  This paper investigates lift, the likelihood ratio between the posterior and
prior belief about sensitive features in a dataset. Maximum and minimum lifts
over sensitive features quantify the adversary's knowledge gain and should be
bounded to protect privacy. We demonstrate that max and min lifts have a
distinct range of values and probability of appearance in the dataset, referred
to as \emph{lift asymmetry}. We propose asymmetric local information privacy
(ALIP) as a compatible privacy notion with lift asymmetry, where different
bounds can be applied to min and max lifts. We use ALIP in the watchdog and
optimal random response (ORR) mechanisms, the main methods to achieve
lift-based privacy. It is shown that ALIP enhances utility in these methods
compared to existing local information privacy, which ensures the same
(symmetric) bounds on both max and min lifts. We propose subset merging for the
watchdog mechanism to improve data utility and subset random response for the
ORR to reduce complexity. We then investigate the related lift-based measures,
including $\ell_1$-norm, $\chi^2$-privacy criterion, and $\alpha$-lift. We
reveal that they can only restrict max-lift, resulting in significant min-lift
leakage. To overcome this problem, we propose corresponding lift-inverse
measures to restrict the min-lift. We apply these lift-based and lift-inverse
measures in the watchdog mechanism. We show that they can be considered as
relaxations of ALIP, where a higher utility can be achieved by bounding only
average max and min lifts.
