Suppose we can construct a valid $(1-\delta)$-confidence interval (CI) for each of the $K$ parameters of potential interest. If the data analyst selects a subset of parameters using arbitrary data-dependent criteria, the aforementioned Her CIs for the selected parameters are no longer valid due to selection bias. We design a new method of adjusting the interval to control the false coverage rate (FCR). The main established method is the “BY procedure” by Benjamini and Yekutieli (JASA, 2005). Unfortunately, the BY guarantee requires certain restrictions on the dependencies between selection criteria and CIs. We propose a natural and much simpler method that is valid under arbitrary dependency structures among the original CIs and arbitrary (unknown) selection criteria, but only applies to a special yet broad class of CIs. To do. Our procedure reports $(1-\delta|\mathcal{S}|/K)$-CI for the selected parameters, with $\delta$ for confidence intervals that implicitly invert the e-values Prove to control her FCR. Examples include those built via supermartingale methods, universal reasoning, Chernoff-style bounds on moment-generating functions, and more. The e-BY procedure is acceptable and recovers the BY procedure as a special case via calibration. Since our study applies under stopping time, continuously monitored confidence sequences, and bandit sampling, it also has implications for post-selection reasoning in the sequential setting.
