# Gibbs: Gibbs In BBSSL: Bayesian Bootstrap Spike-and-Slab LASSO

## Description

This function runs SSVS for linear regression with Spike-and-Slab LASSO prior. By default, this function uses the speed-up trick in Bhattacharya et al. (2016) when p > n.

## Usage

 `1` ```Gibbs(y, X, a, b, lambda, maxiter, burn.in, initial.beta = NULL, sigma = 1) ```

## Arguments

 `y` A vector of continuous responses (n x 1). `X` The design matrix (n x p), without an intercept. `a, b` Parameters of the prior. `lambda` A two-dim vector = c(lambda0, lambda1). `maxiter` An integer which specifies the maximum number of iterations for MCMC. `burn.in` An integer which specifies the maximum number of burn-in iterations for MCMC. `initial.beta` A vector of initial values of beta to used. If set to NULL, the LASSO solution with 10-fold cross validation is used. Default is NULL. `sigma` Noise standard deviation. Default is 1.

## Value

A list, including matrix 'beta' ((maxiter-burn.in) x p), matrix 'tau2' ((maxiter-burn.in) x p), matrix 'gamma' ((maxiter-burn.in) x p), vector 'theta' ((maxiter-burn.in) x 1).

## Author(s)

Lizhen Nie lizhen@statistics.uchicago.edu, Veronika Rockova Veronika.Rockova@chicagobooth.edu

## References

Nie, L., & Ročková, V. (2020). Bayesian Bootstrap Spike-and-Slab LASSO. arXiv:2011.14279.

Bhattacharya, A., Chakraborty, A., & Mallick, B. K. (2016). Fast sampling with Gaussian scale mixture priors in high-dimensional regression. Biometrika, 103(4):985.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```n = 50; p = 12; truth.beta = c(1.3, 1.3, 1.3, 1.3); truth.sigma = 1 data = Generate_data(truth.beta, p, n, truth.sigma = 1, rho = 0.6,"block",4) y = data\$y; X = data\$X; beta = data\$beta # --------------- set parameters ----------------- lambda0 = 7; lambda1 = 0.15; lambda = c(lambda0, lambda1) a = 1; b = p #beta prior for theta # this is for demonstration of usage only # in practice, you may want to use more iterations! MCchain1 = Gibbs(y, X, a, b, lambda, maxiter = 1000, burn.in = 100) ```

BBSSL documentation built on April 27, 2021, 9:06 a.m.