Integrative Methods for Post-Selection Inference Under Convex Constraints

TL;DR

This paper introduces a Bayesian approach for post-selection inference in linear models that uses a novel likelihood approximation, offering flexibility and practical applicability, demonstrated through simulations and real data analysis.

Contribution

It presents a new Bayesian framework with an approximate likelihood for post-selection inference, bridging theory and practice in linear models.

Findings

Approximate likelihood is asymptotically consistent with the exact likelihood.

Framework effectively handles different data models without case-by-case adjustments.

Method demonstrates practical utility in simulations and HIV data analysis.

Abstract

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.