  Hierarchical Bayesian methods enable information sharing across multiple
related regression problems. While standard practice is to model regression
parameters (effects) as (1) exchangeable across datasets and (2) correlated to
differing degrees across covariates, we show that this approach exhibits poor
statistical performance when the number of covariates exceeds the number of
datasets. For instance, in statistical genetics, we might regress dozens of
traits (defining datasets) for thousands of individuals (responses) on up to
millions of genetic variants (covariates). When an analyst has more covariates
than datasets, we argue that it is often more natural to instead model effects
as (1) exchangeable across covariates and (2) correlated to differing degrees
across datasets. To this end, we propose a hierarchical model expressing our
alternative perspective. We devise an empirical Bayes estimator for learning
the degree of correlation between datasets. We develop theory that demonstrates
that our method outperforms the classic approach when the number of covariates
dominates the number of datasets, and corroborate this result empirically on
several high-dimensional multiple regression and classification problems.
