Several Representations of $\alpha$-Mutual Information and Interpretations as Privacy Leakage Measures

In this paper, we present several novel representations of $\alpha$-mutual information ($\alpha$-MI) in terms of Rényi divergence and conditional Rényi entropy. The representations are based on the variational characterizations of $\alpha$-MI using a reverse channel. Based on these representations, we provide several interpretations of the $\alpha-\text{MI}$ as privacy leakage measures using generalized mean and gain functions. Further, as byproducts of the representations, we propose novel conditional Rényi entropies that satisfy the property that conditioning reduces entropy and data-processing inequality.