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 wide areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book will serve as a suitable text for a topics course for advanced undergraduates and graduate students.

Information geometry, which began as an investigation of the natural differential geometric structure possessed by families of probability distributions, is offered here in a complete treatment, translated from the original 1993 work in Japanese. The topic is applicable to information theory, stochastic processes, and systems, including neurocomputing. The authors, who assume some knowledge of statistics, systems theory and information theory, begin with an introduction to differential geometry and the theory of dual connections, before describing statistical inference, geometry of time series and linear systems, multiterminal information theory and statistical inference, information geometry for quantum systems, concluding with a grab-bag of applications including geometry of convex analysis, linear programming and gradient flows, neuro-manifolds and nonlinear systems, lie groups and transformation models.
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There has been much progress in the field of information geometry in recent years and yet there are few textbooks which reflect these developments, except for those which deal with statistics. This text deals with the various methods of information geometry.