Differential Geometrical Foundations Of Information Geometry: Geometry Of Statistical Manifolds And Divergences

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.