Laplace approximation in high-dimensional Bayesian regression

Rina Foygel Barber, Mathias Drton, Kean Ming Tan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

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.

Original languageEnglish (US)
Title of host publicationStatistical Analysis for High-Dimensional Data - The Abel Symposium, 2014
EditorsPeter Buhlmann, Ingrid K. Glad, Mette Langaas, Sylvia Richardson, Arnoldo Frigessi, Marina Vannucci
PublisherSpringer Heidelberg
Pages15-36
Number of pages22
ISBN (Print)9783319270975
DOIs
StatePublished - Jan 1 2016
Event11th Abel Symposium on Statistical Analysis for High-Dimensional Data, 2014 - Kabelvag, Norway
Duration: May 5 2014May 9 2014

Publication series

NameAbel Symposia
Volume11
ISSN (Print)2193-2808
ISSN (Electronic)2197-8549

Other

Other11th Abel Symposium on Statistical Analysis for High-Dimensional Data, 2014
CountryNorway
CityKabelvag
Period5/5/145/9/14

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Barber, R. F., Drton, M., & Tan, K. M. (2016). Laplace approximation in high-dimensional Bayesian regression. In P. Buhlmann, I. K. Glad, M. Langaas, S. Richardson, A. Frigessi, & M. Vannucci (Eds.), Statistical Analysis for High-Dimensional Data - The Abel Symposium, 2014 (pp. 15-36). (Abel Symposia; Vol. 11). Springer Heidelberg. https://doi.org/10.1007/978-3-319-27099-9_2