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Positive and Unlabeled (PU) learning is a type of semi-supervised binaryclassification where the machine learning algorithm differentiates between aset of positive instances (labeled) and a set of both positive and negativeinstances (unlabeled). PU learning has broad applications in settings whereconfirmed negatives are unavailable or difficult to obtain, and there is valuein discovering positives among the unlabeled (e.g., viable drugs among untestedcompounds). Most PU learning algorithms make the selected completely at random(SCAR) assumption, namely that positives are selected independently of theirfeatures. However, in many real-world applications, such as healthcare,positives are not SCAR (e.g., severe cases are more likely to be diagnosed),leading to a poor estimate of the proportion, $\alpha$, of positives amongunlabeled examples and poor model calibration, resulting in an uncertaindecision threshold for selecting positives. PU learning algorithms can estimate$\alpha$ or the probability of an individual unlabeled instance being positiveor both. We propose two PU learning algorithms to estimate $\alpha$, calculatecalibrated probabilities for PU instances, and improve classification metrics:i) PULSCAR (positive unlabeled learning selected completely at random), and ii)PULSNAR (positive unlabeled learning selected not at random). PULSNAR uses adivide-and-conquer approach that creates and solves several SCAR-likesub-problems using PULSCAR. In our experiments, PULSNAR outperformedstate-of-the-art approaches on both synthetic and real-world benchmarkdatasets.

Positive Unlabeled Learning Selected Not At Random (PULSNAR): class proportion estimation when the SCAR assumption does not hold