R\'enyi Pufferfish Privacy with Gaussian-based Priors: From Single Gaussian to Mixture Model

R\'{e}nyi Pufferfish Privacy (RPP) provides a R\'{e}nyi divergence-based privacy framework for correlated data, but existing $\infty$-Wasserstein mechanisms are often conservative and sacrifice data utility. We study Gaussian mechanisms for RPP under Gaussian and Gaussian-mixture priors. For single Gaussian priors, we derive the exact R\'{e}nyi divergence after Gaussian perturbation, obtain a relaxed closed-form sufficient condition for $(\alpha,\epsilon)$-RPP, and characterize the monotonicity of the calibrated noise with respect to the privacy budget $\epsilon$ and the R\'{e}nyi order $\alpha$. To handle more general non-Gaussian and multimodal priors, we approximate secret-conditioned outputs with Gaussian mixture models and introduce an optimal-transport-based sufficient condition for RPP. Experiments on three UCI datasets with statistical (\textsc{RAW}, \textsc{MEAN}) and model-output (\textsc{BNN}, \textsc{GP}) queries show that our prior-aware mechanisms consistently require less noise than a recent RPP additive-noise baseline, achieving an average noise reduction of 48.9\%. These results show that our mechanisms can substantially improve the privacy-utility trade-off under RPP.