Title: A Flexible Procedure for Mixture Proportion Estimation in Positive--Unlabeled Learning,
Abstract: Positive--unlabeled (PU) learning considers two samples, a positive set P
with observations from only one class and an unlabeled set U with observations
from two classes. The goal is to classify observations in U. Class mixture
proportion estimation (MPE) in U is a key step in PU learning. Blanchard et al.
[2010] showed that MPE in PU learning is a generalization of the problem of
estimating the proportion of true null hypotheses in multiple testing problems.
Motivated by this idea, we propose reducing the problem to one dimension via
construction of a probabilistic classifier trained on the P and U data sets
followed by application of a one--dimensional mixture proportion method from
the multiple testing literature to the observation class probabilities. The
flexibility of this framework lies in the freedom to choose the classifier and
the one--dimensional MPE method. We prove consistency of two mixture proportion
estimators using bounds from empirical process theory, develop tuning parameter
free implementations, and demonstrate that they have competitive performance on
simulated waveform data and a protein signaling problem.