April 2023 Canonical noise distributions and private hypothesis tests
Jordan Awan, Salil Vadhan
Author Affiliations +
Ann. Statist. 51(2): 547-572 (April 2023). DOI: 10.1214/23-AOS2259


f-DP has recently been proposed as a generalization of differential privacy allowing a lossless analysis of composition, post-processing, and privacy amplification via subsampling. In the setting of f-DP, we propose the concept of a canonical noise distribution (CND), the first mechanism designed for an arbitrary f-DP guarantee. The notion of CND captures whether an additive privacy mechanism perfectly matches the privacy guarantee of a given f. We prove that a CND always exists, and give a construction that produces a CND for any f. We show that private hypothesis tests are intimately related to CNDs, allowing for the release of private p-values at no additional privacy cost, as well as the construction of uniformly most powerful (UMP) tests for binary data, within the general f-DP framework.

We apply our techniques to the problem of difference-of-proportions testing, and construct a UMP unbiased (UMPU) “semiprivate” test which upper bounds the performance of any f-DP test. Using this as a benchmark, we propose a private test based on the inversion of characteristic functions, which allows for optimal inference on the two population parameters and is nearly as powerful as the semiprivate UMPU. When specialized to the case of (ϵ,0)-DP, we show empirically that our proposed test is more powerful than any (ϵ/2)-DP test and has more accurate type I errors than the classic normal approximation test.

Funding Statement

This work was supported by Cooperative Agreement CB16ADR0160001 from the U.S. Census Bureau. The first author was also supported in part by NSF Award Numbers SES-1534433, SES-1853209, and SES-2150615. The second author was also supported in part by a Simons Investigator Award.


The authors thank the reviewers, the Associate Editor, and Editor Mammen for their careful reading and constructive comments, which greatly improved the quality of this paper. The first author is grateful for the hospitality of the Center for Research on Computation and Society at Harvard University, where part of this work was completed.


Download Citation

Jordan Awan. Salil Vadhan. "Canonical noise distributions and private hypothesis tests." Ann. Statist. 51 (2) 547 - 572, April 2023. https://doi.org/10.1214/23-AOS2259


Received: 1 November 2021; Revised: 1 November 2022; Published: April 2023
First available in Project Euclid: 13 June 2023

zbMATH: 07714171
MathSciNet: MR4600992
Digital Object Identifier: 10.1214/23-AOS2259

Primary: 68P27
Secondary: 62F03

Keywords: differential privacy , frequentist inference , uniformly most powerful test

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 2 • April 2023
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