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.
@cite_16: 
@cite_12: This analysis brings together two long-standing research traditions in political science. One tradition, beginning with Key (1949), focuses on the negative reaction white voters exhibit when located in areas of relatively higher levels of black voter density as measured at the county and or state level. The second tradition, which can be traced back to Tingsten (1963) and Berelson, Lazarsfeld, and McPhee (1954), illustrates how the immediate social context within which voters find themselves exerts a positive influence on their electoral behavior. This study links these two traditions by examining the effect of black density at the local level on white voter behavior. The analysis finds that the level of black voter density at the local level positively influences the probability that a white voter in that locale will vote for a black candidate. The implications of this finding are discussed in the conclusion.
@cite_17: Abstract Several recent studies of voter choice in multiparty elections point to the advantages of multinomial probit (MNP) relative to multinomial conditional logit (MNL). We compare the MNP and MNL models and argue that the simpler logit is often preferable to the more complex probit for the study of voter choice in multi-party elections. Our argument rests on three areas of comparison between MNP and MNL. First, within the limits of typical data—a small sample of revealed voter choices among a few candidates or parties—neither model will clearly appear to have generated the observed data. Second, MNP is susceptible to a number of estimation problems, the most serious of which is that the MNP is often weakly identified in application. Weak identification is difficult to diagnose and may lead to plausible, yet arbitrary or misleading inferences. Finally, the logit model is criticized because it imposes the independence of irrelevant alternatives (IIA) property on voter choice. For most applications the IIA property is neither relevant nor particularly restrictive. We illustrate our arguments using data from recent US and French presidential elections.
There is a richer body of literature on modeling voter preferences. Most of this research separates into two primary methodologies. In the first, researchers are interested in the relationship between voter characteristics and ballot preferences. To obtain labeled datasets, researchers use voter surveys @cite_16 and exit polls @cite_12 . They then fit models via simple GLMs, like the multinomial probit or multinomial logit model, which represent voter choice as a decision among unordered alternatives (parties, candidates) as a function of chooser (voter) attributes" @cite_17 .