GEOMETRY-AWARE BAYESIAN QUANTIFICATION VIA COMPOSITIONAL DATA ANALYSIS

Quantification, or class prevalence estimation, is a critical task for adapting machine learning models to label shift. While recent advances have utilized kernel density estimation (KDE) to model the distribution of classifier posteriors, these methods typically operate in Euclidean space. This creates a geometric mismatch because posterior probabilities are compositional data constrained to a probability simplex. The authors propose a geometry-aware KDE model that maps posterior vectors into log-ratio coordinates, aligning the density estimation with Aitchison geometry. To address numerical instability near the simplex boundaries, they introduce a shrinkage-based regularization that pulls compositions toward the barycenter, ensuring robust density estimation.

The proposed method, KDEy(Ait-λ), is evaluated across 42 datasets spanning tabular, text, and image domains. It demonstrates superior performance compared to standard Euclidean Gaussian KDE baselines, achieving higher accuracy in point estimation and more reliable uncertainty quantification. The authors also derive a Bayesian formulation for this geometry-aware model, enabling principled posterior inference over class prevalences. Experimental results show that the method is highly competitive with state-of-the-art quantifiers, particularly in regimes where target prevalences are near the simplex boundary, where the geometric benefits of the log-ratio representation are most pronounced.

This work bridges the gap between compositional data analysis and machine learning quantification. By providing a principled way to handle the simplex geometry of classifier posteriors, the authors offer a more robust framework for real-world deployment where label shift is common. The inclusion of a Bayesian formulation allows for better uncertainty estimation, which is essential for safety-critical applications where knowing the confidence of a prevalence estimate is as important as the estimate itself.

Key Terms

Quantification

The task of estimating the class distribution (prevalence) in a target population, distinct from predicting individual labels.

Label Shift

A scenario where the class prior probabilities change between training and test distributions, while the class-conditional distributions remain constant.

Probability Simplex

The geometric space defined by vectors whose components are non-negative and sum to one, representing the valid space for posterior probability distributions.

Aitchison Geometry

A geometric framework for compositional data that uses log-ratio transformations to map data from the simplex into a Euclidean space, respecting the relative nature of the components.

Shrinkage

A regularization technique used here to move compositions toward the simplex barycenter, preventing numerical instability in log-ratio transformations near the boundary.

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