# Beyond the Selected Completely At Random Assumption for Learning from Positive and Unlabeled Data

arXiv: https://arxiv.org/abs/1809.03207 · HTML: https://mlsanity.com/paper/1809.03207/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2018.

## Built on by (most cited first, 30 of 36)

- [Doubly-Robust Estimation for Correcting Position-Bias in Click Feedback for Unbiased Learning to Rank](https://mlsanity.com/paper/2203.17118/) (2022)
- [Correcting Exposure Bias for Link Recommendation](https://mlsanity.com/paper/2106.07041/) (2021)
- [Unbiased Learning to Rank Meets Reality: Lessons from Baidu's Large-Scale Search Dataset](https://mlsanity.com/paper/2404.02543/) (2024)
- [Mixture Proportion Estimation Beyond Irreducibility](https://mlsanity.com/paper/2306.01253/) (2023)
- [Quantifying disparities in intimate partner violence: a machine learning method to correct for underreporting](https://mlsanity.com/paper/2110.04133/) (2021)
- [Semi-Supervised Disease Classification based on Limited Medical Image Data](https://mlsanity.com/paper/2405.04295/) (2024)
- [Learning from positive and unlabeled examples -Finite size sample bounds](https://mlsanity.com/paper/2507.07354/) (2025)
- [From Lazy to Prolific: Tackling Missing Labels in Open Vocabulary Extreme Classification by Positive-Unlabeled Sequence Learning](https://mlsanity.com/paper/2408.08981/) (2024)
- [From Biased Selective Labels to Pseudo-Labels: An Expectation-Maximization Framework for Learning from Biased Decisions](https://mlsanity.com/paper/2406.18865/) (2024)
- [Double logistic regression approach to biased positive-unlabeled data](https://mlsanity.com/paper/2209.07787/) (2022)
- [Positive and Unlabeled Data: Model, Estimation, Inference, and Classification](https://mlsanity.com/paper/2407.09735/) (2024)
- [Botcha: Detecting Malicious Non-Human Traffic in the Wild](https://mlsanity.com/paper/2103.01428/) (2021)
- [Stratified Learning: A General-Purpose Statistical Method for Improved Learning under Covariate Shift](https://mlsanity.com/paper/2106.11211/) (2021)
- [Learning from Positive and Unlabeled Data with Augmented Classes](https://mlsanity.com/paper/2207.13274/) (2022)
- [Automatic Debiased Learning from Positive, Unlabeled, and Exposure Data](https://mlsanity.com/paper/2303.04797/) (2023)
- [Single-sample versus case-control sampling scheme for Positive Unlabeled data: the story of two scenarios](https://mlsanity.com/paper/2312.02095/) (2023)
- [Verifying the Selected Completely at Random Assumption in Positive-Unlabeled Learning](https://mlsanity.com/paper/2404.00145/) (2024)
- [Novel Node Category Detection Under Subpopulation Shift](https://mlsanity.com/paper/2404.01216/) (2024)
- [Unidentified and Confounded? Understanding Two-Tower Models for Unbiased Learning to Rank](https://mlsanity.com/paper/2506.20501/) (2025)
- [Accessible, Realistic, and Fair Evaluation of Positive-Unlabeled Learning Algorithms](https://mlsanity.com/paper/2509.24228/) (2025)
- [A proposal for PU classification under Non-SCAR using clustering and logistic model](https://mlsanity.com/paper/2604.17130/) (2026)
- [Quantifying and Auditing LLM Evaluation via Positive--Unlabeled Learning](https://mlsanity.com/paper/2606.19057/) (2026)
- [Detecting corruption in single-bidder auctions via positive-unlabelled learning](https://mlsanity.com/paper/2102.05523/) (2021)
- [Risk bounds for PU learning under Selected At Random assumption](https://mlsanity.com/paper/2201.06277/) (2022)
- [Positive Unlabeled Learning Selected Not At Random (PULSNAR): class proportion estimation when the SCAR assumption does not hold](https://mlsanity.com/paper/2303.08269/) (2023)
- [Soft Label PU Learning](https://mlsanity.com/paper/2405.01990/) (2024)
- [Augmented prediction of a true class for Positive Unlabeled data under selection bias](https://mlsanity.com/paper/2407.10309/) (2024)
- [Learning from Concealed Labels](https://mlsanity.com/paper/2412.02230/) (2024)
- [ESA: Example Sieve Approach for Multi-Positive and Unlabeled Learning](https://mlsanity.com/paper/2412.02240/) (2024)
- [Open-Set Domain Adaptation Under Background Distribution Shift: Challenges and A Provably Efficient Solution](https://mlsanity.com/paper/2512.01152/) (2025)

## Abstract (arXiv)

Most positive and unlabeled data is subject to selection biases. The labeled examples can, for example, be selected from the positive set because they are easier to obtain or more obviously positive. This paper investigates how learning can be ena BHbled in this setting. We propose and theoretically analyze an empirical-risk-based method for incorporating the labeling mechanism. Additionally, we investigate under which assumptions learning is possible when the labeling mechanism is not fully understood and propose a practical method to enable this. Our empirical analysis supports the theoretical results and shows that taking into account the possibility of a selection bias, even when the labeling mechanism is unknown, improves the trained classifiers.