A Bayesian 'sandwich' for variance estimation and hypothesis testing

Many frequentist methods have large-sample Bayesian analogs, but widely-used "sandwich" or "robust" covariance estimates are an exception. We propose such an analog, as the Bayes rule under a form of balanced loss function, that combines elements of standard parametric inference with fidelity of the data to the model. Our development is general, for essentially any regression setting with independent outcomes. Besides being the large-sample equivalent of its frequentist counterpart, we show by simulation that the Bayesian robust standard error can faithfully quantify the variability of parameter estimates even under model misspecification -- thus retaining the major attraction of the original frequentist version. We demonstrate some advantages of our Bayesian analog's standard error estimate when studying the association between age and systolic blood pressure in NHANES.