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We consider a problem of ecological inference, in which individual-levelcovariates are known, but labeled data is available only at the aggregatelevel. The intended application is modeling voter preferences in elections. InRosenman and Viswanathan (2018), we proposed modeling individual voterprobabilities via a logistic regression, and posing the problem as a maximumlikelihood estimation for the parameter vector beta. The likelihood is aPoisson binomial, the distribution of the sum of independent but notidentically distributed Bernoulli variables, though we approximate it with aheteroscedastic Gaussian for computational efficiency. Here, we extend theprior work by proving results about the existence of the MLE and the curvatureof this likelihood, which is not log-concave in general. We further demonstratethe utility of our method on a real data example. Using data on voters inMorris County, NJ, we demonstrate that our approach outperforms otherecological inference methods in predicting a related, but known outcome:whether an individual votes.

Some New Results for Poisson Binomial Models