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We study the problem of privacy preservation in data sharing, where $S$ is asensitive variable to be protected and $X$ is a non-sensitive useful variablecorrelated with $S$. Variable $X$ is randomized into variable $Y$, which willbe shared or released according to $p_{Y|X}(y|x)$. We measure privacy leakageby \emph{information privacy} (also known as \emph{log-lift} in theliterature), which guarantees mutual information privacy and differentialprivacy (DP). Let $\Xepsc \subseteq \X$ contain elements n the alphabet of $X$for which the absolute value of log-lift (abs-log-lift for short) is greaterthan a desired threshold $\eps$. When elements $x\in \Xepsc$ are randomizedinto $y\in \Y$, we derive the best upper bound on the abs-log-lift across theresultant pairs $(s,y)$. We then prove that this bound is achievable via an\emph{$X$-invariant} randomization $p(y|x) = R(y)$ for $x,y\in\Xepsc$. However,the utility measured by the mutual information $I(X;Y)$ is severely damaged inimposing a strict upper bound $\eps$ on the abs-log-lift. To remedy this andinspired by the probabilistic ($\eps$, $\delta$)-DP, we propose a relaxed($\eps$, $\delta$)-log-lift framework. To achieve this relaxation, we introducea greedy algorithm which exempts some elements in $\Xepsc$ from randomization,as long as their abs-log-lift is bounded by $\eps$ with probability $1-\delta$.Numerical results demonstrate efficacy of this algorithm in achieving a betterprivacy-utility tradeoff.

On Properties and Optimization of Information-theoretic Privacy Watchdog