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 $$\varepsilon $$ 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 $$g_\varepsilon (X,Y)$$.