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

We refer to an arbitrary family of events (hypotheses), i.e., has neither any particular algebraic structure nor is a partition of the certain event Ω. We detect logical relations among the given events (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_i's (“partial likelihood”). If we assess (prior) probabilities for the events H_i's, then the ensuing problems are: (i) is this assessment coherent? (ii) is the partial likelihood coherent “per se”? (iii) is the global assignment (the initial one together 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_i|E). This is an instance of a more general issue, the problem of coherent extensions: a very particular case is Bayes' updating for exhaustive and mutually exclusive hypotheses, in which this extension is unique. In the general case the lack of uniqueness gives rise to upper and lower updated probabilities, and we could now update again the latter, given a new event F and a corresponding (possibly partial) likelihood. In this paper, many relevant features of this problem are discussed, keeping an eye on the distinction between semantic and syntactic aspects.