Learning Latent Structures via Hierarchical Nonparametric Bayes: A Look at the Posterior Asymptotics

• Activity Number: 291
• Type: Topic Contributed
• Date/Time: Tuesday, August 6, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Long Nguyen*+
• Keywords: Dirichlet process ; hierarchical Dirichlet process ; optimal transportation ; Wasserstein distance ; posterior asymptotics

Abstract

Hierarchical models represent a powerful tool in statistics and machine learning. In a hierarchical model statistical dependence can be expressed via latent variables, which may also be objects of inference. Latent hierarchies also enable "borrowing of strength" between different data sets through shared random parameters in the hierarchies. In this talk I will discuss some recent progress on the posterior asymptotics for hierarchical models, taking a view that places latent variables at the center of inference. By analyzing the posterior concentration behavior of latent variables (measures) that arise in a number of Bayesian nonparametric models, including the Dirichlet process mixture and the hierarchical Dirichlet processes, we show how to quantify in a precise sense the benefits of borrowing strength in a hierarchical model. The increased efficiency can be quite dramatic, improving from standard nonparametric rates to parametric rate of convergence. The notion of transportation distances (e.g., Wasserstein distance) plays a key role in this theory.