Beliefs, Belief Revision, and Noisy Sensors

In logical AI, an agent’s beliefs are typically categorical, in that they are specified by a set of formulas. An agent may change its beliefs as a result of being informed in one fashion or another about some aspect of the world, or following the execution of some action. The areas of belief revision and reasoning about action deal with just such change in belief. However, most information about the real world is not categorical. While there are well-established accounts for accommodating noncategorical information via probability theory, it is worth asking whether probabilistic information may be reconciled with the logical accounts of belief change. We present such an account in this paper. An agent receives uncertain information as input and its probabilities, expressed as probabilities on possible worlds, are updated via Bayesian conditioning. A set of formulas among the (noncategorical) beliefs is identified as the agent’s (categorical) belief set. This set is defined in terms of the most probable worlds such that the summed probability of these worlds exceeds a given threshold. The effect of this updating on the belief set is examined with respect to its appropriateness as a revision operator. It proves to be the case that a subset of the classical AGM belief revision postulates are satisfied. Most significantly, the success postulate is not guaranteed to hold. However it does hold after a sufficient number of iterations. Not is it the case that in revising by a formula consistent with the agent’s beliefs, revision corresponds to expansion. On the other hand, limiting cases of the presented approach correspond to specific approaches to revision that have appeared in the literature. It is a great pleasure to dedicate this paper to Hector Levesque on the occasion of his 60th birthday. While Hector’s work has broadly focussed on representational aspects of an agent’s beliefs together with accounts of reasoning – whether epistemic, nonmonotonic, limited, in a theory of action, or otherwise – it has certainly touched on many other areas over the years. This paper outlines a possible linking of two such areas, that of reasoning about noisy sensors on the one hand [Bacchus, Halpern, and Levesque 1999], and revision in the presence of (categorical) observations on the other [Shapiro, Pagnucco, Lespérance, and Levesque 2011].