The comparison between probabilistic approaches (not only Bayesian programming) and possibility theories has been debated for a long time and is, unfortunately, a very controversial matter.
Possibility theories (like, for instance, fuzzy sets [Zadeh, 1965], fuzzy logic [Zadeh, 1974, 1975] and possibility theory [Dubois and Prade, 2001]) propose different alternatives to probability to model uncertainty. They argue that probability is insufficient or inconvenient to model certain aspects of incomplete and uncertain knowledge.
The defense of probability is mainly based on Cox’s theorem (see post Cox Theorem) which, starting from four postulates concerning rational reasoning in the presence of uncertainty, demonstrates that the only mathematical framework that satisfies these postulates is probability theory.
The argument then goes like this: if you use a different approach than probability, then you necessarily infringe on one of these postulates. Let us see which one and discuss its utility. The debate is still open…