  Conformal prediction (CP) is a method for constructing a prediction interval
around the output of a fitted model, whose validity does not rely on the model
being correct--the CP interval offers a coverage guarantee that is
distribution-free, but relies on the training data being drawn from the same
distribution as the test data. A recent variant, weighted conformal prediction
(WCP), reweights the method to allow for covariate shift between the training
and test distributions. However, WCP requires knowledge of the nature of the
covariate shift-specifically,the likelihood ratio between the test and training
covariate distributions. In practice, since this likelihood ratio is estimated
rather than known exactly, the coverage guarantee may degrade due to the
estimation error. In this paper, we consider a special scenario where
observations belong to a finite number of groups, and these groups determine
the covariate shift between the training and test distributions-for instance,
this may arise if the training set is collected via stratified sampling. Our
results demonstrate that in this special case, the predictive coverage
guarantees of WCP can be drastically improved beyond the bounds given by
existing estimation error bounds.
