Probabilistic abduction without priors

This paper considers the simple problem of abduction in the framework of Bayes theorem, when the prior probability of the hypothesis is not available, either because there are no statistical data to rely on, or simply because a human expert is reluctant to provide a subjective assessment of this prior probability. This abduction problem remains an open issue since a simple sensitivity analysis on the value of the unknown prior yields empty results. This paper tries to propose some criteria a solution to this problem should satisfy. It then surveys and comments on various existing or new solutions to this problem: the use of likelihood functions (as in classical statistics), the use of information principles like maximum entropy, Shapley value, maximum likelihood. The formal setting includes de Finetti’s 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 maximum entropy solution is significantly different.