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

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  • Leon-Novelo, Luis
  • Casella, George

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.

Suggested Citation

  • Leon-Novelo, Luis & Casella, George, 2012. "Prior influence in linear regression when the number of covariates increases to infinity," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 438-445.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:3:p:438-445
    DOI: 10.1016/j.spl.2011.10.018
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    Keywords

    Linear models; Regression; Bayes; Model selection;

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