# Class-prior Estimation for Learning from Positive and Unlabeled Data

arXiv: https://arxiv.org/abs/1611.01586 · HTML: https://mlsanity.com/paper/1611.01586/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2016.

## Built on by (most cited first, 30 of 46)

- [Dist-PU: Positive-Unlabeled Learning from a Label Distribution Perspective](https://mlsanity.com/paper/2212.02801/) (2022)
- [Positive-Unlabeled Reward Learning](https://mlsanity.com/paper/1911.00459/) (2019)
- [Adapting to Online Label Shift with Provable Guarantees](https://mlsanity.com/paper/2207.02121/) (2022)
- [Learnware: Small Models Do Big](https://mlsanity.com/paper/2210.03647/) (2022)
- [Revisiting the Negative Data of Distantly Supervised Relation Extraction](https://mlsanity.com/paper/2105.10158/) (2021)
- [Positive-Unlabeled Learning with Adversarial Data Augmentation for Knowledge Graph Completion](https://mlsanity.com/paper/2205.00904/) (2022)
- [Beyond Myopia: Learning from Positive and Unlabeled Data through Holistic Predictive Trends](https://mlsanity.com/paper/2310.04078/) (2023)
- [Unsupervised Learning under Latent Label Shift](https://mlsanity.com/paper/2207.13179/) (2022)
- [SPADE: Semi-supervised Anomaly Detection under Distribution Mismatch](https://mlsanity.com/paper/2212.00173/) (2022)
- [Quantifying disparities in intimate partner violence: a machine learning method to correct for underreporting](https://mlsanity.com/paper/2110.04133/) (2021)
- [Positive Unlabeled Contrastive Learning](https://mlsanity.com/paper/2206.01206/) (2022)
- [Positive-Unlabeled Learning using Random Forests via Recursive Greedy Risk Minimization](https://mlsanity.com/paper/2210.08461/) (2022)
- [Positive-Unlabeled Classification under Class-Prior Shift: A Prior-invariant Approach Based on Density Ratio Estimation](https://mlsanity.com/paper/2107.05045/) (2021)
- [Learning from positive and unlabeled examples -Finite size sample bounds](https://mlsanity.com/paper/2507.07354/) (2025)
- [Robust Positive-Unlabeled Learning via Noise Negative Sample Self-correction](https://mlsanity.com/paper/2308.00279/) (2023)
- [Cell Detection from Imperfect Annotation by Pseudo Label Selection Using P-classification](https://mlsanity.com/paper/2107.09289/) (2021)
- [Uncertainty-aware Pseudo-label Selection for Positive-Unlabeled Learning](https://mlsanity.com/paper/2201.13192/) (2022)
- [Positive-Unlabeled Domain Adaptation](https://mlsanity.com/paper/2202.05695/) (2022)
- [Double logistic regression approach to biased positive-unlabeled data](https://mlsanity.com/paper/2209.07787/) (2022)
- [Offline Reinforcement Learning with Domain-Unlabeled Data](https://mlsanity.com/paper/2404.07465/) (2024)
- [LeQua@CLEF2022: Learning to Quantify](https://mlsanity.com/paper/2111.11249/) (2021)
- [Leveraging Contaminated Datasets to Learn Clean-Data Distribution with Purified Generative Adversarial Networks](https://mlsanity.com/paper/2302.01722/) (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)
- [A Positive/Unlabeled Approach for the Segmentation of Medical Sequences using Point-Wise Supervision](https://mlsanity.com/paper/2107.08394/) (2021)
- [Learning from Positive and Unlabeled Data with Augmented Classes](https://mlsanity.com/paper/2207.13274/) (2022)
- [PU GNN: Chargeback Fraud Detection in P2E MMORPGs via Graph Attention Networks with Imbalanced PU Labels](https://mlsanity.com/paper/2211.08604/) (2022)
- [Classification from Positive and Biased Negative Data with Skewed Labeled Posterior Probability](https://mlsanity.com/paper/2203.05749/) (2022)
- [A Boosting Algorithm for Positive-Unlabeled Learning](https://mlsanity.com/paper/2205.09485/) (2022)

## Abstract (arXiv)

We consider the problem of estimating the class prior in an unlabeled dataset. Under the assumption that an additional labeled dataset is available, the class prior can be estimated by fitting a mixture of class-wise data distributions to the unlabeled data distribution. However, in practice, such an additional labeled dataset is often not available. In this paper, we show that, with additional samples coming only from the positive class, the class prior of the unlabeled dataset can be estimated correctly. Our key idea is to use properly penalized divergences for model fitting to cancel the error caused by the absence of negative samples. We further show that the use of the penalized L1-distance gives a computationally efficient algorithm with an analytic solution. The consistency, stability, and estimation error are theoretically analyzed. Finally, we experimentally demonstrate the usefulness of the proposed method.