Methods of Information Geometry (Translations of Mathematical Monographs)

Description

Elementary differential geometry -- Differentiable manifolds -- Tangent vectors and tangent spaces -- Vector fields and tensor fields -- Submanifolds -- Riemannian metrics -- Affine connections and covariant derivatives -- Flatness -- Autoparallel submanifolds -- Projection of connections and embedding curvature -- Riemannian connection -- The geometric structure of statistical models -- Statistical models -- The Fisher metric -- The [alpha]-connection -- Chentsov's theorem and some historical remarks -- The geometry of P (X) -- [alpha]-affine manifolds and [alpha]-families -- Dual connections -- Duality of connections -- Divergences: general contrast functions -- Dually flat spaces -- Canonical divergence -- The dualistic structure of exponential families -- The dualistic structure of [alpha]-affine manifolds and [alpha]-families -- Mutually dual foliations -- A further look at the triangular relation -- Statistical inference and differential geometry -- Estimation based on independent observations -- Exponential families and observed points -- Curved exponential families -- Consistency and first-order efficiency -- Higher-order asymptotic theory of estimation -- Asymptotics of Fisher information -- Higher-order asymptotic theory of tests -- The theory of estimating functions and fiber bundles -- The fiber bundle of local exponential families -- Hilbert bundles and estimating functions -- The geometry of time series and linear systems -- The space of systems and time series