Bayesian variable selection for high dimensional generalized linear models : Convergence rates of the fitted densities

By Wenxin Jiang. 2007

103 citations recorded by the source corpus.

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 xj'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 -½ , 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/02aa36295c4306a9b4e62a02094cdc48d4aab6ac

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Citations in the corpus

Bayesian Subset Modeling for High-Dimensional Generalized Linear Models (2013, background): “From the perspective of Bayesian theory, a remarkable result was obtained by Jiang (2006, 2007), who showed that the true density (1) can be estimated consistently, based on the models sampled from the posterior distribution, through a Bayesian variable selection procedure.”
Bayesian model choice and information criteria in sparse generalized linear models (2011, background): “Jiang (2007) points out that in applied settings, the question of variable selection is not always well-defined due to many coefficients that are approximately rather than exactly zero, and considers the problem of estimating the true parameter vector under the assumption of approximate sparsity.”