Pseudo-Likelihoods for Bayesian Inference

The interplay between Bayesian and frequentist inference can play a remarkable role in order to address some theoretical and computational drawbacks, due to the complexity or misspecification of the model, or to the presence of many nuisance parameters. In this respect, in this paper we review the properties and applications of the so-called pseudo-posterior distributions, i.e., posterior distributions derived from the combination of a pseudo-likelihood function with suitable prior information. In particular, we illustrate the various notions of pseudo-likelihood highlighting their use in the Bayesian framework. Moreover, we show the simple but effective application of pseudo-posterior distributions in three challenging examples.