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Journal of Data Science


Privacy-Preserving Inference on the Ratio of Two Gaussians Using Sums
Volume 21, Issue 1 (2023), pp. 27–42
Pub. online: 16 June 2022      Type: Data Science In Action      Open accessOpen Access
Received
14 April 2022
Accepted
21 May 2022
Published
16 June 2022

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.

Supplementary material

 Supplementary Material
Our simulation code is hosted on GitHub in a Jupyter Notebook file. Replication instructions can be found in the Supplementary Material.

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2023 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.
Open access article under the CC BY license.

Keywords
calibration ratio Gaussian mechanism Laplace mechanism

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