# Double logistic regression approach to biased positive-unlabeled data

arXiv: https://arxiv.org/abs/2209.07787 · HTML: https://mlsanity.com/paper/2209.07787/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 37 references that are in the map.
- First posted on arXiv 16 Sep 2022.

## Builds on (most cited first, 8 of 8)

- [Positive-Unlabeled Learning with Non-Negative Risk Estimator](https://mlsanity.com/paper/1703.00593/) (2017)
- [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)
- [Class-prior Estimation for Learning from Positive and Unlabeled Data](https://mlsanity.com/paper/1611.01586/) (2016)
- [Beyond the Selected Completely At Random Assumption for Learning from Positive and Unlabeled Data](https://mlsanity.com/paper/1809.03207/) (2018)
- [Estimating the class prior and posterior from noisy positives and unlabeled data](https://mlsanity.com/paper/1606.08561/) (2016)
- [Instance-Dependent PU Learning by Bayesian Optimal Relabeling](https://mlsanity.com/paper/1808.02180/) (2018)
- [Efficient Training for Positive Unlabeled Learning](https://mlsanity.com/paper/1608.06807/) (2016)

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

- [Verifying the Selected Completely at Random Assumption in Positive-Unlabeled Learning](https://mlsanity.com/paper/2404.00145/) (2024)
- [A proposal for PU classification under Non-SCAR using clustering and logistic model](https://mlsanity.com/paper/2604.17130/) (2026)
- [PU classification under Non-SCAR: clustering-assisted logistic model with oversampling enhancement](https://mlsanity.com/paper/2609.14675/) (2026)

## Abstract (arXiv)

Positive and unlabelled learning is an important problem which arises naturally in many applications. The significant limitation of almost all existing methods lies in assuming that the propensity score function is constant (SCAR assumption), which is unrealistic in many practical situations. Avoiding this assumption, we consider parametric approach to the problem of joint estimation of posterior probability and propensity score functions. We show that under mild assumptions when both functions have the same parametric form (e.g. logistic with different parameters) the corresponding parameters are identifiable. Motivated by this, we propose two approaches to their estimation: joint maximum likelihood method and the second approach based on alternating maximization of two Fisher consistent expressions. Our experimental results show that the proposed methods are comparable or better than the existing methods based on Expectation-Maximisation scheme.