We extend some of the concepts from the behavioural theory of imprecise probabilities, such as weak and strong coherence, to non-linear spaces of gambles, and we study the relations between them. Then, we see some simplifications under the assumption of epistemic independence of experiments and we study the independent natural extension from given marginals.
Arianna Casanova, Enrique Miranda, Marco Zaffalon. Joint desirability foundations of social choice and opinion pooling. Submitted.
We develop joint foundations for the fields of social choice and opinion pooling using coherent sets of desirable gambles. This leads on the one hand to a new perspective of traditional results of social choice (in particular Arrow's theorem as well as sufficient conditions for the existence of an oligarchy and democracy) and on the other hand to using the same framework to analyse opinion pooling. In particular, we argue that weak Pareto (unanimity) should be given the status of a rationality requirement and use this to discuss the aggregation of experts' opinions based on probability and (state-independent) utility, showing some inherent limitation of this framework, with implications for statistics.
Enrique Miranda, Ignacio Montes, Paolo Vicig. On the elicitation of an optimal outer approximation of a coherent lower probability. Submitted.
Coherent lower probabilities are one of the most general tools within Imprecise Probability Theory, and can be used to model the available information about an unknown or partially known precise probability. In spite of their generality, coherent lower probabilities are sometimes difficult to deal with. For this reason, in previous papers we studied the problem of outer approximating a given coherent lower probability by a more tractable model, such as a 2- or completely-monotone lower probability. Unfortunately, such an outer approximation is not unique in general, even if we restrict our attention to those that are undominated by other models from the same family. In this paper, we investigate whether a number of approaches may help in eliciting a unique undominated outer approximation. These are based on minimising a distance with respect to the initial model, maximising the specificity, or preserving the same preferences as the original model. We apply them to 2- and completely monotone approximating lower probabilities, and also to the particular cases of possibility measures and p-boxes.