期刊
AMERICAN STATISTICIAN
卷 69, 期 3, 页码 165-173出版社
AMER STATISTICAL ASSOC
DOI: 10.1080/00031305.2015.1031827
关键词
Bayesian model averaging; Linear regression; Marginal inclusion probability; Median probability model; Multimodality; Zellner's g-prior
资金
- NSA Young Investigator grant [H98230-14-1-0126]
In this article, we highlight some interesting facts about Bayesian variable selection methods for linear regression models in settings where the design matrix exhibits strong collinearity. We first demonstrate via real data analysis and simulation studies that summaries of the posterior distribution based on marginal and joint distributions may give conflicting results for assessing the importance of strongly correlated covariates. The natural question is which one should be used in practice. The simulation studies suggest that posterior inclusion probabilities and Bayes factors that evaluate the importance of correlated covariates jointly are more appropriate, and some priors may be more adversely affected in such a setting. To obtain a better understanding behind the phenomenon, we study some toy examples with Zellner's g-prior. The results show that strong collinearity may lead to a multimodal posterior distribution over models, in which joint summaries are more appropriate than marginal summaries. Thus, we recommend a routine examination of the correlation matrix and calculation of the joint inclusion probabilities for correlated covariates, in addition to marginal inclusion probabilities, for assessing the importance of covariates in Bayesian variable selection.
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