Robust Bayesian Inference via Variational Approximations of Generalized Rho-Posteriors

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.