On the Relation Between Identifiability, Differential Privacy and Mutual-Information Privacy

This paper investigates the relation between three different notions of privacy: identifiability, differential privacy and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the Hamming distance of the input and output databases, we establish some fundamental connections between these three privacy notions. Given a distortion level $D$, define $\epsilon_{\mathrm{i}}^*(D)$ to be the smallest (best) identifiability level, and $\epsilon_{\mathrm{d}}^*(D)$ to be the smallest differential privacy level. We characterize $\epsilon_{\mathrm{i}}^*(D)$ and $\epsilon_{\mathrm{d}}^*(D)$, and prove that $\epsilon_{\mathrm{i}}^*(D)-\epsilon_X\le\epsilon_{\mathrm{d}}^*(D)\le\epsilon_{\mathrm{i}}^*(D)$ for $D$ in some range, where $\epsilon_X$ is a constant depending on the distribution of the original database $X$, and diminishes to zero when the distribution of $X$ is uniform. Furthermore, we show that identifiability and mutual-information privacy are consistent in the sense that given distortion level $D$, the mechanism that optimizes the mutual-information privacy also minimizes the identifiability level.