We present confidence bounds on the false positives contained in subsets of selected null hypotheses. As the coverage probability holds simultaneously over all possible subsets, these bounds can be applied to an arbitrary number of possibly datadriven subsets.
These bounds are built via a two-step approach. First, build a family of candidate rejection subsets together with associated bounds, holding uniformly on the number of false positives they contain (call this a reference family). Then, interpolate from this reference family to find a bound valid for any subset.
This general program is exemplified for two particular types of reference families: (i) when the bounds are fixed and the subsets are p-value level sets (ii) when the subsets are fixed, spatially structured and the bounds are estimated.[-]