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

arXiv: https://arxiv.org/abs/2407.09735 · HTML: https://mlsanity.com/paper/2407.09735/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 13 Jul 2024.

## Builds on (most cited first, 10 of 10)

- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Learning from positive and unlabeled data: a survey](https://mlsanity.com/paper/1811.04820/) (2018)
- [Mixture Proportion Estimation via Kernel Embedding of Distributions](https://mlsanity.com/paper/1603.02501/) (2016)
- [Mixture Proportion Estimation and PU Learning: A Modern Approach](https://mlsanity.com/paper/2111.00980/) (2021)
- [Beyond the Selected Completely At Random Assumption for Learning from Positive and Unlabeled Data](https://mlsanity.com/paper/1809.03207/) (2018)
- [Domain Adaptation under Open Set Label Shift](https://mlsanity.com/paper/2207.13048/) (2022)
- [Understanding new tasks through the lens of training data via exponential tilting](https://mlsanity.com/paper/2205.13577/) (2022)
- [ELSA: Efficient Label Shift Adaptation through the Lens of Semiparametric Models](https://mlsanity.com/paper/2305.19123/) (2023)
- [PULasso: High-dimensional variable selection with presence-only data](https://mlsanity.com/paper/1711.08129/) (2017)
- [ReTaSA: A Nonparametric Functional Estimation Approach for Addressing Continuous Target Shift](https://mlsanity.com/paper/2401.16410/) (2024)

## Built on by (most cited first, 3 of 3)

- [Beyond Aggregation: Guiding Clients in Heterogeneous Federated Learning](https://mlsanity.com/paper/2509.23049/) (2025)
- [Semiparametric Learning from Open-Set Label Shift Data](https://mlsanity.com/paper/2509.14522/) (2025)
- [In-Context Positive-Unlabeled Learning](https://mlsanity.com/paper/2605.05591/) (2026)

## Abstract (arXiv)

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.