Abstract
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.
Highlights
For many models arising in various fields of statistical analysis, working with the likelihood function can be undesirable
A natural model for such data is a hierarchical Poisson regression 8 with a random effect specified as a spatio-temporal Gaussian process with 9 unknown covariance parameters
We look at 6114 observations of the Northern Cardinal in a section of the eastern United States over a period from 2004 to 2007 (Figure 1). Inspection of the data suggests an overdispersed Poisson model
Summary
For many models arising in various fields of statistical analysis, working with the likelihood function can be undesirable. 1 post hoc, these authors propose an adjustment to the Metropolis likelihood 2 ratio within the sampler itself Their goal, like ours, is to achieve desir[3] able frequentist coverage properties of credible intervals computed based on 4 MCMC. Chernozhukov and Hong (2003) elucidates the asymptotic behavior of πM,n(θ|yn), which motivates the open-face sandwich adjustment These results are di[22] rect analogues of well-known asymptotic properties of true posterior distri[23] butions. Their Theorem 2, which we re-state below, states that the asymp[24] totic distribution of the quasi-Bayes estimator θQB,n is the same as that of 25 the extremum estimator θM,n. 26 Theorem 1 Assuming sufficient regularity of M,n(θ; yn), J1n/2(θQB − θ0) −D→ N (0, I)
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