TY - CHAP

T1 - Conditional Random Quantities and Iterated Conditioning in the Setting of Coherence

AU - Sanfilippo, Giuseppe

AU - Gilio, Angelo

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Bayesian updating

KW - Coherence

KW - betting scheme

KW - conditional previsions

KW - conditional random quantities

KW - iterated conditioning.

KW - Bayesian updating

KW - Coherence

KW - betting scheme

KW - conditional previsions

KW - conditional random quantities

KW - iterated conditioning.

UR - http://hdl.handle.net/10447/97510

UR - http://link.springer.com/chapter/10.1007%2F978-3-642-39091-3_19

M3 - Chapter

SN - 978-3-642-39090-6

T3 - LECTURE NOTES IN COMPUTER SCIENCE

SP - 218

EP - 229

BT - Symbolic and Quantitative Approaches to Reasoning with Uncertainty

ER -