  Assessment of disclosure risk is of paramount importance in the research and
applications of data privacy techniques. The concept of differential privacy
(DP) formalizes privacy in probabilistic terms and provides a robust concept
for privacy protection without making assumptions about the background
knowledge of adversaries. Practical applications of DP involve development of
DP mechanisms to release results at a pre-specified privacy budget. In this
paper, we generalize the widely used Laplace mechanism to the family of
generalized Gaussian (GG) mechanism based on the $l_p$ global sensitivity of
statistical queries. We explore the theoretical requirement for the GG
mechanism to reach DP at prespecified privacy parameters, and investigate the
connections and differences between the GG mechanism and the Exponential
mechanism based on the GG distribution We also present a lower bound on the
scale parameter of the Gaussian mechanism of $(\epsilon,\delta)$-probabilistic
DP as a special case of the GG mechanism, and compare the statistical utility
of the sanitized results in the tail probability and dispersion in the Gaussian
and Laplace mechanisms. Lastly, we apply the GG mechanism in 3 experiments (the
mildew, Czech, adult data), and compare the accuracy of sanitized results via
the $l_1$ distance and Kullback-Leibler divergence and examine how sanitization
affects the prediction power of a classifier constructed with the sanitized
data in the adult experiment.
