Prior influence in linear regression when the number of covariates increases to infinity

Luis Leon-Novelo and George Casella

Statistics & Probability Letters, 2012, vol. 82, issue 3, 438-445

Abstract: It is becoming more typical in regression problems today to have the situation where “p>n”, that is, where the number of covariates is greater than the number of observations. Approaches to this problem include such strategies as model selection and dimension reduction, and, of course, a Bayesian approach. However, the discrepancy between p and n can be so large, especially in genomic data, that examining the limiting case where p→∞ can be a relevant calculation. Here we look at the effect of a prior distribution on the coefficients, and in particular characterize the conditions under which, as p→∞, the prior does not overwhelm the data. Specifically, we find that the prior variance on the growing number of covariates must approach zero at rate 1/p, otherwise the prior will overwhelm the data and the posterior distribution of the regression coefficient will equal the prior distribution.

Keywords: Linear models; Regression; Bayes; Model selection (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spl.2011.10.018

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