  Testing whether the observed data conforms to a purported model (probability
distribution) is a basic and fundamental statistical task, and one that is by
now well understood. However, the standard formulation, identity testing, fails
to capture many settings of interest; in this work, we focus on one such
natural setting, identity testing under promise of permutation. In this
setting, the unknown distribution is assumed to be equal to the purported one,
up to a relabeling (permutation) of the model: however, due to a systematic
error in the reporting of the data, this relabeling may not be the identity.
The goal is then to test identity under this assumption: equivalently, whether
this systematic labeling error led to a data distribution statistically far
from the reference model.
