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

arXiv: https://arxiv.org/abs/2509.03365 · HTML: https://mlsanity.com/paper/2509.03365/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 65 references that are in the map.
- First posted on arXiv 3 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Semi-Supervised Learning with Normalizing Flows](https://mlsanity.com/paper/1912.13025/) (2019)
- [Classifier Calibration: A survey on how to assess and improve predicted class probabilities](https://mlsanity.com/paper/2112.10327/) (2021)
- [Evaluating probabilistic classifiers: Reliability diagrams and score decompositions revisited](https://mlsanity.com/paper/2008.03033/) (2020)
- [Deep Linear Discriminant Analysis](https://mlsanity.com/paper/1511.04707/) (2015)
- [Evaluating Posterior Probabilities: Decision Theory, Proper Scoring Rules, and Calibration](https://mlsanity.com/paper/2408.02841/) (2024)
- [The PAV algorithm optimizes binary proper scoring rules](https://mlsanity.com/paper/1304.2331/) (2013)
- [The Privacy ZEBRA: Zero Evidence Biometric Recognition Assessment](https://mlsanity.com/paper/2005.09413/) (2020)
- [The distribution of calibrated likelihood-ratios in speaker recognition](https://mlsanity.com/paper/1304.1199/) (2013)
- [A bridge between features and evidence for binary attribute-driven perfect privacy](https://mlsanity.com/paper/2110.05840/) (2021)
- [Hiding speaker's sex in speech using zero-evidence speaker representation in an analysis/synthesis pipeline](https://mlsanity.com/paper/2211.16065/) (2022)
- [Explaining a probabilistic prediction on the simplex with Shapley compositions](https://mlsanity.com/paper/2408.01382/) (2024)

## Abstract (arXiv)

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.