Comparison of Poisson-gamma and Laplace mechanisms for differential privacy

CDC WONDER is a web-based tool for the dissemination of epidemiologic data such as the number of deaths stratified by geographic region, year, demographic variables, and specific causes of death. Motivated by the lack of formal privacy protections in place on CDC WONDER, recent work has proposed the creation of a Synthetic CDC WONDER based on a differentially private Poisson-gamma modeling framework, which samples values from the posterior predictive distribution associated with modeling event count data with a Poisson-likelihood and assuming a gamma prior on the underlying event rate. Unlike more traditional approaches for generating differentially private synthetic data, the Poisson-gamma framework incorporates (and relies on) publicly available information such as the estimates of the underlying population sizes and event rates to improve its utility and protects the sensitive data by increasing the informativeness of the prior distribution. While the Poisson-gamma framework has been shown to be capable of producing synthetic data with high utility, its performance has yet to be compared to more conventional techniques for satisfying differential privacy, such as those that rely on output perturbation. The goal of this work is to present a comparison of the Poisson-gamma framework and the Laplace mechanism for the purpose of generating a synthetic dataset comprised of the 26,000 cancer-related deaths in Pennsylvania counties from 1980. Here, we demonstrate that while the Poisson-gamma framework preserves inference on the urban/rural disparity in death rates, the Laplace mechanism -- when forced to produce non-negative values -- produces synthetic data that fail to preserve both the magnitude of the disparity and its direction.