Possibilistic and probabilistic likelihood functions and their extensions: Common features and specific characteristics

We deal with conditional probability in the sense of de Finetti and with T-conditional possibility (with T a triangular norm). We prove that Dubois and Prade conditional possibility is a particular min-conditional possibility and then we compare the two notions of conditioning by an inferential point of view. Moreover, we study T-conditional possibilities as functions of the conditioning event, putting in evidence analogies and differences with conditional probabilities. This allows to characterize likelihood functions (and their aggregations) consistent either with a T-conditional possibility or a conditional probability. This analysis highlights many syntactical coincidences. Nevertheless the main difference is a weak form of monotonicity, which arises only in the possibilistic case.