects of implemented neural network performance, provide some measure of explainability to results. In a practical setting this explainability, in conjunction with quantitative targets for error, can be leveraged to generate confidence that a model has achieved a satisfactory level of performance, and allows regulators to understand the basis for results. A consistent feature of the plots in Figure 3 is a spike in error rates somewhere above the 96 th MSE percentile. Figure 4 may explain this by providing insight into the quality of the data being added to the nominal population at the high MSE threshold range. It is at just above the 96 th MSE percentile that the qualitative nature of the intersection population growth curves changes. We note that at the lower range of threshold values, changes in intersection membership occur in large steps. At the upper range of threshold values, the change in membership occurs more smoothly. This speaks directly to the uniqueness, or alternatively to the amount of information, in the data being added to the intersection. As per information theory, the more unique a datum, the more information it carries relative to the data. So, we observe that as intersection set membership additions become increasingly unique, the model error in Figure 3 increases dramatically. This suggests that the trend of the highly informative data are not as easily encoded as the more common low MSE data in the weights of the neural networks implementing the DPM. From the perspective of the CSDI conditional loss function, the information content of the unique outliers is largely irrelevant. An outlying time of flight has very little information to relate about the largely continuous nominal signal in and around which it occurs. More over with their high information content the outliers are likely to generate some number of spurious correlations with the rest of the dataset. So the conditional CSDI model struggles, colloquially in two directions, to make sense of the highly informative anomalous information. Conversely, when on average there is less information per element in the nominal partition the CSDI model is enabled to learn the possibly continuous, possibly well-defined structure of the data. In the case of a highly structured dataset, each element bears a strong, identifiable, possibly causal conditional relation to neighboring data. CSDI leverages this conditional structure and given a largely nominal signal in the dataset readily learns the nominal distribution. DDPM models do not take advantage of this conditional perspective on the nominal information and, as borne out in the results, are slightly impaired in learning the nominal distribution. DDPM models are also more tolerant of outliers when they occur. This is directly observed in Figure 3 where at thresholds less than the 96 th MSE percentile CSDI models generally outperform DDPM models, but in the range above the 96 th, their error ramps up quickly and often exceeds that of DDPM models. Figure 4: The plots on the first row show the growth in the cardinality \(\chi_{gt}^{d}\cap\chi_{MSE_{n}}^{n}\) generated on the range of percentile values \([0\%,100\%]\) and high range \([95\%,100\%]\). The second row shows the same plots for the growth in the cardinality of intersection \(\chi_{gt}^{d}\cap\chi_{MSE_{n}}^{n}\) generated on like ranges. Figure 5 show the results of a second set of experiments where we test the effect of enlarging a dataset with predominantly nominal signal.