Mathematics & Statistics Colloquium: Dr. Xiongzhi Chen

About the event

Title: The concept of “Expectation”: Part I

Speaker: Xiongzhi Chen, Department of Mathematics and Statistics, Washington State University

Abstract: The concept of “expectation”, also known as “mathematical expectation” or “mean”, is one of the most fundamental concept in statistical learning and probability theory. It is a core for inference and modelling, on which so many methods and theories in statistical learning and a significant part of probability theory have been built. These include one-/two-sample tests, regression models (as special cases of estimating stratified spaces that are formed by “conditional expectation”), laws of large numbers and central limit theorems (as special cases of concentration of measures around “expectation” for empirical processes), etc.

This is the first part of a two-talk series on fundamental aspects of “expectation”, their connections with other branches of mathematics, and some open problems related to the uniqueness of “expectation”.  These talks are motivated by the following two questions:

(1) If for a probability distribution, its “expectation” cannot be uniquely defined, how should we conduct statistical learning using “expectation”?

(2) When does a probability measure on a metric space of non-negative sectional curvature have a uniquely defined “expectation”?

In Part 1, let us go over the key properties of “expectation” in a Euclidean space, Banach space, and compact topological group, their associated laws of large numbers and central limit theorems, and their connections with functional analysis, group and representation theory, and harmonic analysis.

In Part 2, let us discuss the concept of “expectation” in a metric space, with a focus on “expectation” in a Riemannian manifold of non-negative sectional curvature. The latter is the realm that I am much more familiar with than “expectation” in a Banach space or compact group. Let us go over some concrete examples where “expectation” can be defined by not uniquely defined, discuss potential strategies to address the uniqueness problem, and explain their connections with the theory of differential equations (in particular, singularity of the differential of exponential mapping on a Riemannian manifold), algebraic and differential topology (in particular, critical points and level sets of a Lipschitz function on a Riemannian manifold), and harmonic analysis (in particular, on a Riemannian homogeneous space or on the isometry group of a Riemannian manifold).