  We provide a general solution to a fundamental open problem in Bayesian
inference, namely poor uncertainty quantification, from a frequency standpoint,
of Bayesian methods in misspecified models. While existing solutions are based
on explicit Gaussian approximations of the posterior, or computationally
onerous post-processing procedures, we demonstrate that correct uncertainty
quantification can be achieved by replacing the usual posterior with an
intuitive approximate posterior. Critically, our solution is applicable to
likelihood-based, and generalized, posteriors as well as cases where the
likelihood is intractable and must be estimated. We formally demonstrate the
reliable uncertainty quantification of our proposed approach, and show that
valid uncertainty quantification is not an asymptotic result but occurs even in
small samples. We illustrate this approach through a range of examples,
including linear, and generalized, mixed effects models.
