A Flexible Procedure for Mixture Proportion Estimation in Positive-Unlabeled Learning

TL;DR

This paper introduces a flexible, classifier-based framework for estimating mixture proportions in positive-unlabeled learning, demonstrating theoretical consistency and competitive empirical performance across different datasets.

Contribution

It proposes a novel, adaptable approach combining probabilistic classification with one-dimensional mixture proportion estimation, with proven consistency and practical implementations.

Findings

The proposed estimators are consistent under certain conditions.

The method performs competitively on simulated and real data.

It offers tuning parameter free implementations for practical use.

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.