## 2 Materials and Methods
### Proposed Method
Let the estimated parameters be denoted as \(\hat{\theta}=(\hat{\mu}_{m},\hat{\sigma}_{m}^{2})\), which are expressed below as  
\[\hat{\mu}_{m}=\frac{1}{T}\sum_{i=1}^{T}\hat{y}_{i}\]  
\[\hat{\sigma}_{m}=\sqrt{\frac{1}{T}\sum_{i=1}^{T}(\hat{y}_{i}-\hat{\mu}_{m})^{ 2}}\]  
As the prior probability \(\pi(f_{m})\) for the \(m^{th}\) model is directly proportional to the performance of that model across all metrics, it is arguably relevant to estimate as follows:  
Figure 9: The overall schematic diagram of the proposed method. \[\pi(f_{m})=\frac{Acc_{m}+Spec_{m}+Sens_{m}+F1_{m}}{\sum_{l=1}^{3}(Acc_{l}+ Spec_{l}+Sens_{l}+F1_{l})} \tag{5}\]  
where \(Acc_{m}\), \(Spec_{m}\), \(Sens_{m}\) and \(F1_{m}\) represent, respectively, the accuracy, specificity, sensitivity, and F1-score of the \(m^{th}\) base model. Equation (5) shows that the prior probability \(\pi(f_{m})\) is estimated as the normalized performance of all the individual base models. Also, it indicates that, while the performance of a model increases, the prior probability of that particular model is improved. Now, the estimated values of \(p(D|f_{m})\) and \(\pi(f_{m})\) obtained for all three base learners are substituted in Equation (3). The values of \(p(f_{1}|D)\), \(p(f_{2}|D)\) and \(p(f_{3}|D)\), which are denoted as \(w_{1}\), \(w_{2}\), and \(w_{3}\), are the weights of WRN-40, DenseNet-121, and EfficientNet-B7, respectively. To obtain the ensemble prediction, the weights are multiplied by the corresponding model's prediction and summing them across all the models as follows:  
\[E_{i}^{n}=\sum_{m=1}^{3}w_{m}p_{mi}^{n}\quad i=0,1,\quad m=1,2,3 \tag{6}\]  
where \(E_{i}^{n}\) represents the ensemble probability prediction for the \(n^{th}\) test image of \(i^{th}\) class. And, \(p_{mi}^{n}\) represents the probability predicted by the \(m^{th}\) base model for the \(n^{th}\) test image and \(i^{th}\) class. Finally, using the ensemble probability obtained in Equation (6), the updated predicted class \(C_{n}\) for the \(n^{th}\) test image is obtained as follows:  
\[C_{n}=\arg\max_{i\in\{0,1\}}(E_{i}^{n})\]  
The final predicted class, using the proposed method, is then compared with the true label to generate the confusion matrix. And, based on the confusion matrix, the performance metrics accuracy, sensitivity, specificity, and F1-score are calculated. To investigate the impact of noise on the prediction accuracy, we selected a test image sample and created two versions, one with added noise and another in its original form, and generated heatmaps for both images. The first row in Figure 10 shows the original CT image (without noise) and its heatmaps obtained from WRN-40, DenseNet-121, and EfficientNet-B7, respectively. The subsequent two rows demonstrate the noisy images with different noise levels and their heatmaps. The noisy images are generated by introducing moderate and high Gaussian noise with SD \(\sigma\) = 25 and SD \(\sigma\) = 50, respectively, in the chosen sample image. The feature activation heatmaps are generated using Grad-CAM from the last convolutional layer of all the three base models separately. The heatmaps generated by the base models reveal a consistent focus on the bone regions of the vertebrae, which are the most prominent feature. When moderate noise is introduced, the base models remain effective in concentrating on the bone structure, with minimal impact from the noise. However, as evident in the last row of Figure 10, when the high noise is introduced, the models' confidence in their predictions decreases. The reason is that the vertebral bone structure becomes less apparent due to noise, making it difficult for the models to correctly identify the prominent features. This analysis highlights the robustness of the base models, which have learned strong feature representations that enable them to withstand moderate levels of noise. Nevertheless, it also highlights the limitations of these models when confronted with high levels of noise.