IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v169y2019icp61-80.html
   My bibliography  Save this article

Fast Monte Carlo Markov chains for Bayesian shrinkage models with random effects

Author

Listed:
  • Abrahamsen, Tavis
  • Hobert, James P.

Abstract

When performing Bayesian data analysis using a general linear mixed model, the resulting posterior density is almost always analytically intractable. However, if proper conditionally conjugate priors are used, there is a simple two-block Gibbs sampler that is geometrically ergodic in nearly all practical settings, including situations where p>n (Abrahamsen and Hobert, 2017). Unfortunately, the (conditionally conjugate) multivariate Gaussian prior on β does not perform well in the high-dimensional setting where p≫n. In this paper, we consider an alternative model in which the multivariate Gaussian prior is replaced by the normal-gamma shrinkage prior developed by Griffin and Brown (2010). This change leads to a much more complex posterior density, and we develop a simple MCMC algorithm for exploring it. This algorithm, which has both deterministic and random scan components, is easier to analyze than the more obvious three-step Gibbs sampler. Indeed, we prove that the new algorithm is geometrically ergodic in most practical settings.

Suggested Citation

  • Abrahamsen, Tavis & Hobert, James P., 2019. "Fast Monte Carlo Markov chains for Bayesian shrinkage models with random effects," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 61-80.
    • Handle: RePEc:eee:jmvana:v:169:y:2019:i:c:p:61-80
    DOI: 10.1016/j.jmva.2018.08.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X17306863
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2018.08.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. Johnson, Alicia A. & Jones, Galin L., 2015. "Geometric ergodicity of random scan Gibbs samplers for hierarchical one-way random effects models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 325-342.
    4. Xiang Zhou & Peter Carbonetto & Matthew Stephens, 2013. "Polygenic Modeling with Bayesian Sparse Linear Mixed Models," PLOS Genetics, Public Library of Science, vol. 9(2), pages 1-14, February.
    5. Jürg Schelldorfer & Peter Bühlmann & Sara Van De Geer, 2011. "Estimation for High‐Dimensional Linear Mixed‐Effects Models Using ℓ 1 ‐Penalization," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(2), pages 197-214, June.
    6. Rohart, Florian & San Cristobal, Magali & Laurent, Béatrice, 2014. "Selection of fixed effects in high dimensional linear mixed models using a multicycle ECM algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 209-222.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Niloy Biswas & Anirban Bhattacharya & Pierre E. Jacob & James E. Johndrow, 2022. "Coupling‐based convergence assessment of some Gibbs samplers for high‐dimensional Bayesian regression with shrinkage priors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(3), pages 973-996, July.
    2. Fan, Jianqing & Jiang, Bai & Sun, Qiang, 2022. "Bayesian factor-adjusted sparse regression," Journal of Econometrics, Elsevier, vol. 230(1), pages 3-19.
    3. Martin Feldkircher & Florian Huber & Gary Koop & Michael Pfarrhofer, 2022. "APPROXIMATE BAYESIAN INFERENCE AND FORECASTING IN HUGE‐DIMENSIONAL MULTICOUNTRY VARs," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 63(4), pages 1625-1658, November.
    4. Chan, Joshua C.C., 2021. "Minnesota-type adaptive hierarchical priors for large Bayesian VARs," International Journal of Forecasting, Elsevier, vol. 37(3), pages 1212-1226.
    5. Debamita Kundu & Riten Mitra & Jeremy T. Gaskins, 2021. "Bayesian variable selection for multioutcome models through shared shrinkage," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 295-320, March.
    6. Hu, Guanyu, 2021. "Spatially varying sparsity in dynamic regression models," Econometrics and Statistics, Elsevier, vol. 17(C), pages 23-34.
    7. Michael Pfarrhofer, 2024. "Forecasts with Bayesian vector autoregressions under real time conditions," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 43(3), pages 771-801, April.
    8. Dufays, Arnaud & Rombouts, Jeroen V.K., 2020. "Relevant parameter changes in structural break models," Journal of Econometrics, Elsevier, vol. 217(1), pages 46-78.
    9. 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.
    10. Posch, Konstantin & Arbeiter, Maximilian & Pilz, Juergen, 2020. "A novel Bayesian approach for variable selection in linear regression models," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    11. Dimitris Korobilis & Kenichi Shimizu, 2022. "Bayesian Approaches to Shrinkage and Sparse Estimation," Foundations and Trends(R) in Econometrics, now publishers, vol. 11(4), pages 230-354, June.
    12. Gregor Kastner & Florian Huber, 2020. "Sparse Bayesian vector autoregressions in huge dimensions," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 39(7), pages 1142-1165, November.
    13. Tathagata Basu & Matthias C. M. Troffaes & Jochen Einbeck, 2023. "A Robust Bayesian Analysis of Variable Selection under Prior Ignorance," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 1014-1057, February.
    14. Uddin, Md Nazir & Gaskins, Jeremy T., 2023. "Shared Bayesian variable shrinkage in multinomial logistic regression," Computational Statistics & Data Analysis, Elsevier, vol. 177(C).
    15. Florian Huber & Tam'as Krisztin & Michael Pfarrhofer, 2018. "A Bayesian panel VAR model to analyze the impact of climate change on high-income economies," Papers 1804.01554, arXiv.org, revised Feb 2021.
    16. Se Yoon Lee & Bani K. Mallick, 2022. "Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 1-43, May.
    17. 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.
    18. Arnaud Dufays & Zhuo Li & Jeroen V.K. Rombouts & Yong Song, 2021. "Sparse change‐point VAR models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 36(6), pages 703-727, September.
    19. Monica Billio & Roberto Casarin & Matteo Iacopini & Sylvia Kaufmann, 2023. "Bayesian Dynamic Tensor Regression," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(2), pages 429-439, April.
    20. David Kohns & Tibor Szendrei, 2021. "Decoupling Shrinkage and Selection for the Bayesian Quantile Regression," Papers 2107.08498, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:169:y:2019:i:c:p:61-80. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.