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Information geometry provides the mathematical sciences with a new
framework of analysis. It has emerged from the investigation of the
natural differential geometric structure on manifolds of
probability distributions, which consists of a Riemannian metric
defined by the Fisher information and a one-parameter family of
affine connections called the $\alpha$-connections. The duality
between the $\alpha$-connection and the $(-\alpha)$-connection
together with the metric play an essential role in this geometry.
This kind of duality, having emerged from manifolds of probability
distributions, is ubiquitous, appearing in a variety of problems
which might have no explicit relation to probability theory.
Through the duality, it is possible to analyze various fundamental
problems in a unified perspective. The first half of this book is
devoted to a comprehensive introduction to the mathematical
foundation of information geometry, including preliminaries from
differential geometry, the geometry of manifolds or probability
distributions, and the general theory of dual affine
connections.The second half of the text provides an overview of
many areas of applications, such as statistics, linear systems,
information theory, quantum mechanics, convex analysis, neural
networks, and affine differential geometry. The book can serve as a
suitable text for a topics course for advanced undergraduates and
graduate students.
A fascianting account of the life and methods of Ino (1745 - 1818),
pioneer surveyor of Japan. A successful brewer, he learned his art
after age fifty. The accuracy of his maps of the Japanese coasts
astonished Europeans and rendered detailed mapping by the British
Navy unnecessary. Ito is a much prized cult figure of Japanese
engineers even in these days.
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