Single-sample versus case-control sampling scheme for Positive Unlabeled
  data: the story of two scenarios
Jan Mielniczuk Adam Wawrzeńczyk
In the paper we argue that performance of the classifiers based on Empirical
Risk Minimization (ERM) for positive unlabeled data, which are designed for
case-control sampling scheme may significantly deteriorate when applied to a
single-sample scenario. We reveal why their behavior depends, in all but very
specific cases, on the scenario. Also, we introduce a single-sample case
analogue of the popular non-negative risk classifier designed for case-control
data and compare its performance with the original proposal. We show that the
significant differences occur between them, especiall when half or more
positive of observations are labeled. The opposite case when ERM minimizer
designed for the case-control case is applied for single-sample data is also
considered and similar conclusions are drawn. Taking into account difference of
scenarios requires a sole, but crucial, change in the definition of the
Empirical Risk.