  This article investigates uncertainty quantification of the generalized
linear lasso~(GLL), a popular variable selection method in high-dimensional
regression settings. In many fields of study, researchers use data-driven
methods to select a subset of variables that are most likely to be associated
with a response variable. However, such variable selection methods can
introduce bias and increase the likelihood of false positives, leading to
incorrect conclusions. In this paper, we propose a post-selection inference
framework that addresses these issues and allows for valid statistical
inference after variable selection using GLL. We show that our method provides
accurate $p$-values and confidence intervals, while maintaining high
statistical power. In a second stage, we focus on the sparse logistic
regression, a popular classifier in high-dimensional statistics. We show with
extensive numerical simulations that SIGLE is more powerful than
state-of-the-art PSI methods. SIGLE relies on a new method to sample states
from the distribution of observations conditional on the selection event. This
method is based on a simulated annealing strategy whose energy is given by the
first order conditions of the logistic lasso.
