An Exact Sampler for Inference after Polyhedral Model Selection

Abstract

Inference after model selection presents computational challenges when dealing with intractable conditional distributions. Markov chain Monte Carlo (MCMC) is a common method for sampling from these distributions, but its slow convergence often limits its practicality. In this work, we introduce a method tailored for selective inference in cases where the selection event can be characterized by a polyhedron. The method transforms the variables constrained by a polyhedron into variables within a unit cube, allowing for efficient sampling using conventional numerical integration techniques. Compared to MCMC, the proposed sampling method is highly accurate and equipped with an error estimate. Additionally, we introduce an approach to use a single batch of samples for hypothesis testing and confidence interval construction across multiple parameters, reducing the need for repetitive sampling. Furthermore, our method facilitates fast and precise computation of the maximum likelihood estimator based on the selection-adjusted likelihood, enhancing the reliability of MLE-based inference. Numerical results demonstrate the superior performance of the proposed method compared to alternative approaches for selective inference