From Bayes Rule to Bayes Rules: Optimal Information Processing and Axiomatic Foundations Beyond Probability

Jul 16, 2026

This paper develops principled updating rules for possibilistic inference, representing uncertainty about a fixed parameter with a possibility function, the maxitive analogue of a probability distribution. Using a pointwise partial-order comparison, two complementary foundations—an information-conservation viewpoint and an axiomatic viewpoint—yield the same canonical update: the posterior equals the prior-likelihood product followed by supremum normalisation. The learning-rate parameter enters differently, showing its control of epistemic strength and its non-identifiability from normalising evidence alone.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages17

Reading Time43 min

Introduction

Optimal information processing

Axiomatic approach

From Bayes’ Rule to Bayes Rules: Optimal Information Processing and Axiomatic Foundations Beyond Probability Jeremie Houssineau 1 Badr-Eddine Chérief-Abdellatif *2 1Nanyang Technological University, Singapore 2 CNRS, LPSM, Sorbonne Université, France arXiv :2607 .08019v1 [math . ST] 9 Jul 2026 Abstract This paper develops principled updating rules for possibilistic inference, where uncertainty about a fixed parameter is represented by a possibility function, the maxitive analogue of a probability distribution, and comparisons are made pointwise via a partial order. From two complementary foundations, an information-conservation viewpoint and an axiomatic viewpoint, we derive the same canonical update: the posterior is the prior-likelihood product followed by supremum normalisation. The two derivations agree for an arbitrary loss, differing only in where the learning-rate parameter enters. This parameter controls epistemic strength and is not identifiable from the normalising evidence alone, clarifying the role of analogous learning-rate parameters in generalised Bayesian updating. 1 INTRODUCTION Bayes’ rule occupies a unique position in statistical inference: it is at once an operational framework for updating beliefs and a benchmark against which alternative update rules are judged. Yet both classical statistics and modern machine learning routinely depart from standard Bayesian updating. Classical examples include inverse-probability methods and alternative postdata constructions, while modern examples include loss-based “generalised Bayes” updates, tempered posteriors, and evidence-fusion schemes in which uncertainty is not naturally additive. These departures raise a basic question that is easy to state but conceptually difficult: what exactly makes Bayes’ rule the “right” way to process information, and how does that answer depend on the mathematical representation of information? One way to revisit this question is to treat inference as * [Corresponding author: badr-eddine.cherief-abdellatif@cnrs.fr](Corresponding author: badr-eddine.cherief-abdellatif@cnrs.fr) information processing. Starting from two inputs, i) prior information about an unknown but fixed quantity of interest and ii) information supplied by data through a likelihood or loss, one asks for a principled information processing rule (IPR) that outputs a postdata representation of the state of knowledge. In the probabilistic setting, such a perspective is often associated with the work of Zellner [1988] on the information conservation principle (ICP), where Bayes’rule arises as the unique IPR that conserves a chosen scalar measure of information. In parallel, the framework of Bissiri et al. [2016] shows that, when data enter through losses rather than a data-generating mechanism (DGM), a Gibbs (exponentiated-loss) posterior is forced by coherence requirements on how losses should accumulate and how updating should behave under restriction. In this work, we argue that these “foundational” stories are not exclusive to probability, but translate naturally when uncertainty about the parameter is represented by possibility functions. Possibility theory [Zadeh, 1978] is designed to model epistemic uncertainty about a fixed state of nature. A prior state of knowledge about θ ∈ Θ is represented by a possibility function π : Θ → [0 , 1] with supθ∈Θ π (θ) = 1, where larger values indicate greater compatibility with the available information. Crucially, the set of possibility functions F(Θ) on a set Θ is naturally ordered pointwise: one possibility function can directly be said to be (weakly) more informative than another without first compressing them into a single number such as the (cross-)entropy. This order structure has two immediate consequences for information processing: i) it becomes meaningful to distinguish information loss from information creation at the level of information objects themselves, and ii) combining independent inf

FAQs

How does the paper represent uncertainty in possibilistic inference?

Uncertainty about a fixed parameter is represented by a possibility function over the parameter space, with values indicating compatibility with available information.

What is the canonical possibilistic update rule derived in the paper?

The posterior is given by the prior-likelihood product followed by supremum normalisation, expressed using a supremum over the parameter space.

What role does the learning-rate parameter play, and is it identifiable from normalising evidence?

The learning-rate parameter controls epistemic strength; it is not identifiable from the normalising evidence alone, clarifying how analogous parameters affect generalised Bayesian updating.

Jul 16, 2026

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