Positive and Unlabeled Data: Model, Estimation, Inference, and Classification

TL;DR

This paper presents the double exponential tilting model (DETM) for positive and unlabeled data, effectively handling complex scenarios where labeled and unlabeled positives come from different distributions, with theoretical and practical validation.

Contribution

Introduces DETM, a novel model for PU data that accommodates different distributions of labeled and unlabeled positives, with theoretical foundations and inference methods.

Findings

DETM effectively models complex PU data scenarios.

Theoretical properties including identifiability and asymptotics are established.

Practical performance demonstrated through classification and inference tasks.

Abstract

This study introduces a new approach to addressing positive and unlabeled (PU) data through the double exponential tilting model (DETM). Traditional methods often fall short because they only apply to selected completely at random (SCAR) PU data, where the labeled positive and unlabeled positive data are assumed to be from the same distribution. In contrast, our DETM's dual structure effectively accommodates the more complex and underexplored selected at random PU data, where the labeled and unlabeled positive data can be from different distributions. We rigorously establish the theoretical foundations of DETM, including identifiability, parameter estimation, and asymptotic properties. Additionally, we move forward to statistical inference by developing a goodness-of-fit test for the SCAR condition and constructing confidence intervals for the proportion of positive instances in the target domain. We leverage an approximated Bayes classifier for classification tasks, demonstrating DETM's robust performance in prediction. Through theoretical insights and practical applications, this study highlights DETM as a comprehensive framework for addressing the challenges of PU data.