Some results on generalized coherence of conditional probability bounds

Based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence), we adopt a probabilistic approach to uncertainty based on conditional probability bounds. Our notion of g-coherence is equivalent to the 'avoiding uniform loss' property for lower and upper probabilities (a la Walley). Moreover, given a g-coherent imprecise assessment by our algorithms we can correct it obtaining the associated coherent assessment (in the sense of Walley and Williams). As is well known, the problems of checking g-coherence and propagating tight g-coherent intervals are NP and FP^NP complete, respectively, and thus NP-hard. Two notions which may be helpful to reduce co…

Algebraic aspects and coherence conditions for conjunctions among conditional events

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, w…

Generalized Logical Operations among Conditional Events

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of $n$ conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan's Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the Fr'echet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction $mathc…

Logical Conditions for Coherent Qualitative and Numerical Probability Assessments

Algorithms for coherence checking and propagation of conditional probability bounds

In this paper, we propose some algorithms for the checking of generalized coherence (g-coherence) and for the extension of imprecise conditional probability assessments. Our concept of g-coherence is a generalization of de Finetti’s coherence principle and is equivalent to the ”avoiding uniform loss” property for lower and upper probabilities (a la Walley). By our algorithms we can check the g-coherence of a given imprecise assessment and we can correct it in order to obtain the associated coherent assessment (in the sense of Walley and Williams). Exploiting some properties of the random gain we show how, in the linear systems involved in our algorithms, we can work with a reduced set of va…