Integrative methods for post-selection inference under convex constraints

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.