Infer-and-widen, or not?

Abstract: In recent years, there has been substantial interest in the task of selective inference: inference on a parameter that is selected from the data. Many of the existing proposals fall into what we refer to as the \emph{infer-and-widen} framework: they produce symmetric confidence intervals whose midpoints do not account for selection and therefore are biased; thus, the intervals must be wide enough to account for this bias. In this paper, we investigate infer-and-widen approaches in three vignettes: the winner's curse, maximal contrasts, and inference after the lasso. In each of these examples, we show that a state-of-the-art infer-and-widen proposal leads to confidence intervals that are much wider than simple alternatives when all methods are tuned to yield \emph{identical} randomized selection events. Furthermore, even an ``oracle'' infer-and-widen confidence interval -- the narrowest possible interval that could be theoretically attained via infer-and-widen -- is often wider than these alternatives.