  Conformal methods create prediction bands that control average coverage under
no assumptions besides i.i.d. data. Besides average coverage, one might also
desire to control conditional coverage, that is, coverage for every new testing
point. However, without strong assumptions, conditional coverage is
unachievable. Given this limitation, the literature has focused on methods with
asymptotical conditional coverage. In order to obtain this property, these
methods require strong conditions on the dependence between the target variable
and the features. We introduce two conformal methods based on conditional
density estimators that do not depend on this type of assumption to obtain
asymptotic conditional coverage: Dist-split and CD-split. While Dist-split
asymptotically obtains optimal intervals, which are easier to interpret than
general regions, CD-split obtains optimal size regions, which are smaller than
intervals. CD-split also obtains local coverage by creating a data-driven
partition of the feature space that scales to high-dimensional settings and by
generating prediction bands locally on the partition elements. In a wide
variety of simulated scenarios, our methods have a better control of
conditional coverage and have smaller length than previously proposed methods.
