Uncertainty quantification in lasso-type regularization problems

Regularization techniques, which sit at the interface of statistical modeling and machine learning, are often used in the engineering or other applied sciences to tackle high dimensional regression (type) problems. While a number of regularization methods are commonly used, the 'Least Absolute Shrinkage and Selection Operator' or simply LASSO is popular because of its efficient variable selection property. This property of the LASSO helps to deal with problems where the number of predictors is larger than the total number of observations, as it shrinks the coefficients of non-important parameters to zero. In this chapter, both frequentist and Bayesian approaches for the LASSO are discussed, with particular attention to the problem of uncertainty quantification of regression parameters. For the frequentist approach, we discuss a refit technique as well as the classical bootstrap method, and for the Bayesian method, we make use of the equivalent LASSO formulation using a Laplace prior on the model parameters.