  We show that two popular selective inference procedures, namely data carving
(Fithian et al., 2017) and selection with a randomized response (Tian et al.,
2018b), when combined with the polyhedral method (Lee et al., 2016), result in
confidence intervals whose length is bounded. This contrasts results for
confidence intervals based on the polyhedral method alone, whose expected
length is typically infinite (Kivaranovic and Leeb, 2020). Moreover, we show
that these two procedures always dominate corresponding sample-splitting
methods in terms of interval length.
