  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.
