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

2026-07-08

AI Summary

This study addresses the problem of constructing principled information update rules within a non-probabilistic (possibilistic) framework of uncertainty. By integrating the principle of information conservation with an axiomatic approach, the work achieves the first unified formulation in possibilistic reasoning, yielding an optimal update rule: pointwise multiplication of the prior and likelihood followed by supremum normalization. This rule is shown to be valid for arbitrary loss functions, thereby establishing a coherent foundation for generalized Bayesian-style inference beyond classical probability theory. Furthermore, the analysis reveals that while the learning rate parameter is inherently unidentifiable, it serves as a meaningful measure of cognitive strength. The results provide rigorous theoretical support for uncertainty reasoning that transcends traditional probabilistic paradigms.

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.

Problem

• possibilistic inference
• possibility function
• Bayesian updating
• learning-rate parameter
• uncertainty representation

Innovation

• possibilistic inference
• Bayesian updating
• maxitive measures
• axiomatic foundations
• information conservation