A Probabilistic Approach for Learning with Label Proportions Applied to the US Presidential Election

Ecological inference (EI) is a classical problem from political science to model voting behavior of individuals given only aggregate election results. Flaxman et al. recently formulated EI as machine learning problem using distribution regression, and applied it to analyze US presidential elections. However, distribution regression unnecessarily aggregates individual-level covariates available from census microdata, and ignores known structure of the aggregation mechanism. We instead formulate the problem as learning with label proportions (LLP), and develop a new, probabilistic, LLP method to solve it. Our model is the straightforward one where individual votes are latent variables. We use cardinality potentials to efficiently perform exact inference over latent variables during learning, and introduce a novel message-passing algorithm to extend cardinality potentials to multivariate probability models for use within multiclass LLP problems. We show experimentally that LLP outperforms distribution regression for predicting individual-level attributes, and that our method is as good as or better than existing state-of-the-art LLP methods.