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We study the maximal mutual information about a random variable Y (representing non-private information) displayed through an additive Gaussian channel when guaranteeing that only (varepsilon ) bits of information is leaked about a random variable X (representing private information) that is correlated with Y. Denoting this quantity by (g_varepsilon (X,Y)), we show that for perfect privacy, i.e., (varepsilon =0), one has (g_0(X,Y)=0) for any pair of absolutely continuous random variables (X, Y) and then derive a second-order approximation for (g_varepsilon (X,Y)) for small (varepsilon ). This approximation is shown to be related to the strong data processing inequality for mutual information under suitable conditions on the joint distribution (P_{XY}). Next, motivated by an operational interpretation of data privacy, we formulate the privacy-utility tradeoff in the same setup using estimation-theoretic quantities and obtain explicit bounds for this tradeoff when (varepsilon ) is sufficiently small using the approximation formula derived for (g_varepsilon (X,Y)).

Almost Perfect Privacy for Additive Gaussian Privacy Filters