Comment

Abstract

We congratulate Efron for his stimulating and timely work which addresses an important issue on estimation after model selection. In practice, it is typical to ignore the variability of the variable selection step, which could result in inaccurate post-selection inference. Although the flaw of such practice is widely recognized, finding a general solution is extremely challenging. The model selection step is often a complex decision process and can involve collecting expert opinions, preprocessing, applying a variable selection rule, data-driven choice of one or more tuning parameters, among others. Except in simple cases, explicitly characterizing the form of the post-selection estimator is itself difficult. The key result of this paper is a closed-form formula for obtaining the standard deviation of a “bootstrap smoothed” (or “bagged”) estimator. This elegant formula is not only simple to implement but also versatile. It indeed provides a general approach for obtaining a confidence interval for a class of parameters of interest while incorporating the variability of variable selection. Our discussions will focus on two aspects: (1) the generality of the method, and (2) further insight into the performance of the proposed method in a simple but hopefully informative example.