3 Uncertainty Quantification in Lasso-Type Regularization Problems
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Fig. 3.13 Relative frequency of occurrence of variables, for refit-LASSO applied on the Sonar
dataset
family instead of the normal distribution. The graph in Fig. 3.14 shows the bootstrap
distribution of the estimated parameters.
3.5.2.3 Bayesian Approach
We obtained the posterior distribution of the parameters using the MCMClogit
function from the MCMCpack [19] package in R. We took the Laplace priors
for parameter estimation. We have taken 100,000 MCMC samples with a thinning
interval length of 10 and a Metropolis tuning parameter set at 0.05, yielding 10,000
posterior samples for the assessment of the coefficient distribution. It can be seen
that for the Bayesian approach the variability is almost same as that of bootstrap
method (Fig. 3.15).
For a better comparison between each parameter estimation method, we have
shown the standard errors for the coefficient estimates of each important variable in
Fig. 3.16 indexed according to the refit-LASSO method.