活动地点：腾讯会议：498-102-752

活动类型：学术报告

主讲人：张俊教授

活动时间：2021-12-03 09:00:00--2021-12-03 10:00:00

活动内容：

张俊教授专题学术报告

 

报告题目：Conjugate Connections and Statistical Mirror Symmetry

报告人：张俊  教授（美国密歇根大学）

时间：2021年12月3日（周五）上午 9:00

腾讯会议：498-102-752

 

摘要：A parametric statistical model is a family of probability density functions over a given sample space, whereby each function is indexed by a parameter taking value in some subset of Rn. Treating parameterization as a local coordinate chart, the family forms a manifold M endowed with a Riemannian metric g given by the Fisher-information (the well-known Fisher-Rao metric). The classical theory of information geometry prescribes a family of dualistic, torsion-free alpha-connections constructed from Amari-Chensov tensor as deformation from the Levi-Civita connection associated with g. Here we prescribe an alternative geometric framework of the manifold M by treating the parameter as an affine parameter of a flat connection and then prescribing its dual connection (with respect to g) as one that is curvature-free but carries torsion. From our new model, we are able to construct a pair of distinct objects on the tangent bundle TM based on the Sasaki lift of g and a canonical split using data from the base manifold M. The pair consists of a Hermitian structure and an almost Kahler structure simultaneously constructed that are in “mirror correspondence.'' In analogous to mirror symmetry in string theory between a complex manifold on the one side and a symplectic manifold on the other, we call this “statistical mirror-symmetry,'' and speculate its meaning in the context of statistical inference.  (Joint work with Gabriel Khan).