Measures of Information Leakage for Incomplete Statistical Information: Application to a Binary Privacy Mechanism

Information leakage is usually defined as the logarithmic increment in the adversary's probability of correctly guessing the legitimate user's private data or some arbitrary function of the private data when presented with the legitimate user's publicly disclosed information. However, this definition of information leakage implicitly assumes that both the privacy mechanism and the prior probability of the original data are entirely known to the attacker. In reality, the assumption of complete knowledge of the privacy mechanism for an attacker is often impractical. The attacker can usually have access to only an approximate version of the correct privacy mechanism, computed from a limited set of the disclosed data, for which they can access the corresponding un-distorted data. In this scenario, the conventional definition of leakage no longer has an operational meaning. To address this problem, in this article, we propose novel meaningful information-theoretic metrics for information leakage when the attacker has incomplete information about the privacy mechanism-we call them average subjective leakage, average confidence boost, and average objective leakage, respectively. For the simplest, binary scenario, we demonstrate how to find an optimized privacy mechanism that minimizes the worst-case value of either of these leakages.