  It has been widely understood that differential privacy (DP) can guarantee
rigorous privacy against adversaries with arbitrary prior knowledge. However,
recent studies demonstrate that this may not be true for correlated data, and
indicate that three factors could influence privacy leakage: the data
correlation pattern, prior knowledge of adversaries, and sensitivity of the
query function. This poses a fundamental problem: what is the mathematical
relationship between the three factors and privacy leakage? In this paper, we
present a unified analysis of this problem. A new privacy definition, named
\textit{prior differential privacy (PDP)}, is proposed to evaluate privacy
leakage considering the exact prior knowledge possessed by the adversary. We
use two models, the weighted hierarchical graph (WHG) and the multivariate
Gaussian model to analyze discrete and continuous data, respectively. We
demonstrate that positive, negative, and hybrid correlations have distinct
impacts on privacy leakage. Considering general correlations, a closed-form
expression of privacy leakage is derived for continuous data, and a chain rule
is presented for discrete data. Our results are valid for general linear
queries, including count, sum, mean, and histogram. Numerical experiments are
presented to verify our theoretical analysis.
