## 3 Materials and Methods
### Shapley Values Implementation
Additionally, the Background (B) is treated as an additional sector. The image depicts the configuration for a right eye; for a left eye, the sectors are mirrored horizontally. For practical purposes, another important problem to solve is how to obtain an image in which only a subset \(S\subset N\) of sectors is present, i.e., how to simulate the absence of one or more sectors to form the coalitions required for the calculation of the Shapley values. In our case, we have solved this by occlusion. When we talk about occlusion in this context, we mean it in the same sense as the occlusion method in [40], i.e., hiding or covering parts of the image so that the neural network model does not have access to that information. Specifically, we have chosen to occlude the sectors with black, i.e., setting all their pixels to 0 in the three RGB channels, which gives us a total of 128 coalitions (\(2^{7}\)), including the occlusion of the image background. Figure 3 shows an example of occluded sectors in an image. Note that we are alternatively using \"coalition\" and \"combination of occluded sectors\", although formally a coalition \(C\subseteq N\) would represent the sectors that enter the game, while the corresponding combination \(O\) of occluded sectors would be the sectors that are not present in that coalition, \(O=N\setminus C\). Note also that the calculation of the Shapley value also includes in particular \(C=\emptyset\) and \(C=N\), i.e., respectively, an image in which all sectors are occluded or an image in which none are occluded (the original image). The Shapley values, as previously mentioned, are the contribution of each sector to the difference between the model prediction for the original image and the prediction for the empty set. In our case, since our models return two output probabilities (normal and glaucoma classes), where the sum of both will be 1 by the application of the softmax activation function, we are only considering the probability that the image corresponds to a glaucoma case. Therefore, to calculate the Shapley value for each sector \(j\in F\) we have proceeded as follows:  
1. For each input image \(I\), generate all possible combinations of occluded sectors for \(I\), taking into account whether the image belongs to a left or right eye to properly locate the sectors. 2. For each sector \(j\) of \(I\), calculate the probability of the glaucoma class of the model \(f_{S}\) and \(f_{S\cup\{j\}}\) for each possible coalition \(S\subseteq F\setminus\{j\}\) of sectors, using as input \(I_{S}\) and \(I_{S\cup\{j\}}\), the images generated from \(I\) with that coalition of sectors present in the image. 3. With the above probabilities, calculate the Shapley value \(\phi_{j}\) for sector \(j\) following Equation (2). Figure 3: Example of occluded sectors in an image of the left eye of a normal subject extracted from RIM-ONE DL. The nasal inferior (NI) and nasal superior (NS) sectors have been occluded. (**a**) Original color image. (**b**) Example of occlusion of the NI and NS sectors. The most straightforward way to calculate the output probabilities for all possible coalitions of sectors would be to use the original trained models, as they stand. However, as discussed in [41], one of the basic assumptions in machine learning is that training and testing data come from the same distribution, so if we occlude a portion of the sectors in the image when evaluating our models, we could be changing the distribution from which the testing data comes. In this scenario, it would be unclear whether the prediction of a model is affected because the removed features were important or because of the change in the distribution of the data. Therefore, it seems appropriate to re-train the models to accommodate these new images with occluded sectors. From an algorithm design perspective, we considered two alternative retraining strategies that reflect a trade-off between model-centric and data-centric fidelity:  
1.