Bayesian inference with a pairwise likelihood: an approach based on empirical likelihood

In several applications, the model of interest is such that its likelihood function is difficult, or even impractical, to compute. For these situations, it is common to substitute the likelihood with a surrogate, which resembles the full likelihood but is easier to calculate. Among these surrogates are composite likelihoods and in particular pairwise likelihoods. Their properties in classical inference have been widely discussed in the literature; their use within a Bayesian approach has been seldom considered. The substitution of the likelihood with a surrogate is not straightforward in Bayesian inference: the posterior distribution which is obtained must be validated on a case by case basis, as general results are not available. We propose a Bayesian procedure in which the surrogate is the empirical likelihood derived from the pairwise score equation. This pseudo-likelihood has standard asymptotic properties, so the validation of the posterior distribution is based on its asymptotic behavior.