Package 'l1ball'

Fit the L1 prior

Description

This package provides an implementation of the Gibbs sampler, for using l1-ball prior with the regression likelihood $y_i = X_i\theta+ \epsilon_i, \epsilon_i\sim {N}(0,\sigma^2)$.

Arguments

y	A data vector, n by 1
X	A design matrix, n by p
b_w	The parameter in $Beta(1, p^{b_w})$ for $w$, default $b_w=1$
step	Number of steps to run the Markov Chain Monte Carlo
burnin	Number of burn-ins
b_lam	The parameter in $\lambda_i \sim Inverse-Gamma(1, b_\lambda)$, default $b_\lambda=10^{-3}$. To increase the level of shrinkage, use smaller $b_\lambda$.

Value

The posterior sample collected from the Markov Chain:

• trace_theta: $\theta$

• trace_NonZero: The non-zero indicator $1(\theta_i\neq 0)$

• trace_Lam: $\lambda_i$

• trace_Sigma: $\sigma^2$

Examples

n = 60
p = 100
X <- matrix(rnorm(n*p),n,p)
d = 5
w0 <- c(rep(0, p-d), rnorm(d)*0.1+1)
y = X%*% w0 + rnorm(n,0,.1)
trace <- l1ball(y,X,steps=2000,burnin = 2000)
plot(colMeans(trace$trace_theta))

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