  Selective inference is the problem of giving valid answers to statistical
questions chosen in a data-driven manner. A standard solution to selective
inference is simultaneous inference, which delivers valid answers to the set of
all questions that could possibly have been asked. However, simultaneous
inference can be unnecessarily conservative if this set includes many questions
that were unlikely to be asked in the first place. We introduce a less
conservative solution to selective inference that we call locally simultaneous
inference, which only answers those questions that could plausibly have been
asked in light of the observed data, all the while preserving rigorous type I
error guarantees. For example, if the objective is to construct a confidence
interval for the "winning" treatment effect in a clinical trial with multiple
treatments, and it is obvious in hindsight that only one treatment had a chance
to win, then our approach will return an interval that is nearly the same as
the uncorrected, standard interval. Locally simultaneous inference is
implemented by refining any method for simultaneous inference of interest.
Under mild conditions satisfied by common confidence intervals, locally
simultaneous inference strictly dominates its underlying simultaneous inference
method, meaning it can never yield less statistical power but only more.
Compared to conditional selective inference, which demands stronger guarantees,
locally simultaneous inference is more easily applicable in nonparametric
settings and is more numerically stable.
