  Composite likelihoods are increasingly used in applications where the full
likelihood is analytically unknown or computationally prohibitive. Although the
maximum composite likelihood estimator has frequentist properties akin to those
of the usual maximum likelihood estimator, Bayesian inference based on
composite likelihoods has yet to be explored. In this paper we investigate the
use of the Metropolis--Hastings algorithm to compute a pseudo-posterior
distribution based on the composite likelihood. Two methodologies for adjusting
the algorithm are presented and their performance on approximating the true
posterior distribution is investigated using simulated data sets and real data
on spatial extremes of rainfall.
