Differentially Private Inference for Binomial Data

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.

INSPECTRE: Privately Estimating the Unseen

“…Perhaps most relevant to this work are Cai et al (2017); Acharya et al (2018b); Aliakbarpour et al (2018), which study the sample complexity of differentially privately performing classical distribution testing problems, including identity and closeness testing. Some recent work focuses on the testing of simple hypotheses: Canonne et al (2019) studies the sample complexity of this problem, while Awan and Slavkovic (2018) provides a uniformly most powerful (UMP) test for binomial data (though Brenner and Nissim (2014) shows that UMP tests can not exist in general). Other works investigating private hypothesis testing include Wang et al (2015a); Gaboardi et al (2016); Kifer and Rogers (2017); Kakizaki et al (2017); ; Campbell et al (2018); Swanberg et al (2019); Couch et al (2019), which focus less on characterizing the finite-sample guarantees of such tests, and more on understanding their asymptotic properties and applications to computing p-values.…”

A Survey of Differentially Private Regression for Clinical and Epidemiological Research

“…Despite the preponderance of differentially private algorithms applicable to point estimation, only recently have studies begun to address other aspects of statistical inference, namely, confidence intervals and hypothesis testing. Several studies have developed differentially private tests for simple models, including for the mean of a normal distribution (Solea, 2014), the difference between means (D'orazio et al , 2015), the independent samples t ‐test (Ding et al , 2018), the Wilcoxon signed‐rank test (Couch et al , 2018; Task & Clifton, 2016) and the binomial test (Awan & Slavkovic, 2018; Vu & Slavkovic, 2009). Differentially private confidence intervals for the mean of a normal distribution were developed by Karwa & Vadhan (2017).…”

The structure of optimal private tests for simple hypotheses

“…That is, the advantage of the optimal tester must be matched by some coupling of P n and Q n . [AS18] gives a universally optimal test when the domain size is two, however Brenner and Nissim [BN14] shows that such universally optimal tests cannot exist when the domain has more than two elements. A complementary research direction, initiated by Cai et al [CDK17], studies the minimax sample complexity of private hypothesis testing.…”