Estimating Bayes factors via posterior simulation with the Laplace-Metropolis estimator

The key quantity needed for Bayesian hypothesis testing and model selection is the integrated, or marginal, likelihood of a model. We describe a way to use posterior simulation output to estimate integrated likelihoods. We describe the basic Laplace-Metropolis estimator for models without random effects. For models with random effects, we introduce the compound Laplace-Metropolis estimator. We apply this estimator to data from the World Fertility Survey and show it to give accurate results. Batching of simulation output is used to assess the uncertainty involved in using the compound Laplace-Metropolis estimator. The method allows us to test for the effects of independent variables in a random-effects model and also to test for the presence of the random effects.