A probabilistic analysis of compound and iterated conditionals under coherence: theoretical aspects, applications, and related topics

This dissertation analyses various aspects on conditional events, compound conditionals, and coherence in the framework of subjective probability. We investigate logical and probabilistic properties of iterated conditionals defined as trivalent objects and as suitable conditional random quantities. We check the validity of two generalised versions of Bayes' Rule for iterated conditionals and we study the p-validity of generalised versions of Modus Ponens and two-premise centering for iterated conditionals. Then, we study a generalisation of the operations of conjunction and disjunction of conditional events in the context of conditional random quantities, where these new objects, instead of depending on the probabilities of the two conditional events, depend on two arbitrary values a,b in the unit interval. We show that they are connected by a generalised version of the De Morgan's law and, by means of a geometrical approach, we compute the lower and upper bounds on these new objects both in the precise and the imprecise case. Moreover, some particular cases, obtained for specific values of a and b or in case of some logical relations, are analysed. We also show how the subjective approach to conditional probability and compound conditionals can be applied to the theory of fuzzy sets. Then, we continue the analysis of the algebraic approach to conditionals by defining the Boolean algebra of iterated conditionals, and with the study of some (probabilistic) properties related to the concept of “canonical extension”. Finally we show how, thanks to coherence and proper scoring rules, the concepts of entropy and extropy can be extended from partitions to arbitrary family of events, and we find the associated Bregman divergences.