LONG NGUYEN 
Department of Statistics
University of Michigan
“An Optimal Transport Based Theory of Inference with Hierarchical Models”
ABSTRACT
Hierarchical models present a powerful tool in statistics and numerous applied fields. They allow for statistical dependence to be conveniently expressed via latent variables. They also enable the “borrowing of strength” — a heuristic argument hinting at the benefit of combining multiple data sets by making them share some latent variables in common. To understand this phenomenon, one needs to understand the convergence behavior of latent variables that arise in hierarchical models. In this talk I will describe some progress in our study of such questions. We will start with classic models such as location-scale Gaussian mixtures, shape-scale Gamma mixtures, and establish new results on rates of convergence of mixing distributions arising from such models. We will then present results for several hierarchical models widely used in Bayesian nonparametrics, including the Dirichlet process mixtures, and the hierarchical Dirichlet processes. The main tool in our theory is the use of a class of optimal transport based distances (i.e., Wasserstein distances) of probability measures. They turn out to be particularly natural for analyzing the geometry of mixture and hierarchical models.
