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Bayesian Variable Selection and Estimation Based on Global-Local Shrinkage Priors

Author

Listed:
  • Xueying Tang

    (University of Florida)

  • Xiaofan Xu

    (Stubhub Inc.)

  • Malay Ghosh

    (University of Florida)

  • Prasenjit Ghosh

    (Presidency University)

Abstract

We consider in this paper simultaneous Bayesian variable selection and estimation for linear regression models with global-local shrinkage priors on the regression coefficients. We propose a variable selection procedure that selects a variable if the ratio of the posterior mean of its regression coefficient to the corresponding ordinary least square estimate is greater than a half. The regression coefficient is estimated by the posterior mean or zero depending on whether the corresponding variable is selected or not. Under the assumption of orthogonal designs, we prove that if the local parameters have polynomial-tailed priors, the proposed method enjoys the oracle property in the sense that it can achieve variable selection consistency and optimal estimation rate at the same time. However, if, instead, an exponential-tailed prior is used for the local parameters, the proposed method has variable selection consistency but not the optimal estimation rate. We show via simulation and real data examples that our proposed selection mechanism works for nonorthogonal designs as well.

Suggested Citation

  • Xueying Tang & Xiaofan Xu & Malay Ghosh & Prasenjit Ghosh, 2018. "Bayesian Variable Selection and Estimation Based on Global-Local Shrinkage Priors," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 215-246, August.
    • Handle: RePEc:spr:sankha:v:80:y:2018:i:2:d:10.1007_s13171-017-0118-2
    DOI: 10.1007/s13171-017-0118-2
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    References listed on IDEAS

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    6. Anirban Bhattacharya & Debdeep Pati & Natesh S. Pillai & David B. Dunson, 2015. "Dirichlet--Laplace Priors for Optimal Shrinkage," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1479-1490, December.
    7. P. Richard Hahn & Carlos M. Carvalho, 2015. "Decoupling Shrinkage and Selection in Bayesian Linear Models: A Posterior Summary Perspective," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 435-448, March.
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    Cited by:

    1. Ghosh Malay, 2020. "Rejoinder," Statistics in Transition New Series, Polish Statistical Association, vol. 21(4), pages 59-67, August.
    2. Kshitij Khare & Malay Ghosh, 2022. "MCMC Convergence for Global-Local Shrinkage Priors," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 20(1), pages 211-234, September.
    3. Malay Ghosh, 2020. "Rejoinder," Statistics in Transition New Series, Polish Statistical Association, vol. 21(4), pages 59-67, August.
    4. Zhang, Ruoyang & Ghosh, Malay, 2022. "Ultra high-dimensional multivariate posterior contraction rate under shrinkage priors," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    5. Hu, Guanyu, 2021. "Spatially varying sparsity in dynamic regression models," Econometrics and Statistics, Elsevier, vol. 17(C), pages 23-34.
    6. Mauro Bernardi & Daniele Bianchi & Nicolas Bianco, 2022. "Variational inference for large Bayesian vector autoregressions," Papers 2202.12644, arXiv.org, revised Jun 2023.

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