Consistency of spike and slab regression

Hemant Ishwaran, J. Sunil Rao

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

Spike and slab models are a popular and attractive variable selection approach in regression settings. Applications for these models have blossomed over the last decade and they are increasingly being used in challenging problems. At the same time, theory for spike and slab models has not kept pace with the applications. There are many gaps in what we know about their theoretical properties. An important property known to hold in these models is selective shrinkage: a unique property whereby the posterior mean is shrunk toward zero for non-informative variables only. This property has been shown to hold under orthogonality for continuous priors under the modified class of rescaled spike and slab models. In this paper, we extend this result to the general case and prove an oracle property for the posterior mean under a discrete two-component prior. An immediate consequence is that a strong selective shrinkage property holds. Interestingly, the conditions needed for our result to hold in the non-orthogonal setting are more stringent than in the orthogonal case and amount to a type of enforced sparsity condition that must be met by the prior.

Original languageEnglish (US)
Pages (from-to)1920-1928
Number of pages9
JournalStatistics and Probability Letters
Volume81
Issue number12
DOIs
StatePublished - Dec 2011
Externally publishedYes

Bibliographical note

Funding Information:
This work was partially funded by the National Science Foundation grant DMS-0705037 . We thank the Associate Editor and two referees for helpful comments.

Keywords

  • Oracle property
  • Posterior mean
  • Rescaling
  • Shrinkage
  • Two-component prior

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