The open-faced sandwich adjustment for MCMC using estimating functions

TL;DR

This paper introduces the open-faced sandwich adjustment, a method to incorporate non-likelihood objective functions into Bayesian MCMC to obtain valid uncertainty estimates, especially when likelihoods are computationally difficult or unavailable.

Contribution

It presents a novel adjustment technique that enables Bayesian inference using non-likelihood functions, expanding the applicability of MCMC methods.

Findings

Provides accurate frequentist uncertainty estimates in simulations

Successfully applied to a Poisson spatio-temporal model for ornithology data

Demonstrates robustness when likelihood functions are problematic or unknown

Abstract

The 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. Here we describe open-faced sandwich adjustment, a way to incorporate a wide class of non-likelihood 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.