Differentially Private Uniformly Most Powerful Tests for Binomial Data

TL;DR

This paper develops differentially private hypothesis tests for binomial data that are uniformly most powerful, leveraging a novel distribution called Tulap, and demonstrates their superior performance and exact error control through simulations.

Contribution

The paper introduces DP-UMP tests for binomial data based on a new Tulap distribution, providing exact p-values and improved power over existing methods.

Findings

Tests have exact type I error control.

DP-UMP tests outperform current techniques in power.

Applicable to continuous data with distribution-free properties.

Abstract

We derive uniformly most powerful (UMP) tests for simple and one-sided hypotheses for a population proportion within the framework of Differential Privacy (DP), optimizing finite sample performance. We show that in general, DP hypothesis tests for exchangeable data can always be expressed as a function of the empirical distribution. Using this structure, we prove a `Neyman-Pearson lemma' for binomial data under DP, where the DP-UMP only depends on the sample sum. Our tests can also be stated as a post-processing of a random variable, whose distribution we coin "Truncated-Uniform-Laplace" (Tulap), a generalization of the Staircase and discrete Laplace distributions. Furthermore, we obtain exact p-values, which are easily computed in terms of the Tulap random variable. We show that our results also apply to distribution-free hypothesis tests for continuous data. Our simulation results demonstrate that our tests have exact type I error, and are more powerful than current techniques.