Stochastic Independence for Upper and Lower Probabilities in a Coherent Setting
In this paper we extend to upper and lower probabilities our approach to independence, based solely on conditional probability (in a coherent setting). Most difficulties arise (when this notion is put forward in the classical framework) either from the introduction of marginals for upper and lower probabilities (often improperly called, in the relevant literature, “imprecise” probabilities) when trying to extend to them the “product rule”, or from the different ways of introducing conditioning for upper and lower probabilities. Our approach to conditioning in the context of “imprecise” probabilities is instead the most natural: in fact its starting point refers to a direct definition (through coherence) of the “enveloping” conditional “precise” probabilities. The discussion of some critical examples seems to suggest that the intuitive aspects of independence are better captured by referring to just one (precise) probability than to a family (such as that one singling-out a lower or upper probability).
KeywordsConditional Probability Lower Probability Boolean Algebra Product Rule Conditional Event
Unable to display preview. Download preview PDF.
- 2.Coletti, G., Scozzafava, R. (1998) Conditional measures: old and new. In: Proc. of “New Trends in Fuzzy Systems”, Napoli 1996 (eds. D.Mancini, M.Squillante, and A.Ventre ), World Scientific, 107–120.Google Scholar
- 3.Coletti, G., Scozzafava, R. (2000) Zero Probabilities in Stochastical Independence. In Information, Uncertainty, Fusion (Eds. B. Bouchon-Meunier, R. R. Yager, and L. A. Zadeh), Kluwer, Dordrecht, 185–196. (Selected papers from IPMU ‘88, Paris)Google Scholar
- 6.Coletti, G., Scozzafava, R. (2001) Stochastic Independence in a Coherent Setting. Annals of Mathematics and Artificial Intelligence (Special Issue on “Partial Knowledge and Uncertainty: Independence, Conditioning, Inference”), to appearGoogle Scholar
- 7.Coletti, G., Scozzafava, R., Vantaggi, B. (2001) Probabilistic Reasoning as a General Inferential Tool, ECSQARU 2001, submitted.Google Scholar
- 8.Couso, I., Moral, S., Walley, P. (1999) Examples of independence for imprecise probabilities. In: Int. Symp. on Imprecise Probabilities and their applications (ISIPTA ‘89), Ghent, Belgium, 121–130Google Scholar
- 9.De Campos, L.M., Moral, S. (1995) Independence concepts for convex sets of probabilities. In: Uncertainty in Artificial Intelligence (UAI ‘85), Morgan & Kaufmann, San Mateo, 108–115Google Scholar
- 10.de Finetti, B. (1949) Sull’Impostazione Assiomatica del Calcolo delle Probabilità. Annali Univ. Trieste 19, 3–55. (Engl. transl.: Ch.5 in Probability, Induction, Statistics, Wiley, London, 1972 )Google Scholar
- 14.Spohn, W. (1999) Ranking Functions, AGM Style. Research Group “Logic in Philosophy ” Preprint 28 Google Scholar
- 15.Vantaggi, B. (2001) Conditional Independence in a Finite Coherent Setting. Annals of Mathematics and Artificial Intelligence, to appearGoogle Scholar