The Open-Faced Sandwich Adjustment for MCMC Using Estimating Functions

A situation frequently arises where working with the likelihood function is problematic. This can happen for several reasons—perhaps the likelihood is prohibitively computationally expensive, perhaps it lacks some robustness property, or perhaps it is simply not known for the model under consideration. In these cases, it is often possible to specify alternative functions of the parameters and the data that can be maximized to obtain asymptotically normal estimates. However, these scenarios present obvious problems if one is interested in applying Bayesian techniques. This article describes open-faced sandwich adjustment, a way to incorporate a wide class of nonlikelihood objective functions within Bayesian-like models to obtain asymptotically valid parameter estimates and inference via MCMC. Two simulation examples show that the method provides accurate frequentist uncertainty estimates. The open-faced sandwich adjustment is applied to a Poisson spatio-temporal model to analyze an ornithology dataset from the citizen science initiative eBird. An online supplement contains an appendix with additional figures, tables, and discussion, as well as R code.