Information Geometry
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Beschreibung
The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated.
This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo.
The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems.
Will become standard reference of the field Introduces new mathematical treatment of parametrised measure models Includes general results on sufficient statistics, Cramér-Rao inequality, uniqueness of Fisher metric and Amari-Chentsov tensor Provides new applications of information geometry Includes supplementary material: sn.pub/extras
Autor*in
Nihat Ay
Themen in »Information Geometry«
60A10, 62B05, 62B10, 62G05, 53B21 53B05, 46B20, 94A15, 94A17, 94B27 Information Geometry Fisher Metric Amari-Chentsov Tensor Alpha Connections Divergences data structures
Stimmen zu »Information Geometry«
“The monograph is very timely and welcome. It will become a standard textbook and reference for the field. It is not only accessible and useful for scientists who work in probability or statistics, but also for all graduate students of mathematics.” (Andreas Knauf, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 121, 2019)
“This book also provides a general framework that integrates the differential geometry into the functional analysis, it is an easy read for graduate students in mathematical statistics. I believe that it could be standard reference for information geometry.” (Etsuo Kumagai Hamada, Mathematical Reviews, June, 2018)
“The book as well as [Shun-ichi Amari’s Information Geometry and its Applications] will remain standard textbooks on information geometry in the foreseeable future and will become the classics in the arena afterwards.” (Hirokazu Nishimura, zbMATH 1383.53002)
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