Pufferfish Privacy: An Information-Theoretic Study

Pufferfish privacy (PP) is a generalization of differential privacy (DP), that offers flexibility in specifying sensitive information and integrates domain knowledge into the privacy definition. Inspired by the illuminating equivalent formulation of DP in terms of mutual information due to Cuff and Yu, this work explores PP through the lens of information theory. We provide an information-theoretic formulation of PP, termed mutual information PP (MI-PP), in terms of the conditional mutual information between the mechanism and the secret, given the public information. We show that MI-PP is implied by the regular PP and characterize conditions under which the reverse implication is also true, recovering the DP information-theoretic equivalence result as a special case. We establish convexity, composability, and post-processing properties for MI-PP mechanisms and derive noise levels for the Gaussian and Laplace mechanisms. The obtained mechanisms are applicable under relaxed assumptions and provide improved noise levels in some regimes, compared to classic, sensitivity-based approaches. Lastly, applications of MI-PP to auditing privacy frameworks, statistical inference tasks, and algorithm stability are explored.