Belief Revision from Probability

Abstract

In previous work ([5, 6]), we develop a question-relative, probabilistic account of belief. On this account, what someone believes relative to a given question is (i) closed under entailment, (ii) sufficiently probable given their evidence, and (iii) sensitive to the relative probabilities of the answers to the question. Here we explore the implications of this account for the dynamics of belief. We show that the principles it validates are much weaker than those of orthodox theories of belief revision like AGM [1], but still stronger than those valid according to the popular Lockean theory of belief [4], which equates belief with high subjective probability. We then consider a restricted class of models, suitable for many but not all applications, and identify some further natural principles valid on this class. We conclude by arguing that the present framework compares favorably to the rival probabilistic accounts of belief developed by Leitgeb [13,14] and Lin and Kelly [17]. Probability StructuresWe will work with the following simplification of the models in [5]: Definition 1.1. A probability structure is a tuple S, E , Q, Pr,t such that:1. S is a non-empty set (of states), 2. E ⊆ P(S)\{ / 0} (the possible bodies of evidence), 3. Q (the question) is a partition of S, 4. Pr (the prior) is a probability distribution over S, and 5. t ∈ [0, 1] (the threshold)Propositions are modeled as subsets of S, where p is true in s if and only if s ∈ p. We say that E ′ ∈ E is the result of discovering p in E ∈ E just in case E ′ = E ∩ p; this will allow us to talk about how beliefs evolve in response to changes in one's evidence.Which propositions an agent believes is a function of their evidence and is also given by a set of states, so that an agent with evidence E believes p if and only if B(E) ⊆ p. This ensures that their beliefs are closed under entailment, and thus already marks a departure from popular 'Lockean' accounts of belief [4], according to which one believes a proposition if and only if its probability exceeds a particular threshold. But it is compatible with the more plausible direction of Lockeanism, namely: THRESHOLD: You believe p only if p is sufficiently probable given your evidence.If B(E) ⊆ p, then Pr(p|E) ≥ t.We can think of the members of the question Q as its answers; we write [s] Q for the member of Q containing s. The proposal in [5] then boils down to claiming that s ∈ B(E) if and only if s ∈ E and the answers to Q that are more probable than [s] Q have total probability less than the threshold t. Writing Pr E for Pr(•|E), this can be formalized as follows: