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Demo of the bayeslm package

In this vignette, we show how to use bayeslm to sample coefficients for a Gaussian linear regression with a number of different prior distributions.

library(bayeslm)

Generate data

set.seed(200)
p = 20
n = 100
kappa = 1.25
beta_true = c(c(1,2,3),rnorm(p-3,0,0.01))
sig_true = kappa*sqrt(sum(beta_true^2))
x = matrix(rnorm(p*n),n,p)
y = x %*% beta_true + sig_true * rnorm(n)
x = as.matrix(x)
y = as.matrix(y)
data = data.frame(x = x, y = y)

Modeling \(Y \mid X = x\) using linear regression

OLS

First, we run OLS and inspect the estimated coefficients

fitOLS = lm(y~x-1)
coef(fitOLS)
#>          x1          x2          x3          x4          x5          x6 
#>  1.14415549  1.99442217  2.38953931 -0.73055959 -0.71840494  0.25239729 
#>          x7          x8          x9         x10         x11         x12 
#>  0.39044226  0.41366834 -0.39830816  0.04170669 -0.20976664 -0.24878570 
#>         x13         x14         x15         x16         x17         x18 
#>  0.63260575  0.07410838  0.40714461 -0.18807163  0.79535530  0.32748131 
#>         x19         x20 
#>  0.59214904  0.51461938

bayeslm

Updating coefficients individually

The bayeslm sampler can group coefficients into blocks and sample several coefficients at once. Below, we simply place every coefficient into its own block.

block_vec = rep(1, p)

Now, we run bayeslm on six different priors and store their estimated coefficients

# Horseshoe prior
fit1 = bayeslm(y, x, prior = 'horseshoe', icept = FALSE, 
               block_vec = block_vec, N = 10000, burnin=2000)
beta_est1 = colMeans(fit1$beta)

# Laplace prior
fit2 = bayeslm(y, x, prior = 'laplace', icept = FALSE, 
               block_vec = block_vec, N = 10000, burnin=2000)
beta_est2 = colMeans(fit2$beta)

# Ridge prior
fit3 = bayeslm(y, x, prior = 'ridge', icept = FALSE, 
               block_vec = block_vec, N = 10000, burnin=2000)
beta_est3 = colMeans(fit3$beta)

# "Sharkfin" prior
fit4 = bayeslm(y, x, prior = 'sharkfin', icept = FALSE, 
               block_vec = block_vec, N = 10000, burnin=2000)
beta_est4 = colMeans(fit4$beta)

# "Non-local" prior
fit5 = bayeslm(y, x, prior = 'nonlocal', icept = FALSE, 
               block_vec = block_vec, N = 10000, burnin=2000)
beta_est5 = colMeans(fit5$beta)

# Inverse laplace prior
fit6 = bayeslm(y, x, prior = 'inverselaplace', lambda = 0.01, icept = FALSE, 
               block_vec = block_vec, N = 10000, burnin=2000)
beta_est6 = colMeans(fit6$beta)

And we plot the posterior distribution of the regression coefficients, along with the OLS estimates, against the true simulated coefficients.

plot(NULL,xlim=range(beta_true),ylim=range(beta_true), 
     xlab = "beta true", ylab = "estimation", )
points(beta_true,beta_est1,pch=20)
points(beta_true,fitOLS$coef,col='red')
points(beta_true,beta_est2,pch=20,col='cyan')
points(beta_true,beta_est3,pch=20,col='orange')
points(beta_true,beta_est4,pch=20,col='pink')
points(beta_true,beta_est5,pch=20,col='lightgreen')
points(beta_true,beta_est6,pch=20,col='grey')
legend("topleft", c("OLS", "horseshoe", "laplace", "ridge", "sharkfin", 
  "nonlocal", "inverselaplace"), col = c("red", "black", "cyan", "orange", 
    "pink", "lightgreen", "grey"), pch = rep(1, 7))
abline(0,1,col='red')

We can also compare the root mean squared error (RMSE) for each prior

rmseOLS = sqrt(sum((fitOLS$coef-beta_true)^2))
rmse1 = sqrt(sum((beta_est1-beta_true)^2))
rmse2 = sqrt(sum((beta_est2-beta_true)^2))
rmse3 = sqrt(sum((beta_est3-beta_true)^2))
rmse4 = sqrt(sum((beta_est4-beta_true)^2))
rmse5 = sqrt(sum((beta_est5-beta_true)^2))
rmse6 = sqrt(sum((beta_est6-beta_true)^2))
print(cbind(ols = rmseOLS, hs = rmse1,laplace = rmse2, ridge = rmse3, 
            sharkfin = rmse4,nonlocal = rmse5, inverselaplace = rmse6))
#>           ols       hs  laplace    ridge sharkfin nonlocal inverselaplace
#> [1,] 2.013695 1.082114 1.463458 1.954048 1.990372 2.291407       1.058102

Updating coefficients in blocks

Here, we demonstrate:

  1. How to place several coefficients in the same block for the slice-within-Gibbs sampler
  2. How to use formula notation (y ~ x) in the bayeslm library
# Put the first two coefficients in one elliptical sampling block
block_vec2 = c(2, rep(1, p-2))
fitb = bayeslm(y ~ x - 1, data = data, prior = 'horseshoe', 
               block_vec = block_vec2, N = 10000, burnin = 2000)
#> horseshoe prior 
#> fixed running time 0.000402834
#> sampling time 0.223155
summary(fitb)
#> Average number of rejections before one acceptance : 
#> 31.2156 
#> Summary of beta draws 
#>    based on 8000 valid draws (number of burn in is 2000) 
#> Summary of Posterior draws 
#>  Moments 
#>        mean std dev eff sample size    
#> x1   0.6403    0.52             764    
#> x2   2.1377    0.68             618 ***
#> x3   2.6466    0.47            2362 ***
#> x4  -0.2519    0.39            1529    
#> x5  -0.2663    0.37            1662    
#> x6   0.0351    0.24            5648    
#> x7   0.1552    0.33            2617    
#> x8   0.1484    0.32            2675    
#> x9  -0.2187    0.36            1703    
#> x10  0.0558    0.25            4862    
#> x11 -0.0727    0.26            4570    
#> x12 -0.0597    0.26            4983    
#> x13  0.2759    0.37            1184    
#> x14  0.0089    0.29            7268    
#> x15  0.1151    0.28            2807    
#> x16 -0.0355    0.28            6747    
#> x17  0.3939    0.46            1102    
#> x18  0.0882    0.25            3785    
#> x19  0.2470    0.37            1680    
#> x20  0.1576    0.32            2567    
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
#> 
#> Summary of sigma draws 
#> Mean of standard deviation is  4.555075 
#> S.d. of standard deviation samples is  0.3624706 
#> Effective sample size of s.d. is  3357.107

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