Package SafeBayes. October 20, 2016

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1 Type Package Package SafeBayes October 20, 2016 Title Generalized and Safe-Bayesian Ridge and Lasso Regression Version 1.1 Date Depends R (>= 3.1.2), stats Description Functions for Generalized and Safe- Bayesian Ridge and Lasso Regression models with both fixed and varying variance. License GPL-2 NeedsCompilation yes LazyLoad true Author Rianne de Heide [aut, cre], Gustavo de los Campos [ctb], Paulino Perez Rodriguez [ctb], Bob Wheeler [ctb] Maintainer Rianne de Heide <r.de.heide@cwi.nl> Repository CRAN Date/Publication :04:18 R topics documented: GBLasso GBLassoFV GBRidge GBRidgeFV metroplambda rinvgauss SBLassoIlog SBLassoISq SBLassoRlog SBLassoRSq SBRidgeIlog SBRidgeISq

2 2 GBLasso SBRidgeRlog SBRidgeRSq Index 34 GBLasso Generalized Bayesian Lasso Description The function GBLasso (Generalized Bayesian Lasso) provides a Gibbs sampler to sample the posterior of generalized Bayesian lasso regression models with learning rate η. Usage GBLasso(y, X = NULL, eta = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, weights = NULL, piter = TRUE) Arguments y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. eta Learning rate η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$lambda: prior for the penalty parameter λ with three items. $value Initial value for λ. Default 50 $type Can be fixed : initial value is used as fixed penalty parameter or random, in which case a prior for λ is specified. Default random. For a Gamma prior on λ 2 : $shape for shape parameter and $rate for the rate parameter; for a Beta prior on λ: $shape1, $shape2 and $max for λ proportional to Beta(λ/max, shape1, shape2). Default: Gamma(0, 0). niter Number of iterations, integer. Default burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta weights piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1. Print iterations, logical. Default TRUE.

3 GBLasso 3 Details Value Details on the generalized Bayesian lasso can be found in Chapter 2 of (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009). Several authors have brought forward the idea of equipping Bayesian updating with a learning rate η, resulting in an η-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the η-generalized posterior, the likelihood is raised to the power η in order to trade the relative weight of the likelihood and the prior, where η = 1 corresponds to standard Bayes. $y Vector of original outcome variables. $weights $mu $vare $yhat $SD.yHat Vector of weights. Posterior mean of the intercept. Posterior mean of of the variance. Posterior mean of mu + X*beta + epsilon. Corresponding standard deviation. $whichna Vector with indices of missing values of y. $fit$pd $fit$dic Estimated number of effective parameters. Deviance Information Criterion. $lambda Posterior mean of λ. $bl Posterior mean of β. $SD.bL Corresponding standard deviation. $tau2 Posterior mean of τ 2. $prior $niter $burnin $thin List containing the priors used. Number of iterations. $eta Learning rate η. Author(s) R. de Heide References Number of iterations for burn-in. Number of iterations for thinning. de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Audibert, J.Y Bayesian generalized double pareto shrinkage. Statistica Sinica

4 4 GBLassoFV Catoni, O PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS. Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings Springer Berlin Heidelberg McAllester, D PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21 Vovk, V.G Aggregating strategies. In Proc. COLT Zhang, T Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press Examples rm(list=ls()) library(safebayes) # Simulate data x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1 y <- NULL # for each x, a y that is 0 + Gaussian noise for (i in 1:100) { y[i] <- 0 + rnorm(1, mean=0, sd=1/4) } # Now sample 100 zero's and ones (coin toss) cointoss <- sample(0:1, 100, replace=true) # indices of the ones indices <- which(cointoss==1) # we replace x and y with (0,0) for the indices the cointoss # landed tail (1) x[indices] <- 0 y[indices] <- 0 plot(x,y) # Determine the generalized posterior for eta = 0.25 obj <- GBLasso(y, x, eta=0.25) # posterior means of the coefficients beta and intercept mu betafour <- obj$bl mufour <- obj$mu GBLassoFV Generalized Bayesian Lasso with fixed variance

5 GBLassoFV 5 Description Usage The function GBLassoFV (Generalized Bayesian Lasso with Fixed Variance) provides a Gibbs sampler to sample the posterior of generalized Bayesian lasso regression models with fixed variance and with learning rate η. GBLassoFV(y, X = NULL, sigma2 = NULL, eta = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, weights = NULL, piter = TRUE) Arguments Details y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. sigma2 Fixed variance parameter σ 2, numeric. Default NULL, in which case the variance will be estimated from the data. eta Learning rate η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$lambda: prior for the penalty parameter λ with three items. $value Initial value for λ. Default 50 $type Can be fixed : initial value is used as fixed penalty parameter or random, in which case a prior for λ is specified. Default random. For a Gamma prior on λ 2 : $shape for shape parameter and $rate for the rate parameter; for a Beta prior on λ: $shape1, $shape2 and $max for λ proportional to Beta(λ/max, shape1, shape2). Default: Gamma(0, 0). niter Number of iterations, integer. Default burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta weights piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1. Print iterations, logical. Default TRUE. Details on the generalized Bayesian lasso can be found in Chapter 2 of (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009). Several authors have brought forward the idea of equipping Bayesian updating with a learning rate η, resulting in an η-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal

6 6 GBLassoFV Value with model misspecification. In the η-generalized posterior, the likelihood is raised to the power η in order to trade the relative weight of the likelihood and the prior, where η = 1 corresponds to standard Bayes. $y Vector of original outcome variables. $weights $mu $vare $yhat $SD.yHat Vector of weights. Posterior mean of the intercept. Posterior mean of of the variance. Posterior mean of mu + X*beta + epsilon. Corresponding standard deviation. $whichna Vector with indices of missing values of y. $fit$pd $fit$dic Estimated number of effective parameters. Deviance Information Criterion. $lambda Posterior mean of λ. $bl Posterior mean of β. $SD.bL Corresponding standard deviation. $tau2 Posterior mean of τ 2. $prior $niter $burnin $thin List containing the priors used. Number of iterations. $eta Learning rate η. Author(s) R. de Heide References Number of iterations for burn-in. Number of iterations for thinning. de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Audibert, J.Y Bayesian generalized double pareto shrinkage. Statistica Sinica Catoni, O PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS. Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings Springer Berlin Heidelberg McAllester, D PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21

7 GBRidge 7 Vovk, V.G Aggregating strategies. In Proc. COLT Zhang, T Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press Examples rm(list=ls()) library(safebayes) # Simulate data x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1 y <- NULL # for each x, a y that is 0 + Gaussian noise for (i in 1:100) { y[i] <- 0 + rnorm(1, mean=0, sd=1/4) } # Now sample 100 zero's and ones (coin toss) cointoss <- sample(0:1, 100, replace=true) # indices of the ones indices <- which(cointoss==1) # we replace x and y with (0,0) for the indices the cointoss # landed tail (1) x[indices] <- 0 y[indices] <- 0 plot(x,y) # Determine the generalized posterior for eta = 0.25 obj <- GBLassoFV(y, x, eta=0.25) # posterior means of the coefficients beta and intercept mu betafour <- obj$bl mufour <- obj$mu GBRidge Generalized Bayesian Ridge Regression Description Usage The function GBRidge (Generalized Bayesian Ridge Regression) provides a Gibbs sampler to sample the posterior of generalized Bayesian ridge regression models with learning rate η. GBRidge(y, X = NULL, eta = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, weights = NULL, piter = TRUE)

8 8 GBRidge Arguments Details Value y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. eta Learning rate η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$varbr: prior for the variance of the Gaussian prior for the coefficients β, 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 burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta weights piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1. Print iterations, logical. Default TRUE. Details on generalized Bayesian regression can be found in (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009). Several authors have brought forward the idea of equipping Bayesian updating with a learning rate η, resulting in an η-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the η-generalized posterior, the likelihood is raised to the power η in order to trade the relative weight of the likelihood and the prior, where η = 1 corresponds to standard Bayes. $y Vector of original outcome variables. $weights $mu $vare $yhat $SD.yHat Vector of weights. Posterior mean of the intercept. Posterior mean of of the variance. Posterior mean of mu + X*beta + epsilon. Corresponding standard deviation. $whichna Vector with indices of missing values of y. $fit$pd $fit$dic Estimated number of effective parameters. Deviance Information Criterion.

9 GBRidge 9 $br Posterior mean of β. $SD.bR $prior $niter $burnin $thin Corresponding standard deviation. List containing the priors used. Number of iterations. $eta Learning rate η. Author(s) R. de Heide References Number of iterations for burn-in. Number of iterations for thinning. de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Audibert, J.Y Bayesian generalized double pareto shrinkage. Statistica Sinica Catoni, O PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS. Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings Springer Berlin Heidelberg McAllester, D PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21 Vovk, V.G Aggregating strategies. In Proc. COLT Zhang, T Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press Examples rm(list=ls()) library(safebayes) # Simulate data x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1 y <- NULL # for each x, a y that is 0 + Gaussian noise for (i in 1:100) { y[i] <- 0 + rnorm(1, mean=0, sd=1/4) } # Now sample 100 zero's and ones (coin toss) cointoss <- sample(0:1, 100, replace=true)

10 10 GBRidgeFV # indices of the ones indices <- which(cointoss==1) # we replace x and y with (0,0) for the indices the cointoss # landed tail (1) x[indices] <- 0 y[indices] <- 0 plot(x,y) # Determine the generalized posterior for eta = 0.25 obj <- GBLasso(y, x, eta=0.25) # posterior means of the coefficients beta and intercept mu betafour <- obj$bl mufour <- obj$mu GBRidgeFV Generalized Bayesian Ridge Regression with fixed variance Description Usage The function GBRidgeFV (Generalized Bayesian Ridge Regression with Fixed Variance) provides a Gibbs sampler to sample the posterior of generalized Bayesian ridge regression models with fixed variance and with learning rate η. GBRidgeFV(y, X = NULL, sigma2 = NULL, eta = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, weights = NULL, piter = TRUE) Arguments y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. sigma2 Fixed variance parameter σ 2, numeric. Default NULL, in which case the variance will be estimated from the data. eta Learning rate η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$varbr: prior for the variance of the Gaussian prior for the coefficients β, 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.

11 GBRidgeFV 11 burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta weights piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1. Print iterations, logical. Default TRUE. Details Details on generalized Bayesian regression can be found in (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009). Several authors have brought forward the idea of equipping Bayesian updating with a learning rate η, resulting in an η-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the η-generalized posterior, the likelihood is raised to the power η in order to trade the relative weight of the likelihood and the prior, where η = 1 corresponds to standard Bayes. Value $y Vector of original outcome variables. $weights Vector of weights. $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 β. $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. $eta Learning rate η. Author(s) R. de Heide

12 12 GBRidgeFV References de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Audibert, J.Y Bayesian generalized double pareto shrinkage. Statistica Sinica Catoni, O PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS. Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings Springer Berlin Heidelberg McAllester, D PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21 Vovk, V.G Aggregating strategies. In Proc. COLT Zhang, T Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press Examples rm(list=ls()) library(safebayes) # Simulate data x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1 y <- NULL # for each x, a y that is 0 + Gaussian noise for (i in 1:100) { y[i] <- 0 + rnorm(1, mean=0, sd=1/4) } # Now sample 100 zero's and ones (coin toss) cointoss <- sample(0:1, 100, replace=true) # indices of the ones indices <- which(cointoss==1) # we replace x and y with (0,0) for the indices the cointoss # landed tail (1) x[indices] <- 0 y[indices] <- 0 plot(x,y) # Determine the generalized posterior for eta = 0.25 obj <- GBLassoFV(y, x, eta=0.25) # posterior means of the coefficients beta and intercept mu betafour <- obj$bl mufour <- obj$mu

13 metroplambda 13 metroplambda Metropolis-Hastings algorithm to sample lambda with a Beta prior for the Bayesian Lasso Description Metropolis-Hastings algorithm to sample lambda with a Beta prior from (de los Campos et al., 2009) for the Bayesian Lasso regression model. Usage metroplambda(tau2, lambda, shape1 = 1.2, shape2 = 1.2, max = 200, ncp = 0) Arguments tau2 lambda shape1 shape2 max ncp Latent parameter tau-squared to form the Laplace prior on the coefficients of the Lasso from a normal-mixture. Initial value for lambda. First shape parameter for the Beta distribution. Second shape parameter for the Beta distribution. Maximum value of lambda. Dummy parameter. Details Metropolis-Hastings algorithm to sample lambda with a Beta prior from (de los Campos et al., 2009) for the Bayesian Lasso regression model. Value Returns a value for lambda to use in the Gibbs samplers of the functions in the SafeBayes package. Author(s) Copied from (de los Campos et al., 2009). References de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182:

14 14 rinvgauss Examples rm(list=ls()) library(safebayes) tau2 <- 1/4 lambda <- 50 metroplambda(tau2=tau2, lambda=lambda) rinvgauss The inverse Gaussian and Wald distributions Description Random generator for the inverse Gaussian and Wald distributions. Usage rinvgauss(n, nu, lambda) Arguments n nu lambda vector of numbers of observations vector real and non-negative parameter the Wald distribution results when nu=1 vector real and non-negative parameter Details This function is copied from the SuppDists package by Bob Wheeler. I have copied this function here, because the SuppDists pacakge is no longer maintained, so that I can maintain the rinvgauss function for use in the functions in this package. Probability functions: ] λ f(x, ν, λ) = [ λ 2πx 3 exp (x ν)2 2ν 2 the density x F (x, ν, λ) = Φ [ λ ( x ) ] [ λ ( x ) ] x ν 1 + e 2λ/ν Φ x ν + 1 the distribution function where Φ[] is the standard normal distribution function. The calculations are accurate to at least seven significant figures over an extended range - much larger than that of any existing tables. We have tested them for λ/ν = 10e 20, and λ/ν = 10e4. Accessible tables are those of Wassan and Roy (1969), which unfortunately, are sometimes good to only two significant digits. Much better tables are available in an expensive CRC Handbook (1989), which are accurate to at least 7 significant digits for λ/ν 0.02 to λ/ν 4000.

15 rinvgauss 15 Value These are first passage time distributions of Brownian motion with positive drift. See Whitmore and Seshadri (1987) for a heuristic derivation. The Wald (1947) form represents the average sample number in sequential analysis. The distribution has a non-monotonic failure rate, and is of considerable interest in lifetime studies: Ckhhikara and Folks (1977). A general reference is Seshadri (1993). This is an extremely difficult distribution to treat numerically, and it would not have been possible without some extraordinary contributions. An elegant derivation of the distribution function is to be found in Shuster (1968). The first such derivation seems to be that of Zigangirov (1962), which because of its inaccessibility, the author has not read. The method of generating random numbers is due to Michael, Schucany, and Haas (1976). The approximation of Whitmore and Yalovsky (1978) makes it possible to find starting values for inverting the distribution. All three papers are short, elegant, and non- trivial. The output values conform to the output from other such functions in R. rinvgauss() generates random numbers. Author(s) Bob Wheeler <bwheelerg@gmail.com> References Ckhhikara, R.S. and Folks, J.L. (1977) The inverse Gaussian distribution as a lifetime model. Technometrics CRC Handbook. (1989). Percentile points of the inverse Gaussian distribution. J.A. Koziol (ed.) Boca Raton, FL. Michael, J.R., Schucany, W.R. and Haas, R.W. (1976). Generating random variates using transformations with multiple roots. American Statistician Seshadri, V. (1993). The inverse Gaussian distribution. Clarendon, Oxford Shuster, J. (1968). On the inverse Gaussian distribution function. Jour. Am. Stat. Assoc Wasan, M.T. and Roy, L.K. (1969). Tables of inverse Gaussian percentage points. Technometrics Wald, A. (1947). Sequential analysis. Wiley, NY Whitmore, G.A. and Seshadri, V. (1987). A heuristic derivation of the inverse Gaussian distribution. American Statistician Whitmore, G.A. and Yalovsky, M. (1978). A normalizing logarithmic transformation for inverse Gaussian random variables. Technometrics Zigangirov, K.S. (1962). Expression for the Wald distribution in terms of normal distribution. Radiotech.Electron Examples rinvgauss(1, 1, 16)

16 16 SBLassoIlog SBLassoIlog I-log-Safe-Bayesian Lasso Description The function SBLassoIlog (I-log-Safe-Bayesian Lasso) provides a Gibbs sampler together with the I-log-Safe-Bayesian algorithm for Bayesian lasso regression models with varying variance. Usage SBLassoIlog(y, X = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE) Arguments Details y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$lambda: prior for the penalty parameter λ with three items. $value Initial value for λ. Default 50 $type Can be fixed : initial value is used as fixed penalty parameter or random, in which case a prior for λ is specified. Default random. For a Gamma prior on λ 2 : $shape for shape parameter and $rate for the rate parameter; for a Beta prior on λ: $shape1, $shape2 and $max for λ proportional to Beta(λ/max, shape1, shape2). Default: Gamma(0, 0). niter Number of iterations, integer. Default burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Print iterations, logical. Default TRUE. Details on the Safe-Bayesian lasso can be found in Chapter 2 of (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.

17 SBLassoIlog 17 Value $y Vector of original outcome variables. $mu $vare $yhat $SD.yHat Posterior mean of the intercept. Posterior mean of of the variance. Posterior mean of mu + X*beta + epsilon. Corresponding standard deviation. $whichna Vector with indices of missing values of y. $fit$pd $fit$dic Estimated number of effective parameters. Deviance Information Criterion. $lambda Posterior mean of λ. $bl Posterior mean of β. $SD.bL Corresponding standard deviation. $tau2 Posterior mean of τ 2. $prior $niter $burnin $thin List containing the priors used. Number of iterations. Number of iterations for burn-in. Number of iterations for thinning. $CEallen List of cumulative eta-in-model-log-loss per η. $eta.min Author(s) R. de Heide References Learning rate η minimizing the cumulative eta-in-model-log-loss. de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings 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)

18 18 SBLassoISq } plot(x,y) ## Not run: # Let I-log-SafeBayes learn the learning rate sbobj <- SBLassoIlog(y, x, etaseq=c(1, 0.5, 0.25)) # eta sbobj$eta.min ## End(Not run) SBLassoISq I-square-Safe-Bayesian Lasso Description Usage The function SBLassoISq (I-square-Safe-Bayesian Lasso) provides a Gibbs sampler together with the I-square-Safe-Bayesian algorithm for Bayesian lasso regression models with fixed variance. SBLassoISq(y, X = NULL, sigma2 = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE) Arguments y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. sigma2 Fixed variance parameter σ 2, numeric. Default NULL, in which case the variance will be estimated from the data per addition of new data point in the Safe- Bayesian algorithm. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$lambda: prior for the penalty parameter λ with three items. $value Initial value for λ. Default 50 $type Can be fixed : initial value is used as fixed penalty parameter or random, in which case a prior for λ is specified. Default random. For a Gamma prior on λ 2 : $shape for shape parameter and $rate for the rate parameter; for a Beta prior on λ: $shape1, $shape2 and $max for λ proportional to Beta(λ/max, shape1, shape2). Default: Gamma(0, 0). niter Number of iterations, integer. Default 1100.

19 SBLassoISq 19 burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Print iterations, logical. Default TRUE. Details Details on the Safe-Bayesian lasso can be found in Chapter 2 of (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. $lambda Posterior mean of λ. $bl Posterior mean of β. $SD.bL Corresponding standard deviation. $tau2 Posterior mean of τ 2. $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-square-loss per η. $eta.min Learning rate η minimizing the cumulative eta-in-model-square-loss. Author(s) R. de Heide

20 20 SBLassoRlog References de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings 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-square-SafeBayes learn the learning rate sbobj <- SBLassoISq(y, x, etaseq=c(1, 0.5, 0.25)) # eta sbobj$eta.min ## End(Not run) SBLassoRlog R-log-Safe-Bayesian Lasso Description The function SBLassoRlog (R-log-Safe-Bayesian Lasso) provides a Gibbs sampler together with the R-log-Safe-Bayesian algorithm for Bayesian lasso regression models. Usage SBLassoRlog(y, X = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE)

21 SBLassoRlog 21 Arguments Details Value y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$lambda: prior for the penalty parameter λ with three items. $value Initial value for λ. Default 50 $type Can be fixed : initial value is used as fixed penalty parameter or random, in which case a prior for λ is specified. Default random. For a Gamma prior on λ 2 : $shape for shape parameter and $rate for the rate parameter; for a Beta prior on λ: $shape1, $shape2 and $max for λ proportional to Beta(λ/max, shape1, shape2). Default: Gamma(0, 0). niter Number of iterations, integer. Default burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Print iterations, logical. Default TRUE. Details on the Safe-Bayesian lasso can be found in Chapter 2 of (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. $y Vector of original outcome variables. $mu $vare $yhat $SD.yHat Posterior mean of the intercept. Posterior mean of of the variance. Posterior mean of mu + X*beta + epsilon. Corresponding standard deviation. $whichna Vector with indices of missing values of y. $fit$pd $fit$dic Estimated number of effective parameters. Deviance Information Criterion. $lambda Posterior mean of λ. $bl Posterior mean of β. $SD.bL Corresponding standard deviation.

22 22 SBLassoRlog $tau2 Posterior mean of τ 2. $prior $niter $burnin $thin List containing the priors used. Number of iterations. Number of iterations for burn-in. Number of iterations for thinning. $CMRlogEallen List of cumulative posterior-expected posterior-randomized log-loss per η. $eta.min Learning rate η minimizing the cumulative posterior-expected posterior-randomized log-loss. Author(s) R. de Heide References de Heide, R The Safe-Bayesian Lasso. Master Thesis, Leiden University. de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings Springer Berlin Heidelberg Examples rm(list=ls()) # Simulate data x <- runif(10, -1, 1) # 10 random uniform x's between -1 and 1 y <- NULL # for each x, an y that is 0 + Gaussian noise for (i in 1:10) { y[i] <- 0 + rnorm(1, mean=0, sd=1/4) } plot(x,y) ## Not run: # Let R-log-SafeBayes learn the learning rate sbobj <- SBLassoRlog(y, x, etaseq=c(1, 0.5, 0.25)) # eta sbobj$eta.min ## End(Not run)

23 SBLassoRSq 23 SBLassoRSq R-square-Safe-Bayesian Lasso Description The function SBLassoRSq (R-square-Safe-Bayesian Lasso) provides a Gibbs sampler together with the R-square-Safe-Bayesian algorithm for Bayesian lasso regression models with fixed variance. Usage SBLassoRSq(y, X = NULL, sigma2 = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE) Arguments Details y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. sigma2 Fixed variance parameter σ 2, numeric. Default NULL, in which case the variance will be estimated from the data per addition of new data point in the Safe- Bayesian algorithm. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$lambda: prior for the penalty parameter λ with three items. $value Initial value for λ. Default 50 $type Can be fixed : initial value is used as fixed penalty parameter or random, in which case a prior for λ is specified. Default random. For a Gamma prior on λ 2 : $shape for shape parameter and $rate for the rate parameter; for a Beta prior on λ: $shape1, $shape2 and $max for λ proportional to Beta(λ/max, shape1, shape2). Default: Gamma(0, 0). niter Number of iterations, integer. Default burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Print iterations, logical. Default TRUE. Details on the Safe-Bayesian lasso can be found in Chapter 2 of (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.

24 24 SBLassoRSq Value $y Vector of original outcome variables. $mu $vare $yhat $SD.yHat Posterior mean of the intercept. Posterior mean of of the variance. Posterior mean of mu + X*beta + epsilon. Corresponding standard deviation. $whichna Vector with indices of missing values of y. $fit$pd $fit$dic Estimated number of effective parameters. Deviance Information Criterion. $lambda Posterior mean of λ. $bl Posterior mean of β. $SD.bL Corresponding standard deviation. $tau2 Posterior mean of τ 2. $prior $niter $burnin $thin List containing the priors used. Number of iterations. Number of iterations for burn-in. Number of iterations for thinning. $CMRSEallen List of cumulative posterior-expected posterior-randomized square-loss per η. $eta.min Author(s) R. de Heide References Learning rate η minimizing the cumulative posterior-expected posterior-randomized square-loss. de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings 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

25 SBRidgeIlog 25 for (i in 1:10) { y[i] <- 0 + rnorm(1, mean=0, sd=1/4) } plot(x,y) ## Not run: # Let R-square-SafeBayes learn the learning rate sbobj <- SBLassoRSq(y, x, etaseq=c(1, 0.5, 0.25)) # eta sbobj$eta.min ## End(Not run) SBRidgeIlog I-log-Safe-Bayesian Ridge Regression Description Usage The function SBRidgeIlog (I-log-Safe-Bayesian Ridge Regression) provides a Gibbs sampler together with the I-log-Safe-Bayesian algorithm for Ridge regression models with varying variance. SBRidgeIlog(y, X = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE) Arguments y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$varbr: prior for the variance of the Gaussian prior for the coefficients β, 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 burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default Print iterations, logical. Default TRUE.

26 26 SBRidgeIlog 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 β. $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 η. $eta.min Learning rate η minimizing the cumulative eta-in-model-log-loss. Author(s) R. de Heide References de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings Springer Berlin Heidelberg

27 SBRidgeISq 27 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$eta.min ## End(Not run) SBRidgeISq I-square-Safe-Bayesian Ridge Regression Description Usage The function SBRidgeISq (I-square-Safe-Bayesian Ridge Regression) provides a Gibbs sampler together with the I-square-Safe-Bayesian algorithm for Ridge regression models with fixed variance. SBRidgeISq(y, X = NULL, sigma2 = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE) Arguments y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. sigma2 Fixed variance parameter σ 2, numeric. Default NULL, in which case the variance will be estimated from the data per addition of new data point in the Safe- Bayesian algorithm. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0).

28 28 SBRidgeISq prior$varbr: prior for the variance of the Gaussian prior for the coefficients β, 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 burnin Number of iterations for burn-in, integer. Default 100. thin Number of iterations for thinning, integer. Default 10. minabsbeta piter Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default 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 β. $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-square-loss per η. $eta.min Learning rate η minimizing the cumulative eta-in-model-square-loss. Author(s) R. de Heide

29 SBRidgeRlog 29 References de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings 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-square-SafeBayes learn the learning rate sbobj <- SBRidgeISq(y, x, etaseq=c(1, 0.5, 0.25)) # eta sbobj$eta.min ## End(Not run) SBRidgeRlog R-log-Safe-Bayesian Ridge Regression Description The function SBRidgeRlog (R-log-Safe-Bayesian Ridge Regression) provides a Gibbs sampler together with the R-log-Safe-Bayesian algorithm for Ridge regression models with varying variance. Usage SBRidgeRlog(y, X = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE)

30 30 SBRidgeRlog Arguments y Vector of outcome variables, numeric, NA allowed, length n. X Design matrix, numeric, dimension n p, n 2. etaseq Vector of learning rates η, numeric, 0 η 1. Default 1. prior List containing the following elements prior$vare: prior for the variance parameter σ 2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chisquare distribution. Default (0, 0). prior$varbr: prior for the variance of the Gaussian prior for the coefficients β, 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 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 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 β. $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. $CMRlogEallen List of cumulative posterior-expected posterior-randomized log-loss per η. $eta.min Learning rate η minimizing the cumulative posterior-expected posterior-randomized log-loss.

31 SBRidgeRSq 31 Author(s) R. de Heide References de Heide, R 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 Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: Grunwald, P.D chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, Proceedings 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 R-log-SafeBayes learn the learning rate sbobj <- SBRidgeRlog(y, x, etaseq=c(1, 0.5, 0.25)) # eta sbobj$eta.min ## End(Not run) SBRidgeRSq R-square Safe-Bayesian Ridge Regression Description Usage The function SBRidgeRSq (R-square-Safe-Bayesian Ridge Regression) provides a Gibbs sampler together with the R-square-Safe-Bayesian algorithm for Ridge regression models with fixed variance. SBRidgeRSq(y, X = NULL, sigma2 = NULL, etaseq = 1, prior = NULL, niter = 1100, burnin = 100, thin = 10, minabsbeta = 1e-09, piter = TRUE)