RlogSafeBayesian Ridge Regression
Description
The function SBRidgeRlog (RlogSafeBayesian Ridge Regression) provides a Gibbs sampler together with the RlogSafeBayesian algorithm for Ridge regression models with varying variance.
Usage
1 2 
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

nIter 
Number of iterations, integer. Default 1100. 
burnIn 
Number of iterations for burnin, 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 SafeBayesian 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 burnin. 
$thin 
Number of iterations for thinning. 
$CMRlogEallen 
List of cumulative posteriorexpected posteriorrandomized logloss per eta. 
$eta.min 
Learning rate eta minimizing the cumulative posteriorexpected posteriorrandomized logloss. 
Author(s)
R. de Heide
References
de Heide, R. 2016. The SafeBayesian Lasso. Master Thesis, Leiden University.
de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes. 2009. Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: 375385.
Grunwald, P.D. 2012. chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 2931, 2012. Proceedings. 169183. Springer Berlin Heidelberg
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  rm(list=ls())
# Simulate data
x < runif(10, 1, 1) # 10 random uniform x's between 1 and 1
y < NULL
# for each x, an y that is 0 + Gaussian noise
for (i in 1:10) {
y[i] < 0 + rnorm(1, mean=0, sd=1/4)
}
plot(x,y)
## Not run:
# Let RlogSafeBayes learn the learning rate
sbobj < SBRidgeRlog(y, x, etaseq=c(1, 0.5, 0.25))
# eta
sbobj$eta.min
## End(Not run)

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