# Canonical Noise Distributions and Private Hypothesis Tests

arXiv: https://arxiv.org/abs/2108.04303 · HTML: https://mlsanity.com/paper/2108.04303/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2021.

## Built on by (most cited first, 8 of 8)

- [Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies](https://mlsanity.com/paper/2210.06140/) (2022)
- [A Statistical Viewpoint on Differential Privacy: Hypothesis Testing, Representation and Blackwell's Theorem](https://mlsanity.com/paper/2409.09558/) (2024)
- [Gaussian Certified Unlearning in High Dimensions: A Hypothesis Testing Approach](https://mlsanity.com/paper/2510.13094/) (2025)
- [Beyond Laplace and Gaussian: Exploring the Generalized Gaussian Mechanism for Private Machine Learning](https://mlsanity.com/paper/2506.12553/) (2025)
- [Optimal Survey Design for Private Mean Estimation](https://mlsanity.com/paper/2501.18121/) (2025)
- [Infinitely divisible privacy and beyond I: resolution of the s2=2k conjecture](https://mlsanity.com/paper/2512.00734/) (2025)
- [Near-Optimal Private Tests for Simple and MLR Hypotheses](https://mlsanity.com/paper/2601.21959/) (2026)
- [Differentially Private Truncation of Unbounded Data via Public Second Moments](https://mlsanity.com/paper/2602.22282/) (2026)

## Abstract (arXiv)

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) "semi-private" 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 for the two population parameters and is nearly as powerful as the semi-private 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.