Operational definitions for some common information leakage metrics

Maximal leakage from a random variable X to a random variable Y is defined as the multiplicative increase, upon observing Y, of the probability of correctly guessing a randomized function of X, maximized over all such functions [1]. Herein, this guessing framework is used to give operational definitions to common information leakage metrics, including Shannon capacity, maximal correlation, and local differential privacy. Shannon capacity is shown to capture the multiplicative increase of the probability of correct guessing over the restricted set of functions of X that can be reliably reconstructed from Y, hence underestimating leakage. Maximal correlation is shown to capture the multiplicative change in the variance of functions of X, rather than the guessing probability. Local differential privacy is shown to capture the multiplicative increase of the guessing probability of functions of X, maximized over realizations of Y and over distributions P
_x
. Moreover, maximizing over realizations of Y for a fixed P
_x
is shown to yield a valid leakage measure, which is equal to the maximum information rate.