Inflated density ratio and its variation and generalization for computing marginal likelihoods

In the Bayesian framework, the marginal likelihood plays an important role in variable selection and model comparison. The marginal likelihood is the marginal density of the data after integrating out the parameters over the parameter space. However, this quantity is often analytically intractable due to the complexity of the model. In this paper, we first examine the properties of the inflated density ratio (IDR) method, which is a Monte Carlo method for computing the marginal likelihood using a single MC or Markov chain Monte Carlo (MCMC) sample. We then develop a variation of the IDR estimator, called the dimension reduced inflated density ratio (Dr.IDR) estimator. We further propose a more general identity and then obtain a general dimension reduced (GDr) estimator. Simulation studies are conducted to examine empirical performance of the IDR estimator as well as the Dr.IDR and GDr estimators. We further demonstrate the usefulness of the GDr estimator for computing the normalizing constants in a case study on the inequality-constrained analysis of variance.