# Quantification under prior probability shift: the ratio estimator and its extensions

arXiv: https://arxiv.org/abs/1807.03929 · HTML: https://mlsanity.com/paper/1807.03929/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2018.

## Built on by (most cited first, 15 of 15)

- [Measuring Fairness Under Unawareness of Sensitive Attributes: A Quantification-Based Approach](https://mlsanity.com/paper/2109.08549/) (2021)
- [A unified framework for dataset shift diagnostics](https://mlsanity.com/paper/2205.08340/) (2022)
- [Binary Quantification and Dataset Shift: An Experimental Investigation](https://mlsanity.com/paper/2310.04565/) (2023)
- [Minimising quantifier variance under prior probability shift](https://mlsanity.com/paper/2107.08209/) (2021)
- [Bayesian Quantification with Black-Box Estimators](https://mlsanity.com/paper/2302.09159/) (2023)
- [Quantification using Permutation-Invariant Networks based on Histograms](https://mlsanity.com/paper/2403.15123/) (2024)
- [Importance Weighting Correction of Regularized Least-Squares for Target Shift](https://mlsanity.com/paper/2210.09709/) (2022)
- [Classification under Nuisance Parameters and Generalized Label Shift in Likelihood-Free Inference](https://mlsanity.com/paper/2402.05330/) (2024)
- [Prior shift estimation for positive unlabeled data through the lens of kernel embedding](https://mlsanity.com/paper/2502.21194/) (2025)
- [Estimating prevalence with precision and accuracy](https://mlsanity.com/paper/2507.06061/) (2025)
- [Fairness Under Group-Conditional Prior Probability Shift: Invariance, Drift, and Target-Aware Post-Processing](https://mlsanity.com/paper/2602.05144/) (2026)
- [Multi-Label Quantification](https://mlsanity.com/paper/2211.08063/) (2022)
- [Comments on Friedman's Method for Class Distribution Estimation](https://mlsanity.com/paper/2405.16666/) (2024)
- [Theory-inspired Label Shift Adaptation via Aligned Distribution Mixture](https://mlsanity.com/paper/2411.02047/) (2024)
- [Geometry-Aware Bayesian Quantification via Compositional Data Analysis](https://mlsanity.com/paper/2607.04977/) (2026)

## Abstract (arXiv)

The quantification problem consists of determining the prevalence of a given label in a target population. However, one often has access to the labels in a sample from the training population but not in the target population. A common assumption in this situation is that of prior probability shift, that is, once the labels are known, the distribution of the features is the same in the training and target populations. In this paper, we derive a new lower bound for the risk of the quantification problem under the prior shift assumption. Complementing this lower bound, we present a new approximately minimax class of estimators, ratio estimators, which generalize several previous proposals in the literature. Using a weaker version of the prior shift assumption, which can be tested, we show that ratio estimators can be used to build confidence intervals for the quantification problem. We also extend the ratio estimator so that it can: (i) incorporate labels from the target population, when they are available and (ii) estimate how the prevalence of positive labels varies according to a function of certain covariates.