  We study approaches to improve fine-grained short answer Question Answering
models by integrating coarse-grained data annotated for paragraph-level
relevance and show that coarsely annotated data can bring significant
performance gains. Experiments demonstrate that the standard multi-task
learning approach of sharing representations is not the most effective way to
leverage coarse-grained annotations. Instead, we can explicitly model the
latent fine-grained short answer variables and optimize the marginal
log-likelihood directly or use a newly proposed \emph{posterior distillation}
learning objective. Since these latent-variable methods have explicit access to
the relationship between the fine and coarse tasks, they result in
significantly larger improvements from coarse supervision.
