Bayesian inference from composite likelihoods, with an application to spatial extremes

Composite likelihoods are increasingly used in applications where the full likelihood is analytically unknown or computationally prohibitive. Although some frequentist properties of the maximum composite likelihood estimator are akin to those of the maximum likelihood estimator, Bayesian inference based on composite likelihoods is in its early stages. This paper discusses inference when one uses composite likelihood in Bayes' formula. We establish that using a composite likelihood results in a proper posterior density, though it can differ considerably from that stemming from the full likelihood. Building on previous work on composite likelihood ratio tests, we use asymptotic theory for misspecified models to propose two adjustments to the composite likelihood to obtain appropriate inference. We also investigate use of the Metropolis Hastings algorithm and two implementations of the Gibbs sampler for obtaining draws from the composite posterior. We test the methods on simulated data and apply them to a spatial extreme rainfall dataset. For the simulated data, we find that posterior credible intervals yield appropriate empirical coverage rates. For the extreme precipitation data, we are able to both effectively model marginal behavior throughout the study region and obtain appropriate measures of spatial dependence.