  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: $\epsilon$-BDP
$\Longleftarrow$ $\big(\frac{1}{2}-\frac{1}{e^{\epsilon}+1}\big)$-BSP, and
$\epsilon$-BDP $\Longrightarrow$ $(e^{2\epsilon}-1)$-BSP $\Longrightarrow$
$(e^{2\epsilon}-1)$-SP. In addition, we obtain a minor result $\epsilon$-DP
$\Longleftarrow$ $\big(\frac{1}{2}-\frac{1}{e^{\epsilon}+1}\big)$-SP, which
improves the result of Kasiviswanathan and Smith [3] stating $\epsilon$-DP
$\Longleftarrow$ $\epsilon/6$-SP for $\epsilon \leq 1.35$. In part two, we
establish the relations between BDP and MP. First, $\epsilon$-BDP
$\Longrightarrow$ $\epsilon$-MP. Second, for a family of distributions that are
downward scalable in the sense of Li et al. [4], it is shown that
$\epsilon$-BDP $\Longleftarrow$ $\epsilon$-MP.
