Bayesian variable selection for GLM
I consider the problem of variable selection for Generalized Linear Models (GLM). A great deal of effort has been expended in variable selection for linear regression models and many selection criteria have been proposed and well known in practice. However, for GLM, the standard practice is to use criteria AIC or BIC, or use Chi-square tests for nested models. Due to great difficulty in achieving analytical tractability, much less research in variable selection has been done for GLM, even if it is parallel to linear regression models. In this dissertation, I present a comprehensive Bayesian solution to this problem, which extends the hierarchical formulation of George and Foster (2000) to GLM. It involves choosing priors for parameters and models that bring in hyperparameters, integrating model-specific parameters out of the likelihood function, estimating the values of the hyperparameters from data or choosing hyperpriors for the hyperparameters and finally obtaining posterior probabilities of models as selection criteria. Unlike most previous research in this eld, the model posterior achieved in this work can be calculated easily and accurately without resorting to simulation methods like the Gibbs sampling, Reversible Jump MCMC, etc., hence bypassing the high-dimensional convergence problem. I achieve analytical tractability for GLM by proposing an Integrated Laplace Approximation that has been shown better than classical Laplace's method in this context. I describe two approaches for developing selection criteria: Empirical Bayes (EB), and Fully Bayes (FB), which are different in the way of handling hyperparameters. I also present an alternative FB approach, Conditional Fully Bayes (CFB), based on a different hyper-parameterization. In addition, I propose a method of restricting the integration region of the hyperparameters to improve FB selection performance. For each approach, various criteria are developed and their performance is evaluated through simulation.