Independence in Generalized Interval Probability

Recently we proposed a new form of imprecise probability based on the generalized interval, where the probabilistic calculus structure resembles the traditional one in the precise probability because of the Kaucher arithmetic. In this paper, we study the independence properties of the generalized interval probability. It resembles the stochastic independence with proper and improper intervals and supports logic interpretation. The graphoid properties of the independence are investigated. INTRODUCTION Probability theory provides the common ground to quantify uncertainty. However, it has limitations in representing epistemic uncertainty that is due to lack of knowledge. It does not differentiate the total ignorance from other probability distributions, which leads to the Bertrand-style paradoxes such as the Van Fraasen's cube factory (van Fraassen 1989). Probability theory with precise measure also has limitation in capturing indeterminacy and inconsistency. When beliefs from different people are inconsistent, a range of opinions or estimations cannot be represented adequately without assuming some consensus of precise values on the distribution of opinions. Therefore imprecise probabilities have been proposed to quantify aleatory and epistemic uncertainty simultaneously. Instead of a precise value of the probability ( ) P E p = associated with an event E , a pair of lower and upper probabilities ( ) [ , ] P E p p = are used to include a set of probabilities and quantify epistemic uncertainty. The range of the interval [ , ] p p captures the epistemic uncertainty component and indeterminacy. [0,1] P = accurately represents the total ignorance. When p p = , the degenerated interval probability becomes a precise one. In a general sense, imprecise probability is a generalization of precise probability. Many representations of imprecise probabilities have been developed. For example, the Dempster-Shafer evidence theory (Dempster 1967; Shafer 1990) characterizes evidence with discrete probability masses associated with a power set of values, where Belief-Plausibility pairs are used to measure uncertainties. The behavioral imprecise probability theory (Walley 1991) models uncertainties with the lower prevision (supremum acceptable buying price) and the upper prevision (infimum acceptable selling price) following the notations of de Finetti's subjective probability theory. The possibility theory (Dubois and Prade 1988) represents