  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 $\epsilon$ bits of information is leaked
about a random variable $X$ (representing private information) that is
correlated with $Y$. Denoting this quantity by $g_\epsilon(X,Y)$, we show that
for perfect privacy, i.e., $\epsilon=0$, 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 $g_\epsilon(X,Y)$ for small $\epsilon$. 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 $\epsilon$ is
sufficiently small using the approximation formula derived for
$g_\epsilon(X,Y)$.
