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August 2007 Bayesian variable selection for high dimensional generalized linear models: Convergence rates of the fitted densities
Wenxin Jiang
Ann. Statist. 35(4): 1487-1511 (August 2007). DOI: 10.1214/009053607000000019

Abstract

Bayesian variable selection has gained much empirical success recently in a variety of applications when the number K of explanatory variables $(x_1,\ldots,x_K)$ 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 $K\gg n$. 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.

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Wenxin Jiang. "Bayesian variable selection for high dimensional generalized linear models: Convergence rates of the fitted densities." Ann. Statist. 35 (4) 1487 - 1511, August 2007. https://doi.org/10.1214/009053607000000019

Information

Published: August 2007
First available in Project Euclid: 29 August 2007

zbMATH: 1123.62026
MathSciNet: MR2351094
Digital Object Identifier: 10.1214/009053607000000019

Subjects:
Primary: 62F99
Secondary: 62J12

Keywords: Convergence rates , generalized linear models , high dimensional data , posterior distribution , prior distribution , Sparsity , Variable selection

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 4 • August 2007
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