We study the privacy of releasing posterior sample paths from a Gaussian process (GP) when the entire training set including covariates and responses is private. Unlike standard differential-privacy (DP) mechanisms that add external noise, posterior sampling is random by construction. We show that this intrinsic randomness yields DP guarantees by deriving explicit Rényi-DP bounds for GP posterior sample-path release. The bounds separate posterior-mean leakage from data-dependent posterior-covariance leakage showing that meaningful privacy depends sharply on effective ridge regularisation. We apply membership-inference attacks to show that empirical leakage follows the predicted dependence on regularisation, posterior variance and the number of released posterior sample-paths. Utility experiments on downstream posterior-sampling tasks identify noisy-observation regimes where privacy-compatible regularisation preserves useful decisions with modest utility loss. When stronger privacy is needed, the intrinsic guarantee can be sharpened by adding calibrated GP noise, providing an explicit additional privacy knob.

翻译：我们研究了当整个训练集（包括协变量和响应变量）均为私有时，从高斯过程中释放后验样本路径的隐私问题。与通过添加外部噪声实现标准差分隐私的机制不同，后验采样本身具有随机性。我们通过推导高斯过程后验样本路径释放的显式Rényi差分隐私界，证明这种内在随机性能够产生差分隐私保证。该边界将后验均值泄露与依赖于数据的后验协方差泄露分离开来，表明有意义的隐私保护高度依赖于有效的岭正则化。我们采用成员推断攻击表明，经验泄露遵循正则化、后验方差以及释放的后验样本路径数量的预测依赖性。在下游后验采样任务上的效用实验识别出含噪观测范式：在该范式中，兼容隐私的正则化能够以适度的效用损失保留有用决策。当需要更强的隐私保护时，可通过添加校准的高斯噪声强化内在隐私保证，从而提供显式的额外隐私调节旋钮。