  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 model selection in the linear model.
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.
At the core of our methods is a new approximation to the exact likelihood
conditional on selection, the latter being generally intractable. We prove
that, under appropriate conditions, our approximation is asymptotically
consistent with the exact truncated likelihood. The advantages of our methods
in practical data analysis are demonstrated in simulations and in application
to HIV drug-resistance data.
