Rényi Pufferfish Privacy with Gaussian-based Priors: From Single Gaussian to Mixture Model

Rényi Pufferfish Privacy (RPP) provides a Rényi divergence-based privacy framework for correlated data, but existing $\mathrm{\infty }$-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 Rényi 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 Rényi 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 ($\text{textsc}RAW$, $\text{textsc}MEAN$) and model-output ($\text{textsc}BNN$, $\text{textsc}GP$) 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.