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Interpretations of probability. (English) Zbl 0998.81508
Utrecht: VSP. 228 pp. (1999).
Publisher’s description: Like geometry, probability can not be reduced to just one model to describe all physical and biological phenomena. Each model has a restricted range of applications. Quantum physics demonstrated that the use of conventional probability models induces some paradoxes. Such paradoxes can be resolved by using non-Kolmogorov probability models, developed on the basis of purely classical interpretations of probability: frequency and ensemble. Frequency models describe violations of the law of large numbers. Ensemble models are models with infinitely small probabilities.
This is the first fundamental book devoted to non-Kolmogorov probability models. It provides the first mathematical theory of negative probabilities – with numerous applications to quantum physics, information theory, complexity, biology and psychology. Natural models with negative (frequency and ensemble) probabilities are developed in the framework of so called \(p\)-adic analysis. The book also contains an extremely interesting model of cognitive information reality with flows of information probabilities, describing the process of thinking, social and psychological phenomena.
This book will be of value and interest to specialists in probability theory, statistics, functional analysis, quantum physics and (partly) specialists in cognitive sciences and psychology.

81P05 General and philosophical questions in quantum theory
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
60A05 Axioms; other general questions in probability
28Axx Classical measure theory
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
46N50 Applications of functional analysis in quantum physics
91E10 Cognitive psychology