Marginal Likelihood for a Class of Bayesian Generalized Linear Models

The Bayes factor has become an important tool for model selection. The marginal likelihoods are also important because they can be used to rank the models. In fact, the Bayes factor is the ratio of the marginal likelihoods for two models with proper prior densities. We discuss the marginal likelihood for a class of generalized linear models used in small area estimation for mortality data analysis. Computation in these models is intensive and requires the implementation of Markov chain Monte Carlo (MCMC) methods. A sophisticated method for computing the marginal likelihoods for generalized linear models using reduced Metropolis-Hastings (M-H) samplers has recently been introduced. Also, a much simpler method that uses the Laplace approximation has been proposed. Our method lies between these two in simplicity, and it uses importance sampling via a simple output analysis from a MCMC sampler. We also show that the new method can be approximated without using the MCMC sampler. We illustrate our methods for two Poisson regression models which have been used for mapping mortality data, and we compare them with the methods based on reduced M-H samplers and the Laplace approximation. The method based on the Laplace approximation gives smaller marginal log-likelihoods than the other three methods, but they select the same model. We use a simulation study in which the marginal likelihood is used to investigate the improvement in the goodness of fit of one hierarchical Poisson regression model over the other.

[1]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo Methods for Computing Bayes Factors , 2001 .

[2]  S. Chib,et al.  Marginal Likelihood From the Metropolis–Hastings Output , 2001 .

[3]  Balgobin Nandram,et al.  Bayesian Analysis and Mapping of Mortality Rates for Chronic Obstructive Pulmonary Disease , 2000 .

[4]  B. Mallick,et al.  Generalized Linear Models : A Bayesian Perspective , 2000 .

[5]  R. Kass,et al.  Reference Bayesian Methods for Generalized Linear Mixed Models , 2000 .

[6]  Jun S. Liu,et al.  The Multiple-Try Method and Local Optimization in Metropolis Sampling , 2000 .

[7]  M. Daniels.  A prior for the variance in hierarchical models , 1999 .

[8]  Alan E. Gelfand,et al.  Model choice: A minimum posterior predictive loss approach , 1998, AISTATS.

[9]  Bradley P. Carlin,et al.  Generalized Linear Models for Small-Area Estimation , 1998 .

[10]  L. Wasserman,et al.  Computing Bayes Factors by Combining Simulation and Asymptotic Approximations , 1997 .

[11]  Ming-Hui Chen,et al.  On Monte Carlo methods for estimating ratios of normalizing constants , 1997 .

[12]  Bradley P. Carlin,et al.  Hierarchical Spatio-Temporal Mapping of Disease Rates , 1997 .

[13]  A. Raftery.  Approximate Bayes factors and accounting for model uncertainty in generalised linear models , 1996 .

[14]  A. Gelfand,et al.  Hierarchical Spatio-Temporal Mapping of Disease , 2007 .

[15]  Xiao-Li Meng,et al.  SIMULATING RATIOS OF NORMALIZING CONSTANTS VIA A SIMPLE IDENTITY: A THEORETICAL EXPLORATION , 1996 .

[16]  S. Chib.  Marginal Likelihood from the Gibbs Output , 1995 .

[17]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[18]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[19]  B. Carlin,et al.  Bayesian Model Choice Via Markov Chain Monte Carlo Methods , 1995 .

[20]  L. Wasserman,et al.  Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio , 1995 .

[21]  Walter R. Gilks,et al.  BUGS - Bayesian inference Using Gibbs Sampling Version 0.50 , 1995 .

[22]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[23]  A. Gelfand,et al.  Bayesian Model Choice: Asymptotics and Exact Calculations , 1994 .

[24]  M. Tanner,et al.  Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler , 1992 .

[25]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[26]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .