Conditional random quantities and compounds of conditionals

Abstract

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as XH + P(X|H)Hc; in particular (the indicator of) a conditional event E|H is looked at as EH +P(E|H)Hc. This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a meaning to the sum X|H +Y |K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold.