Showing papers on "Information geometry published in 1983"

A foundation of information geometry

Abstract: 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.

An urnful of blending functions

Abstract: The author explores the link between probability and geometry. In the process, he shows how to exploit simple probabilistic arguments to derive many of the classical geometric properties of the parametric curves and surfaces currently in vogue in computer-aided geometric design. He also uses this probabilistic approach to introduce many new types of curves and surfaces into computer-aided geometric design, and demonstrates how probability theory can be used to simplify, unify, and generalize many well-known results. He concludes that urn models are a powerful tool for generating discrete probability distributions, and built into these special distributions are many propitious properties essential to the blending functions of computer-aided geometric design. This fact allows mathematicians to use probabilistic arguments to simplify, unify, and generalize many geometric results. He believes that this link between probability and geometry will ultimately prove beneficial to both disciplines, and expects that it will continue to be a productive area for future inspiration and research.

Thorpe–Hitchin inequality for compact Einstein 4‐manifolds of metric signature (++−−) and the generalized Hirzebruch index formula

Abstract: It is proved that the Euler characteristic and the Hirzebruch index of a compact oriented Einstein 4‐manifold of metric signature (++−−) satisfy an inequality which is well known as the Thorpe–Hitchin inequality for the case of a Riemannian metric. To derive the inequality, a generalized Hirzebruch formula relating the index to the first pseudo‐Pontrjagin number of the manifold is proved. This formula may be contrasted with Chern’s generalized Gauss–Bonnet formula for a pseudo‐Riemannian manifold.

A note on the geometry of kullback-leibler information numbers

Abstract: Csiszar (1975) has shown thdt kullhack-Leibler information numbers possess some geometrical properties much like those in Euclidean geometry. Thls paper extends these results by characterizing the shortest line between two ditributions as well as the midpoit of the line.

The cardinality of manifold atlases

Abstract: It is shown that any finite dimensionalC0 manifold (connected and Hausdorff but otherwise unrestricted) has an atlas of cardinality not greater than that of the continuum; while if it has a Holder continuous pseudo-Riemannian metric then there is a countable atlas.