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Global-Local Mixtures: A Unifying Framework

Author

Listed:
  • Anindya Bhadra

    (Purdue University)

  • Jyotishka Datta

    (University of Arkansas)

  • Nicholas G. Polson

    (The University of Chicago Booth School of Business)

  • Brandon T. Willard

    (The University of Chicago Booth School of Business)

Abstract

Global-local mixtures, including Gaussian scale mixtures, have gained prominence in recent times, both as a sparsity inducing prior in p ≫ n problems as well as default priors for non-linear many-to-one functionals of high-dimensional parameters. Here we propose a unifying framework for global-local scale mixtures using the Cauchy-Schlömilch and Liouville integral transformation identities, and use the framework to build a new Bayesian sparse signal recovery method. This new method is a Bayesian counterpart of the Lasso $\sqrt {\text {Lasso}}$ (Belloni et al., Biometrika 98, 4, 791–806, 2011) that adapts to unknown error variance. Our framework also characterizes well-known scale mixture distributions including the Laplace density used in Bayesian Lasso, logit and quantile via a single integral identity. Finally, we derive a few convolutions that commonly arise in Bayesian inference and posit a conjecture concerning bridge and uniform correlation mixtures.

Suggested Citation

  • Anindya Bhadra & Jyotishka Datta & Nicholas G. Polson & Brandon T. Willard, 2020. "Global-Local Mixtures: A Unifying Framework," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 426-447, August.
    • Handle: RePEc:spr:sankha:v:82:y:2020:i:2:d:10.1007_s13171-019-00191-2
    DOI: 10.1007/s13171-019-00191-2
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    References listed on IDEAS

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    1. Carlos M. Carvalho & Nicholas G. Polson & James G. Scott, 2010. "The horseshoe estimator for sparse signals," Biometrika, Biometrika Trust, vol. 97(2), pages 465-480.
    2. Kai Zhang & Lawrence D. Brown & Edward George & Linda Zhao, 2014. "Uniform Correlation Mixture of Bivariate Normal Distributions and Hypercubically Contoured Densities That Are Marginally Normal," The American Statistician, Taylor & Francis Journals, vol. 68(3), pages 183-187, March.
    3. N. G. Polson & J. G. Scott, 2013. "Data augmentation for non-Gaussian regression models using variance-mean mixtures," Biometrika, Biometrika Trust, vol. 100(2), pages 459-471.
    4. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    5. Nicholas G. Polson & James G. Scott & Jesse Windle, 2014. "The Bayesian bridge," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(4), pages 713-733, September.
    6. Park, Trevor & Casella, George, 2008. "The Bayesian Lasso," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 681-686, June.
    7. Chatterjee, A. & Lahiri, S. N., 2011. "Bootstrapping Lasso Estimators," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 608-625.
    8. Nicholas G. Polson & James G. Scott & Jesse Windle, 2013. "Bayesian Inference for Logistic Models Using Pólya--Gamma Latent Variables," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(504), pages 1339-1349, December.
    9. Anindya Bhadra & Jyotishka Datta & Nicholas G. Polson & Brandon Willard, 2016. "Default Bayesian analysis with global-local shrinkage priors," Biometrika, Biometrika Trust, vol. 103(4), pages 955-969.
    10. Tingni Sun & Cun-Hui Zhang, 2012. "Scaled sparse linear regression," Biometrika, Biometrika Trust, vol. 99(4), pages 879-898.
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