## 4 Discussion  
As regards the comparison with the previous classifier [21] where no pre-processing was implemented, we obtained better results in terms of the figures of merit and activation maps. In fact, the CNN reaches an accuracy of 82.0%. Moreover, the CNN compares very well with the literature [34] where an accuracy of 77% is obtained with a classifier trained on about 60,000 exams. Compared to [12; 13], the CNN based classifier achieves better performances in terms of accuracy (respectively 47% and 71.4%). As regards the study by [PERSON] et al. [14], our classifier works better also in terms of Cohen (Kappa) coefficient on the four classes problem, since we reach a k equal to 0.76 with respect to 0.67. Other \begin{table}
\begin{tabular}{l c c c c} \hline  & **A** & **B** & **C** & **D** \\ \hline
**A** & 1 & \(p\) = 0.43 & \(p\) \(<\) 0.05 & \(p\) = 0.12 \\
**B** & \(p\) = 0.43 & 1 & \(p\) \(<\) 0.05 & \(p\) = 0.16 \\
**A** & 1 & \(p\) \(<\) 0.05 & \(p\) \(<\) 0.05 & \(p\) \(<\) 0.05 \\
**B** & \(p\) \(<\) 0.05 & 1 & \(p\) \(<\) 0.05 & \(p\) \(<\) 0.05 \\
**C** & \(p\) \(<\) 0.05 & \(p\) \(<\) 0.05 & 1 & \(p\) = 0.20 \\
**D** & \(p\) \(<\) 0.05 & \(p\) \(<\) 0.05 & \(p\) = 0.20 & 1 \\ \hline \end{tabular}
\end{table} 
Table 4: Top: results of the Dunn test, corrected for multiple comparison, computed on CC projections. Bottom: results of the Dunn test, corrected for multiple comparison, computed on MLO projections. studies [35, 36] reach better accuracy on the classification of two classes: dense versus non-dense and BI-RADS 2 versus BI-RADS 3, respectively. It is, hence, not possible to compare our results with theirs. In [37], the accuracy on the 4 BI-RADS classes is equal to 98.75% which is higher than the accuracy reached in this work. However, their method is not explained and not explainable. As a general consideration, the comparison with other different methods is not performed on the same dataset, making it not completely fair. Since our breast density classifier was trained on digital mammograms, the method cannot be applied to MIAS and DDSM which contain analog digitized mammograms. Moreover, analog mammography is not used in hospitals anymore. We found that pre-processing has a crucial impact not only on the accuracy, but also on the explainability of the classifier. In fact, the grad-CAM activation maps showed a good localization capability once the pectoral muscle has been removed from the image. For this reason, we believe that CNN classifiers should be trained on real medical images paying particular attention not only to the classification performances but also to obtaining reasonable activation maps. As regards the training on different sample compositions, the discussion on the more appropriate strategy to be used in training ML algorithms in case of unbalanced data set is highly debated [38]. Both the balanced and the natural distribution approach can be actually used [39]. We found out that the CNN performs better on the BI-RADS distribution in terms of accuracy, precision and recall. This distribution is the closest to the natural one reported on the BI-RADS Atlas. This result was not unexpected as the native distribution is strongly unbalanced over the density classes, while the uniform one forces us to use far fewer images than the other two. We then visualized the saliency maps computed on the test set to check whether the classifier is looking at the dense part of the breast to perform the classification. We found out that for the medio-lateral oblique projections the saliency maps highlighted more the regions of the pectoral muscle than the dense parenchyma. For this reason, we segmented the pectoral muscle and retrained the classifier. Then, we compared the saliency maps obtained with and without the muscle and we found out that segmentation helps in identifying the correct dense region as shown in Figure 6. Furthermore, the performance in terms of figures of merit increases for the classifier trained with the segmentation.