  Suppose that one can construct a valid $(1-\delta)$-confidence interval (CI)
for each of $K$ parameters of potential interest. If a data analyst uses an
arbitrary data-dependent criterion to select some subset $S$ of parameters,
then the aforementioned CIs for the selected parameters are no longer valid due
to selection bias. We design a new method to adjust the intervals in order to
control the false coverage rate (FCR). The main established method is the "BY
procedure" by Benjamini and Yekutieli (JASA, 2005). The BY guarantees require
certain restrictions on the selection criterion and on the dependence between
the CIs. We propose a new simple method which, in contrast, is valid under any
dependence structure between the original CIs, and any (unknown) selection
criterion, but which only applies to a special, yet broad, class of CIs that we
call e-CIs. To elaborate, our procedure simply reports $(1-\delta|S|/K)$-CIs
for the selected parameters, and we prove that it controls the FCR at $\delta$
for confidence intervals that implicitly invert e-values; examples include
those constructed via supermartingale methods, via universal inference, or via
Chernoff-style bounds, among others. The e-BY procedure is admissible, and
recovers the BY procedure as a special case via a particular calibrator. Our
work also has implications for post-selection inference in sequential settings,
since it applies at stopping times, to continuously-monitored confidence
sequences, and under bandit sampling. We demonstrate the efficacy of our
procedure using numerical simulations and real A/B testing data from Twitter.
