An extension of chaotic probability models to real-valued variables

Abstract

In a recent series of papers, Fine and colleagues [P.I. Fierens, T.L. Fine, Towards a frequentist interpretation of sets of measures, in: G. de Cooman, T.L. Fine, T. Seidenfeld (Eds.), Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing, 2001; P.I. Fierens, T.L. Fine, Towards a chaotic probability model for frequentist probability, in: J. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, Carleton Scientific, 2003; L.C. Rêgo, T.L. Fine, Estimation of chaotic probabilities, in: Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, 2005] have presented the first steps towards a frequentist understanding of sets of measures as imprecise probability models which have been called chaotic models. Simulation of the chaotic variables is an integral part of the theory. Previous models, however, dealt only with sets of probability measures on finite algebras, that is, probability measures which can be related to variables with a finite number of possible values. In this paper, an extension of chaotic models is proposed in order to deal with the more general case of real-valued variables. This extension is based on the introduction of real-valued test functions which generalize binary-valued choices in the previous work.