Conditional Random Quantities and Iterated Conditioning in the Setting of Coherence

We consider conditional random quantities (c.r.q.’s) in the setting of coherence. Given a numerical r.q. X and a non impossible event H, based on betting scheme we represent the c.r.q. X|H as the unconditional r.q. XH + μH, where μ is the prevision assessed for X|H. We develop some elements for an algebra of c.r.q.’s, by giving a condition under which two c.r.q.’s X|H and Y |K coincide. We show that X|HK coincides with a suitable c.r.q. Y |K and we apply this representation to Bayesian updating of probabilities, by also deepening some aspects of Bayes’ formula. Then, we introduce a notion of iterated c.r.q. (X|H)|K, by analyzing its relationship with X|HK. Our notion of iterated conditional cannot formalize Bayesian updating but has an economic rationale. Finally, we define the coherence for prevision assessments on iterated c.r.q.’s and we give an illustrative example.