Rates of Convergence of Generalised Variational Inference Posteriors under Prior Misspecification

We prove rates of convergence and robustness to prior misspecification within a Generalised Variational Inference (GVI) framework with bounded divergences. This addresses a significant open challenge for GVI and Federated GVI that employ a different divergence to the Kullback-Leibler under prior misspecification, operate within a subset of possible probability measures, and result in intractable posteriors. Our theoretical contributions extend to misspecified priors that lead to inconsistent Bayes posteriors. In particular, we are able to establish sufficient conditions for existence and uniqueness of GVI posteriors on arbitrary Polish spaces, prove that the GVI posterior measure concentrates on a neighbourhood of loss minimisers, and extend this to rates of convergence regardless of the prior measure.