*Abstract: Exponential families are families of probability distributions which are important in statistics. When viewed geometrically, they lead to Riemannian manifolds equipped with a pair of flat connections that are dual to each other in a natural sense. When the two connections are flat, we say that the Riemannian manifold is dually flat. In this talk we will show, after Mathieu Molitor, that a Riemannian manifold is dually flat if and only if its tangent bundle is a Kahler manifold in a canonical way. As examples, we discuss the Kahler structure associated to a probability simplex, which corresponds to an open dense subset of complex projective space, and of the two-dimensional normal distribution, which is associated to the Seigel-Jacobi space.*

*Host: Xiang Tang*