Mathematical Characterizations of Measurement Statistics

Henry Krips

Shiro Ishikawa

It is a matter of course that Kolmogorov’s probability theory is a very useful mathematical tool for the analysis of statistics. However, this fact never means that statistics is based on Kolmogorov’s probability theory, since it is not guaranteed that mathematics and our world are connected. In order that mathematics asserts some statements concerning our world, a certain theory (so called “world view”) mediates between mathematics and our world. Recently we propose measurement theory (i.e., the theory of the quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. In this paper, we assert that statistics is based on measurement theory. And, for example, we show, from the pure theoretical point of view (i.e., from the measurement theoretical point of view), that regression analysis can not be justified without Bayes’ theorem. This may imply that even the conventional classification of (Fisher’s) statistics and Bayesian statistics should be reconsidered.

Roger Balian

Stanley Gudder, Jean-Paul Marchand, Walter Wyss

We have previously constructed an entropy functional which characterizes statistical inference from partial measurement by maximum relative entropy. Here we discuss the mathematical properties of this concept in greater detail and establish its relation to the Bures distance and the Uhlmann transition probability.

Huzihiro Araki

The Kubo-Martin-Schwinger boundary condition, which had first been introduced as a property characterizing the canonical ensemble average in quantum statistical mechanics, was later recoginized as a property characterizing the modular automorphisms in the so-called Tomita-Takesaki theory of von Neumann algebras. This miraculous link made it possible to apply powerful mathematical technique supplied by the Tomita-Takesaki theory to a study of some problems in mathematical formulation of statistical mechanics. More recently, the KMS condition in classical statistical mechanics has also been investigated. The aim of this exposition is to review for non-specialists those development of the “rigorous” treatment of statistical mechanics which are related to the KMS condition.