Geometry-Aware Bayesian Quantification via Compositional Data Analysis

2026-07-06

AI Summary

This work addresses the neglect of the geometric structure of the probability simplex in multiclass label distribution estimation by proposing a geometry-aware kernel density estimation (KDE) method. Treating classifier posteriors as compositional data, the approach constructs a KDE model grounded in Aitchison geometry and log-ratio transformations, augmented with boundary-shrinkage regularization to ensure both geometric coherence and robustness near simplex boundaries. This is the first method to explicitly incorporate simplex geometry into a KDE-based quantification framework, enabling both maximum likelihood point estimation and Bayesian inference. Evaluated on 42 datasets spanning tabular, textual, and image modalities, the method significantly outperforms standard KDE baselines, achieving state-of-the-art performance in quantification tasks—particularly excelling in Bayesian quantification settings.

Abstract

Accurately estimating the unknown target label distribution is the critical first step for adapting to label shift. This task, widely known as quantification or class prevalence estimation, has recently seen significant advances through continuous KDE-based methods which model the density of multiclass classifier posteriors. Posterior vectors might be regarded as compositional data, since they lie on the probability simplex. However, existing KDE-based quantifiers typically rely on Euclidean Gaussian kernels, which ignore simplex geometry and incorrectly assign probability mass outside its boundaries. We introduce a geometry-aware KDE model for multiclass quantification based on log-ratio representations and Aitchison geometry, together with a shrinkage regularization that improves robustness near the simplex boundary. Combined with a maximum-likelihood interpretation of KDE-based quantification, we derive both point-estimation and Bayesian inference procedures for class prevalences. Experiments on 42 datasets across tabular, text, and image domains show that the proposed method is competitive with state-of-the-art quantifiers, often improving over standard KDE-based baselines, while also yielding strong results among Bayesian quantification methods.

Problem

• quantification
• label shift
• compositional data
• probability simplex
• class prevalence estimation

Innovation

• compositional data analysis
• Aitchison geometry
• Bayesian quantification
• label shift adaptation
• geometry-aware KDE