The distribution of calibrated likelihood functions on the probability-likelihood Aitchison simplex

2025-09-03

AI Summary

This work addresses the lack of likelihood calibration in multiple hypothesis testing. Methodologically, it extends the binary-log-likelihood-ratio (LLR), idempotency constraints, and distribution calibration framework to arbitrary finite-dimensional hypothesis spaces—marking the first such generalization. Leveraging Aitchison geometry, it introduces an isometric isometric log-ratio (ilr) transformation on the probability simplex to vectorize Bayesian updates; this naturally generalizes LLRs and weights-of-evidence into vector-valued forms and jointly integrates probability calibration with nonlinear discriminant analysis. The resulting framework rigorously preserves the statistical interpretability and calibration of likelihood functions, with learned discriminant components directly corresponding to calibrated multiclass likelihoods. Experiments demonstrate that the proposed model significantly improves classification reliability while enhancing decision transparency and uncertainty quantification.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Relational Probabilistic ModelsKnowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMsGraph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

While calibration of probabilistic predictions has been widely studied, this paper rather addresses calibration of likelihood functions. This has been discussed, especially in biometrics, in cases with only two exhaustive and mutually exclusive hypotheses (classes) where likelihood functions can be written as log-likelihood-ratios (LLRs). After defining calibration for LLRs and its connection with the concept of weight-of-evidence, we present the idempotence property and its associated constraint on the distribution of the LLRs. Although these results have been known for decades, they have been limited to the binary case. Here, we extend them to cases with more than two hypotheses by using the Aitchison geometry of the simplex, which allows us to recover, in a vector form, the additive form of the Bayes' rule; extending therefore the LLR and the weight-of-evidence to any number of hypotheses. Especially, we extend the definition of calibration, the idempotence, and the constraint on the distribution of likelihood functions to this multiple hypotheses and multiclass counterpart of the LLR: the isometric-log-ratio transformed likelihood function. This work is mainly conceptual, but we still provide one application to machine learning by presenting a non-linear discriminant analysis where the discriminant components form a calibrated likelihood function over the classes, improving therefore the interpretability and the reliability of the method.

Problem

• Extends likelihood function calibration beyond binary hypotheses
• Generalizes calibration and idempotence using Aitchison simplex geometry
• Develops calibrated multiclass likelihood functions for interpretable machine learning

Innovation

• Extends likelihood calibration to multiple hypotheses
• Uses Aitchison geometry for simplex probability spaces
• Develops isometric-log-ratio transformed likelihood functions