Statistical Properties of Sanitized Results from Differentially Private Laplace Mechanism with Bounding Constraints

Protection of individual privacy is a common concern when releasing and sharing data and information. Differential privacy (DP) formalizes privacy in probabilistic terms without making assumptions about the background knowledge of data intruders, and thus provides a robust concept for privacy protection. Practical applications of DP involve development of differentially private mechanisms to generate sanitized results at a pre-specified privacy budget. In the sanitization of bounded statistics such as proportions and correlation coefficients, the bounding constraints will need to be incorporated in the differentially private mechanisms. There has been little work on examining the consequences of the bounding constraints on the accuracy of sanitized results and the statistical inferences of the population parameters based on the sanitized results. In this paper, we formalize the differentially private truncated and boundary inflated truncated (BIT) mechanisms for releasing statistics with bounding constraints. The impacts of the truncated and BIT Laplace mechanisms on the statistical accuracy and validity of sanitized statistics are evaluated both theoretically and empirically via simulation studies. We also provide an upper bound for the MSE between the sanitized and original results for a given $n$ and its convergence rate toward 0 in the truncated and BIT Laplace mechanisms.