pqrBayes

Bayesian Penalized Quantile Regression

Bayesian regularized quantile regression utilizing two major classes of shrinkage priors (the spike-and-slab priors and the horseshoe family of priors) leads to efficient Bayesian shrinkage estimation, variable selection and valid 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)). Besides, robust sparse Bayesian regression with the horseshoe family of (horseshoe, horseshoe+ and regularized horseshoe) priors has also been implemented and yielded valid inference results under heavy-tailed model errors (Fan et al.(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 with Spike-and-Slab Priors)

Data Generation for Sparse 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 sparse 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 ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL ,robust = TRUE, prior = "SS", 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,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "Laplace", 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 ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL ,robust = FALSE, prior = "SS", model = "linear", hyper=NULL,debugging=FALSE)
coverage2 = coverage(fit2,coefficient,u.grid=NULL, model = "linear")

# Bayesian LASSO  (Park and Casella, JASA, 2008)
fit3 = pqrBayes(g, y,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "Laplace", 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_RBL    =  colMeans(CI_RBL)[1:3]
cp_BLSS   =  colMeans(CI_BLSS)[1:3] # 95% empirical coverage probabilities for coefficients under the non-robust linear model
cp_BL     =  colMeans(CI_BL)[1:3]

Bayesian shrinkage estimaton via robust Bayesian LASSO with spike-and-slab priors

n=100; p=500;
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

# RBLSS: robust Bayesian LASSO with spike-and-slab priors (Ren et al., Biometrics, 2023)

fit = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL ,robust = TRUE, prior = "SS", model = "linear", hyper=NULL,debugging=FALSE)

fit$coefficients$GS.beta # posterior samples for regression coefficients
estimation_1 = estimation.pqrBayes(fit,coefficient,model="linear")
coeff_est_1 = estimation_1$coeff.est    
mse_1 = estimation_1$error$MSE

# RBL: Bayesian quantile LASSO (Li, Xi & Lin, Bayesian Analysis, 2010)

fit1 = pqrBayes(g, y,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "Laplace", model = "linear", hyper=NULL,debugging=FALSE)

fit1$coefficients$GS.beta # posterior samples for regression coefficients
estimation_2 = estimation.pqrBayes(fit1,coefficient,model="linear")
coeff_est_2 = estimation_2$coeff.est    
mse_2 = estimation_2$error$MSE

# BLSS: Bayesian LASSO with spike-and-slab priors (Ren et al., Biometrics, 2023)

fit2 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL ,robust = FALSE, prior = "SS", model = "linear", hyper=NULL,debugging=FALSE)

estimation_3 = estimation.pqrBayes(fit2,coefficient,model="linear")
coeff_est_3 = estimation_3$coeff.est
mse_3 = estimation_3$error$MSE

# Bayesian LASSO  (Park and Casella, JASA, 2008)

fit3 = pqrBayes(g, y,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "Laplace", model = "linear", hyper=NULL,debugging=FALSE)

estimation_4 = estimation.pqrBayes(fit3,coefficient,model="linear")
coeff_est_4 = estimation_4$coeff.est
mse_4 = estimation_4$error$MSE

Example 2 (Robust Bayesian Inference for Sparse Linear Regression with the Horseshoe Family of Priors)

Data Generation for Sparse 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_RBHS = CI_RBHS_plus = CI_RBRHS = CI_BHS= CI_BHS_plus = CI_BRHS =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


# RBHS: robust Bayesian Regression with horseshoe priors
fit1 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "HS", model = "linear", hyper=NULL,debugging=FALSE)
coverage1 = coverage(fit1,coefficient,u.grid=NULL, model = "linear")

# RBHS+: robust Bayesian Regression with horseshoe plus priors
fit2 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "HS+", model = "linear", hyper=NULL,debugging=FALSE)
coverage2 = coverage(fit2,coefficient,u.grid=NULL, model = "linear")

# RBRHS: robust Bayesian Regression with regularized horseshoe priors
fit3 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "RHS", model = "linear", hyper=NULL,debugging=FALSE)
coverage3 = coverage(fit3,coefficient,u.grid=NULL, model = "linear")

# BHS: Bayesian Regression with horseshoe priors
fit4 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "HS", model = "linear", hyper=NULL,debugging=FALSE)
coverage4=coverage(fit4,coefficient,u.grid=NULL, model = "linear")

# BHS+: Bayesian Regression with horseshoe plus priors
fit5 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "HS+", model = "linear", hyper=NULL,debugging=FALSE)
coverage5 = coverage(fit5,coefficient,u.grid=NULL, model = "linear")

# BRHS: Bayesian Regression with regularized horseshoe priors
fit6 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "RHS", model = "linear", hyper=NULL,debugging=FALSE)
coverage6 = coverage(fit6,coefficient,u.grid=NULL, model = "linear")


CI_RBHS[h,]   = coverage1
CI_RBHS_plus[h,]  = coverage2
CI_RBRHS[h,]   = coverage3
CI_BHS[h,]    = coverage4
CI_BHS_plus[h,]   = coverage5
CI_BRHS[h,]  = coverage6

cat("Replicate = ", h, "\n")

}
# the intercept has not been regularized

cp_RBHS  =  colMeans(CI_RBHS)[1:3]
cp_RBHS_plus   =  colMeans(CI_RBHS_plus)[1:3]
cp_RBRHS    =  colMeans(CI_RBRHS)[1:3]
cp_BHS    =  colMeans(CI_BHS)[1:3]
cp_BHS_plus   =  colMeans(CI_BHS_plus)[1:3]
cp_BRHS   =  colMeans(CI_BRHS)[1:3]

robust Bayesian shrinkage with the horseshoe family of priors

n=100; p=500;
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

# RBHS: robust Bayesian Regression with horseshoe priors

fit1 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "HS", model = "linear", hyper=NULL,debugging=FALSE)

fit1$coefficients$GS.beta # posterior samples for regression coefficients
estimation_1 = estimation.pqrBayes(fit1,coefficient,model="linear")
coeff_est_1 = estimation_1$coeff.est    
mse_1 = estimation_1$error$MSE

# RBHS+: robust Bayesian Regression with horseshoe plus priors

fit2 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "HS+", model = "linear", hyper=NULL,debugging=FALSE)

fit2$coefficients$GS.beta # posterior samples for regression coefficients
estimation_2 = estimation.pqrBayes(fit2,coefficient,model="linear")
coeff_est_2 = estimation_2$coeff.est    
mse_2 = estimation_2$error$MSE

# RBRHS: robust Bayesian Regression with regularized horseshoe priors

fit3 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "RHS", model = "linear", hyper=NULL,debugging=FALSE)

fit3$coefficients$GS.beta # posterior samples for regression coefficients
estimation_3 = estimation.pqrBayes(fit3,coefficient,model="linear")
coeff_est_3 = estimation_3$coeff.est    
mse_3 = estimation_3$error$MSE

# BHS: Bayesian Regression with horseshoe priors
fit4 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "HS", model = "linear", hyper=NULL,debugging=FALSE)

estimation_4 = estimation.pqrBayes(fit4,coefficient,model="linear")
coeff_est_4 = estimation_4$coeff.est
mse_4 = estimation_4$error$MSE

# BHS+: Bayesian Regression with horseshoe plus priors
fit5 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "HS+", model = "linear", hyper=NULL,debugging=FALSE)

estimation_5 = estimation.pqrBayes(fit5,coefficient,model="linear")
coeff_est_5 = estimation_5$coeff.est
mse_5 = estimation_5$error$MSE

# BRHS: Bayesian Regression with regularized horseshoe priors
fit6 = pqrBayes(g, y ,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "RHS", model = "linear", hyper=NULL,debugging=FALSE)

estimation_6 = estimation.pqrBayes(fit6,coefficient,model="linear")
coeff_est_6 = estimation_6$coeff.est
mse_6 = estimation_6$error$MSE

Example 3 (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]
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,e=NULL, d = NULL, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior="SS", 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,e=NULL, d =NULL,quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "Laplace", 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,e=NULL, d =NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "SS", 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,e=NULL, d =NULL, quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "Laplace", 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 4 (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 with spike and slab prior 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, e=NULL, d=3, quant=quant, iterations=10000, burn.in = NULL,robust = TRUE, prior = "SS", model = "group", hyper=NULL,debugging=FALSE)

fit$coefficients$GS.beta # posterior samples for regression coefficients 
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, e=NULL,d=3, quant=quant, iterations=10000, burn.in = NULL, robust = TRUE, prior = "Laplace", model = "group", hyper=NULL,debugging=FALSE)

fit1$coefficients$GS.beta # posterior samples for regression coefficients
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, d=3, e=NULL,quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "SS", 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,d=3, e=NULL,quant=quant, iterations=10000, burn.in = NULL, robust = FALSE, prior = "Laplace", 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.
• Fan, K., Srijana, S., Dissanayake, V. and Wu, C. (2025). Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors. arXiv:2507.10975