A Bayesian 'sandwich' for variance estimation

TL;DR

This paper introduces a Bayesian analog to the frequentist 'sandwich' variance estimator, providing a robust method for variance estimation in regression models that remains effective under model misspecification.

Contribution

It proposes a new Bayesian approach for sandwich variance estimation applicable to any regression with independent outcomes, combining parametric inference with data fidelity.

Findings

Bayesian sandwich estimator performs well under model misspecification.

Simulation studies confirm accurate quantification of variability.

Application to NHANES data demonstrates practical utility.

Abstract

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.