## Abstract

Bayesian variable selection has gained much empirical success recently in a variety of applications when the number K of explanatory variables (x _{1}, . . . , x_{K}) is possibly much larger than the sample size n. For generalized linear models, if most of the x_{j}'s have very small effects on the response y, we show that it is possible to use Bayesian variable selection to reduce overfitting caused by the curse of dimensionality K ≫ n. In this approach a suitable prior can be used to choose a few out of the many x_{j}'s to model y, so that the posterior will propose probability densities p that are "often close" to the true density p in some sense. The closeness can be described by a Hellinger distance between p and p that scales at a power very close to n^{-1/2}, which is the "finite-dimensional rate" corresponding to a low-dimensional situation. These findings extend some recent work of Jiang [Technical Report 05-02 (2005) Dept. Statistics, Northwestern Univ.] on consistency of Bayesian variable selection for binary classification.

Original language | English (US) |
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Pages (from-to) | 1487-1511 |

Number of pages | 25 |

Journal | Annals of Statistics |

Volume | 35 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2007 |

## Keywords

- Convergence rates
- Generalized linear models
- High dimensional data
- Posterior distribution
- Prior distribution
- Sparsity
- Variable selection

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty