Ningyi Liu | Searcharxiv

arXiv2025

Bayesian Elastic Net Regression with Structured Prior Dependence

Many regularization priors for Bayesian regression assume the regression coefficients are a priori independent. In particular this is the case for standard Bayesian treatments of the lasso and the elastic net. While independence may be reasonable in some data-analytic settings, incorporating dependence in these prior distributions provides greater modeling flexibility. This paper introduces the orthant normal distribution in its general form and shows how it can be used to structure prior dependence in the Bayesian elastic net regression model. An L1-regularized version of Zellner's g prior is introduced as a special case, creating a new link between the literature on penalized optimization and an important class of regression priors. Computation is challenging due to an intractable normalizing constant in the prior. We avoid this issue by modifying slightly a standard prior of convenience for the hyperparameters in such a way to enable simple and fast Gibbs sampling of the posterior distribution. The benefit of including structured prior dependence in the Bayesian elastic net regression model is demonstrated through simulation and a near-infrared spectroscopy data example.

stat.ME↗

arXiv2024

Sampling the Bayesian Elastic Net

The Bayesian elastic net regression model is characterized by the regression coefficient prior distribution, the negative log density of which corresponds to the elastic net penalty function. While Markov chain Monte Carlo (MCMC) methods exist for sampling from the posterior of the regression coefficients given the penalty parameters, full Bayesian inference that incorporates uncertainty about the penalty parameters remains a challenge due to an intractable integrable in the posterior density function. Though sampling methods have been proposed that avoid computing this integral, all correctly-specified methods for full Bayesian inference that have appeared in the literature involve at least one "Metropolis-within-Gibbs" update, requiring tuning of proposal distributions. The computational landscape is complicated by the fact that two forms of the Bayesian elastic net prior have been introduced, and two representations (with and without data augmentation) of the prior suggest different MCMC algorithms. We review the forms and representations of the prior, discuss all combinations of these different treatments for the first time, and introduce one combination of form and representation that has yet to appear in the literature. We introduce MCMC algorithms for full Bayesian inference for all treatments of the prior. The algorithms allow for direct sampling of all parameters without any "Metropolis-within-Gibbs" steps. The key to the new approach is a careful transformation of the parameter space and an analysis of the resulting full conditional density functions that allows for efficient rejection sampling. We make empirical comparisons between our approaches and existing MCMC samplers for different data structures.

stat.CO↗