Abstract: The exploratory and interactive nature of modern data analysis often introduces selection bias, posing challenges for traditional statistical inference methods. A common strategy to address this bias is by conditioning on the selection event. However, this often results in a conditional distribution that is intractable and requires Markov chain Monte Carlo (MCMC) sampling for inference. Notably, some of the most widely used selection algorithms yield selection events that can be characterized as polyhedra, such as the lasso for variable selection and the epsilon-greedy algorithm for multi-armed bandit problems. This talk will present a method that is tailored for conducting inference following polyhedral selection. The method transforms the variables constrained within a polyhedron into variables within a unit cube, allowing for exact sampling. Compared to MCMC, the proposed method offers superior speed and accuracy, providing a practical and efficient approach for conditional selective inference. Additionally, it facilitates the computation of the selection-adjusted maximum likelihood estimator, enabling MLE-based inference. Numerical results demonstrate the enhanced performance of the proposed method compared to alternative approaches for selective inference.