  While traditional multiple testing procedures prohibit adaptive analysis
choices made by users, Goeman and Solari (2011) proposed a simultaneous
inference framework that allows users such flexibility while preserving
high-probability bounds on the false discovery proportion (FDP) of the chosen
set. In this paper, we propose a new class of such simultaneous FDP bounds,
tailored for nested sequences of rejection sets. While most existing
simultaneous FDP bounds are based on closed testing using global null tests
based on sorted p-values, we additionally consider the setting where side
information can be leveraged to boost power, the variable selection setting
where knockoff statistics can be used to order variables, and the online
setting where decisions about rejections must be made as data arrives. Our
finite-sample, closed form bounds are based on repurposing the FDP estimates
from false discovery rate (FDR) controlling procedures designed for each of the
above settings. These results establish a novel connection between the parallel
literatures of simultaneous FDP bounds and FDR control methods, and use proof
techniques employing martingales and filtrations that are new to both these
literatures. We demonstrate the utility of our results by augmenting a recent
knockoffs analysis of the UK Biobank dataset.
