Application Of Principal Bundle On Statistical Manifold

The concept of principal bundles was introduced on statistical manifolds.First,some basic knowledge of information geometry and exponential family was given,in-cluding the concepts of Fisher information matrix and ?-connection.Also the expo-nential family was explained by some examples.Then the concept of principal bundles and the geometry of principal bundles were introduced,where the concepts of local trivialization,transition function and connection were given.During the process of introducing the geometry of principal bundles,the torsion and curvature were mainly explained.After that,the ?-structure of principal bundles was studied,and the crucial theorem was given.Finally,these methods were used to calculate the manifold of bivariate normal distributions.Use the method of information geometry at first,taking the logarithm of the density function of bivariate normal distribution.Then calculate the Fisher information matrix,?-connection coefficients and a-connection forms.After that,use the property of principal bundles and calculate more,such as horizontal space and horizontal lift.Through the ?-structure of principal bundles,?X(?)X was get.Similarly.geometric properties such as curvature could be derived.The advantage of the paper was introducing principal bundles on statistical man-ifolds.In this way,all operations were transferred to linear operations so that calcula-tions became simpler.