Conditional events, conditioning, and random sets

A central problem in the Dempster/Shafer theory of evidence is conditioning. This paper presents a new approach to a solution of this problem by establishing a link between conditional events and discrete random sets. Conditional events are introduced as sets of equivalent events under conditioning. These sets may become targets of a multivalued mapping. Thus, conditional belief functions can be introduced. Both Bayesian and pure random set conditioning rules are derived and discussed. Random set conditioning allows expressing conditional degrees of belief when marginal beliefs are unknown. Finally, an updating rule is introduced that is equivalent to the law of total probability (Jeffrey´s rule) if all beliefs are probabilities