Abstract
Evaluating the marginal likelihood in Bayesian analysis is essential for model selection. Estimators based on a single Markov chain Monte Carlo sample from the posterior distribution include 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 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 optimal weights. Simulation studies were 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.
Highlights
The Bayes factor quantifying evidence of one model over a competing model is commonly used for model comparison or variable selection in Bayesian inference
An Markov chain Monte Carlo (MCMC) sample from the posterior distribution is readily available from Bayesian computing software
We propose a new algorithm, partition weighted kernel (PWK), for estimating the marginal likelihood based on this single MCMC sample and its corresponding kernel values
Summary
Yu-Bo Wang*, Ming-Hui Chen†, Lynn Kuo‡, and Paul O. Lewis§ *Department of Statistics, University of Connecticut, Storrs, CT 06269, USA †Department of Statistics, University of Connecticut, Storrs, CT 06269, USA ‡Department of Statistics, University of Connecticut, Storrs, CT 06269, USA §Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs, CT 06269, USA
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