Title: Learning from Positive and Unlabeled Data with a Selection Bias. Abstract: We consider the problem of learning a binary classifier only from positive data and unlabeled data (PU learning). Recent methods of PU learning commonly assume that the labeled positive data are identically distributed as the unlabeled positive data. However, this assumption is unrealistic in many instances of PU learning because it fails to capture the existence of a selection bias in the labeling process. When the data has a selection bias, it is difficult to learn the Bayes optimal classifier by conventional methods of PU learning. In this paper, we propose a method to partially identify the classifier. The proposed algorithm learns a scoring function that preserves the order induced by the class posterior under mild assumptions, which can be used as a classifier by setting an appropriate threshold. Through experiments, we show that the method outperforms previous methods for PU learning on various real-world datasets.