  Given $K$ uncertainty sets that are arbitrarily dependent -- for example,
confidence intervals for an unknown parameter obtained with $K$ different
estimators, or prediction sets obtained via conformal prediction based on $K$
different algorithms on shared data -- we address the question of how to
efficiently combine them in a black-box manner to produce a single uncertainty
set. We present a simple and broadly applicable majority vote procedure that
produces a merged set with nearly the same error guarantee as the input sets.
We then extend this core idea in a few ways: we show that weighted averaging
can be a powerful way to incorporate prior information, and a simple
randomization trick produces strictly smaller merged sets without altering the
coverage guarantee. Further improvements can be obtained if the sets are
exchangeable. We also show that many modern methods, like split conformal
prediction, median of means, HulC and cross-fitted ``double machine learning'',
can be effectively derandomized using these ideas.
