  This paper studies how to approximate pufferfish privacy when the adversary's
prior belief of the published data is Gaussian distributed. Using Monge's
optimal transport plan, we show that $(\epsilon, \delta)$-pufferfish privacy is
attained if the additive Laplace noise is calibrated to the differences in mean
and variance of the Gaussian distributions conditioned on every discriminative
secret pair. A typical application is the private release of the summation (or
average) query, for which sufficient conditions are derived for approximating
$\epsilon$-statistical indistinguishability in individual's sensitive data. The
result is then extended to arbitrary prior beliefs trained by Gaussian mixture
models (GMMs): calibrating Laplace noise to a convex combination of differences
in mean and variance between Gaussian components attains
$(\epsilon,\delta)$-pufferfish privacy.
