The role of coherence in eliciting and handling imprecise probabilities and its application to medical diagnosis

We refer to an arbitrary family H = {H-1,H-2,...,H-n} of events (hypotheses ), i.e.,H has neither any particular algebraic structure nor is a partition of the certain event Omega. We detect logical relations among the given ev ents (the latter could represent some possible diseases), and some further information is carried by probability assessments, relative to an event E ( e.g., a symptom) conditionally to some of the H-t's ("partial likelihood"). If we assess (prior) probabilities for the events Hi's, then the ensuing p roblems are: (i) is this assessment coherent? (ii) is the partial likelihoo d coherent "per se"? (iii) is the global assignment (the initial one togeth er with the likelihood) coherent? If the relevant answers are all YES, then we may try to "update" (coherently) the priors P(H-i) into the posteriors P(H,IE). This is an instance of a more general issue, the problem of cohere nt extensions: a very particular case is Bayes' updating for exhaustive and mutually exclusive hypotheses, in which this extension is unique. In the g eneral case the lack of uniqueness gives rise to upper and lower updated pr obabilities, and we could now update again the latter, given a new event P and a corresponding (possibly partial) likelihood. In this paper, many rele vant features of this problem are discussed, keeping an eye on the distinct ion between semantic and syntactic aspects. (C) 2000 Elsevier Science Inc. All rights reserved.