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 c, 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. © 2013 Springer-Verlag Berlin Heidelberg.
Conditional Random Quantities and Iterated Conditioning in the Setting of Coherence / Gilio, Angelo; Giuseppe, Sanfilippo. - STAMPA. - 7958 LNAI(2013), pp. 218-229. ((Intervento presentato al convegno 12th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2013 tenutosi a Utrecht nel 8 July 2013 through 10 July 2013. - LECTURE NOTES IN COMPUTER SCIENCE. [10.1007/978-3-642-39091-3-19].
Conditional Random Quantities and Iterated Conditioning in the Setting of Coherence
GILIO, ANGELO;
2013
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 + μH c, 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. © 2013 Springer-Verlag Berlin Heidelberg.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
