Bayesian Bootstrap Spike-and-Slab LASSO

Abstract

The impracticality of posterior sampling has prevented the widespread adoption of spike-and-slab priors in high-dimensional applications. To alleviate the computational burden, optimization strategies have been proposed that quickly find local posterior modes. Trading off uncertainty quantification for computational speed, these strategies have enabled spike-and-slab deployments at scales that would be previously unfeasible. We build on one recent development in this strand of work: the Spike-and-Slab LASSO procedure of Ročková and George (2018). Instead of optimization, however, we explore multiple avenues for posterior sampling, some traditional and some new. Intrigued by the speed of Spike-and-Slab LASSO mode detection, we explore the possibility of sampling from an approximate posterior by performing MAP optimization on many independently perturbed datasets. To this end, we explore Bayesian bootstrap ideas and introduce a new class of jittered Spike-and-Slab LASSO priors with random shrinkage targets. These priors are a key constituent of the Bayesian Bootstrap Spike-and-Slab LASSO (BB-SSL) method proposed here. BB-SSL turns fast optimization into approximate posterior sampling. Beyond its scalability, we show that BB-SSL has a strong theoretical support. Indeed, we find that the induced pseudo-posteriors contract around the truth at a near-optimal rate in sparse normal-means and in high-dimensional regression. We compare our algorithm to the traditional Stochastic Search Variable Selection (under Laplace priors) as well as many state-of-the-art methods for shrinkage priors. We show, both in simulations and on real data, that our method fares superbly in these comparisons, often providing substantial computational gains.

References in corpus (6)

• Bayesian inference and the parametric bootstrap
• Simultaneous Variable and Covariance Selection with the Multivariate Spike-and-Slab Lasso
• Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector
• Nonparametric learning from Bayesian models with randomized objective functions
• Bayes Shrinkage at GWAS scale: Convergence and Approximation Theory of a Scalable MCMC Algorithm for the Horseshoe Prior
• Neuronized Priors for Bayesian Sparse Linear Regression

Cited by in corpus (4)

• Spike-and-Slab Meets LASSO: A Review of the Spike-and-Slab LASSO
• Introducing prior information in Weighted Likelihood Bootstrap with applications to model misspecification
• Asymptotics of cut distributions and robust modular inference using Posterior Bootstrap
• Nonparametric posterior learning for emission tomography with multimodal data