Classical Bayesian updating was strictly confined to additive probability distributions, struggling when information was qualitative or partially specified; axiomatic possibilistic updating rules extend optimal information processing beyond classical probability theory.
Since the eighteenth century, Thomas Bayes' rule has reigned as the supreme mathematical law of rational learning: prior beliefs are updated in light of evidence through conditional probability.
However, classical probability demands that an agent assign precise numeric values to every possible state of the world, a requirement that collapses when evidence is purely qualitative, ordinal, or fraught with severe epistemic ignorance.
This foundational paper in statistical decision theory derives principled updating rules for possibilistic inference, establishing axiomatic foundations for processing information encoded in possibility distributions. The authors prove that generalized possibilistic updating satisfies core rational axioms while handling partial ignorance without arbitrary prior assumptions.
Expanding Bayesian rationality into non-probabilistic domains provides robust mathematical tools for safety-critical AI, geopolitical risk forecasting, and automated reasoning in data-scarce environments.
From Bayes' Rule to Bayes Rules: Optimal Information Processing and Axiomatic Foundations Beyond Probability
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.
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