Differential Geometrical Foundations of Information Geometry

This monograph gives foundations of information geometry from the viewpoint of differential geometry. In information geometry, a statistical manifold structure is important, which is related to geometry of a pair of dual affine connections and an asymmetric distance called divergence. First, we summarize geometry of statistical manifolds. As applications, we explain statistical inferences and information criterions from the viewpoint of differential geometry.

Information geometry suggests generalized conformal structures on statistical manifold, which unify conformal structures and projective structures, etc. Hence we summarize these conformal geometries. In addition, such generalized conformal structure is useful for geometry of deformed exponential families, which is related to theory of complex systems and anomalous statistics. Hence we study geometry of deformed exponential families.

We also summarize recent developments of information geometry. In particular, we study geometry of Bayesian statistics, and infinite dimensional information geometry, etc.

• Statistical Manifolds and Hessian Manifolds
• Divergences
• Geometry of Statistical Inferences
• Generalized Conformal Structures
• Conformal Divergences
• Geometry of Deformed Exponential Families
• Generalizations of Expectation and Independence of Random Variables
• Equaffine Structures and Tchebychev Forms, Geometry of Bayesian Statistics
• Infinite Dimensional Information Geometry