Relations among different privacy notions
This work provides foundational insights for privacy researchers by formalizing connections between key privacy definitions, though it is incremental in nature.
The paper tackles the problem of clarifying relationships among different privacy notions, establishing quantitative equivalences and implications between Bayesian differential privacy, Bayesian semantic privacy, semantic privacy, and membership privacy, with specific bounds such as ε-BDP implying (e^{2ε}-1)-BSP.
We present a comprehensive view of the relations among several privacy notions: differential privacy (DP) [1], Bayesian differential privacy (BDP) [2], semantic privacy (SP) [3], and membership privacy (MP) [4]. The results are organized into two parts. In part one, we extend the notion of semantic privacy (SP) to Bayesian semantic privacy (BSP) and show its essential equivalence with Bayesian differential privacy (BDP) in the quantitative sense. We prove the relations between BDP, BSP, and SP as follows: $ε$-BDP $\Longleftarrow$ $\big(\frac{1}{2}-\frac{1}{e^ε+1}\big)$-BSP, and $ε$-BDP $\Longrightarrow$ $(e^{2ε}-1)$-BSP $\Longrightarrow$ $(e^{2ε}-1)$-SP. In addition, we obtain a minor result $ε$-DP $\Longleftarrow$ $\big(\frac{1}{2}-\frac{1}{e^ε+1}\big)$-SP, which improves the result of Kasiviswanathan and Smith [3] stating $ε$-DP $\Longleftarrow$ $ε/6$-SP for $ε\leq 1.35$. In part two, we establish the relations between BDP and MP. First, $ε$-BDP $\Longrightarrow$ $ε$-MP. Second, for a family of distributions that are downward scalable in the sense of Li et al. [4], it is shown that $ε$-BDP $\Longleftarrow$ $ε$-MP.