Duality Theorem Research Of Statistical Manifold

Information Geometry was first mentioned by the Japanese professor Amari. Its initial thinking is to build Differential Geometry which has Riemannian metric and dual contact. At the same time, it referenced divergence as a function of distance in order to establish the framework of information geometry. Its duality is very special but also very important part of it. Under the invariance principle, the manifold is proved to have a unique Riemannian metric given by the Fisher information matrix, and a dual pairs of affine connection. When the manifold is dually flat, an invariant divergence is defined between two probability distributions. The generalized Pythagorean theorem holds, where the e-geodesic and m-geodesic defined by the two dual affine connections play a fundamental role. This leads to the projection theorem and its dual. In this paper, Mainly on the basis of differential geometry, probability and statistics knowledge to solve the theory of probability and statistics, integrated use of Information Geometry knowledge to research the nature and application of dual structure and dual flat space on statistical manifold.especially study and research the dual nature between distribution families and the dual problem of divergence function on statistical manifold.1. Introduces about the background of this study, presented the basic knowledge about differentiable manifolds and related statistical manifold.2. Introduces dual structure, dual flat space and dual coordinate system.3. The definitions and dual nature of exponential family, mixture family and divergence in statistical manifold.