Extended Bayesian information criteria for model selection with large model spaces

The ordinary Bayesian information criterion is too liberal formodel selection when the model space is large. In this paper,we re-examine the Bayesian paradigm for model selection andpropose an extended family of Bayesian information criteria,which take into account both the number of unknown parametersand the complexity of the model space. Their consistency isestablished, in particular allowing the number of covariatesto increase to infinity with the sample size. Their performancein various situations is evaluated by simulation studies. Itis demonstrated that the extended Bayesian information criteriaincur a small loss in the positive selection rate but tightlycontrol the false discovery rate, a desirable property in manyapplications. The extended Bayesian information criteria areextremely useful for variable selection in problems with a moderatesample size but with a huge number of covariates, especiallyin genome-wide association studies, which are now an activearea in genetics research.