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It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent model selector (in the frequentist sense). Here we go beyond the usual consistency for nested pairwise models, and show that for a wide class of prior distributions, including intrinsic priors, the corresponding Bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models. We find that the asymptotics of the Bayes factors for intrinsic priors are equivalent to those of the Schwarz (BIC) criterion. Also, recall that the Jeffreys--Lindley paradox refers to the well-known fact that a point null hypothesis on the normal mean parameter is always accepted when the variance of the conjugate prior goes to infinity. This implies that some limiting forms of proper prior distributions are not necessarily suitable for testing problems. Intrinsic priors are limits of proper prior distributions, and for finite sample sizes they have been proved to behave extremely well for variable selection in regression; a consequence of our results is that for intrinsic priors Lindleys paradox does not arise.
We study the problem of high-dimensional variable selection via some two-step procedures. First we show that given some good initial estimator which is $ell_{infty}$-consistent but not necessarily variable selection consistent, we can apply the nonne
Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance
The dual problem of testing the predictive significance of a particular covariate, and identification of the set of relevant covariates is common in applied research and methodological investigations. To study this problem in the context of functiona
We consider a nonparametric additive model of a conditional mean function in which the number of variables and additive components may be larger than the sample size but the number of nonzero additive components is small relative to the sample size.
Discussion of Latent variable graphical model selection via convex optimization by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].