Information Geometry And Applications

Information geometry emerged from studies on invariant properties of a manifold of probability distributions. It includes convex analysis and its duality as a special but important part. 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 connections. 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 the present paper, we also consider Bayesian statistics from a viewpoint of differential geometry, and elucidate rela-tions between equiaffine geometry and Bayesian statistics. A prior distribution in Bayesian statistics is regarded as a volume form on a statistical manifold. Apply-ing equiaffine geometry to Bayesian statistics, the relation between alpha-parallel priors and the Jeffreys prior is given. A typical example of information geometry is a curved exponential family. We consider the classical inference of estima-tion and its Bayesian statistics from the geometric point of view. In particular, we analyze the asymptotic theory of estimation and evaluate various efficient estimators. In this thesis, as applications, we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of con-nections introduced into it. Using the theory we derive interesting geometrical interpretations for matrix approximation.