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Semiparametric Bayesian linear model

Source: R/source_sba.R
sblm.Rd

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

Usage

sblm(
  y,
  X,
  X_test = X,
  psi = length(y),
  laplace_approx = TRUE,
  fixedX = (length(y) >= 500),
  approx_g = FALSE,
  nsave = 1000,
  ngrid = 100,
  verbose = TRUE
)

Arguments

y

n x 1 response vector

X

n x p matrix of predictors (no intercept)

X_test

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

psi

prior variance (g-prior)

laplace_approx

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

fixedX

logical; if TRUE, treat the design as fixed (non-random) when sampling the transformation; otherwise treat covariates as random with an unknown distribution

approx_g

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

nsave

number of Monte Carlo simulations

ngrid

number of grid points for inverse approximations

verbose

logical; if TRUE, print time remaining

Value

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.

Details

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. By default, fixedX is set to FALSE for smaller datasets (n < 500) and TRUE for larger datasets (n >= 500).

Note

The location (intercept) and scale (sigma_epsilon) are not identified, so any intercepts in X and X_test will be removed. The model-fitting *does* include an internal location-scale adjustment, but the function only outputs inferential summaries for the identifiable parameters.

Examples

# \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] "3 sec remaining"
#> [1] "2 sec remaining"
#> [1] "Total time: 3 seconds"
names(fit) # what is returned
#>  [1] "coefficients"   "fitted.values"  "post_theta"     "post_ypred"    
#>  [5] "post_g"         "model"          "y"              "X"             
#>  [9] "X_test"         "psi"            "laplace_approx" "fixedX"        
#> [13] "approx_g"      

# Note: this is Monte Carlo sampling...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] 0.913

# Check: correlation with true coefficients
cor(dat$beta_true, coef(fit))
#> [1] 0.9349882

# 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$post_g), lwd = 3) # posterior mean
lines(y, dat$g_true, type='p', pch=2) # true transformation
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)

# }