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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
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 relationship between DP and its information-theoretic
variant 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. Lastly,
applications to auditing privacy frameworks, statistical inference tasks, and
algorithm stability are explored.
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