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Predictive performance of Dirichlet process shrinkage methods in linear regression

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  • Nott, David J.
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    Abstract

    An obvious Bayesian nonparametric generalization of ridge regression assumes that coefficients are exchangeable, from a prior distribution of unknown form, which is given a Dirichlet process prior with a normal base measure. The purpose of this paper is to explore predictive performance of this generalization, which does not seem to have received any detailed attention, despite related applications of the Dirichlet process for shrinkage estimation in multivariate normal means, analysis of randomized block experiments and nonparametric extensions of random effects models in longitudinal data analysis. We consider issues of prior specification and computation, as well as applications in penalized spline smoothing. With a normal base measure in the Dirichlet process and letting the precision parameter approach infinity the procedure is equivalent to ridge regression, whereas for finite values of the precision parameter the discreteness of the Dirichlet process means that some predictors can be estimated as having the same coefficient. Estimating the precision parameter from the data gives a flexible method for shrinkage estimation of mean parameters which can work well when ridge regression does, but also adapts well to sparse situations. We compare our approach with ridge regression, the lasso and the recently proposed elastic net in simulation studies and also consider applications to penalized spline smoothing.

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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 52 (2008)
    Issue (Month): 7 (March)
    Pages: 3658-3669

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    Handle: RePEc:eee:csdana:v:52:y:2008:i:7:p:3658-3669

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    Web page: http://www.elsevier.com/locate/csda

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    1. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320.
    2. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768.
    3. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108.
    4. Smith, M. & Kohn, R., . "Nonparametric Regression using Bayesian Variable Selection," Statistics Working Paper _009, Australian Graduate School of Management.
    5. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, April.
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    Cited by:
    1. Lian, Heng, 2010. "Sparse Bayesian hierarchical modeling of high-dimensional clustering problems," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1728-1737, August.

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