  Conformal prediction, and split conformal prediction as a specific
implementation, offer a distribution-free approach to estimating prediction
intervals with statistical guarantees. Recent work has shown that split
conformal prediction can produce state-of-the-art prediction intervals when
focusing on marginal coverage, i.e. on a calibration dataset the method
produces on average prediction intervals that contain the ground truth with a
predefined coverage level. However, such intervals are often not adaptive,
which can be problematic for regression problems with heteroskedastic noise.
This paper tries to shed new light on how prediction intervals can be
constructed, using methods such as normalized and Mondrian conformal
prediction, in such a way that they adapt to the heteroskedasticity of the
underlying process. Theoretical and experimental results are presented in which
these methods are compared in a systematic way. In particular, it is shown how
the conditional validity of a chosen conformal predictor can be related to
(implicit) assumptions about the data-generating distribution.
