## 4 The Proposed Model
### Evidential Deep Learning Module  
#### 4.3.1. Evidential Deep Learning for Classification  
In deep learning related classification tasks, it is common to utilize the softmax function to transform the models' output values into class probabilities. However, the softmax function can lead to overconfidence in the model even when it produces incorrect results [27]. Recently, evidential deep learning (EDL) based on second-order uncertainty modeling to parameterize the Dirichlet distribution of class probabilities provides the model with the ability to \"know unknown\". Specifically, given a \(K\)-class classification problem, EDL assumes that the class probabilities, denoted as \(\mathbf{p}=(p_{1},p_{2},\ldots,p_{k})\), follow a prior Dirichlet distribution \(D(\mathbf{p}\mid\mathbf{\omega})\), where \(\mathbf{\omega}=(\omega_{1},\omega_{2},\ldots,\omega_{k})\). Building upon the Dempster-Shafer Theory (DST) [28] and Subjective Logic (SL) [29], the \(\mathbf{\omega}\) is connected to the acquired evidence \(\mathbf{e}\) through the equation \(\mathbf{\omega}=\mathbf{e}+1\). It is worth noting that to ensure a non-negative output that serves as the evidence vector for the predicted Dirichlet distribution, the previous softmax layer is substituted with a ReLU layer. Then, the predicted uncertainty \(u\) and the class probability of category \(j\) are obtained as:  
\[u=\frac{K}{S} \tag{4}\]  
\[p_{j}=\frac{\omega_{j}}{S} \tag{5}\]  
where \(S=\sum_{i}^{K}(e_{i}+1)\) is also referred to as the Dirichlet strength.  
For each \(i\)-th sample, we adopt the following form of the cross-entropy loss function during the training process:  
\[\mathcal{L}_{i}^{CE}=\int\left[\sum_{j=1}^{3}-s_{ij}log\big{(}p_{ij}\big{)} \right]\frac{1}{B(\mathbf{\omega_{i}})}\prod_{j=1}^{3}\;p_{ij}^{\omega_{ij}-1}d\bm {p}_{i} \tag{6}\]  
where \(p_{ij}\) denotes the predicted probability of the \(i\)-th sample being diagnosed as the \(j\)-th stage, \(s_{ij}\) is the \(j\)-th component of the disease diagnosis vector represented by one-hot encoding, and \(B(\mathbf{\omega_{i}})\) represents the \(K\)-dimensional multinomial beta function. In this task, the value of \(K\) is 3, indicating the three stages of diagnosis.  
In addition, to address the issue of misleading evidence generated for incorrect labels, we propose the introduction of a KL divergence term. This term aims to enforce the cumulative evidence to converge towards 0 when a sample is misclassified.  
\[\mathcal{L}_{i}^{KL}=KL[D(\mathbf{p}_{i}\mid\mathbf{\tilde{\omega}_{i}})\parallel D( \mathbf{p}_{i}\mid 1)] \tag{7}\]  
where \(\mathbf{\tilde{\omega_{i}}}=\mathbf{s_{i}}+(1-\mathbf{s_{i}})\odot\mathbf{\omega_{i}}\) is the adjusted Dirichlet distribution parameter.  
In conclusion, for the classification problem, the overall loss function for the \(i\)-th sample can be defined as follows:  
\[\mathcal{L}_{i}^{c}=\mathcal{L}_{i}^{CE}+\lambda_{epoch}\;\mathcal{L}_{i}^{KL} \tag{8}\]  
where \(\lambda_{epoch}=\min(1.0,epoch/10)\in[0,1]\) is the annealing coefficient. \(epoch\) is the index of the current training epoch. By gradually increasing \(\lambda_{epoch}\), the network can assign relatively lower importance to the KL divergence term at the beginning of training, which has a positive effect on the exploration of the parameter space and avoids outputting a flat uniform distribution.