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Monte Carlo sampling for Bayesian linear regression with an unknown (nonparametric) transformation. A g-prior is assumed for the regression coefficients.


  X_test = X,
  psi = length(y),
  laplace_approx = TRUE,
  approx_g = FALSE,
  nsave = 1000,
  ngrid = 100,
  verbose = TRUE



n x 1 response vector


n x p matrix of predictors


n_test x p matrix of predictors for test data; default is the observed covariates X


prior variance (g-prior)


logical; if TRUE, use a normal approximation to the posterior in the definition of the transformation; otherwise the prior is used


logical; if TRUE, apply large-sample approximation for the transformation


number of Monte Carlo simulations


number of grid points for inverse approximations


logical; if TRUE, print time remaining


a list with the following elements:

  • coefficients the posterior mean of the regression coefficients

  • fitted.values the posterior predictive mean at the test points X_test

  • post_theta: nsave x p samples from the posterior distribution of the regression coefficients

  • post_ypred: nsave x n_test samples from the posterior predictive distribution at test points X_test

  • post_g: nsave posterior samples of the transformation evaluated at the unique y values

  • model: the model fit (here, sblm)

as well as the arguments passed in.


This function provides fully Bayesian inference for a transformed linear model using Monte Carlo (not MCMC) sampling. The transformation is modeled as unknown and learned jointly with the regression coefficients (unless approx_g = TRUE, which then uses a point approximation). This model applies for real-valued data, positive data, and compactly-supported data (the support is automatically deduced from the observed y values). The results are typically unchanged whether laplace_approx is TRUE/FALSE; setting it to TRUE may reduce sensitivity to the prior, while setting it to FALSE may speed up computations for very large datasets.


# \donttest{
# Simulate some data:
dat = simulate_tlm(n = 100, p = 5, g_type = 'step')
y = dat$y; X = dat$X # training data
y_test = dat$y_test; X_test = dat$X_test # testing data

hist(y, breaks = 25) # marginal distribution

# Fit the semiparametric Bayesian linear model:
fit = sblm(y = y, X = X, X_test = X_test)
#> [1] "4 seconds remaining"
#> [1] "3 seconds remaining"
#> [1] "1 seconds remaining"
#> [1] "Total time:  5 seconds"
names(fit) # what is returned
#>  [1] "coefficients"  "fitted.values" "post_theta"    "post_ypred"   
#>  [5] "post_g"        "model"         "y"             "X"            
#>  [9] "X_test"        "psi"           "approx_g"      "sigma_epsilon"

# Note: this is Monte Carlo sampling, so no need for MCMC diagnostics!

# Evaluate posterior predictive means and intervals on the testing data:
plot_pptest(fit$post_ypred, y_test,
            alpha_level = 0.10) # coverage should be about 90%

#> [1] 0.889

# Check: correlation with true coefficients
    coef(fit)[-1]) # excluding the intercept
#> [1] 0.975411

# Summarize the transformation:
y0 = sort(unique(y)) # posterior draws of g are evaluated at the unique y observations
plot(y0, fit$post_g[1,], type='n', ylim = range(fit$post_g),
     xlab = 'y', ylab = 'g(y)', main = "Posterior draws of the transformation")
temp = sapply(1:nrow(fit$post_g), function(s)
  lines(y0, fit$post_g[s,], col='gray')) # posterior draws
lines(y0, colMeans(fit$post_g), lwd = 3) # posterior mean

# Add the true transformation, rescaled for easier comparisons:
      scale(dat$g_true)*sd(colMeans(fit$post_g)) + mean(colMeans(fit$post_g)), type='p', pch=2)
legend('bottomright', c('Truth'), pch = 2) # annotate the true transformation

# Posterior predictive checks on testing data: empirical CDF
y0 = sort(unique(y_test))
plot(y0, y0, type='n', ylim = c(0,1),
     xlab='y', ylab='F_y', main = 'Posterior predictive ECDF')
temp = sapply(1:nrow(fit$post_ypred), function(s)
  lines(y0, ecdf(fit$post_ypred[s,])(y0), # ECDF of posterior predictive draws
        col='gray', type ='s'))
lines(y0, ecdf(y_test)(y0),  # ECDF of testing data
     col='black', type = 's', lwd = 3)

# }