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Abstract
Penalized regression methods, such as L1 regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In contrast to the frequentist literature, little is known about the properties of such priors and the convergence and concentration of the corresponding posterior distribution. In this article, we propose a new class of Dirichlet–Laplace priors, which possess optimal posterior concentration and lead to efficient posterior computation. Finite sample performance of Dirichlet–Laplace priors relative to alternatives is assessed in simulated and real data examples.
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Notes on contributors
Anirban Bhattacharya
Anirban Bhattacharya is Assistant Professor, Department of Statistics, Texas A&M University, College Station, TX 77843 (E-mail: [email protected]), Debdeep Pati is Assistant Professor, Department of Statistics, Florida State University, Tallahassee, FL 32306 (E-mail: [email protected]), Natesh S. Pillai is Associate Professor, Department of Statistics, Harvard University, Cambridge, MA 02138 (E-mail: [email protected]), David B. Dunson is Arts and Sciences Distinguished Professor, Department of Statistical Science, Duke University, Durham, NC 27708 (E-mail: [email protected]) The authors thank Dr. Ismael Castillo and Dr. James Scott for sharing source code. Dr. Anirban Bhattacharya, Dr. Debdeep Pati and Dr. Natesh S. Pillai acknowledge support from the Office of Naval Research (ONR BAA 14-0001). The authors would like to thank ONR Program Officer Predrag Neskovic for his interest in this work. Dr. Pillai is also partially supported by NSF-DMS 1107070. Dr. David B. Dunson is partially supported by the DARPA MSEE program and the grant number R01 ES017240-01 from the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH).
Debdeep Pati
Anirban Bhattacharya is Assistant Professor, Department of Statistics, Texas A&M University, College Station, TX 77843 (E-mail: [email protected]), Debdeep Pati is Assistant Professor, Department of Statistics, Florida State University, Tallahassee, FL 32306 (E-mail: [email protected]), Natesh S. Pillai is Associate Professor, Department of Statistics, Harvard University, Cambridge, MA 02138 (E-mail: [email protected]), David B. Dunson is Arts and Sciences Distinguished Professor, Department of Statistical Science, Duke University, Durham, NC 27708 (E-mail: [email protected]) The authors thank Dr. Ismael Castillo and Dr. James Scott for sharing source code. Dr. Anirban Bhattacharya, Dr. Debdeep Pati and Dr. Natesh S. Pillai acknowledge support from the Office of Naval Research (ONR BAA 14-0001). The authors would like to thank ONR Program Officer Predrag Neskovic for his interest in this work. Dr. Pillai is also partially supported by NSF-DMS 1107070. Dr. David B. Dunson is partially supported by the DARPA MSEE program and the grant number R01 ES017240-01 from the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH).
Natesh S. Pillai
Anirban Bhattacharya is Assistant Professor, Department of Statistics, Texas A&M University, College Station, TX 77843 (E-mail: [email protected]), Debdeep Pati is Assistant Professor, Department of Statistics, Florida State University, Tallahassee, FL 32306 (E-mail: [email protected]), Natesh S. Pillai is Associate Professor, Department of Statistics, Harvard University, Cambridge, MA 02138 (E-mail: [email protected]), David B. Dunson is Arts and Sciences Distinguished Professor, Department of Statistical Science, Duke University, Durham, NC 27708 (E-mail: [email protected]) The authors thank Dr. Ismael Castillo and Dr. James Scott for sharing source code. Dr. Anirban Bhattacharya, Dr. Debdeep Pati and Dr. Natesh S. Pillai acknowledge support from the Office of Naval Research (ONR BAA 14-0001). The authors would like to thank ONR Program Officer Predrag Neskovic for his interest in this work. Dr. Pillai is also partially supported by NSF-DMS 1107070. Dr. David B. Dunson is partially supported by the DARPA MSEE program and the grant number R01 ES017240-01 from the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH).
David B. Dunson
Anirban Bhattacharya is Assistant Professor, Department of Statistics, Texas A&M University, College Station, TX 77843 (E-mail: [email protected]), Debdeep Pati is Assistant Professor, Department of Statistics, Florida State University, Tallahassee, FL 32306 (E-mail: [email protected]), Natesh S. Pillai is Associate Professor, Department of Statistics, Harvard University, Cambridge, MA 02138 (E-mail: [email protected]), David B. Dunson is Arts and Sciences Distinguished Professor, Department of Statistical Science, Duke University, Durham, NC 27708 (E-mail: [email protected]) The authors thank Dr. Ismael Castillo and Dr. James Scott for sharing source code. Dr. Anirban Bhattacharya, Dr. Debdeep Pati and Dr. Natesh S. Pillai acknowledge support from the Office of Naval Research (ONR BAA 14-0001). The authors would like to thank ONR Program Officer Predrag Neskovic for his interest in this work. Dr. Pillai is also partially supported by NSF-DMS 1107070. Dr. David B. Dunson is partially supported by the DARPA MSEE program and the grant number R01 ES017240-01 from the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH).