Methods of information geometry. Transl. from the Japanese by Daishi Harada.

*(English)*Zbl 0960.62005
Translations of Mathematical Monographs. 191. Providence, RI: American Mathematical Society (AMS). Oxford: Oxford University Press, x, 206 p. (2000).

The field of information geometry has developed from investigations of the natural structures inherent in spaces of probability distributions. The structure of Riemannian spaces with dual connections which emerged from these investigations gives insight not only into fields directly related to probability theory, but also into a wide range of fields by providing a framework within which to analyze the underlying structures of the field. Information geometry opens a new paradigm useful for elucidation of information systems, intelligent systems, control systems, physical systems, mathematical systems, and so on. The utility of information geometry, however, is not limited to these fields. It has, for example, been productively applied to areas such as statistical physics and the mathematical theory underlying neural networks. Further, dualistic differential geometric structure is a general concept not inherently tied to probability distributions. For example, the interior method for linear programming may be analyzed from this point of view, and this suggests its relation to completely integrable dynamical systems. Finally, the investigation of the information geometry of quantum systems may lead to even further developments.

This English translation presents for the first time the entirety of the emerging field of information geometry. The first three chapters of the book – Elementary differential geometry; The geometric structure of statistical models; and Dual connections – are a concise but comprehensive introduction into the mathematical foundations of information geometry. The book introduces only the amount of differential geometry necessary for the remaining chapters, and endeavors to do so in a manner which, while consistent with the conventional definitions in mathematical texts, allows the intuition underlying the concepts to be comprehended most immediately.

The other five chapters – Statistical inference and differential geometry; The geometry of time series and linear systems; Multiterminal information theory and statistical inference; Information geometry for quantum systems; and Miscellaneous topics – are devoted to an overview of wide areas of applications. The authors consider classical inference of estimation and hypothesis testing from the geometric point of view, and then move towards nonparametric and semi-parametric inference, where the shape of the underlying distribution is unknown. The differential geometry of systems and time series is surveyed, and the importance of dual connections in the analysis of their properties is shown. One of the principal problems motivating information theory is the faithful communication of a message given the constraints of channel capacity, the standard approach is to analyze the probabilistic structure of the message and from this construct a code. In contrast, the goal of statistics is to infer the underlying probabilistic structure generating the message. Hence, although these two fields share the foundations of probabilistic structures on which they build their theories, because of the differences in their objectives their analyses tend to follow separate theoretical paths.

A problem which binds these two fields together is considered in this book. The problem of how the dualistic structure of the Fisher metric and the \(\alpha\)-connections on statistical models are extended to manifolds of quantum states is investigated. It is expected that new theoretical developments will arise from the applications of information geometric ideas to these disparate areas, and this book will play a key role for the further progress of information geometry.

This English translation presents for the first time the entirety of the emerging field of information geometry. The first three chapters of the book – Elementary differential geometry; The geometric structure of statistical models; and Dual connections – are a concise but comprehensive introduction into the mathematical foundations of information geometry. The book introduces only the amount of differential geometry necessary for the remaining chapters, and endeavors to do so in a manner which, while consistent with the conventional definitions in mathematical texts, allows the intuition underlying the concepts to be comprehended most immediately.

The other five chapters – Statistical inference and differential geometry; The geometry of time series and linear systems; Multiterminal information theory and statistical inference; Information geometry for quantum systems; and Miscellaneous topics – are devoted to an overview of wide areas of applications. The authors consider classical inference of estimation and hypothesis testing from the geometric point of view, and then move towards nonparametric and semi-parametric inference, where the shape of the underlying distribution is unknown. The differential geometry of systems and time series is surveyed, and the importance of dual connections in the analysis of their properties is shown. One of the principal problems motivating information theory is the faithful communication of a message given the constraints of channel capacity, the standard approach is to analyze the probabilistic structure of the message and from this construct a code. In contrast, the goal of statistics is to infer the underlying probabilistic structure generating the message. Hence, although these two fields share the foundations of probabilistic structures on which they build their theories, because of the differences in their objectives their analyses tend to follow separate theoretical paths.

A problem which binds these two fields together is considered in this book. The problem of how the dualistic structure of the Fisher metric and the \(\alpha\)-connections on statistical models are extended to manifolds of quantum states is investigated. It is expected that new theoretical developments will arise from the applications of information geometric ideas to these disparate areas, and this book will play a key role for the further progress of information geometry.

Reviewer: Serguey M.Pokas (Odessa)

##### MSC:

62B10 | Statistical aspects of information-theoretic topics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62F12 | Asymptotic properties of parametric estimators |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53A15 | Affine differential geometry |

53B05 | Linear and affine connections |

81Q70 | Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory |

94A15 | Information theory (general) |