  Model selection is an important activity in modern data analysis and the
conventional Bayesian approach to this problem involves calculation of marginal
likelihoods for different models, together with diagnostics which examine
specific aspects of model fit. Calculating the marginal likelihood is a
difficult computational problem. Our article proposes some extensions of the
Laplace approximation for this task that are related to copula models and which
are easy to apply. Variations which can be used both with and without
simulation from the posterior distribution are considered, as well as use of
the approximations with bridge sampling and in random effects models with a
large number of latent variables. The use of a t-copula to obtain higher
accuracy when multivariate dependence is not well captured by a Gaussian copula
is also discussed.
