On the length of confidence intervals with conditional coverage
Inference after model selection is currently a very active area of research. The polyhedral method (Lee et al., 2016) allows for valid inference after model selection if the model selection event can be described by polyhedral constraints. In that reference, the method is exemplified by constructing two valid confidence intervals when the Lasso estimator is used to select a model. We here study the length of such intervals. We first show that the polyhedral method gives confidence intervals whose expected length is typically infinite. However, the polyhedral method combined with data carving (Fithian et al., 2017) or with selection with a randomized response (Lee et al., 2016) gives confidence intervals whose length is always bounded. Our upper bound is easy to compute, easy to interpret and, in the interesting case where the polyhedral method alone gives intervals with infinite expected length, also sharp.
*This is joint work with Danijel Kivaranovic
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