# Estimating the class prior and posterior from noisy positives and unlabeled data

arXiv: https://arxiv.org/abs/1606.08561 · HTML: https://mlsanity.com/paper/1606.08561/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2016.

## Built on by (most cited first, 25 of 25)

- [A Self-supervised Riemannian GNN with Time Varying Curvature for Temporal Graph Learning](https://mlsanity.com/paper/2208.14073/) (2022)
- [Distantly Supervised Named Entity Recognition via Confidence-Based Multi-Class Positive and Unlabeled Learning](https://mlsanity.com/paper/2204.09589/) (2022)
- [Binary Classification from Multiple Unlabeled Datasets via Surrogate Set Classification](https://mlsanity.com/paper/2102.00678/) (2021)
- [Learning Graph Neural Networks with Positive and Unlabeled Nodes](https://mlsanity.com/paper/2103.04683/) (2021)
- [Quantifying disparities in intimate partner violence: a machine learning method to correct for underreporting](https://mlsanity.com/paper/2110.04133/) (2021)
- [When Less is Enough: Positive and Unlabeled Learning Model for Vulnerability Detection](https://mlsanity.com/paper/2308.10523/) (2023)
- [One-Class Risk Estimation for One-Class Hyperspectral Image Classification](https://mlsanity.com/paper/2210.15457/) (2022)
- [Learning from positive and unlabeled examples -Finite size sample bounds](https://mlsanity.com/paper/2507.07354/) (2025)
- [Double logistic regression approach to biased positive-unlabeled data](https://mlsanity.com/paper/2209.07787/) (2022)
- [Leveraging Contaminated Datasets to Learn Clean-Data Distribution with Purified Generative Adversarial Networks](https://mlsanity.com/paper/2302.01722/) (2023)
- [Noisy Positive-Unlabeled Learning with Self-Training for Speculative Knowledge Graph Reasoning](https://mlsanity.com/paper/2306.07512/) (2023)
- [Deep Positive-Unlabeled Anomaly Detection for Contaminated Unlabeled Data](https://mlsanity.com/paper/2405.18929/) (2024)
- [Positive-Unlabeled Diffusion Models for Preventing Sensitive Data Generation](https://mlsanity.com/paper/2503.03789/) (2025)
- [A Novel Perspective for Positive-Unlabeled Learning via Noisy Labels](https://mlsanity.com/paper/2103.04685/) (2021)
- [Verifying the Selected Completely at Random Assumption in Positive-Unlabeled Learning](https://mlsanity.com/paper/2404.00145/) (2024)
- [Prior shift estimation for positive unlabeled data through the lens of kernel embedding](https://mlsanity.com/paper/2502.21194/) (2025)
- [Value Bonuses using Ensemble Errors for Exploration in Reinforcement Learning](https://mlsanity.com/paper/2602.12375/) (2026)
- [Risk bounds for PU learning under Selected At Random assumption](https://mlsanity.com/paper/2201.06277/) (2022)
- [A Boosting Algorithm for Positive-Unlabeled Learning](https://mlsanity.com/paper/2205.09485/) (2022)
- [Regression with Sensor Data Containing Incomplete Observations](https://mlsanity.com/paper/2304.13415/) (2023)
- [Learning from Ambiguous Data with Hard Labels](https://mlsanity.com/paper/2501.01844/) (2025)
- [Skill-based Safe Reinforcement Learning with Risk Planning](https://mlsanity.com/paper/2505.01619/) (2025)
- [Semiparametric Learning from Open-Set Label Shift Data](https://mlsanity.com/paper/2509.14522/) (2025)
- [AUCp: Pseudo-AUC for Inference Model Selection with Unlabeled Validation Data in Abnormality Detection](https://mlsanity.com/paper/2606.08742/) (2026)
- [Importance Weighting for Unlabeled-unlabeled Learning under Distribution Shift](https://mlsanity.com/paper/2609.10994/) (2026)

## Abstract (arXiv)

We develop a classification algorithm for estimating posterior distributions from positive-unlabeled data, that is robust to noise in the positive labels and effective for high-dimensional data. In recent years, several algorithms have been proposed to learn from positive-unlabeled data; however, many of these contributions remain theoretical, performing poorly on real high-dimensional data that is typically contaminated with noise. We build on this previous work to develop two practical classification algorithms that explicitly model the noise in the positive labels and utilize univariate transforms built on discriminative classifiers. We prove that these univariate transforms preserve the class prior, enabling estimation in the univariate space and avoiding kernel density estimation for high-dimensional data. The theoretical development and both parametric and nonparametric algorithms proposed here constitutes an important step towards wide-spread use of robust classification algorithms for positive-unlabeled data.