Conditional Versus Unconditional Approaches to Selective Inference

We investigate a class of methods for selective inference that condition on a selection event. Such methods operate in a two-stage process. First, a (sub)collection of hypotheses is determined in a data-driven way from some large universe of hypotheses. Second, inference is done within the data-driven collection, conditional on the information that was used for the selection. Examples of such methods include basic data splitting, as well as modern data carving methods and post-selection inference methods based on the polyhedral lemma. In this paper, we adopt a holistic view on such methods, viewing the selection, conditioning and final error control steps together as a single method. From this perspective, we show that selective inference methods based on selection and conditioning are always dominated by multiple testing methods defined directly on the full universe of hypotheses. This result even holds when this universe is potentially infinite and only defined implicitly, such as in data splitting. We investigate four case studies of potential power gain from switching to a non-selective and/or an unconditional perspective.