# Identifying At-Risk Students for Early Intervention--A Probabilistic Machine Learning Approach
## 3 Materials and Methods
### Parameter Uncertainty and Model Evaluation  
A forest and a posterior plot were used at every stage, s, with a 94% highest density interval (HDI) for parameter uncertainty. The HDI is one of the ways of defining a credible interval. The HDI credible interval was used to indicate which distribution points are most credible and which cover most of the distribution. Credible intervals are the uncertainty levels in the model's parameters.  
The evaluation of the predictive models' performance was conducted utilizing several established metrics, namely, accuracy, F1-score, precision, and recall. These metrics were derived from the false negative (FN), false positive (FP), true negative (TN), and true positive (TP) values obtained from the confusion matrix. Such a comprehensive evaluation ensures a thorough and objective assessment of the models' predictive capabilities in identifying at-risk students for early intervention.  
\[\text{Accuracy}=\frac{\text{TP}+\text{TN}}{\text{TP}+\text{FP}+\text{FN}+ \text{TN}} \tag{11}\]  
Accuracy is a measure of how correctly the model identifies the at-risk students. It is defined as the ratio of the number of correctly identified at-risk students to the total number of students in the dataset. High accuracy means that the model can accurately identify at-risk students.  
\[\text{Precision}=\frac{\text{TP}}{\text{TP}+\text{FP}} \tag{12}\]  
Precision is the ratio of true positives (students who are identified as at-risk and are actually at-risk) to the total number of students identified as at-risk (true positives plus false positives). A high precision indicates that when the model identifies a student as at-risk, it is likely to be correct.  
\[\text{Recall}=\frac{\text{TP}}{\text{TP}+\text{FN}} \tag{13}\]  
Recall is the ratio of true positives to the total number of at-risk students in the dataset (true positives plus false negatives). A high recall indicates that the model can identify most of the at-risk students in the dataset.  
\[\text{F1}-\text{Score}=\frac{2\text{TP}}{2\text{TP}+\text{FP}+\text{FN}} \tag{14}\]  
F1-score is a measure of the model's accuracy that considers both precision and recall. It is the harmonic mean of precision and recall. A high F1-score indicates that the model performs well in identifying at-risk students.  
The proposed study presents several significant societal benefits. Firstly, it can improve academic outcomes by identifying at-risk students earlier, allowing for timely interventions and support to improve academic achievement. This can ultimately lead to higher student retention rates and reduce costs associated with student attrition. Secondly, it can enable more efficient use of resources by allowing targeted interventions, reducing the need for resources to be allocated to students who may not need them. Thirdly, this study highlights the potential of probabilistic machine learning for incorporating domain knowledge and uncertainty, thereby improving the accuracy and usefulness of predictive models in various fields beyond education. Fourthly, it can enhance teachers' effectiveness by enabling them to tailor their teaching approach and resources to meet the needs of each student. Fifthly, it can promote equity by ensuring that all students receive appropriate support, regardless of background or demographic characteristics. Overall, this study offers far-reaching benefits for society, showcasing the potential of probabilistic machine learning for improving academic outcomes, reducing educational disparities, and enhancing the overall efficiency of the education system.