Sampling models for selective inference

TL;DR

This paper examines the theoretical foundations of constructing valid inferential models after data-driven selection processes, addressing ambiguity in conditioning strategies through principles like ancillarity and the Conditionality Principle.

Contribution

It introduces two principles to resolve conditioning ambiguity in selective inference, linking classical and modern statistical paradigms, with practical guidance for common scenarios.

Findings

Ancillarity notions are preserved after conditioning on selection.

The Conditionality Principle supports valid inference post-selection.

Guidance provided for choosing appropriate inferential methods in practice.

Abstract

This paper explores the challenges of constructing suitable inferential models in scenarios where the parameter of interest is determined in light of the data, such as regression after variable selection. Two compelling arguments for conditioning converge in this context, whose interplay can introduce ambiguity in the choice of conditioning strategy: the Conditionality Principle, from classical statistics, and the `condition on selection' paradigm, central to selective inference. We discuss two general principles that can be employed to resolve this ambiguity in some recurrent contexts. The first one refers to the consideration of how information is processed at the selection stage. The second one concerns an exploration of ancillarity in the presence of selection. We demonstrate that certain notions of ancillarity are preserved after conditioning on the selection event, supporting the application of the Conditionality Principle. We illustrate these concepts through examples and provide guidance on the adequate inferential approach in some common scenarios.