  The field of distribution-free predictive inference provides tools for
provably valid prediction without any assumptions on the distribution of the
data, which can be paired with any regression algorithm to provide accurate and
reliable predictive intervals. The guarantees provided by these methods are
typically marginal, meaning that predictive accuracy holds on average over both
the training data set and the test point that is queried. However, it may be
preferable to obtain a stronger guarantee of training-conditional coverage,
which would ensure that most draws of the training data set result in accurate
predictive accuracy on future test points. This property is known to hold for
the split conformal prediction method. In this work, we examine the
training-conditional coverage properties of several other distribution-free
predictive inference methods, and find that training-conditional coverage is
achieved by some methods but is impossible to guarantee without further
assumptions for others.
