Bayesian Analysis

Selection of Tuning Parameters, Solution Paths and Standard Errors for Bayesian Lassos

Vivekananda Roy and Sounak Chakraborty

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Penalized regression methods such as the lasso and elastic net (EN) have become popular for simultaneous variable selection and coefficient estimation. Implementation of these methods require selection of the penalty parameters. We propose an empirical Bayes (EB) methodology for selecting these tuning parameters as well as computation of the regularization path plots. The EB method does not suffer from the “double shrinkage problem” of frequentist EN. Also it avoids the difficulty of constructing an appropriate prior on the penalty parameters. The EB methodology is implemented by efficient importance sampling method based on multiple Gibbs sampler chains. Since the Markov chains underlying the Gibbs sampler are proved to be geometrically ergodic, Markov chain central limit theorem can be used to provide asymptotically valid confidence band for profiles of EN coefficients. The practical effectiveness of our method is illustrated by several simulation examples and two real life case studies. Although this article considers lasso and EN for brevity, the proposed EB method is general and can be used to select shrinkage parameters in other regularization methods.

Article information

Bayesian Anal., Volume 12, Number 3 (2017), 753-778.

First available in Project Euclid: 7 September 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference 62J07: Ridge regression; shrinkage estimators
Secondary: 60J05: Discrete-time Markov processes on general state spaces

Bayesian lasso elastic net empirical Bayes geometric ergodicity importance sampling Markov chain Monte Carlo shrinkage standard errors

Creative Commons Attribution 4.0 International License.


Roy, Vivekananda; Chakraborty, Sounak. Selection of Tuning Parameters, Solution Paths and Standard Errors for Bayesian Lassos. Bayesian Anal. 12 (2017), no. 3, 753--778. doi:10.1214/16-BA1025.

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Supplemental materials

  • Supplementary Material: Supplementary Material for “Selection of Tuning Parameters, Solution Paths and Standard Errors for Bayesian Lassos”. The online supplementary materials contain the proofs of lemmas. Also a summary of the steps involved in the estimation of the tuning parameters and the solution paths is given in the supplementary materials.