Furthermore, we obtain exactp-values, which are easily computed in terms of the Tulap random variable. We show that our results also apply to distribution-free hypothesis testsforcontinuous data.

Outside the hypothesis testing setting, there is some work on optimal population inference under DP .

Hypothesis tests are one of the most basic and common statistical analyses that analysts perform on data. The goal of a hypothesis test is to see whether some "effect" in the data (e.g., men are taller than women) is plausibly the result of random variation in the sample, rather than a true fact about the population. Hypothesis tests are the bedrock of statistical analysis in the social sciences, medicine, and other fields, and a variety of hypothesis tests are used, depending on the type of data and the sort of effect one is considering. However, data in these fields often consists of private information about individuals. Researchers are under moral and legal obligations to protect the privacy of that data and can often only access that data if they can guarantee their analysis will not violate the privacy of those individuals. Differential privacy has emerged as the most convincing formal definition of privacy protection in this setting.

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. 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.

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. 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.