## 2 Materials and Methods
### Proposed Method
```
Input: \(\text{D}\leftarrow\text{Dataset}\) \(\{M_{1},M_{2},M_{3}\}\leftarrow\text{Base models}\) \(f_{m}\leftarrow\text{Predicted probability of }m^{th}\) model \(y_{T}\leftarrow\text{Ground truth of }T^{th}\) image Output: \(w_{m}\leftarrow\text{weight of }m^{th}\) model \(p_{mi}^{n}\leftarrow\text{probability of }m^{th}\) model for \(n^{th}\) test image from \(i^{th}\) class \(E_{i}^{n}\leftarrow\text{ensemble probability for }n^{th}\) test image of \(i^{th}\) class \(C_{n}\leftarrow\text{predicted class of }n^{th}\) test image Bayesian Probability-based Ensemble Learning: for\(M_{k}\in\text{Base models}\)do  Obtain the prediction for each sample from all \(M_{k}\)  Define the posterior probability distribution \(M_{k}\) for Calculate \(p(f_{m}|D)\) using Bayes theorem do  Express \(p(f_{m}|D)\) in terms of total probability distribution  Calculate \(p(D|f_{m})\) and \(\pi(f_{m})\) while\(\hat{y}\sim\mathcal{N}(\mu_{m},\sigma_{m}^{2})\)do  Obtain \(\hat{\mu}\) and \(\hat{\sigma}\) using MLE  Calculate \(\pi(f_{m})\) using \(Acc_{m},Spec_{m},Sens_{m}\) and \(F1_{m}\)  Calculate weights \(p(D|f_{m})\) using \(\hat{\mu},\hat{\sigma}\) and \(\pi(f_{m})\) FunctionEnsemblePredictions(\(w_{m},p_{m}\)): \(w_{m},p_{m}\gets M_{k}\)for\(m\leftarrow\text{Models}\)do for\(n\in\{0,1\}\)do for\(i\gets i^{th}\) test imagedo \(E_{i}^{n}=\sum_{m=1}^{3}w_{m}\cdot p_{mi}^{n}\) ; return\(E_{n}^{i}\) Class Predictions: \(C_{n}=\arg\max_{i\in\{0,1\}}[E_{i}^{n}]\)
```  
**Algorithm 1**Proposed ensemble learning algorithm  
The dataset \(D=[y_{\text{obs}_{i}},y_{\text{obs}_{j}},\ldots,y_{\text{obs}_{T}}]\) comprises the true labels of the images in the training set, where \(T\) is the total number of images. Let the outcome (probability) of each base model for a given image, belonging to the fracture or non-fracture class, be represented by \(y_{i}\), \(i=0,1\). The three base models, WRN-40, DenseNet-121, and EfficientNet-B7 are denoted by \(M_{1}\), \(M_{2}\) and \(M_{3}\), respectively. Suppose the predictions of \(M_{1}\), \(M_{2}\), and \(M_{3}\) for all the \(T\) images are represented by \(f_{1}\), \(f_{2}\) and \(f_{3}\), respectively. Therefore, each \(f_{m},m=1,2,3\) contains a random sequence of probability values \(y_{i}\), of length \(T\). Then, the intuition  
where \(p(y|f_{m},D)\) is the probability of \(y\) given the \(m^{th}\) model prediction and data \(D\), which also represents the actual prediction obtained by the \(m^{th}\) model. And, \(p(f_{m}|D)\) is the posterior probability of the \(m^{th}\) model given the data \(D\), which also acts as the weight assigned to the \(m^{th}\) model. If the \(m^{th}\) model performs well on the validation set, then the value of \(p(f_{m}|D)\) will also be higher [40]. The posterior distribution \(p(f_{m}|D)\) can also be expressed based on [PERSON] theorem as,  
\[p(f_{m}|D)=\frac{p(D|f_{m})\pi(f_{m})}{\sum_{j=1}^{3}p(D|f_{j})\pi(f_{j})}\quad,\quad m=1,2,3 \tag{3}\]  
where \(p(D|f_{m})\) represents the likelihood of the data given prediction \(f_{m}\) and \(\pi(f_{m})\) is the prior probability of the \(m^{th}\) model. As per the central limit theorem, it is assumed that the combined predicted distribution, \(\hat{y}\), from all three models follows a normal distribution with mean \(\mu_{m}\) and SD \(\sigma_{m}\), i.e., \(\hat{y}\sim\mathcal{N}(\mu_{m},\sigma_{m}^{2})\). The probability distribution function (PDF) of \(\hat{y}\) is expressed as  
\[f(\hat{y})=\frac{1}{\sigma_{m}\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{ \hat{y}-\mu_{m}}{\sigma_{m}}\right)^{2}\right) \tag{4}\]  
The likelihood \(p(D|f_{m})\) in Equation (3) can also be expressed as \(f(\hat{y}\mid\theta_{m})\), where \(\theta_{m}=(\mu_{m},\sigma_{m}^{2})\). Subsequently, the parameters of Equation (4) are estimated using the maximum MLE [41].