Naive Bayes is among the simplest probabilistic classifiers. It often performs surprisingly well in many real world applications, despite the strong assumption that all features are conditionally independent given the class. In the learning process of this classifier with the known structure, class probabilities and conditional probabilities are calculated using training data, and then values of these probabilities are used to classify new observations. In this paper, we introduce three novel optimization models for the naive Bayes classifier where both class probabilities and conditional probabilities are considered as variables. The values of these variables are found by solving the corresponding optimization problems. Numerical experiments are conducted on several real world binary classification data sets, where continuous features are discretized by applying three different methods. The performances of these models are compared with the naive Bayes classifier, tree augmented naive Bayes, the SVM, C4.5 and the nearest neighbor classifier. The obtained results demonstrate that the proposed models can significantly improve the performance of the naive Bayes classifier, yet at the same time maintain its simple structure.

Integrating machine learning (ML) methods in educational research has the potential to greatly impact upon research, teaching, learning and assessment by enabling personalised learning, adaptive assessment and providing insights into student performance, progress and learning patterns. To reveal more about this notion, we investigated ML approaches used for educational data analysis in the last decade and provided recommendations for further research. Using a systematic literature review (SLR), we examined 77 publications from two large and high-impact databases for educational research using bibliometric mapping and evaluative review analysis. Our results suggest that the top five most frequently used keywords were similar in both databases. The majority of the publications (88%) utilised supervised ML approaches for predicting students’ performances and finding learning patterns. These methods include decision trees, support vector machines, random forests, and logistic regression. Semi-supervised learning methods were less frequently used, but also demonstrated promising results in predicting students’ performance. Finally, we discuss the implications of these results for statisticians, researchers, and policymakers in education.

The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.