A generalization of the fundamental theorem of de Finetti for imprecise conditional probability assessments

In this paper, we consider coherent imprecise probability assessments on finite families of conditional events and we study the problem of their extension. With this aim, we adopt a generalized definition of coherence, called g-coherence, which is based on a suitable generalization of the coherence principle of de Finetti. At first, we recall some theoretical results and an algorithm obtained in some previous papers where the case of precise conditional probability assessments has been studied. Then, we extend these results to the case of imprecise probabilistic assessments and we obtain a theorem which can be looked at as a generalization of the version of the fundamental theorem of de Finetti given by some authors for the case of conditional events. Our algorithm can also be exploited to produce lower and upper probabilities which are coherent in the sense of Walley and Williams. Moreover, we compare our approach to similar ones, like probability logic or probabilistic deduction. Finally, we apply our algorithm to some well-known inference rules assuming some logical relations among the given events.