  Most positive and unlabeled data is subject to selection biases. The labeled
examples can, for example, be selected from the positive set because they are
easier to obtain or more obviously positive. This paper investigates how
learning can be ena BHbled in this setting. We propose and theoretically
analyze an empirical-risk-based method for incorporating the labeling
mechanism. Additionally, we investigate under which assumptions learning is
possible when the labeling mechanism is not fully understood and propose a
practical method to enable this. Our empirical analysis supports the
theoretical results and shows that taking into account the possibility of a
selection bias, even when the labeling mechanism is unknown, improves the
trained classifiers.
