  Large-sample Bayesian analogs exist for many frequentist methods, but are
less well-known for the widely-used 'sandwich' or 'robust' variance estimates.
We review existing approaches to Bayesian analogs of sandwich variance
estimates and propose a new 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. Being the large-sample
equivalent of its frequentist counterpart, we show by simulation that Bayesian
robust standard error estimates can faithfully quantify the variability of
parameter estimates even under model misspecification -- thus retaining the
major attraction of the original frequentist version. We demonstrate our
Bayesian analog of standard error estimates when studying the association
between age and systolic blood pressure in NHANES.
