  Recent advances in Markov chain Monte Carlo (MCMC) extend the scope of
Bayesian inference to models for which the likelihood function is intractable.
Although these developments allow us to estimate model parameters, other basic
problems such as estimating the marginal likelihood, a fundamental tool in
Bayesian model selection, remain challenging. This is an important scientific
limitation because testing psychological hypotheses with hierarchical models
has proven difficult with current model selection methods. We propose an
efficient method for estimating the marginal likelihood for models where the
likelihood is intractable, but can be estimated unbiasedly. It is based on
first running a sampling method such as MCMC to obtain samples for the model
parameters, and then using these samples to construct the proposal density in
an importance sampling (IS) framework with an unbiased estimate of the
likelihood. Our method has several attractive properties: it generates an
unbiased estimate of the marginal likelihood, it is robust to the quality and
target of the sampling method used to form the IS proposals, and it is
computationally cheap to estimate the variance of the marginal likelihood
estimator. We also obtain the convergence properties of the method and provide
guidelines on maximizing computational efficiency. The method is illustrated in
two challenging cases involving hierarchical models: identifying the form of
individual differences in an applied choice scenario, and evaluating the best
parameterization of a cognitive model in a speeded decision making context.
Freely available code to implement the methods is provided. Extensions to
posterior moment estimation and parallelization are also discussed.
