Commutativity of probabilistic belief revision

Bayesian updating, also known as belief revision or conditioning, is a core mechanism of probability theory, and of AI. The human mind is very sensitive to the order in which it is being “primed”, but Bayesian updating works commutatively: the order of the evidence does not matter. Thus, there is a mismatch. This paper develops Bayesian updating as an explicit operation on (discrete) probability distributions, so that the commutativity of Bayesian updating can be clearly formulated and made explicit in several examples. The commutativity mismatch is underexplored, but plays a fundamental role, for instance in the move to quantum cognition.