The $ρ$-posterior framework provides universal Bayesian estimation with explicit contamination rates and optimal convergence guarantees, but has remained computationally difficult due to an optimization over reference distributions that precludes intractable posterior computation. We develop a PAC-Bayesian framework that recovers these theoretical guarantees through temperature-dependent Gibbs posteriors, deriving finite-sample oracle inequalities with explicit rates and introducing tractable variational approximations that inherit the robustness properties of exact $ρ$-posteriors. Numerical experiments demonstrate that this approach achieves theoretical contamination rates while remaining computationally feasible, providing the first practical implementation of $ρ$-posterior inference with rigorous finite-sample guarantees.

翻译：$ρ$-后验框架为通用贝叶斯估计提供了明确的污染率与最优收敛性保证，但由于需要对参考分布进行优化而难以进行后验计算，导致其计算一直较为困难。我们提出了一个PAC-贝叶斯框架，通过温度依赖的吉布斯后验恢复这些理论保证，推导出具有显式收敛速率的有限样本oracle不等式，并引入可处理的变分近似方法，这些方法继承了精确$ρ$-后验的鲁棒性特性。数值实验表明，该方法在保持计算可行性的同时达到了理论污染率，首次实现了具有严格有限样本保证的$ρ$-后验推断的实用化方案。