  Under model misspecification, the MLE generally converges to the pseudo-true
parameter, the parameter corresponding to the distribution within the model
that is closest to the distribution from which the data are sampled. In many
problems, the pseudo-true parameter corresponds to a population parameter of
interest, and so a misspecified model can provide consistent estimation for
this parameter. Furthermore, the well-known sandwich variance formula of
Huber(1967) provides an asymptotically accurate sampling distribution for the
MLE, even under model misspecification. However, confidence intervals based on
a sandwich variance estimate may behave poorly for low sample sizes, partly due
to the use of a plug-in estimate of the variance. From a Bayesian perspective,
plug-in estimates of nuisance parameters generally underrepresent uncertainty
in the unknown parameters, and averaging over such parameters is expected to
give better performance. With this in mind, we present a Bayesian sandwich
posterior distribution, whose likelihood is based on the sandwich sampling
distribution of the MLE. This Bayesian approach allows for the incorporation of
prior information about the parameter of interest, averages over uncertainty in
the nuisance parameter and is asymptotically robust to model misspecification.
In a small simulation study on estimating a regression parameter under
heteroscedasticity, the addition of accurate prior information and the
averaging over the nuisance parameter are both seen to improve the accuracy and
calibration of confidence intervals for the parameter of interest.
