Class Prior Estimation with Biased Positives and Unlabeled Examples

Positive-unlabeled learning is often studied under the assumption that the labeled positive sample is drawn randomly from the true distribution of positives. In many application domains, however, certain regions in the support of the positive class-conditional distribution are over-represented while others are under-represented in the positive sample. Although this introduces problems in all aspects of positive-unlabeled learning, we begin to address this challenge by focusing on the estimation of class priors, quantities central to the estimation of posterior probabilities and the recovery of true classification performance. We start by making a set of assumptions to model the sampling bias. We then extend the identifiability theory of class priors from the unbiased to the biased setting. Finally, we derive an algorithm for estimating the class priors that relies on clustering to decompose the original problem into subproblems of unbiased positive-unlabeled learning. Our empirical investigation suggests feasibility of the correction strategy and overall good performance.