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.
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