Differentially Private Computation of the Gini Index for Income Inequality

TL;DR

This paper introduces a differentially private method for computing the Gini index, balancing privacy and accuracy by using smooth sensitivity and Bayesian post-processing, applicable to income inequality data.

Contribution

It develops a novel differentially private mechanism for the Gini index using smooth sensitivity, improving accuracy over global sensitivity-based methods.

Findings

The mechanism reduces noise errors compared to global sensitivity approaches.

Smooth sensitivity provides highly accurate estimates in certain data settings.

Bayesian post-processing offers interval estimates for the Gini index.

Abstract

The Gini index is a widely reported measure of income inequality. In some settings, the underlying data used to compute the Gini index are confidential. The organization charged with reporting the Gini index may be concerned that its release could leak information about the underlying data. We present an approach for bounding this information leakage by releasing a differentially private version of the Gini index. In doing so, we analyze how adding, deleting, or altering a single observation in any specific dataset can affect the computation of the Gini index; this is known as the local sensitivity. We then derive a smooth upper bound on the local sensitivity. Using this bound, we define a mechanism that adds noise to the Gini index, thereby satisfying differential privacy. Using simulated and genuine income data, we show that the mechanism can reduce the errors from noise injection substantially relative to differentially private algorithms that rely on the global sensitivity, that is, the maximum of the local sensitivities over all possible datasets. We characterize settings where using smooth sensitivity can provide highly accurate estimates, as well as settings where the noise variance is simply too large to provide reliably useful results. We also present a Bayesian post-processing step that provides interval estimates about the value of the Gini index computed with the confidential data.