A Model of Minimal Probabilistic Belief Revision

Abstract

A probabilistic belief revision function assigns to every initial probabilistic belief and every observable event some revised probabilistic belief that only attaches positive probability to states in this event. We propose three axioms for belief revision functions: (1) linearity, meaning that if the decision maker observes that the true state is in {a,b}, and hence state c is impossible, then the proportions of c''s initial probability that are shifted to a and b, respectively, should be independent of c''s initial probability; (2) transitivity, stating that if the decision maker deems belief β equally similar to states a and b, and deems β equally similar to states b and c, then he should deem β equally similar to states a and c; (3) information-order independence, stating that the way in which information is received should not matter for the eventual revised belief. We show that a belief revision function satisfies the three axioms above if and only if there is some linear one-to-one function ϕ, transforming the belief simplex into a polytope that is closed under orthogonal projections, such that the belief revision function satisfies minimal belief revision with respect to ϕ. By the latter, we mean that the decision maker, when having initial belief β₁ and observing the event E, always chooses the revised belief β₂ that attaches positive probability only to states in E and for which ϕ(β₂) has minimal Euclidean distance to ϕ(β₁).microeconomics ;