Algebraic aspects and coherence conditions for conjunctions among conditional events

Abstract

We deepen the study of a notion of conjunction among conditional events, introduced in previous papers in theframework of coherence. This notion of conjunction, differently from other approaches, is given in the setting ofconditional random quantities. We show that some well known properties which are satisfied by conjunctionsof unconditional events are also satisfied by conjunctions of conditional events. In particular we examine anadditive property and a decomposition formula, by also obtaining a generalized inclusion-exclusion formula. Then,by exploiting the notion of conjunction, we introduce the set of constituents generated bynconditional events.Moreover, under logical independence, we give a necessary and sufficient condition of coherence for the previsionassessments on a familyFconstituted bynconditional events and all possible conjunctions among some of them.This condition of coherence has a simple geometrical characterization in terms of a suitable convex hull. Such acharacterization amounts to the solvability of a linear system as in the case of unconditional events. Then, weillustrate the set of all coherent assessments on the familyFby a list of linear inequalities on the componentsof the prevision assessment. Finally, given a coherent assessmentMonF, we show that every possible value ofthe random vector associated withFis itself a particular coherent assessment onF.