  It is well known that it is impossible to construct useful confidence
intervals (CIs) about the mean or median of a response $Y$ conditional on
features $X = x$ without making strong assumptions about the joint distribution
of $X$ and $Y$. This paper introduces a new framework for reasoning about
problems of this kind by casting the conditional problem at different levels of
resolution, ranging from coarse to fine localization. In each of these
problems, we consider local quantiles defined as the marginal quantiles of $Y$
when $(X,Y)$ is resampled in such a way that samples $X$ near $x$ are
up-weighted while the conditional distribution $Y \mid X$ does not change. We
then introduce the Weighted Quantile method, which asymptotically produces the
uniformly most accurate confidence intervals for these local quantiles no
matter the (unknown) underlying distribution. Another method, namely, the
Quantile Rejection method, achieves finite sample validity under no assumption
whatsoever. We conduct extensive numerical studies demonstrating that both of
these methods are valid. In particular, we show that the Weighted Quantile
procedure achieves nominal coverage as soon as the effective sample size is in
the range of 10 to 20.
