  Given a trained model and a data sample, membership-inference (MI) attacks
predict whether the sample was in the model's training set. A common
countermeasure against MI attacks is to utilize differential privacy (DP)
during model training to mask the presence of individual examples. While this
use of DP is a principled approach to limit the efficacy of MI attacks, there
is a gap between the bounds provided by DP and the empirical performance of MI
attacks. In this paper, we derive bounds for the \textit{advantage} of an
adversary mounting a MI attack, and demonstrate tightness for the widely-used
Gaussian mechanism. We further show bounds on the \textit{confidence} of MI
attacks. Our bounds are much stronger than those obtained by DP analysis. For
example, analyzing a setting of DP-SGD with $\epsilon=4$ would obtain an upper
bound on the advantage of $\approx0.36$ based on our analyses, while getting
bound of $\approx 0.97$ using the analysis of previous work that convert
$\epsilon$ to membership inference bounds.
  Finally, using our analysis, we provide MI metrics for models trained on
CIFAR10 dataset. To the best of our knowledge, our analysis provides the
state-of-the-art membership inference bounds for the privacy.
