Simulation Pseudo-Bias Correction to the Harmonic Mean Estimator of Integrated Likelihoods

Abstract

Bayesian model selection depends on the integrated likelihood of the data given the model. Newton and Raftery’s harmonic mean estimator (HME) is simple to implement by computing the likelihood of the data at MCMC draws from the posterior distribution. Alternative methods in the literature require additional simulations or more extensive computations. In theory HME is consistent but can have an infinite variance. In practice, the computed HME often is simulation pseudo-biased. This article identifies the source of the pseudo-bias and recommends several algorithms for adjusting the HME to remove it. The pseudo-bias can be substantial and can negatively affect HME’s ability to select the correct model in Bayesian model selection. The pseudo-bias often causes the computed HME to overestimate the integrated likelihood, and the amount of pseudo-bias tends to be larger for more complex models. When the computed HME errs, it tends to select models that are too complex. Simulation studies of linear and logistic regression models demonstrate that the adjusted HME effectively removes the pseudo-bias, is more accurate, and indicates more reliably the best model.Supplemental materials are available online. These materials include the appendices and Gauss program.