Investigation Of α-families And Its Application

Information geometry has emerged from the study of invariant properties of the manifold of probability distributions. It is regarded as mathematical sciences having vast developing areas of applications and giving new trends in geometrical and topological methods. Information geometry has many applications which are treated in many different branches, for instance, statistical inference, linear systems and time series, neural networks and non-linear systems, linear programming, convex analysis and completely integrable dynamical systems, quantum information geometry and geometric modelling.On the basis of regarding a statistical model as a statistical manifold, In this paper, firstly, it discussed the additivity and convexity of information matrix on the statistical manifold, and promoted it, secondly it talked about the Cramer-Rao inequality and the Pythagorean relationship on the exponential family and the mixed family and its related conclusions, thirdly generalized it to the a-families, and proved it in detail. In this paper, the main content include:The first part we introduced the background of the problem.The second part gives a brief introduction to statistical manifold, the dual connection, dually flat spaces, canonical divergence, the exponential family and the mixed family, the m-representation of the tangent vector X, the e-representation of the tangent vector X and its related theorem.The third fourth part proves the Cramer-Rao inequality of the α-representation of the tangent vector X and the Pythagorean relationship of a-families.The fourth part is summary and outlook.