Fibre Bundles And Lie Groups On Statistical Manifolds

In this paper, fibre bundles and Lie groups on statistical manifolds arestudied. To begin with, the history of diferentiable geometry is reviewed.Furthermore, some fundamental concepts and main results of informationgeometry are reviewed. Furthermore, some preliminaries in theory of fi-bre bundles and diferential geometry on fibre bundles are presented, andthen fibre bundles are introduced to statistical manifolds. The calcula-tion of geometric structure on statistical manifolds can be transferred tothe corresponding frame bundles, which involves only linear calculation,hence is more convenient. This approach is then checked by an example:the manifold of monistic Gaussian distribution. Afterwards, the conceptof statistical Lie groups, as well as some related basic results, are intro-duced. Then three examples: monistic Gaussian distributions, exponentialdistributions, multi-independent-Gaussian distributions, are studied. Al-so, two ways to construct new statistical Lie groups from known ones areproved and an example is presented to check it. In fact, the product ofmanifolds of monistic Gaussian distributions is isomorphic to manifold ofmulti-independent-Gaussian distributions. This paper ends with a briefsummary.