### 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 language | English (US) |
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Title of host publication | Statistical Analysis for High-Dimensional Data - The Abel Symposium, 2014 |

Editors | Peter Buhlmann, Ingrid K. Glad, Mette Langaas, Sylvia Richardson, Arnoldo Frigessi, Marina Vannucci |

Publisher | Springer Heidelberg |

Pages | 15-36 |

Number of pages | 22 |

ISBN (Print) | 9783319270975 |

DOIs | |

State | Published - Jan 1 2016 |

Event | 11th Abel Symposium on Statistical Analysis for High-Dimensional Data, 2014 - Kabelvag, Norway Duration: May 5 2014 → May 9 2014 |

### Publication series

Name | Abel Symposia |
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Volume | 11 |

ISSN (Print) | 2193-2808 |

ISSN (Electronic) | 2197-8549 |

### Other

Other | 11th Abel Symposium on Statistical Analysis for High-Dimensional Data, 2014 |
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Country | Norway |

City | Kabelvag |

Period | 5/5/14 → 5/9/14 |

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### Cite this

*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