Privacy-Preserving Inference on the Ratio of Two Gaussians Using Sums

Abstract

The ratio of two Gaussians is useful in many contexts of statistical inference. We discuss statistically valid inference of the ratio under Differential Privacy (DP). We use the delta method to derive the asymptotic distribution of the ratio estimator and use the Gaussian mechanism to provide (epsilon, delta)-DP guarantees. Like many statistics, quantities involved in the inference of a ratio can be re-written as functions of sums, and sums are easy to work with for many reasons. In the context of DP, the sensitivity of a sum is easy to calculate. We focus on getting the correct coverage probability of 95% confidence intervals (CIs) of the DP ratio estimator. Our simulations show that the no-correction method, which ignores the DP noise, gives CIs that are too narrow to provide proper coverage for small samples. In our specific simulation scenario, the coverage of 95% CIs can be as low as below 10%. We propose two methods to mitigate the under-coverage issue, one based on Monte Carlo simulation and the other based on analytical correction. We show that the CIs of our methods have much better coverage with reasonable privacy budgets. In addition, our methods can handle weighted data, when the weights are fixed and bounded.