A New Monte Carlo Method for Estimating Marginal Likelihoods

Abstract

Evaluating the marginal likelihood in Bayesian analysis is essential in model selection. There are existing estimators based on a single Markov chain Monte Carlo sample from the posterior distribution, including the harmonic mean estimator and the inflated density ratio estimator. We propose a new class of Monte Carlo estimators based on this single Markov chain Monte Carlo sample. This class can be thought of as a generalization of the harmonic mean and inflated density ratio estimators using a partition weighted kernel (likelihood times prior). We also show that our estimator is consistent and has better theoretical properties than the harmonic mean and inflated density ratio estimators. In addition, we provide guidelines on choosing the optimal weights. A simulation study is conducted to examine the empirical performance of the proposed estimator. We further demonstrate the desirable features of the proposed estimator with two real data sets: one is from a prostate cancer study using an ordinal probit regression model with latent variables; the other is for the power prior construction from two Eastern Cooperative Oncology Group phase III clinical trials using the cure rate survival model with similar objectives. This talk is based on the joint work with Yu-Bo Wang, Ming-Hui Chen, and Paul Lewis at University of Connecticut. Key Words: Bayesian model selection; Cure rate model; Harmonic mean estimator; Inflated density estimator; Ordinal probit regression; Power prior.