时间：2024年12月19日（星期四）9:00-11:00
地点：E4-233
主持人：理论科学研究院 陈华一
主讲人:上海数学与交叉学科研究院 张俊
主讲人简介: 张俊，上海数学与交叉学科研究院（SIMIS）教授。从1992年起，他在密歇根大学安娜堡分校心理学系担任助理教授、副教授、教授并在数学系、统计系、密歇根大数据学院挂职，目前休假。曾在澳洲墨尔本大学、法国马赛国立科学研究中心、加拿大滑铁卢大学、日本理化学研究所脑研究院、哈佛大学应用数学中心等学术单位担任访问职位。张俊教授在计算神经科学、认知与行为建模、机器学习、统计学、复杂系统等领域做出学术贡献，在数学心理学领域知名。
讲座主题：Information Geometry and Statistical Mirror Symmetry
讲座摘要: 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 such parameterization as a local coordinate chart, the family forms a manifold M equipped 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 conjugate 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 i) treating the parameter as an affine parameter of a flat connection on M and then ii) prescribing its g-conjugate connection as a curvature-free but torsion-carrying one. This new framework enables the construction of a pair of distinct objects on the tangent bundle TM 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.” To the extent this complex-to-symplectic correspondence can be constructed from any parametric statistical model, we call this “statistical mirror-symmetry'' and speculate its meaning in the context of statistical inference. (Joint work with Gabriel Khan of Iowa State University).
