  Stacking is a widely used model averaging technique that yields
asymptotically optimal prediction among all linear averages. We show that
stacking is most effective when the model predictive performance is
heterogeneous in inputs, so that we can further improve the stacked mixture
with a hierarchical model. With the input-varying yet partially-pooled model
weights, hierarchical stacking improves average and conditional predictions.
Our Bayesian formulation includes constant-weight (complete-pooling) stacking
as a special case. We generalize to incorporate discrete and continuous inputs,
other structured priors, and time-series and longitudinal data. We demonstrate
on several applied problems.
