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Laplace Approximation in High-Dimensional Bayesian Regression

  • Rina Foygel Barber
  • Mathias Drton
  • Kean Ming Tan
Conference paper
Part of the Abel Symposia book series (ABEL, volume 11)

Abstract

We consider Bayesian variable selection in sparse high-dimensional regression, where the number of covariates p may be large relative to the sample size n, but at most a moderate number q of covariates are active. Specifically, we treat generalized linear models. For a single fixed sparse model with well-behaved prior distribution, classical theory proves that the Laplace approximation to the marginal likelihood of the model is accurate for sufficiently large sample size n. We extend this theory by giving results on uniform accuracy of the Laplace approximation across all models in a high-dimensional scenario in which p and q, and thus also the number of considered models, may increase with n. Moreover, we show how this connection between marginal likelihood and Laplace approximation can be used to obtain consistency results for Bayesian approaches to variable selection in high-dimensional regression.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Rina Foygel Barber
    • 1
  • Mathias Drton
    • 2
  • Kean Ming Tan
    • 3
  1. 1.Department of StatisticsThe University of ChicagoChicagoUSA
  2. 2.Department of StatisticsUniversity of WashingtonSeattleUSA
  3. 3.Department of BiostatisticsUniversity of WashingtonSeattleUSA

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