  We present a new modeling technique for solving the problem of ecological
inference, in which individual-level associations are inferred from labeled
data available only at the aggregate level. We model aggregate count data as
arising from the Poisson binomial, the distribution of the sum of independent
but not identically distributed Bernoulli random variables. We relate
individual-level probabilities to individual covariates using both a logistic
regression and a neural network. A normal approximation is derived via the
Lyapunov Central Limit Theorem, allowing us to efficiently fit these models on
large datasets. We apply this technique to the problem of revealing voter
preferences in the 2016 presidential election, fitting a model to a sample of
over four million voters from the highly contested swing state of Pennsylvania.
We validate the model at the precinct level via a holdout set, and at the
individual level using weak labels, finding that the model is predictive and it
learns intuitively reasonable associations.
