Adjusting Posteriors from Composite and Misspecified Likelihoods with Application to Spatial Extremes in R-INLA

Methods such as low-rank approximations, covariance tapering and composite likelihoods are useful for speeding up inference in settings where the true likelihood is unknown or intractable to compute. However, such methods can lead to considerable model misspecification, which is particularly challenging to deal with and can provide misleading results when performing Bayesian inference. Adjustment methods for improving properties of posteriors based on composite and/or misspecified likelihoods have been developed, but these cannot be used in conjunction with certain computationally efficient inference software, such as the \texttt{R-INLA} package. We develop a novel adjustment method aimed at postprocessing posterior distributions to improve the properties of their credible intervals. The adjustment method can be used in conjunction with any software for Bayesian inference that allows for sampling from the posterior. Several simulation studies demonstrate that the adjustment method performs well in a variety of different settings. The method is also applied for performing improved modelling of extreme hourly precipitation from high-resolution radar data in Norway, using the spatial conditional extremes model and \texttt{R-INLA}. The results show a clear improvement in performance for all model fits after applying the posterior adjustment method.