# Nonparametric semi-supervised learning of class proportions

arXiv: https://arxiv.org/abs/1601.01944 · HTML: https://mlsanity.com/paper/1601.01944/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2016.

## Built on by (most cited first, 13 of 13)

- [Mixture Proportion Estimation and PU Learning: A Modern Approach](https://mlsanity.com/paper/2111.00980/) (2021)
- [Domain Adaptation under Open Set Label Shift](https://mlsanity.com/paper/2207.13048/) (2022)
- [Mixture Proportion Estimation Beyond Irreducibility](https://mlsanity.com/paper/2306.01253/) (2023)
- [When Less is Enough: Positive and Unlabeled Learning Model for Vulnerability Detection](https://mlsanity.com/paper/2308.10523/) (2023)
- [Making Binary Classification from Multiple Unlabeled Datasets Almost Free of Supervision](https://mlsanity.com/paper/2306.07036/) (2023)
- [Positive-Unlabeled Domain Adaptation](https://mlsanity.com/paper/2202.05695/) (2022)
- [Unraveling the Impact of Heterophilic Structures on Graph Positive-Unlabeled Learning](https://mlsanity.com/paper/2405.19919/) (2024)
- [Improving State-of-the-Art in One-Class Classification by Leveraging Unlabeled Data](https://mlsanity.com/paper/2203.07206/) (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)
- [PUAL: A Classifier on Trifurcate Positive-Unlabeled Data](https://mlsanity.com/paper/2405.20970/) (2024)
- [Mixture Proportion Estimation and Weakly-supervised Kernel Test for Conditional Independence](https://mlsanity.com/paper/2604.07191/) (2026)
- [Dataset Usage Inference without Shadow Models or Held-out Data](https://mlsanity.com/paper/2606.26257/) (2026)
- [Positive-Unlabeled Preference Optimization For Chest X-ray Report Generation](https://mlsanity.com/paper/2608.05341/) (2026)

## Abstract (arXiv)

The problem of developing binary classifiers from positive and unlabeled data is often encountered in machine learning. A common requirement in this setting is to approximate posterior probabilities of positive and negative classes for a previously unseen data point. This problem can be decomposed into two steps: (i) the development of accurate predictors that discriminate between positive and unlabeled data, and (ii) the accurate estimation of the prior probabilities of positive and negative examples. In this work we primarily focus on the latter subproblem. We study nonparametric class prior estimation and formulate this problem as an estimation of mixing proportions in two-component mixture models, given a sample from one of the components and another sample from the mixture itself. We show that estimation of mixing proportions is generally ill-defined and propose a canonical form to obtain identifiability while maintaining the flexibility to model any distribution. We use insights from this theory to elucidate the optimization surface of the class priors and propose an algorithm for estimating them. To address the problems of high-dimensional density estimation, we provide practical transformations to low-dimensional spaces that preserve class priors. Finally, we demonstrate the efficacy of our method on univariate and multivariate data.