dist.LASSO {LaplacesDemon}  R Documentation 
LASSO Distribution
Description
These functions provide the density and random generation for the Bayesian LASSO prior distribution.
Usage
dlasso(x, sigma, tau, lambda, a=1, b=1, log=FALSE)
rlasso(n, sigma, tau, lambda, a=1, b=1)
Arguments
x 
This is a location vector of length 
n 
This is the number of observations, which must be a positive integer that has length 1. 
sigma 
This is a positiveonly scalar hyperparameter

tau 
This is a positiveonly vector of hyperparameters,

lambda 
This is a positiveonly scalar hyperhyperparameter,

a , b 
These are positiveonly scalar hyperhyperhyperparameters
for gamma distributed 
log 
Logical. If 
Details
Application: Multivariate Scale Mixture
Density:
p(\theta) \sim \mathcal{N}_k(0, \sigma^2 diag(\tau^2))(\frac{1}{sigma^2}) \mathcal{EXP}(\frac{\lambda^2}{2}) \mathcal{G}(a,b)
Inventor: Parks and Casella (2008)
Notation 1:
\theta \sim \mathcal{LASSO}(\sigma, \tau, \lambda, a, b)
Notation 2:
p(\theta) = \mathcal{LASSO}(\theta  \sigma, \tau, \lambda, a, b)
Parameter 1: hyperparameter global scale
\sigma > 0
Parameter 2: hyperparameter local scale
\tau > 0
Parameter 3: hyperhyperparameter global scale
\lambda > 0
Parameter 4: hyperhyperhyperparameter scale
a > 0
Parameter 5: hyperhyperhyperparameter scale
b > 0
Mean:
E(\theta)
Variance:
Mode:
The Bayesian LASSO distribution (Parks and Casella, 2008) is a heavytailed mixture distribution that can be considered a variance mixture, and it is in the family of multivariate scale mixtures of normals.
The LASSO distribution was proposed as a prior distribution, as a
Bayesian version of the frequentist LASSO, introduced by Tibshirani
(1996). It is applied as a shrinkage prior in the presence of sparsity
for J
regression effects. LASSO priors are most appropriate in
largedimensional models where dimension reduction is necessary to
avoid overly complex models that predict poorly.
The Bayesian LASSO results in regression effects that are a compromise between regression effects in the frequentist LASSO and ridge regression. The Bayesian LASSO applies more shrinkage to weak regression effects than ridge regression.
The Bayesian LASSO is an alternative to horseshoe regression and ridge regression.
Value
dlasso
gives the density and
rlasso
generates random deviates.
References
Park, T. and Casella, G. (2008). "The Bayesian Lasso". Journal of the American Statistical Association, 103, p. 672–680.
Tibshirani, R. (1996). "Regression Shrinkage and Selection via the Lasso". Journal of the Royal Statistical Society, Series B, 58, p. 267–288.
See Also
Examples
library(LaplacesDemon)
x < rnorm(100)
sigma < rhalfcauchy(1, 5)
tau < rhalfcauchy(100, 5)
lambda < rhalfcauchy(1, 5)
x < dlasso(x, sigma, tau, lambda, log=TRUE)
x < rlasso(length(tau), sigma, tau, lambda)