We introduce the $\widetildeρ$-posterior, a modified version of the $ρ$-posterior, obtained by replacing the supremum over competitor parameters with a softmax aggregation. This modification allows a PAC-Bayesian analysis of the $\widetildeρ$-posterior. This yields finite-sample oracle inequalities with explicit convergence rates that inherit the key robustness properties of the original framework, in particular, graceful degradation under model misspecification and data contamination. Crucially, the PAC-Bayesian oracle inequalities extend to variational approximations of the $\widetildeρ$-posterior, providing theoretical guarantees for tractable inference. Numerical experiments on exponential families, regression, and real-world datasets confirm that the resulting variational procedures achieve robustness competitive with theoretical predictions at computational cost comparable to standard variational Bayes.

翻译：我们引入$\widetildeρ$-后验，这是对$ρ$-后验的一种修正版本，通过将竞争参数上的上确界替换为softmax聚合得到。该修正使得$\widetildeρ$-后验的PAC-贝叶斯分析成为可能。由此得到的有限样本oracle不等式具有显式收敛速率，继承了原始框架的关键鲁棒性特性，特别是在模型误设和数据污染条件下的性能优雅退化。至关重要的是，PAC-贝叶斯oracle不等式可推广至$\widetildeρ$-后验的变分逼近，为可计算推断提供了理论保证。在指数族、回归及真实数据集上的数值实验表明，所得变分程序在计算成本与标准变分贝叶斯相当的前提下，达到了与理论预测相媲美的鲁棒性。