We study Differential Privacy in the abstract setting of Probability onmetric spaces. Numerical, categorical and functional data can be handled in auniform manner in this setting. We demonstrate how mechanisms based on datasanitisation and those that rely on adding noise to query responses fit withinthis framework. We prove that once the sanitisation is differentially private,then so is the query response for any query. We show how to constructsanitisations for high-dimensional databases using simple 1-dimensionalmechanisms. We also provide lower bounds on the expected error fordifferentially private sanitisations in the general metric space setting.Finally, we consider the question of sufficient sets for differential privacyand show that for relaxed differential privacy, any algebra generating theBorel $\sigma$-algebra is a sufficient set for relaxed differential privacy.
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