Calibrated Generalized Bayesian Inference

We provide a simple and general solution for accurate uncertainty quantification of Bayesian inference in misspecified or approximate models, and for generalized posteriors more generally. While existing solutions are based on explicit Gaussian posterior approximations, or post-processing procedures, we demonstrate that correct uncertainty quantification can be achieved by substituting the usual posterior with an intuitively appealing alternative posterior that conveys the same information. This solution applies to both likelihood-based and loss-based posteriors, and we formally demonstrate the reliable uncertainty quantification of this approach. The new approach is demonstrated through a range of examples, including linear models, and doubly intractable models.