pqrBayes

Bayesian Penalized Quantile Regression

Bayesian regularized quantile regression utilizing sparse priors to promote exact sparsity leads to efficient Bayesian shrinkage estimation, variable selection and statistical inference. In this package, we have implemented robust Bayesian variable selection with spike-and-slab priors under high-dimensional linear regression models (Fan et al. (2024) and Ren et al. (2023)), and regularized quantile varying coefficient models (Zhou et al.(2023)). In
particular, valid robust Bayesian inferences under both models in the presence of heavy-tailed errors can be validated on finite samples. Additional models with spike-and-slab priors include robust Bayesian group LASSO and robust binary Bayesian LASSO (Fan and Wu (2025)). The Markov Chain Monte Carlo (MCMC) algorithms of the proposed and alternative models are implemented in C++.

How to install

• To install from github, run these two lines of code in R

install.packages("devtools")
devtools::install_github("cenwu/pqrBayes")

• Released versions of pqrBayes are available on CRAN , and can be installed within R via

install.packages("pqrBayes")

Example 1 (Robust Bayesian Inference for Sparse Linear Regression)

Data Generation for Linear Model

Data <- function(n,p,quant){
  sig1 = matrix(0,p,p)
  diag(sig1)=1
  for (i in 1: p)
  {
  for (j in 1: p)
  {
  sig1[i,j]=0.5^abs(i-j)
  }
 }
xx = MASS::mvrnorm(n,rep(0,p),sig1)
x = cbind(1,xx)
error=rt(n,2) -quantile(rt(n,2),probs = quant) # can also be changed to normal error for non-robust setting
beta = c(0,1,1.5,2,rep(0,p-3))
betaa = beta[-1]
y = x%*%beta+error
dat = list(y=y, x=xx, beta=betaa)
return(dat)
}

95% empirical coverage probabilities for linear regression coefficients

n=100; p=500; rep=1000;
quant = 0.5; # focus on median for Bayesian inference

CI_RBLSS = CI_RBL = CI_BLSS = CI_BL= matrix(0,rep,p)

for (h in 1:rep) {
dat = Data(n,p,quant)
y = dat$y
g = dat$x
coefficient = dat$beta

# an intercept not subject to regularization is automatically included by the package

# RBLSS: robust Bayesian LASSO with spike-and-slab priors (Ren et al., Biometrics, 2023)
fit = pqrBayes(g, y, u=NULL, d=NULL,e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL,robust = TRUE, sparse=TRUE, model = "linear", hyper=NULL,debugging=FALSE)
coverage = coverage(fit,coefficient,u.grid=NULL, model = "linear")

# RBL: Bayesian quantile LASSO (Li, Xi & Lin, Bayesian Analysis, 2010)
fit1 = pqrBayes(g, y, u=NULL,d=NULL, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL, robust = TRUE, sparse=FALSE, model = "linear", hyper=NULL,debugging=FALSE)
coverage1 = coverage(fit1,coefficient,u.grid=NULL, model = "linear")

# BLSS: Bayesian LASSO with spike-and-slab priors (Ren et al., Biometrics, 2023)
fit2 = pqrBayes(g, y, u=NULL,d=NULL, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL, robust = FALSE, sparse=TRUE, model= "linear", hyper=NULL,debugging=FALSE)
coverage2 = coverage(fit2,coefficient,u.grid=NULL, model = "linear")

# BL: Bayesian LASSO  (Park and Casella, JASA, 2008)
fit3 = pqrBayes(g, y, u=NULL,d=NULL, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL, robust = FALSE, sparse=FALSE, model = "linear", hyper=NULL,debugging=FALSE)
coverage3 = coverage(fit3,coefficient,u.grid=NULL, model = "linear")

CI_RBLSS[h,] = coverage
CI_RBL[h,]   = coverage1
CI_BLSS[h,]  = coverage2
CI_BL[h,]    = coverage3
cat("Replicate = ", h, "\n")

}
# the intercept has not been regularized
cp_RBLSS =  colMeans(CI_RBLSS)[1:3] # 95% empirical coverage probabilities for coefficients under the robust linear model
cp_BLSS  =  colMeans(CI_BLSS)[1:3]
cp_RBL   =  colMeans(CI_RBL)[1:3]
cp_BL    =  colMeans(CI_BL)[1:3]

Example 2 (Robust Bayesian Inference for Sparse Varying Coefficients)

Data Generation for the Varying Coefficient Model

Data <- function(n,p,quant){
  sig1 = matrix(0,p,p)
  diag(sig1)=1
  for (i in 1: p)
  {
  for (j in 1: p)
  {
  sig1[i,j]=0.5^abs(i-j)
  }
 }
x = MASS::mvrnorm(n,rep(0,p),sig1)
x = cbind(1,x)
error=rt(n,2) -quantile(rt(n,2),probs = quant)
u = runif(n,0.01,0.99)
gamma0 = 2+2*sin(u*2*pi)
gamma2 = -6*u*(1-u)
gamma1 = 2*exp(2*u-1)
gamma3 = -4*u^3
y = gamma1*x[,2] + gamma2*x[,3]  + gamma3*x[,4] + gamma0 + error
dat = list(y=y, u=u, x=x, gamma=cbind(gamma0,gamma1,gamma2,gamma3))
return(dat)
}

95% empirical coverage probabilities for sparse varying coefficients

n=250; p=100; # the actual dimension after basis expansion is 505
rep=200;
quant = 0.5; # focus on median for Bayesian inference

CI_BQRVCSS = CI_BQRVC = CI_BVCSS = CI_BVC= c()

for (h in 1:rep) {
dat = Data(n,p,quant)
y = dat$y
u = dat$u
x = dat$x
g = x[,-1]
kn=2
degree=2
u.grid = (1:200)*0.005
gamma_0_grid = 2+2*sin(2*u.grid*pi)
gamma_1_grid = 2*exp(2*u.grid-1)
gamma_2_grid = -6*u.grid*(1-u.grid)
gamma_3_grid = -4*u.grid^3
coefficient = cbind(gamma_0_grid,gamma_1_grid,gamma_2_grid,gamma_3_grid)

# a varying intercept not subject to regularization is automatically included by the package

# BQRVCSS: Bayesian regularized quantile VC model with spike-and-slab priors (Zhou et al., CSDA, 2023)
fit = pqrBayes(g, y, u, d=NULL,e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = list(kn=2,degree=2), robust = TRUE, sparse=TRUE, model = "VC", hyper=NULL,debugging=FALSE)
coverage = coverage(fit,coefficient,u.grid, model = "VC")

# BQRVC: Bayesian regularized quantile VC model (Zhou et al., CSDA, 2023)
fit1 = pqrBayes(g, y, u, d=NULL,e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = list(kn=2, degree=2), robust = TRUE, sparse=FALSE, model = "VC", hyper=NULL,debugging=FALSE)
coverage1 = coverage(fit1,coefficient,u.grid, model = "VC")

# BVCSS: Bayesian regularized VC model with spike-and-slab priors (Zhou et al., CSDA, 2023)
fit2 = pqrBayes(g, y, u, d = NULL,e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = list(kn=2, degree=2), robust = FALSE, sparse=TRUE, model = "VC", hyper=NULL,debugging=FALSE)
coverage2 = coverage(fit2,coefficient,u.grid, model = "VC")

# BVC: Bayesian regularized VC model (Zhou et al., CSDA, 2023)
fit3 = pqrBayes(g, y, u, d=NULL,e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = list(kn=2, degree=2), robust = FALSE, sparse=FALSE, model = "VC", hyper=NULL,debugging=FALSE)
coverage3 = coverage(fit3,coefficient,u.grid,model = "VC")

CI_BQRVCSS = rbind(CI_BQRVCSS,coverage)
CI_BQRVC   = rbind(CI_BQRVC,coverage1)
CI_BVCSS   = rbind(CI_BVCSS,coverage2)
CI_BVC     = rbind(CI_BVC,coverage3)
cat("Replicate = ", h, "\n")

}
# the varying intercept has not been regularized
cp_BQRVCSS =  colMeans(CI_BQRVCSS) # 95% empirical coverage probabilities for the varying coefficients under the default setting
cp_BQRVC   =  colMeans(CI_BQRVC)
cp_BVCSS   =  colMeans(CI_BVCSS)
cp_BVC     =  colMeans(CI_BVC)

Example 3 (Bayesian Shrinkage Estimation for Robust Bayesian Group LASSO)

Data Generation for Group LASSO

Data <- function(n,p,quant){
  sig1 = matrix(0,p,p)
  diag(sig1)=1
  for (i in 1: p)
  {
  for (j in 1: p)
  {
  sig1[i,j]=0.5^abs(i-j)
  }
 }
xx = MASS::mvrnorm(n,rep(0,p),sig1)
x = cbind(1,xx)
error=rt(n,2) -quantile(rt(n,2),probs = quant) # can also be changed to normal error for non-robust setting
beta = c(0,1,1.5,2,0,0,0,0.5,0.55,0.6,rep(0,p-9))
betaa = beta[-1]
y = x%*%beta+error
dat = list(y=y, x=xx, beta=betaa)
return(dat)
}

robust Bayesian shrinkage estimation under group LASSO

n=100; p=300;
quant = 0.5; # focus on median for Bayesian estimation
dat = Data(n,p,quant)
y = dat$y
g = dat$x
coefficient = dat$beta

# an intercept not subject to regularization is automatically included by the package

# RBGLSS, (Ren et al. Biometrics, 2023)
fit = pqrBayes(g, y, u=NULL,d=3, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL,robust = TRUE, sparse=TRUE, model = "group", hyper=NULL,debugging=FALSE)
estimation_1 = estimation.pqrBayes(fit,coefficient,model="group")
coeff_est_1 = estimation_1$coeff.est    
mse_1 = estimation_1$error$MSE

# RBGL: Bayesian Quantile Group LASSO (Li, Xi, & Lin, Bayesian Analysis, 2010)
fit1 = pqrBayes(g, y, u=NULL,d=3, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL, robust = TRUE, sparse=FALSE, model = "group", hyper=NULL,debugging=FALSE)
estimation_2 = estimation.pqrBayes(fit1,coefficient,model="group")
coeff_est_2 = estimation_2$coeff.est    
mse_2 = estimation_2$error$MSE 

# BGLSS: Bayesian group LASSO with spike-and-slab priors (Xu & Ghosh, Bayesian Analysis, 2015) 
fit2 = pqrBayes(g, y, u=NULL,d=3, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL, robust = FALSE, sparse=TRUE, model= "group", hyper=NULL,debugging=FALSE)
estimation_3 = estimation.pqrBayes(fit2,coefficient,model="group")
coeff_est_3 = estimation_3$coeff.est    
mse_3 = estimation_3$error$MSE    

# BGL: Bayesian group LASSO (Casella et al., Bayesian Analysis, 2010)
fit3 = pqrBayes(g, y, u=NULL,d=3, e=NULL,quant=quant, iterations=10000, burn.in = NULL, spline = NULL, robust = FALSE, sparse=FALSE, model = "group", hyper=NULL,debugging=FALSE)
estimation_4 = estimation.pqrBayes(fit3,coefficient,model="group")
coeff_est_4 = estimation_4$coeff.est    
mse_4 = estimation_4$error$MSE    

Methods

This package provides implementation for methods from

• Fan, K., Subedi, S., Yang, G., Lu, X., Ren, J. and Wu, C. (2024). Is Seeing Believing? A Practitioner’s Perspective on High-dimensional Statistical Inference in Cancer Genomics Studies. Entropy, 26(9),794
• Zhou, F., Ren, J., Ma, S. and Wu, C. (2023). The Bayesian Regularized Quantile Varying Coefficient Model. Computational Statistics & Data Analysis, 107808
• Ren, J., Zhou, F., Li, X., Ma, S., Jiang, Y., and Wu, C. (2023). Robust Bayesian variable selection for gene–environment interactions. Biometrics, 79(2), 684-694
• Fan, K. and Wu, C. (2025). A New Robust Binary Bayesian LASSO. Stat, 14(3), e70078.