sir_adjust {SeBR}

Post-processing with importance sampling

Description

Given Monte Carlo draws from the surrogate posterior, apply sampling importance reweighting (SIR) to correct for the true model likelihood.

Usage

sir_adjust(fit, sir_frac = 0.3, nsims_prior = 100, verbose = TRUE)

Arguments

fit

a fitted model object that includes

• coefficients the posterior mean of the regression coefficients

• 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 (sblm or sbsm)

sir_frac

fraction of draws to sample for SIR

nsims_prior

number of draws from the prior

verbose

logical; if TRUE, print time remaining

Details

The Monte Carlo sampling for sblm and sbsm uses a surrogate likelihood for posterior inference, which enables much faster and easier computing. SIR provides a correction for the actual (specified) likelihood. However, this correction step is quite slow and typically does not produce any noticeable discrepancies, even for small sample sizes.

Value

the fitted model object with the posterior draws subsampled based on the SIR adjustment

Note

SIR sampling is done WITHOUT replacement, so sir_frac is typically between 0.1 and 0.5. The nsims_priors draws are used to approximate a prior expectation, but larger values can significantly slow down this function.

Examples

# Simulate some data:
dat = simulate_tlm(n = 50, 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 = 10) # marginal distribution

# Fit the semiparametric Bayesian linear model:
fit = sblm(y = y, X = X, X_test = X_test)
names(fit) # what is returned

# Update with SIR:
fit_sir = sir_adjust(fit)

# Prediction: unadjusted vs. adjusted?

# Point estimates:
y_hat = fitted(fit)
y_hat_sir = fitted(fit_sir)
cor(y_hat, y_hat_sir) # similar

# Interval estimates:
pi_y = t(apply(fit$post_ypred, 2, quantile, c(0.05, .95))) # 90% PI
pi_y_sir = t(apply(fit_sir$post_ypred, 2, quantile, c(0.05, .95))) # 90% PI

# PI overlap (%):
overlaps = 100*sapply(1:length(y_test), function(i){
  # innermost part
  (min(pi_y[i,2], pi_y_sir[i,2]) - max(pi_y[i,1], pi_y_sir[i,1]))/
    # outermost part
    (max(pi_y[i,2], pi_y_sir[i,2]) - min(pi_y[i,1], pi_y_sir[i,1]))
})
summary(overlaps) # mostly close to 100%

# Coverage of PIs on testing data (should be ~ 90%)
mean((pi_y[,1] <= y_test)*(pi_y[,2] >= y_test)) # unadjusted
mean((pi_y_sir[,1] <= y_test)*(pi_y_sir[,2] >= y_test)) # adjusted

# Plot together with testing data:
plot(y_test, y_test, type='n', ylim = range(pi_y, pi_y_sir, y_test),
     xlab = 'y_test', ylab = 'y_hat', main = paste('Prediction intervals: testing data'))
abline(0,1) # reference line
suppressWarnings(
  arrows(y_test, pi_y[,1], y_test, pi_y[,2],
         length=0.15, angle=90, code=3, col='gray', lwd=2)
) # plot the PIs (unadjusted)
suppressWarnings(
  arrows(y_test, pi_y_sir[,1], y_test, pi_y_sir[,2],
         length=0.15, angle=90, code=3, col='darkgray', lwd=2)
) # plot the PIs (adjusted)
lines(y_test, y_hat, type='p', pch=2) # plot the means (unadjusted)
lines(y_test, y_hat_sir, type='p', pch=3) # plot the means (adjusted)