# Instance-Dependent PU Learning by Bayesian Optimal Relabeling

arXiv: https://arxiv.org/abs/1808.02180 · HTML: https://mlsanity.com/paper/1808.02180/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2018.

## Built on by (most cited first, 18 of 18)

- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Multiscale Positive-Unlabeled Detection of AI-Generated Texts](https://mlsanity.com/paper/2305.18149/) (2023)
- [Alleviating Structural Distribution Shift in Graph Anomaly Detection](https://mlsanity.com/paper/2401.14155/) (2024)
- [Fairness-aware Model-agnostic Positive and Unlabeled Learning](https://mlsanity.com/paper/2206.09346/) (2022)
- [Beyond Myopia: Learning from Positive and Unlabeled Data through Holistic Predictive Trends](https://mlsanity.com/paper/2310.04078/) (2023)
- [SPADE: Semi-supervised Anomaly Detection under Distribution Mismatch](https://mlsanity.com/paper/2212.00173/) (2022)
- [Positive-Unlabeled Learning using Random Forests via Recursive Greedy Risk Minimization](https://mlsanity.com/paper/2210.08461/) (2022)
- [Learning from positive and unlabeled examples -Finite size sample bounds](https://mlsanity.com/paper/2507.07354/) (2025)
- [Evaluating the Predictive Performance of Positive-Unlabelled Classifiers: a brief critical review and practical recommendations for improvement](https://mlsanity.com/paper/2206.02423/) (2022)
- [DialogVCS: Robust Natural Language Understanding in Dialogue System Upgrade](https://mlsanity.com/paper/2305.14751/) (2023)
- [Double logistic regression approach to biased positive-unlabeled data](https://mlsanity.com/paper/2209.07787/) (2022)
- [PSPU: Enhanced Positive and Unlabeled Learning by Leveraging Pseudo Supervision](https://mlsanity.com/paper/2407.06698/) (2024)
- [Noisy Positive-Unlabeled Learning with Self-Training for Speculative Knowledge Graph Reasoning](https://mlsanity.com/paper/2306.07512/) (2023)
- [Multi-Label Out-of-Distribution Detection with Spectral Normalized Joint Energy](https://mlsanity.com/paper/2405.04759/) (2024)
- [Risk bounds for PU learning under Selected At Random assumption](https://mlsanity.com/paper/2201.06277/) (2022)
- [Soft Label PU Learning](https://mlsanity.com/paper/2405.01990/) (2024)
- [PUAL: A Classifier on Trifurcate Positive-Unlabeled Data](https://mlsanity.com/paper/2405.20970/) (2024)
- [AUC Maximization from Biased Positive-unlabeled Data with Confidence](https://mlsanity.com/paper/2609.10928/) (2026)

## Abstract (arXiv)

When learning from positive and unlabelled data, it is a strong assumption that the positive observations are randomly sampled from the distribution of X conditional on Y = 1, where X stands for the feature and Y the label. Most existing algorithms are optimally designed under the assumption. However, for many real-world applications, the observed positive examples are dependent on the conditional probability P(Y = 1|X) and should be sampled biasedly. In this paper, we assume that a positive example with a higher P(Y = 1|X) is more likely to be labelled and propose a probabilistic-gap based PU learning algorithms. Specifically, by treating the unlabelled data as noisy negative examples, we could automatically label a group positive and negative examples whose labels are identical to the ones assigned by a Bayesian optimal classifier with a consistency guarantee. The relabelled examples have a biased domain, which is remedied by the kernel mean matching technique. The proposed algorithm is model-free and thus do not have any parameters to tune. Experimental results demonstrate that our method works well on both generated and real-world datasets.