SBRidgeIlog: I-log-Safe-Bayesian Ridge Regression

In SafeBayes: Generalized and Safe-Bayesian Ridge and Lasso Regression

Description

The function SBRidgeIlog (I-log-Safe-Bayesian Ridge Regression) provides a Gibbs sampler together with the I-log-Safe-Bayesian algorithm for Ridge regression models with varying variance.

Usage

SBRidgeIlog(y, X = NULL, etaseq = 1, prior = NULL, nIter = 1100, burnIn = 100, 
	thin = 10, minAbsBeta = 1e-09, pIter = TRUE)

Arguments

y	Vector of outcome variables, numeric, NA allowed, length n.
X	Design matrix, numeric, dimension n x p, n >= 2.
etaseq	Vector of learning rates eta, numeric, 0 <= eta <= 1. Default 1.
prior	List containing the following elements prior$varE: prior for the variance parameter σ^2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chi-square distribution. Default (0,0). prior$varBR: prior for the variance of the Gaussian prior for the coefficients beta, with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chi-square distribution. Default (0,0).
nIter	Number of iterations, integer. Default 1100.
burnIn	Number of iterations for burn-in, integer. Default 100.
thin	Number of iterations for thinning, integer. Default 10.
minAbsBeta	Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default 10^-9.
pIter	Print iterations, logical. Default TRUE.

Details

Details on generalized Bayesian regression can be found in (de Heide, 2016). The implementation of the Gibbs sampler is based on the BLR package of (de los Campos et al., 2009).

The Safe-Bayesian algorithm was proposed by Grunwald (2012) as a method to learn the learning rate for the generalized posterior to deal with model misspecification.

Value

$y	Vector of original outcome variables.
$mu	Posterior mean of the intercept.
$varE	Posterior mean of of the variance.
$yHat	Posterior mean of mu + X*beta + epsilon.
$SD.yHat	Corresponding standard deviation.
$whichNa	Vector with indices of missing values of y.
$fit$pD	Estimated number of effective parameters.
$fit$DIC	Deviance Information Criterion.
$bR	Posterior mean of beta.
$SD.bR	Corresponding standard deviation.
$prior	List containing the priors used.
$nIter	Number of iterations.
$burnIn	Number of iterations for burn-in.
$thin	Number of iterations for thinning.
$CEallen	List of cumulative eta-in-model-log-loss per eta.
$eta.min	Learning rate eta minimizing the cumulative eta-in-model-log-loss.

Author(s)

R. de Heide

References

de Heide, R. 2016. The Safe-Bayesian Lasso. Master Thesis, Leiden University.

de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes. 2009. Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: 375-385.

Grunwald, P.D. 2012. chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings. 169-183. Springer Berlin Heidelberg

Examples

rm(list=ls())
# Simulate data
x <- runif(10, -1, 1) # 10 random uniform x's between -1 and 1
y <- NULL

# for each x, an y that is 0 + Gaussian noise
for (i in 1:10) {
  y[i] <-  0 + rnorm(1, mean=0, sd=1/4) 
  }

plot(x,y)

## Not run: 
# Let I-log-SafeBayes learn the learning rate 
sbobj <- SBRidgeIlog(y, x, etaseq=c(1, 0.5, 0.25))

# eta 
sbobj$eta.min
## End(Not run)

SafeBayes documentation built on May 1, 2019, 9:23 p.m.