Noise-Calibrated Inference from Differentially Private Sufficient Statistics in Exponential Families

TL;DR

This paper proposes a method for differentially private inference in exponential families by releasing DP sufficient statistics, enabling noise-calibrated likelihood inference and synthetic data generation with valid uncertainty quantification.

Contribution

It introduces a general recipe for releasing DP sufficient statistics, develops asymptotic inference tools, and provides a practical pipeline validated on real data.

Findings

Valid confidence intervals for DP MLEs are derived.

A noise-aware likelihood correction supports bootstrap intervals.

The privacy distortion rate is shown to be unavoidable by a minimax lower bound.

Abstract

Many differentially private (DP) data release systems either output DP synthetic data and leave analysts to perform inference as usual, which can lead to severe miscalibration, or output a DP point estimate without a principled way to do uncertainty quantification. This paper develops a clean and tractable middle ground for exponential families: release only DP sufficient statistics, then perform noise-calibrated likelihood-based inference and optional parametric synthetic data generation as post-processing. Our contributions are: (1) a general recipe for approximate-DP release of clipped sufficient statistics under the Gaussian mechanism; (2) asymptotic normality, explicit variance inflation, and valid Wald-style confidence intervals for the plug-in DP MLE; (3) a noise-aware likelihood correction that is first-order equivalent to the plug-in but supports bootstrap-based intervals; and (4) a matching minimax lower bound showing the privacy distortion rate is unavoidable. The resulting theory yields concrete design rules and a practical pipeline for releasing DP synthetic data with principled uncertainty quantification, validated on three exponential families and real census data.