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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 Cramer-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.
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