Bayesian belief network for positive unlabeled learning with uncertainty

•UPTAN, a Bayesian network, for uncertain data under PU learning scenario is given.•Uncertain Conditional Mutual Information (UCMI) is proposed.•The algorithm for learning the structure of the Bayesian network is given.•The approach for estimating parameters of the Bayesian network is given.•UPTAN outperforms UPNB, a state-of-art algorithm, in our experiment. The current state-of-art for tackling the problem of classification of static uncertain data under PU learning (Positive Unlabeled Learning) scenario, is UPNB. It is based on the Bayesian assumption, which does not hold for real-life applications, and hence it may depress the classification performance of UPNB. In this paper, we propose UPTAN (Uncertain Positive Tree Augmented Naive Bayes), a Bayesian network algorithm, so as to utilize the dependence information among uncertain attributes for classification. We propose uncertain conditional mutual information (UCMI) for measuring the mutual information between uncertain attributes, and then use it to learn the tree structure of Bayesian network. Furthermore, we give our approach for estimating the parameters of the Bayesian network for uncertain data without negative training examples. Our experiments on 20 UCI datasets show that UPTAN has excellent classification performance, with average F1 being 0.8257, which outperforms UPNB by 3.73%.