Bayesian Variable Selection and Computation for Generalized Linear Models with Conjugate Priors.

Area of Science:

• Statistics
• Computational Statistics

Background:

• Variable subset selection is crucial for building accurate generalized linear models (GLMs).
• Existing methods for variable selection in GLMs often lack clear theoretical connections.
• Bayesian approaches offer a robust framework for model selection, but computational efficiency can be a challenge.

Purpose of the Study:

• To establish theoretical and computational links between six popular variable subset selection methods in GLMs.
• To investigate the performance of conjugate priors for Bayesian variable selection in GLMs.
• To develop an efficient computational strategy for evaluating multiple Bayesian criteria across all subset models.

Main Methods:

• Derivation of closed-form analytic relationships between Bayesian criteria (Bayes factor, CPO, L measure, DIC, AIC, BIC) under conjugate priors for linear models.
• Examination of computational relationships for arbitrary GLMs using conjugate priors.
• Utilizing Markov chain Monte Carlo (MCMC) sampling from the full model to compute multiple Bayesian criteria for all subset models.

Main Results:

• Established closed-form analytic relationships between Bayes factor, CPO, L measure, DIC, AIC, and BIC for linear models under conjugate priors.
• Demonstrated that four Bayesian criteria can be computed simultaneously for all subset models once MCMC samples from the full model are obtained.
• Validated the performance of Chen and Ibrahim's (2003) conjugate priors in Bayesian variable selection.

Conclusions:

• The study provides a unified theoretical and computational framework for Bayesian variable selection in GLMs.
• The proposed methodology enhances computational efficiency by enabling simultaneous calculation of multiple Bayesian criteria.
• The findings facilitate more robust and efficient model selection in the presence of complex GLMs.