bridge.reg: Bayesian Bridge Regression

In BayesBridge: Bridge Regression

Description

Bayesian Bridge Regression via Gibbs sampling.

Usage

bridge.reg(y, X, nsamp, alpha=0.5,
           sig2.shape=0.0, sig2.scale=0.0, nu.shape=2.0, nu.rate=2.0,
           alpha.a=1.0, alpha.b=1.0,
           sig2.true=0.0, tau.true=0.0,
           burn=500, method="triangle", ortho=FALSE)

bridge.reg.tri(y, X, nsamp, alpha=0.5,
           sig2.shape=0.0, sig2.scale=0.0, nu.shape=2.0, nu.rate=2.0,
           alpha.a=1.0, alpha.b=1.0,	   
           sig2.true=0.0, tau.true=0.0,
           burn=500, ortho=FALSE, betaburn=0, extras=FALSE)

bridge.reg.stb(y, X, nsamp, alpha=0.5,
           sig2.shape=0.0, sig2.scale=0.0, nu.shape=2.0, nu.rate=2.0,
           alpha.a=1.0, alpha.b=1.0, 
           sig2.true=0.0, tau.true=0.0,
           burn=500, ortho=FALSE)

Arguments

y	An N dimensional vector of data.
X	An N x P dimensional design matrix.
nsamp	The number of MCMC samples saved.
alpha	The exponential power parameter; set <= 0 to estimate.
sig2.shape	Shape parameter for sig2's prior.
sig2.scale	Scale parameter for sig2's prior.
nu.shape	Shape parameter for nu's prior.
nu.rate	Rate parameter for nu's prior.
alpha.a	First shape parameter for alpha's prior.
alpha.b	Second shape parameter for alpha's prior.
sig2.true	The variance when it is known.
tau.true	The scale parameter tau when it is known.
burn	The number of MCMC samples discarded.
method	The method to use. Either "triangle" or "stable".
ortho	When the design matrix is orthogonal set to TRUE.
betaburn	Number of burn iterations when sampling beta.
extras	Only used for package testing.

Details

Bridge regression is a regularized regression in which the regression coefficient's prior is an exponential power distribution. Specifically, inference on the regression coefficient beta is made using the model

y = X β + ε, ε \sim N(0, σ^2 \; I),

p(β) \propto \exp(∑_j -(|β_j|/τ)^{α}).

The variance sig2, the scale parameter tau, and alpha may be estimated. This routine uses an inverse-gamma prior for sig2, a gamma prior for ν=τ^{-α}, and a beta prior for alpha, when estimating those parameters. That is

σ^2 \sim IG(\textmd{sig2.shape}, \textmd{sig2.scale}),

ν \sim Ga(\textmd{nu.shape}, \textmd{rate}=\textmd{nu.rate}),

α \sim Be(\textmd{alpha.a}, \textmd{alpha.b}).

The parameters sig2, tau, or alpha are taken to be known when sig2.true > 0, tau.true > 0, or alpha > 0 respectively.

When the design matrix is orthogonal set ortho=TRUE.

References

Nicholas G. Poslon, James G. Scott, and Jesse Windle. The Bayesian Bridge. http://arxiv.org/abs/1109.2279.

See Also

bridge.EM.

Examples

# Load the diabetes data...
data(diabetes, package="BayesBridge");
cov.name = colnames(diabetes$x);
y = diabetes$y;
X = diabetes$x;

# Center the data.
y = y - mean(y);
mX = colMeans(X);
for(i in 1:442){
    X[i,] = X[i,] - mX;
}

# Run the bridge regression when sig2 and tau are unknown.
gb = bridge.reg(y, X, nsamp=10000, alpha=0.5, 
     sig2.shape=0.0, sig2.scale=0.0, nu.shape=2.0, nu.rate=2.0);

Example output

[1] "Variable extras only for Package testing."
Bridge Regression (mix. of triangles): known alpha=0.5
Burn-in: 500, Num. Samples: 10000
Burn-in complete: 0.007207 sec. for 500 iterations.
Expect approx. 0.14414 sec. for 10000 samples.
Sampling complete: 0.137057 sec. for 10000 iterations.

BayesBridge documentation built on May 29, 2017, 10:40 a.m.