Towards "simultaneous selective inference": post-hoc bounds on the false discovery proportion

Some pitfalls of the false discovery rate (FDR) as an error criterion for multiple testing of $n$ hypotheses include (a) committing to an error level $q$ in advance limits its use in exploratory data analysis, and (b) controlling the false discovery proportion (FDP) on average provides no guarantee on its variability. We take a step towards overcoming these barriers using a new perspective we call "simultaneous selective inference." Many FDR procedures (such as Benjamini-Hochberg) can be viewed as carving out a $\textit{path}$ of potential rejection sets $\varnothing = \mathcal R_0 \subseteq \mathcal R_1 \subseteq \cdots \subseteq \mathcal R_n \subseteq [n]$, assigning some algorithm-dependent estimate $\widehat{\text{FDP}}(\mathcal R_k)$ to each one. Then, they choose $k^* = \max\{k: \widehat{\text{FDP}}(\mathcal R_k) \leq q\}$. We prove that for all these algorithms, given independent null p-values and a confidence level $\alpha$, either the same $\widehat{FDP}$ or a minor variant thereof bounds the unknown FDP to within a small explicit (algorithm-dependent) constant factor $c_{\text{alg}}(\alpha)$, uniformly across the entire path, with probability $1-\alpha$. Our bounds open up a middle ground between fully simultaneous inference (guarantees for all $2^n$ possible rejection sets), and fully selective inference (guarantees only for $\mathcal R_{k^*}$). They allow the scientist to $\textit{spot}$ one or more suitable rejection sets (Select Post-hoc On the algorithm's Trajectory), by picking data-dependent sizes or error-levels, after examining the entire path of $\widehat{\text{FDP}}(\mathcal R_k)$ and the uniform upper band on $\text{FDP}$. The price for the additional flexibility of spotting is small, for example the multiplier for BH corresponding to 95% confidence is approximately 2.