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Commutativity of probabilistic belief revision
1iHub, Radboud University, Nijmegen, Netherlands.
Frontiers in Cognition
|June 24, 2026
Summary
Bayesian updating, a key AI and probability concept, is commutative, meaning evidence order doesn't matter. This paper explores the mismatch between this and human cognition, where order is crucial.
Area of Science:
- Artificial Intelligence
- Probability Theory
- Cognitive Science
Background:
- Bayesian updating is fundamental to probability and AI.
- Human cognition is sensitive to the order of information.
- A mismatch exists between commutative Bayesian updating and human sequential processing.
Purpose of the Study:
- To explicitly formulate Bayesian updating as an operation on probability distributions.
- To analyze the commutativity of Bayesian updating.
- To highlight the underexplored commutativity mismatch in cognitive contexts.
Main Methods:
- Developed Bayesian updating as an explicit operation on discrete probability distributions.
- Illustrated the commutativity of Bayesian updating with examples.
- Formulated the commutativity of Bayesian updating clearly.
Main Results:
- Demonstrated that Bayesian updating, as formulated, is commutative.
- Provided explicit examples of commutative Bayesian updating.
- Highlighted the role of the commutativity mismatch in areas like quantum cognition.
Conclusions:
- Bayesian updating is mathematically commutative.
- The mismatch between commutative updating and human sequential processing is significant.
- Understanding this mismatch is crucial for advancing fields like quantum cognition.
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