While achieving exact conditional coverage in conformal prediction is unattainable without
making strong, untestable regularity assumptions, the promise of conformal prediction hinges
on finding approximations to conditional guarantees that are realizable in practice. A promising
direction for obtaining conditional dependence for conformal sets–in particular capturing
heteroskedasticity–is through estimating the conditional density ℙY|X and conformalizing its
level sets. Previous work in this vein has focused on nonconformity scores based on the empirical
cumulative distribution function (CDF). Such scores are, however, computationally costly, typically
requiring expensive sampling methods. To avoid the need for sampling, we observe that the CDF-based
score reduces to a Mahalanobis distance in the case of Gaussian scores, yielding a closed-form
expression that can be directly conformalized. Moreover, the use of a Gaussian-based score opens
the door to a number of extensions of the basic conformal method; in particular, we show how to
construct conformal sets with missing output values, refine conformal sets as partial information
about Y becomes available, and construct conformal sets on transformations of the output space.
Finally, empirical results indicate that our approach produces conformal sets that more closely
approximate conditional coverage in multivariate settings compared to alternative methods.
