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Chapter 8 : Statistical measurements in C∗-algebraic formulation

シロウ イシカワ, Shiro Ishikawa, 史郎 石川

Published Jan 1, 2006• 0 citations
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Abstract

8.1 Statistical measurements (C∗-algebraic formulation) 8.2 General statistical system (Example) 8.3 Bayes theorem in statistical MT 8.4 Kalman filter in noise 8.5 Information and entropy 8.6 Belief measurement theory (=BMT) 8.7 Appendix (Bertrand's paradox)

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I have proposed OST (Observable System Theory) as a foundational theory of system theory.3 This theory adopts two principles, namely, the Born principle and the Causal principle, and furthermore explicitly uses the one-shot measurement principle as a rule of use. OST is one of the most fundamental scientific theories, alongside mathematics and physics, and it can provide a unified framework for discussing quantum mechanics (information), statistics (system theory), practical logic, epistemology (philosophy), statistical mechanics, and so on.In this paper, I clarify that Fisherian statistics, Bayesian statistics, and quantum mechanics share a common mathematical foundation, namely OST. In statistics, an OST-based understanding of the Kalman filter is presented. Moreover, the so-called “interpretation problem” of quantum mechanics after measurement is unnecessary, and the “Wigner’s friend paradox” is resolved within OST. In addition, the FR paradox (= Frauchiger–Renner paradox) is understood as a paradox arising from the violation of the one-shot measurement principle, and thus it suggests the validity of OST. Through these discussions, I claim the following two points:(i) OST-based statistics will, in the future, become the standard understanding of statistics.(ii) OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics