A Tunable Measure for Information Leakage

A tunable measure for information leakage called \textit{maximal $α$-leakage} is introduced. This measure quantifies the maximal gain of an adversary in refining a tilted version of its prior belief of any (potentially random) function of a dataset conditioning on a disclosed dataset. The choice of $α$ determines the specific adversarial action ranging from refining a belief for $α=1$ to guessing the best posterior for $α= \infty$, and for these extremal values this measure simplifies to mutual information (MI) and maximal leakage (MaxL), respectively. For all other $α$ this measure is shown to be the Arimoto channel capacity. Several properties of this measure are proven including: (i) quasi-convexity in the mapping between the original and disclosed datasets; (ii) data processing inequalities; and (iii) a composition property.