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Inference after model selection has been an active research topic in the pastfew years, with numerous works offering different approaches to addressing theperils of the reuse of data. In particular, major progress has been maderecently on large and useful classes of problems by harnessing general theoryof hypothesis testing in exponential families, but these methods have theirlimitations. Perhaps most immediate is the gap between theory and practice:implementing the exact theoretical prescription in realistic situations---forexample, when new data arrives and inference needs to be adjustedaccordingly---may be a prohibitive task. In this paper we propose a Bayesianframework for carrying out inference after model selection in the linear model.Our framework is very flexible in the sense that it naturally accommodatesdifferent models for the data, instead of requiring a case-by-case treatment.At the core of our methods is a new approximation to the exact likelihoodconditional on selection, the latter being generally intractable. We provethat, under appropriate conditions, our approximation is asymptoticallyconsistent with the exact truncated likelihood. The advantages of our methodsin practical data analysis are demonstrated in simulations and in applicationto HIV drug-resistance data.

Integrative Methods for Post-Selection Inference Under Convex Constraints