Probabilistic Abduction Without Priors

This paper considers the simple problem of abduction in the framework of Bayes theorem, i.e. computing a posterior probability of an hypothesis when its prior probability is not available, either because there are no statistical data on which to rely on, or simply because a human expert is reluctant to provide a subjective assessment of this prior probability. The problem remains an open issue since a simple sensitivity analysis on the value of the unknown prior yields empty results. This paper tries to survey and comment on various solutions to this problem: the use of likelihood functions (as in classical statistics), the use of information principles like maximal entropy, Shapley value, maximum likelihood. We also study the problem in the setting of de Finetti coherence approach, which does not exclude conditioning on contingent events with zero probability. We show that the ad hoc likelihood function method, that can be reinterpreted in terms of possibility theory, is consistent with most other formal approaches. However, the maximal entropy solution is significantly different.