Nowadays, Bayesian methods are routinely used for estimating parameters of item response theory (IRT) models. However, the marginal likelihoods are still rarely used for comparing IRT models due to their complexity and a relatively high dimension of the model parameters. In this paper, we review Monte Carlo (MC) methods developed in the literature in recent years and provide a detailed development of how these methods are applied to the IRT models. In particular, we focus on the "best possible" implementation of these MC methods for the IRT models. These MC methods are used to compute the marginal likelihoods under the one-parameter IRT model with the logistic link (1PL model) and the two-parameter logistic IRT model (2PL model) for a real English Examination dataset. We further use the widely applicable information criterion (WAIC) and deviance information criterion (DIC) to compare the 1PL model and the 2PL model. The 2PL model is favored by all of these three Bayesian model comparison criteria for the English Examination data.
Keywords: Bayes factor; CMDE; IWMDE; MCMC; Marginal posterior density.