Almost Perfect Privacy for Additive Gaussian Privacy Filters

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 ε bits of information is leaked about a random variable X (representing private information) that is correlated with Y . Denoting this quantity by gε(X,Y ), we show that for perfect privacy, i.e., ε = 0, one has g0(X,Y ) = 0 for any pair of absolutely continuous random variables (X,Y ) and then derive a second-order approximation for gε(X,Y ) for small ε. This approximation is shown to be related to the strong data processing inequality for mutual information under suitable conditions on the joint distribution PXY . Next, motivated by an operational interpretation of data privacy, we formulate the privacy-utility tradeoff in the same setup using estimationtheoretic quantities and obtain explicit bounds for this tradeoff when ε is sufficiently small using the approximation formula derived for gε(X,Y ).