Inference after model selection has been an active research topic in the past few years, with numerous works offering different approaches to addressing the perils of the reuse of data. In particular, major progress has been made recently on large and useful classes of problems by harnessing general theory of hypothesis testing in exponential families, but these methods have their limitations. Perhaps most immediate is the gap between theory and practice: implementing the exact theoretical prescription in realistic situations—for example, when new data arrives and inference needs to be adjusted accordingly—may be a prohibitive task. In this paper, we propose a Bayesian framework for carrying out inference after variable selection. Our framework is very flexible in the sense that it naturally accommodates different models for the data instead of requiring a case-by-case treatment. This flexibility is achieved by considering the full selective likelihood function where, crucially, we propose a novel and nontrivial approximation to the exact but intractable expression. The advantages of our methods in practical data analysis are demonstrated in an application to HIV drug-resistance data.
J.T. was supported in part by ARO Grant 70940MA. A.W. was partially supported by ERC Grant 030-8944 and by ISF Grant 039-9325.
The idea of providing Bayesian adjusted inference after variable selection was previously proposed by Daniel Yekutieli and Edward George, and presented at the 2012 Joint Statistical Meetings in San Diego. Our interest re-arose with the recent developments on exact post-selection inference in the linear model. A.W. is thankful to Daniel and Ed for helpful conversations and to Ed for pointing out the paper by Bayarri and DeGroot. S.P. is thankful to Xuming He, Liza Levina, Rina Foygel Barber and Yi Wang for reading an initial version of the draft and offering valuable comments and insights. The authors would also like to sincerely thank and acknowledge Chiara Sabatti for the long discussions and for her suggestions along the way.
"Integrative methods for post-selection inference under convex constraints." Ann. Statist. 49 (5) 2803 - 2824, October 2021. https://doi.org/10.1214/21-AOS2057