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Consistency of spike and slab regression


  • Ishwaran, Hemant
  • Sunil Rao, J.


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.

Suggested Citation

  • Ishwaran, Hemant & Sunil Rao, J., 2011. "Consistency of spike and slab regression," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1920-1928.
  • Handle: RePEc:eee:stapro:v:81:y:2011:i:12:p:1920-1928
    DOI: 10.1016/j.spl.2011.08.005

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    References listed on IDEAS

    1. Park, Trevor & Casella, George, 2008. "The Bayesian Lasso," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 681-686, June.
    2. Geweke, John & Meese, Richard, 1981. "Estimating regression models of finite but unknown order," Journal of Econometrics, Elsevier, vol. 16(1), pages 162-162, May.
    3. Ishwaran H. & Rao J.S., 2003. "Detecting Differentially Expressed Genes in Microarrays Using Bayesian Model Selection," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 438-455, January.
    4. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    5. Ishwaran, Hemant & Rao, J. Sunil, 2005. "Spike and Slab Gene Selection for Multigroup Microarray Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 764-780, September.
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    Cited by:

    1. Li, Cheng & Jiang, Wenxin, 2016. "On oracle property and asymptotic validity of Bayesian generalized method of moments," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 132-147.


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