  We consider the problem of distribution-free predictive inference, with the
goal of producing predictive coverage guarantees that hold conditionally rather
than marginally. Existing methods such as conformal prediction offer marginal
coverage guarantees, where predictive coverage holds on average over all
possible test points, but this is not sufficient for many practical
applications where we would like to know that our predictions are valid for a
given individual, not merely on average over a population. On the other hand,
exact conditional inference guarantees are known to be impossible without
imposing assumptions on the underlying distribution. In this work we aim to
explore the space in between these two, and examine what types of relaxations
of the conditional coverage property would alleviate some of the practical
concerns with marginal coverage guarantees while still being possible to
achieve in a distribution-free setting.
