NBEB-PUL: a naïve Bayesian ensemble-based algorithm for positive and unlabeled learning

Existing ensemble-based positive and unlabeled learning (PUL) methods often fail to adequately account for the impact of noise in both positive and unlabeled samples during classifier construction. To address this issue, this paper proposes a naïve Bayesian ensemble-based PUL algorithm (NBEB-PUL), which consists of two stages: label assignment and noise filtering. In the label assignment stage, NBEB-PUL employs naïve Bayesian classifiers as base learners and integrates them using the AdaBoosting ensemble strategy to construct a strong classifier, named Ada-NBC. This strong classifier is then utilized to compute the mean posterior probabilities of a validation set for unlabeled samples, enabling explicit modeling of labeling uncertainty. Based on low-uncertainty predictions a set of reliable positive samples is iteratively identified, along with a residual unlabeled sample set. In the noise filtering stage, NBEB-PUL leverages the ensemble classifier generated in the first stage to reclassify the reliable positive set and the residual unlabeled set, resulting in Ada-NBC-predicted positive and negative sample sets. These sets are then intersected with the first-stage positive and unlabeled sets to extract overlapping high-confidence samples, forming the final positive and negative sample sets. The samples pruned from the dataset during this process are identified as noise. The feasibility, rationality, and effectiveness of NBEB-PUL were validated on 23 benchmark datasets from UCI and KEEL. Experimental results demonstrate that the algorithm exhibits stable convergence during the training process as the number of iterations increases. Moreover, NBEB-PUL outperforms six state-of-the-art PUL algorithms (S-EM, Biased-SVM, Modified-PUL, PU-LP, LP-PUL, and AdaPU) in terms of classification accuracy under varying positive sample proportions of 0.45, 0.40, 0.35, and 0.30. These results confirm that NBEB-PUL provides an effective and robust solution for positive and unlabeled learning in the presence of noise.