Dirichlet–Laplace Priors for Optimal Shrinkage

Penalized regression methods, such as L ₁ 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 pro...

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Bibliographic Details
Published in:Journal of the American Statistical Association 2015-12, Vol.110 (512), p.1479-1490
Main Authors: Bhattacharya, Anirban, Pati, Debdeep, Pillai, Natesh S., Dunson, David B.
Format: Article
Language:English
Subjects:
Bayesian
Bayesian analysis
Bayesian method
Convergence
Convergence rate
High-dimensional
Lasso
Normal distribution
Penalized regression
probability
Regression analysis
Regularization
shrinkage
Shrinkage prior
Simulation
Sparsity
Statistics
Theory and Methods
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Summary:Penalized regression methods, such as L ₁ 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.
ISSN:1537-274X
0162-1459
1537-274X
DOI:10.1080/01621459.2014.960967