Conditional random quantities and iterated conditioning in the setting of coherence

Authors:
Angelo Gilio;Giuseppe Sanfilippo
Affiliations:
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, University of Rome "La Sapienza", Italy;Dipartimento di Matematica e Informatica, University of Palermo, Italy
Venue:
ECSQARU'13 Proceedings of the 12th European conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
Year:
2013

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Abstract

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+μHc, 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.