A foundation of information geometry

This paper proposes a new geometrical basic theory for the field of information science. the traditional theory for the information system is occupied fully by detailed discussions of the properties of a system, and the importance has not been recognized for the mutual relationship within a set of information systems. There are many cases in engineering where a set of information systems must be considered by a single model (for example, in the cases of statistical and system models). to discuss the properties of a model, the mutual relations among the elements must be described. In such a case, the geometrical structure among the information systems, which are elements of the model (e.g., distance, linearity and curvature), is important. This paper recognizes first the family of probability distributions as a manifold. Then it is shown that the structures such as Riemannian metric, α-pseudo-distance including a real parameter α, and α-affine connection, can be introduced naturally. the differential geometrical structure of the space is described. Using the proposed method, the differential geometrical structure inherent in the set of information sources or set of systems can be recognized. the structure is related closely to the system identification problem (parameter estimation), approximation and robustness problems.