Abstract
In the paper we propose non-well-founded probabilities as a kind of fuzzy ones. They are defined on the set of streams. We also show that the set of p-adic numbers can be understood as a set of streams. In the set theory without the axiom of foundation, the powerset is not a Boolean algebra in the general case. Therefore, if we tried to define probabilities on non-well-founded data, i.e. on streams or p-adic numbers, then we couldn’t use the Kolmogorovian approach and we should refer to non-Kolmogorovian models of probabilities. Probabilities on streams have a lot of unexpected properties. For instance, p-adic probabilities may be negative rational numbers as well as rational numbers that are larger than 1. Bayes’ formula doesn’t also hold in the general case for non-well-founded probabilities.
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Schumann, A. (2008). Non-well-Founded Probabilities on Streams. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_8
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DOI: https://doi.org/10.1007/978-3-540-85027-4_8
Publisher Name: Springer, Berlin, Heidelberg
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