Risk and Regret of Hierarchical Bayesian Learners

TL;DR

This paper develops analytical tools and regret bounds for hierarchical Bayesian models, formalizing concepts of robustness and statistical strength sharing, and providing guidance for their practical use in online and batch learning.

Contribution

It introduces a formal framework with regret bounds for hierarchical priors, linking Bayesian robustness and sharing strength to concrete theoretical guarantees.

Findings

Hierarchical priors can be analyzed with regret bounds comparing to the best single model.

Bayesian log-loss regret bounds can be converted into risk bounds for bounded losses.

Hierarchical Gaussian and Student's t priors formalize robustness and statistical sharing benefits.

Abstract

Common statistical practice has shown that the full power of Bayesian methods is not realized until hierarchical priors are used, as these allow for greater "robustness" and the ability to "share statistical strength." Yet it is an ongoing challenge to provide a learning-theoretically sound formalism of such notions that: offers practical guidance concerning when and how best to utilize hierarchical models; provides insights into what makes for a good hierarchical prior; and, when the form of the prior has been chosen, can guide the choice of hyperparameter settings. We present a set of analytical tools for understanding hierarchical priors in both the online and batch learning settings. We provide regret bounds under log-loss, which show how certain hierarchical models compare, in retrospect, to the best single model in the model class. We also show how to convert a Bayesian log-loss regret bound into a Bayesian risk bound for any bounded loss, a result which may be of independent interest. Risk and regret bounds for Student's $t$ and hierarchical Gaussian priors allow us to formalize the concepts of "robustness" and "sharing statistical strength." Priors for feature selection are investigated as well. Our results suggest that the learning-theoretic benefits of using hierarchical priors can often come at little cost on practical problems.