# Spike-and-Slab Group Lassos for Grouped Regression and Sparse Generalized Additive Models

## TL;DR

This paper introduces the spike-and-slab group lasso (SSGL), a Bayesian method for variable selection and estimation in grouped linear regression and sparse generalized additive models, with efficient algorithms and uncertainty quantification.

## Contribution

It extends the spike-and-slab lasso to nonparametric models, develops a fast MAP estimation algorithm, and provides theoretical guarantees for high-dimensional settings.

## Key findings

- Efficient block coordinate ascent algorithm for MAP estimation.
- Theoretical posterior concentration rates in high dimensions.
- Successful application to simulations and real data.

## Abstract

We introduce the spike-and-slab group lasso (SSGL) for Bayesian estimation and variable selection in linear regression with grouped variables. We further extend the SSGL to sparse generalized additive models (GAMs), thereby introducing the first nonparametric variant of the spike-and-slab lasso methodology. Our model simultaneously performs group selection and estimation, while our fully Bayes treatment of the mixture proportion allows for model complexity control and automatic self-adaptivity to different levels of sparsity. We develop theory to uniquely characterize the global posterior mode under the SSGL and introduce a highly efficient block coordinate ascent algorithm for maximum a posteriori (MAP) estimation. We further employ de-biasing methods to provide uncertainty quantification of our estimates. Thus, implementation of our model avoids the computational intensiveness of Markov chain Monte Carlo (MCMC) in high dimensions. We derive posterior concentration rates for both grouped linear regression and sparse GAMs when the number of covariates grows at nearly exponential rate with sample size. Finally, we illustrate our methodology through extensive simulations and data analysis.