  Consider a classification problem where we do not have access to labels for
individual training examples, but only have average labels over subpopulations.
We give practical examples of this setup and show how such a classification
task can usefully be analyzed as a weakly supervised clustering problem. We
propose three approaches to solving the weakly supervised clustering problem,
including a latent variables model that performs well in our experiments. We
illustrate our methods on an analysis of aggregated elections data and an
industry data set that was the original motivation for this research.
