# MIPART: A Partial Decision Tree-Based Method for Multiple-Instance Classification
## 3 Materials and Methods
### Ensemble Multiple Instance Classification  
In ensemble multiple instance classification, multiple classifiers are combined to improve predictive accuracy and generalization. This approach aggregates the predictions of individual classifiers, commonly using techniques such as majority voting for categorical targets or averaging methods for numerical targets (e.g., weighted averages, medians,minimum, or maximum values). The goal is to build a more robust classifier by leveraging the strengths of multiple models, resulting in improved classification performance for multi-instance data. Research has demonstrated that ensemble approaches consistently outperform single-instance classifiers in terms of accuracy and reliability. Ensemble methods typically fall into four main types, namely Boosting [39], Stacking [40], Voting [41], and Bagging, each of which can be adapted for multiple instance classification tasks.  
Among these techniques, bootstrap aggregating (Bagging) [42] is a widely implemented ensemble method that enhances the robustness and accuracy of classifiers. Bagging operates by creating multiple training subsets from the original dataset, where instances are selected randomly with replacement. This means that some instances may appear in multiple subsets, while others may not be included at all. Each subset is then used to train a separate classifier, resulting in an ensemble of diverse models. Once trained, the predictions from these individual classifiers are aggregated, typically through majority voting for categorical outcomes or averaging for numerical outputs. It is supposed that for each bootstrap sample \(S_{b}\), the classifier \(C_{b}\) is trained, resulting in an ensemble of diverse models. Once trained, the predictions from these individual classifiers are aggregated, typically through majority voting for categorical outcomes or averaging for numerical outputs. The final prediction for a new bag \(B_{j}\) is given by Equation (4), as follows:  
\[\hat{C}\big{(}B_{j}\big{)}=argmax_{c}(\sum_{b=1}^{B}I(C_{b}(B_{j})=c)) \tag{4}\]  
where \(I\) is the indicator function that equals 1 if the classifier \(C_{b}\) predicts class \(c\) for bag \(B_{j}\), and 0 otherwise, indicating whether a specific prediction contributes to the count for the class \(c\) when aggregating the predictions from multiple classifiers. Additionally, \(\hat{C}\big{(}B_{j}\big{)}\) represents the predicted class label for bag \(B_{j}\), while \(argmax_{c}\) identifies the class \(c\) that maximizes the aggregated predictions, effectively determining the final classification outcome. This aggregation process mitigates the effects of outliers and reduces variance, ultimately leading to a more stable and accurate final model. The flexibility of Bagging allows it to adapt well to the unique characteristics of multi-instance context, where the variability among bags can complicate the classification task. In our proposed MIPART method, we have selected Bagging as the ensemble technique to enhance the performance of our classifier. By combining the strengths of multiple learners through Bagging, we aim to improve the model's ability to generalize across different multi-instance datasets, thus achieving higher predictive accuracy and reliability.