# Differentially Private Inference for Binomial Data

## TL;DR

This paper develops optimal differentially private hypothesis tests for binomial data, providing exact p-values, confidence intervals, and demonstrating improved power over existing methods.

## Contribution

It introduces a framework for DP hypothesis testing using linear constraints and the Tulap distribution, deriving the first DP-UMP tests for binomial data.

## Key findings

- Tests have exact type I error control.
- Proposed tests outperform current techniques in power.
- Applicable to continuous data hypothesis testing.

## 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 can be written in terms of linear constraints, and 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. Using the above techniques, we show that our tests can be applied to give uniformly most accurate one-sided confidence intervals and optimal confidence distributions. We also derive uniformly most powerful unbiased (UMPU) two-sided tests, which lead to uniformly most accurate unbiased (UMAU) two-sided confidence intervals. We show that our results can be applied to distribution-free hypothesis tests for continuous data. Our simulation results demonstrate that all our tests have exact type I error, and are more powerful than current techniques.